quantum chemical studies of thermochemistry, kinetics and

Quantum Chemical Studies ofThermochemistry, Kinetics and

Molecular Structure.

by

Naomi Louise Haworth

A thesis submitted in fulfilment of the requirements for the

degree of Doctor of Philosophy.

School of Chemistry

University of Sydney

February, 2003

Declaration

I hereby declare that this thesis is my own work and that, to the best of my knowledge, it

contains no material previously published or written by another person nor material which has

been accepted for the award of another degree or diploma at an institute of higher education,

except where due acknowledgement is given.

Naomi Haworth

For the Glory of God

Acknowledgements

I would like to express my extreme gratitude to my supervisor, Dr George Bacskay, for the

wonderful way he has helped and guided me over the past four years. Thankyou particularly

for your kindness, understanding and patience with me in the hard times. I would also like to

thank my associate supervisor, Associate Professor John Mackie, for teaching and advising

me in the kinetics work in this thesis and for proposing the projects on fluorocarbons,

phosphorus compounds and NNH + O.

I am also grateful to the many other academics and students with whom I have shared

collaborative projects, in particular Nathan Owens, Klaas Nauta and Scott Kable (CFClBr2)

and Charles Collyer and Matt Templeton (Phaseolotoxin). Thanks also to all my coworkers

over the years (Jason, Jens, Kausala, Karina, Justin, Debbie, Adam, Keiran, Siobhan and

many more) for their help and advice and for the fun we’ve shared.

I thank my family for their love and for supporting me and believing in me throughout all my

academic career, and also my friends for their support and encouragement. Particular thanks

go to Justine for all the ways, big and small, that you’ve helped me out over the past few

months and for your patience; to Evan for trying to keep me sane; and to Geoff for helping

with the proof reading.

I would like to thank the Australian Partnership for Advanced Computing (APAC) National

Facility for access to the COMPAQ AlphaServer SC system and the Australian Centre for

Advanced Computing and Communications (ac3) for access to their SGI Origin 2400

computer system. Finally, I express my sincere gratitude to the Australian Postgraduate

Association for funding my PhD scholarship.

Abstract

This thesis is concerned with a range of chemical problems which are amenable to theoretical

investigation via the application of current methods of computational quantum chemistry.

These problems include the calculation of accurate thermochemical data, the prediction of

reaction kinetics, the study of molecular potential energy surfaces, and the investigation of

molecular structures and binding.

The heats of formation (from both atomisation energies and isodesmic schemes) of a set of

approximately 120 C1 and C2 fluorocarbons and oxidised fluorocarbons (along with C3F6 and

CF3CHFCF2) were calculated with the Gaussian-3 (G3) method (along with several

approximations thereto). These molecules are of importance in the flame chemistry of

2-H-heptafluoropropane, which has been proposed as a potential fire retardant with which to

replace chloro- and bromofluorocarbons (CFC’s and BFC’s). The calculation of the data

reported here was carried out in parallel with the modelling studies of Hynes et al.1-3 of shock

tube experiments on CF3CHFCF3 and on C3F6 with either hydrogen or oxygen atoms.

G3 calculations were also employed in conjunction with the experimental work of Owens et

al.4 to describe the pyrolysis of CFClBr2 (giving CFCl) at a radiation wavelength of 265 nm.

The theoretical prediction of the dissociation energy of the two C-Br bonds was found to be

consistent with the energy at which carbene production was first observed, thus supporting the

hypothesis that the pyrolysis releases two bromine radicals (rather than a Br2 molecule). On

the basis of this interpretation of the experiments, the heat of formation of CFClBr2 is

predicted to be 184 ± 5 kJ mol−1, in good agreement with the G3 value of 188 ± 5 kJ mol−1.

Accurate thermochemical data was computed for 18 small phosphorus containing molecules

(P2, P4, PH, PH2, PH3, P2H2, P2H4, PO, PO2, PO3, P2O, P2O2, HPO, HPOH, H2POH, H3PO,

HOPO and HOPO2), most of which are important in the reaction model introduced by

Twarowski5 for the combustion of H2 and O2 in the presence of phosphine. Twarowski

reported that the H + OH recombination reaction is catalysed by the combustion products of

PH3 and proposed two catalytic cycles, involving PO2, HOPO and HOPO2, to explain this

observation. Using our thermochemical data we computed the rate coefficients of the most

Abstract

important reactions in these cycles (using transition state and RRKM theories) and confirmed

that at 2000K both cycles have comparable rates and are significantly faster than the

uncatalysed H + OH recombination.

The heats of formation used in this work on phosphorus compounds were calculated using the

G2, G3, G3X and G3X2 methods along with the far more extensive CCSD(T)/CBS type

scheme. The latter is based on the evaluation of coupled cluster energies using the correlation

consistent triple-, quadruple- and pentuple-zeta basis sets and extrapolation to the complete

basis set (CBS) limit along with core-valence correlation corrections (with counterpoise

corrections for phosphorus atoms), scalar relativistic corrections and spin-orbit coupling

effects. The CCSD(T)/CBS results are consistent with the available experimental data and

therefore constitute a convenient set of benchmark values with which to compare the more

approximate Gaussian-n results. The G2 and G3 methods were found to be of comparable

accuracy, however both schemes consistently underestimated the benchmark atomisation

energies. The performance of G3X is significantly better, having a mean absolute deviation

(MAD) from the CBS results of 1.8 kcal mol−1, although the predicted atomisation energies

are still consistently too low. G3X2 (including counterpoise corrections to the core-valence

correlation energy for phosphorus) was found to give a slight improvement over G3X,

resulting in a MAD of 1.5 kcal mol−1. Several molecules were also identified for which the

approximations underlying the Gaussian-n methodologies appear to be unreliable; these

include molecules with multiple or strained P-P bonds.

The potential energy surface of the NNH + O system was investigated using density

functional theory (B3LYP/6-31G(2df,p)) with the aim of determining the importance of this

route in the production of NO in combustion reactions. In addition to the standard reaction

channels, namely decomposition into NO + NH, N2 + OH and H + N2O via the ONNH

intermediate, several new reaction pathways were also investigated. These include the direct

abstraction of H by O and three product channels via the intermediate ONHN, giving N2 +

OH, H + N2O and HNO + N. For each of the species corresponding to stationary points on the

B3LYP surface, valence correlated CCSD(T) calculations were performed with the

aug-cc-pVxZ (x = Q, 5) basis sets and the results extrapolated to the complete basis set limit.

Core-valence correlation corrections, scalar relativistic corrections and spin orbit effects were

also included in the resulting energetics and the subsequent calculation of thermochemical

data. Heats of formation were also calculated using the G3X method. Variational transition

Abstract

state theory was used to determine the critical points for the barrierless reactions and the

resulting B3LYP energetics were scaled to be compatible with the G3X and CCSD(T)/CBS

values. As the results of modelling studies are critically dependent on the heat of formation of

NNH, more extensive CCSD(T)/CBS calculations were performed for this molecule,

predicting the 0298f H∆ to be 60.6 ± 0.5 kcal mol−1. Rate coefficients for the overall reaction

processes were obtained by the application of multi-well RRKM theory. The thermochemical

and kinetic results thus obtained were subsequently used in conjunction with the GRIMech

3.0 reaction data set in modelling studies of combustion systems, including methane / air and

CO / H2 / air mixtures in completely stirred reactors. This study revealed that, contrary to

common belief, the NNH + O channel is a relatively minor route for the production of NO.

The structure of the inhibitor Nδ-(N′-Sulfodiaminophosphinyl)-L-ornithine, PSOrn, and the

nature of its binding to the OTCase enzyme was investigated using density functional

(B3LYP) theory. The B3LYP/6-31G(d) calculations on the model compound, PSO, revealed

that, while this molecule could be expected to exist in an amino form in the gas phase, on

complexation in the active site of the enzyme it would be expected to lose two protons to form

a dinegative imino tautomer. This species is shown to bind strongly to two H3CNHC(NH2)2+

moieties (model compounds for arginine residues), indicating that the strong binding observed

between inhibitor and enzyme is partially due to electrostatic interactions as well as extensive

hydrogen bonding (both model Arg+ residues form hydrogen bonds to two different sites on

PSO). Population analysis and hydrogen bonding studies have revealed that the

intramolecular bonding in this species consists of either single or semipolar bonds (that is, S

and P are not hypervalent) and that terminal oxygens (which, being involved in semipolar

bonds, carry negative charges) can be expected to form up to 4 hydrogen bonds with residues

in the active site.

In the course of this work several new G3 type methods were proposed, including

G3MP4(SDQ) and G3[MP2(Full)], which are less expensive approximations to G3, and

G3X2, which is an extension of G3X designed to incorporate additional electron correlation.

As noted earlier, G3X2 shows a small improvement on G3X; G3MP4(SDQ) and

G3[MP2(Full)], in turn, show good agreement with G3 results, with MAD’s of ~ 0.4 and

~ 0.5 kcal mol−1 respectively.

Abstract

1. R. G. Hynes, J. C. Mackie and A. R. Masri, J. Phys. Chem. A, 1999, 103, 5967.


3. R. G. Hynes, J. C. Mackie and A. R. Masri, Proc. Combust. Inst., 2000, 28, 1557.

4. N. L. Owens, Honours Thesis, School of Chemistry, University of Sydney, 2001.

5. A. Twarowski, Combustion and Flame, 1995, 102, 41.

Publications

Parts of this work have been published or submitted for publication in the following journal

articles:

N. L. Haworth, M. H. Smith, G. B. Bacskay, J. C. Mackie

Heats of Formation of Hydrofluorocarbons Obtained by Gaussian-3 and Related Quantum

Chemical Computations.

J. Phys. Chem. A 2000, 104, 7600.

N. L. Haworth, G. B. Bacskay, J. C. Mackie

The Role of Phosphorus Dioxide in the H + OH Recombination Reaction: Ab Initio Quantum

Chemical Computation of Thermochemical and Rate Parameters.

J. Phys. Chem. A 2002, 106, 1533.

N. L. Haworth, G. B. Bacskay

The Structure of Nδ-(N′-Sulfodiaminophosphinyl)-L-ornithine and its Binding to Ornithine

Transcarbamoylase: A Quantum Chemical Study.

Molecular Simulation 2002, 28, 773.

N. L. Haworth, G. B. Bacskay

Determination of Accurate Quantum Chemical Energies and Heats of Formation for

Phosphorus Compounds.

J. Chem. Phys. 2002, 117, 11175.

N. L. Owens, B. K. Nauta, S. H. Kable, N. L. Haworth, G. B. Bacskay

An Experimental and Theoretical Investigation of the Triple Fragmentation of CFClBr2 by

Photolysis near 250nm.

Chem. Phys. Lett. 2003, 370, 469.

N. L. Haworth, G. B. Bacskay, J. C. Mackie

An Ab Initio Quantum Chemical and Kinetic Study of the NNH + O Reaction Potential

Energy Surface: How Important is this Route to NO in Combustion?

J. Phys. Chem. A, in press.

The following publications are also closely related to the work presented in this thesis.

J. C. Mackie, G. B. Bacskay, N. L. Haworth

Reactions of Phosphorus-Containing Species of Importance in the Catalytic Recombination of

H + OH: Quantum Chemical and Kinetic Studies.

J. Phys. Chem. A 2002, 106, 10825.

J. C. Mackie, N. L. Haworth, G. B. Bacskay

How Important is the NNH + O Route to NO in Combustion?

2003 Australian Symposium on Combustion & The 8th Australian Flames Days, Melbourne,

December 8-9, 2003, submitted.

Table of Contents

1 Introduction ................................................................................................... 1

1.1 References...................................................................................................................6

2 Theoretical Methods of Quantum Chemistry............................................. 8

2.1 Introduction.................................................................................................................9

2.1.1 The Born-Oppenheimer Approximation .........................................................10

2.2 Ab Initio Quantum Chemistry ..................................................................................12

2.2.1 Many-Electron Wavefunctions........................................................................12

2.2.1.1 The Independent Particle Model............................................................13

2.2.1.2 Antisymmetry ........................................................................................14

2.2.1.3 Configuration Interaction Wavefunctions..............................................15

2.2.1.4 The Variation Principle..........................................................................16

2.2.2 Hartree-Fock Self Consistent Field Theory.....................................................18

2.2.2.1 The Self Consistent Field (SCF) Procedure...........................................23

2.2.2.2 Spin Unrestricted Hartree-Fock Theory (UHF).....................................24

2.2.2.3 Spin Restricted Closed Shell Hartree-Fock Theory (RHF) ...................26

2.2.2.4 Spin Restricted Open Shell Hartree-Fock Theory (ROHF)...................27

2.2.3 Electron Correlation ........................................................................................28

2.2.3.1 Multiconfigurational SCF Theory (MCSCF).........................................29

2.2.3.2 Configuration Interaction (CI) ...............................................................30

2.2.3.3 Møller-Plesset Perturbation Theory (MPPT).........................................33

2.2.3.4 Coupled Cluster Theory (CC)................................................................34

2.2.3.5 Quadratic Configuration Interaction (QCI) ...........................................38

2.3 Density Functional Theory .......................................................................................39

2.3.1 The Kohn-Sham Equations..............................................................................39

2.3.2 The Local Density Approximation (LDA) ......................................................41

2.3.3 Corrections to the LDA ...................................................................................43

2.3.4 Implementation of DFT...................................................................................45

2.4 Basis sets...................................................................................................................46

2.4.1 Gaussian Type Orbitals ...................................................................................47

2.4.2 Construction of Contracted Gaussian Basis Sets.............................................48

2.4.3 Pople’s Gaussian Basis Sets ............................................................................49

2.4.4 Correlation Consistent Basis Sets....................................................................49

2.4.5 Basis Set Superposition Error..........................................................................50

2.5 Derivatives of the Energy .........................................................................................52

2.5.1 Analytic Energy Derivatives ...........................................................................52

2.5.2 Geometric Derivatives.....................................................................................54

2.6 Molecular Properties.................................................................................................57

2.6.1 Geometry Optimisation ...................................................................................57

2.6.1.1 Partial Geometry Optimisation ..............................................................60

2.6.2 Normal Mode Analysis....................................................................................60

2.7 Computational Strategies for Chemical Accuracy....................................................63

2.7.1 Isodesmic and Isogyric Reaction Schemes......................................................63

2.7.2 Gaussian-n (Gn) Methods................................................................................65

2.7.2.1 Gaussian-1 (G1) Theory ........................................................................65


2.7.2.2.1 G2-RAD Theory...........................................................................68


2.7.2.3.1 G3-RAD Theory...........................................................................70

2.7.2.4 Gaussian-3X (G3X) Theory...................................................................71

2.7.2.5 G3X2 Theory .........................................................................................71

2.7.3 Complete Basis Set Methods...........................................................................72

2.8 Thermochemistry......................................................................................................75

2.8.1 Partition Functions ..........................................................................................75

2.8.2 Thermodynamic Properties .............................................................................78

2.9 Kinetics .....................................................................................................................79

2.9.1 Transition State Theory (TST) ........................................................................79

2.9.2 Variational Transition State Theory (VTST) ..................................................81

2.9.3 RRKM Theory.................................................................................................81

2.10 Population Analysis ..................................................................................................85

2.11 References.................................................................................................................89

3 Thermochemistry of Fluorocarbons..........................................................96

3.1 Introduction...............................................................................................................97

3.2 Theory and Computational Methods ......................................................................100

3.3 Results and Discussion ...........................................................................................105

3.3.1 Heats of Formation from G3 and Related Atomisation Energies..................105

3.3.2 Heats of Formation from G3 and Related Isodesmic Reaction Enthalpies ...114

3.3.3 Comparison of G2 and G3 Methods: Analysis of Atomisation Energies

of Fluoromethanes .........................................................................................122

3.3.4 Heats of Formation by Complete Basis Set Coupled Cluster

Calculations ...................................................................................................126

3.4 Conclusion ..............................................................................................................130

3.5 References...............................................................................................................131

4 The Role of Phosphorus Compounds in the H + OH Recombination

Reaction......................................................................................................136

4.1 Introduction.............................................................................................................137



4.3.1 G2, G3, and G3X Thermochemistry .............................................................142

4.3.2 Reliability of G3, G3X and Related Methods ...............................................146

4.3.2.1 PO and G3(RAD) Procedures..............................................................147

4.3.2.2 Comparison with QCISD(T,Full) ........................................................150

4.4 Kinetic Parameters..................................................................................................152

4.5 Conclusion ..............................................................................................................161

4.6 References...............................................................................................................162

5 Accurate Thermochemistry of Phosphorus Compounds ......................165

5.1 Introduction.............................................................................................................166



5.3.1 CCSD(T) Benchmark Calculations ...............................................................172

5.3.1.1 Testing the B3LYP Geometry .............................................................172

5.3.1.2 Atomisation Energies and Extrapolation Schemes ..............................175

5.3.1.3 Core-Valence Correlation, BSSE and Scalar Relativistic Effects .......180

5.3.2 G3, G3X and G3X2 Calculations..................................................................182

5.3.2.1 Analysis of Molecules for which G3n Methods Perform Poorly ........188

5.3.2.1.1 P4 ................................................................................................188

5.3.2.1.2 P2O, P2, P2H2 ..............................................................................190

5.3.3 Enthalpies of Formation ................................................................................193

5.4 Conclusion ..............................................................................................................196

5.5 References...............................................................................................................197

6 The Role of the NNH + O Reaction in the Production of NO in

Flames.........................................................................................................201

6.1 Introduction.............................................................................................................202


6.2.1 Quantum Chemical Calculations of Thermochemistry .................................205

6.2.2 Derivation of Rate Coefficients for Individual Reaction Channels...............207


6.3.1 Quantum Chemistry.......................................................................................209

6.3.2 Potential Energy Surfaces and Reaction Paths..............................................215

6.3.3 Kinetic Parameters.........................................................................................226

6.3.4 Comparison with Experiment........................................................................229

6.3.5 Kinetic Modelling..........................................................................................230

6.4 Conclusions.............................................................................................................234

6.5 References...............................................................................................................235

7 The Enthalpy and Mechanism of the Photolysis of CFClBr2................239

7.1 Introduction.............................................................................................................240

7.2 Experimental Methods and Results ........................................................................242

7.2.1 Methodology..................................................................................................242

7.2.2 Results ...........................................................................................................242

7.3 Theoretical Methods and Results............................................................................248

7.3.1 Methodology..................................................................................................248

7.3.2 Results ...........................................................................................................250

7.4 Discussion...............................................................................................................251

7.5 Conclusion ..............................................................................................................254

7.6 References...............................................................................................................255

8 The Molecular Structure and Intra- and Inter-Molecular Bonding

of PSOrn.....................................................................................................257

8.1 Introduction.............................................................................................................258

8.2 Methods ..................................................................................................................260


8.3.1 Free (Model) Inhibitor...................................................................................261

8.3.2 Bound (Model) Inhibitor ...............................................................................264

8.3.3 Charge Distribution and Bonding..................................................................271

8.3.3.1 Population Analysis .............................................................................273

8.3.3.2 Hydrogen Bonding...............................................................................276

8.4 Conclusion ..............................................................................................................280

8.5 References...............................................................................................................281

9 Conclusion..................................................................................................282

Appendix 1 Fluorocarbons Supplementary Material ................................A1-1

Appendix 2 Phosphorus Compounds Supplementary Material ...............A2-1

Appendix 3 NNH + O Supplementary Material.........................................A3-1

1 Introduction

Chapter 1

Introduction

Chapter 1. Introduction

2

Computational quantum chemistry is a cornerstone of modern theoretical chemistry. Research

in this field is concerned with the description of atoms, molecules and solids at a fundamental

electronic level. Such a description enables us to determine various properties of these

systems through computation rather than via experiment; theoretical studies therefore provide

excellent sources of information when experimental data is impossible or difficult to obtain

and when additional data is required for the interpretation or confirmation of experimental

results.

In this thesis the application of computational quantum chemistry to several important

molecular problems is described; in particular, the calculation of accurate thermochemical

data; the prediction of reaction kinetics and hence the modelling of complex chemical

systems; the mapping and interpretation of molecular potential energy surfaces; and the

interpretation of the nature of inter- and intra-molecular binding in various situations. Five

distinct problems have been investigated in this work: the thermochemistry of fluorocarbons;

the flame chemistry of small phosphorus containing molecules and also of diazenyl (NNH);

the photodissociation of CFClBr2; and finally the elucidation of the structure of the inhibitor

Nδ-(N′-Sulfodiaminophosphinyl)-L-ornithine (PSOrn) when bound to the enzyme ornithine

transcarbamoylase (OTCase) and the source of its extremely high binding constant. In the

course of this research we have also investigated the accuracy and reliability of a range of

computational schemes for the calculation of thermochemical data and proposed several

modifications which are intended to provide either improved accuracy or a reduction in

computational expense.

With the introduction of the Montreal Protocol limiting the use of chloro- and bromo-

fluorocarbon molecules (CFC’s and BFC’s), interest has turned to fluorocarbons themselves

as potential replacements for use as fire retardants.1-6 While fluorine atoms do not act

catalytically to quench flames (unlike Cl and Br), their strong binding to hydrogen does result

in rapid flame extinguishment. As this process is not catalytic, fluorine rich molecules, such

as CF3CHFCF3 and C3F6, are favoured for this purpose.4-6 Consequently, these species have

been the subject of a number of recent shock tube experiments in order to elucidate their

reaction and decomposition mechanisms.7-9 Although experimental and/or theoretical

thermochemical data have been reported for many of the species involved in these reactions,

for some molecules, including CF3CHFCF3 itself, prior to this work there were no data


3

available while for others the precision was relatively low. We have used the G3 method (and

two approximate versions thereof) to calculate the molecular energies and heats of formation

for a set of ~ 120 C1 and C2 fluorocarbons and oxidised fluorocarbons as well as CH3CHCH2,

CH3CH2CH2, CF3CFCF2 and CF3CHFCF2. The use of isodesmic reaction schemes in order to

improve the accuracy of these data was also investigated. The results are reported and

discussed in Chapter 3, along with more accurate CCSD(T)/CBS calculations for several

selected molecules for which the G3 results differed significantly from experimental values.

These CBS calculations have confirmed the accuracy of the G3 heats of formation.

Phosphorus containing molecules have also been proposed as potential fire retardants;10 this

was largely inspired by the work of Twarowski11-14 who showed that catalytic amounts of the

decomposition products of phosphine could catalyse the recombination of H and OH radicals.

Two reaction schemes were proposed to explain this catalysis; these involve the

recombination of either H or OH with a PO2 radical to give HOPO or HOPO2 respectively,

followed by abstraction by a hydrogen atom to regenerate the catalytic PO2 and release water

or H2. Unfortunately, at the time of Twarowski’s investigation the available experimental and

theoretical thermochemical data was not sufficiently accurate to allow the reliable prediction

of relative reaction rates. The work presented in Chapters 4 and 5 describes the use of G2,

G3, G3X and CCSD(T)/CBS type schemes to calculate accurate thermochemical data for the

molecules involved in these catalytic cycles and the subsequent prediction of reliable reaction

rates for the catalysis. As phosphorus is a second row element, larger basis sets and more

extensive calculations (higher levels of theory) are required than for first row elements in

order to obtain a comparable level of accuracy. Given the paucity of reliable experimental

data, the accuracy of computational schemes such as G2, G3 and G3X for phosphorus

containing molecules could not be assessed without the generation of a theoretical benchmark,

namely the CCSD(T)/CBS results. As this method represents the highest level of quantum

chemical theory currently available for this class of molecules, the resulting thermochemistry

is important not only as a benchmark against which the performance of G215, G316, G3X17

and G3X218 (proposed as an improvement on G3X) may be assessed but also as a valuable

resource for any future studies of phosphorus compounds.

The flame chemistry of nitrogen compounds is also of considerable recent interest, in

particular with respect to the production of nitrous oxides, NOx. These species act as


4

pollutants in the atmosphere, and thus, as for CFC’s and BFC’s, they have attracted

restrictions on the amounts which can be vented into the environment. The development of

systems which minimise the generation and release of NOx requires accurate modelling of

nitrogen flame chemistry. While NO production via the thermal, prompt-NO, N2O and fuel-

NO routes has been recognised for some time19, more recently another source of NO, from the

reaction of NNH with oxygen atoms, has been proposed20. Although several thermochemical

calculations and modelling studies have been reported for relevant reactions of this system,20-

23 some potentially important reaction channels were not considered. This means that the

results of the modelling studies reported to date may not be reliable, which could, at least in

part, account for the apparent overprediction of NO production observed in several modelling

studies.24,25 Chapter 6 describes a thorough investigation of the NNH + O potential energy

surface, including: the identification of all stationary points and potential reaction paths;

accurate calculations of thermochemical data at each stationary point; and mapping of the

PES along each of the reaction coordinates. The rates for each of these reactions were then

calculated using transition state theory and RRKM, followed by modelling studies to

determine the importance of this route for NO production in combustion systems.

As noted earlier, the use of bromofluorocarbons has been limited by the Montreal Protocol

due to the activity of the bromine atoms (produced by combustion or by UV photolysis in the

atmosphere) in depleting stratospheric ozone. The photolysis mechanisms of these species are

thus of considerable interest. It has been observed that some halomethane species, namely

CF2Br2, CF2I2 and CF2BrI, photolyse via a triple fragmentation pathway (loss of Br and/or I

atoms) at relatively long wavelengths (> 200 nm)26-31; it was also noted that only the

difluoromethanes appear to undergo this triple fragmentation. Recent experiments, however,

have succeeded in producing CFCl from the CFClBr2 dibromomethane at a wavelength of

265 nm.32,33 In order to help determine the heat of formation of CFClBr2 and to aid with the

establishment of the mechanism of carbene production, G3 calculations were performed. Of

particular importance was to resolve whether this photolysis can occur via a triple

fragmentation pathway. The joint experimental and theoretical investigation of this problem is

reported in Chapter 7.

PSOrn is the active component of a natural toxin, phasolotoxin.34,35 This toxin is a powerful

enzyme inhibitor36,37, binding to the enzyme ornithine transcarbamoylase (OTCase) with a


5

dissociation constant of 1.6 × 10−12 M at 37°C and pH = 8.38 OTCase acts to catalyse the

reaction between carbamoyl phosphate and L-ornithine to form L-citrulline and phosphate. As

such it is essential for the biosynthesis of arginine in plants and microbes and acts as part of

the urea cycle in mammals; such strong inhibition of the enzyme therefore results in cell

death. Consequently, it is of great interest to determine the nature of this strong binding. In

addition, PSOrn has a highly unusual molecular structure, containing a P-N-S linkage, thus

the nature of the intramolecular bonding also warrants investigation. The X-ray crystal

structure of PSOrn when bound to OTCase has recently been reported by Langley et al.38

While this shows the positions of enzyme residues around the active site (and thus indicates

possible hydrogen bonds between enzyme and inhibitor) hydrogen atoms themselves are, of

course, not revealed. As a result there is some question over whether the nitrogen of the P-N-

S linkage is protonated in a (chemically expected) amino form or deprotonated to give an

imino structure. The relative stabilities of various amino and imino isomers were investigated

both when bound to selected (model) enzyme residues and in the gas phase using density

functional theory, specifically B3LYP/6-31G(d). Roby-Davidson population analyses were

carried out in an effort to determine whether the bonds in PSOrn were single, double or

semipolar and to estimate the charges carried by the atoms. The hydrogen bonding potential

of some of these atoms was also investigated, so as to aid in the interpretation of the hydrogen

bonding pattern observed in the crystal structure. The results of this work are presented in

Chapter 8.


6

1.2 References

1. M. D. Nyden, G. T. Linteris, D. R. Burgess, Jr., P. R. Westmoreland, W. Tsang and

M. R. Zachariah, Flame Inhibition Chemistry and the Search for Additional Fire

Fighting Chemicals in Evaluation of Alternative In-Flight Fire Suppressants for Full-

Scale Testing in Simulated Aircraft Engine Nacelles and Dry Bays, W. Grosshandler,

R. Gann, and W. Pitts, Eds.; NIST Special Publication 861; National Institute of

Standards and Technology: Washington, D.C., 1994. p. 467.

2. M. R. Zachariah, P. R. Westmoreland, D. R. Burgess, Jr., W. Tsang and C. F. Melius,

J. Phys. Chem., 1996, 100, 8737.

3. D. R. Burgess, Jr., M. R. Zachariah, W. Tsang and P. R. Westmoreland, Prog. Ener.

Comb. Sci., 1995, 21, 453.

4. O. Sanogo, J.-L. Delfau, R. Akrich and C. Vovelle, Combust. Sci. Technol., 1997, 122,

33.

5. G. T. Linteris, D. R. Burgess, Jr., V. Babushok, M. Zachariah, W. Tsang and P.

Westmoreland, Combust. Flame, 1998, 113, 164.

6. R. G. Hynes, J. C. Mackie and A. R. Masri, Combust. Flame, 1998, 113, 554.




10. J. Green, Fire Sciences, 1996, 14, 426.





15. L. A. Curtiss, K. Raghavachari, G. W. Trucks and J. A. Pople, J. Chem. Phys., 1991,

94, 7221.

16. L. A. Curtiss, K. Raghavachari, P. C. Redfern, V. Rassolov and J. A. Pople, J. Chem.

Phys., 1998, 109, 7764.

17. L. A. Curtiss, P. C. Redfern, K. Raghavachari and J. A. Pople, J. Chem. Phys., 2001,

114, 108.

18. N. L. Haworth, G. B. Bacskay and J. C. Mackie, J. Phys. Chem. A, 2002, 106, 1533.

19. J. A. Miller and C. T. Bowman, Prog. Energy Combust. Sci., 1989, 15, 287.


7

20. J. W. Bozzelli and A. M. Dean, Int. J. Chem. Kinet., 1995, 27, 1097.

21. J. A. Harrison and R. G. A. R. MacIagan, J. Chem. Soc. Faraday Trans., 1990, 86,

3519.

22. J. L. Durant, Jr., J. Phys. Chem., 1994, 98, 518.

23. S. P. Walch, J. Chem. Phys., 1993, 98, 1170.

24. D. Charlston-Goch, B. L. Chadwick, R. J. S. Morrison, A. Campisi, D. D. Thomsen

and N. M. Laurendeau, Combust. Flame, 2001, 125, 729.

25. A. A. Konnov and J. De Ruyck, Combust. Flame, 2001, 125, 1258.

26. P. Felder, X. Yang, G. Baum and J. R. Huber, Israel J. Chem., 1993, 34, 33.

27. T. R. Gosnell, A. J. Taylor and J. L. Lyman, J. Chem. Phys., 1991, 94, 5949.

28. J. van Hoeymissen, W. Uten and J. Peeters, Chem. Phys. Lett., 1994, 226, 159.

29. M. R. Cameron, S. A. Johns, G. F. Metha and S. H. Kable, Phys. Chem. Chem. Phys.,

2000, 2, 2539.

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Chem. A, 2001, 105, 5606.

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32. J. S. Guss, O. Votava and S. H. Kable, J. Chem. Phys., 2001, 115, 11118.


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1985, 228, 347.

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227.

37. A. S. Bachmann, P. Matile and A. J. Slusarenko, Physiol. Mol. Plant Pathol., 1998,

53, 287.

38. D. B. Langley, M. D. Templeton, B. A. Fields, R. E. Mitchell and C. A. Collyer, J.

Biol. Chem., 2000, 275, 20012.

2 Theoretical Methods of Quantum Chemistry

Chapter 2

Theoretical Methods

of Quantum

Chemistry

Chapter 2. Theoretical Methods

9

2.1 Introduction

Quantum chemistry is (naturally) based on the principles of quantum physics first developed

in the 1920’s by such pioneers of modern physics as Heisenberg, Bohr, Sommerfeld, Born,

Pauli, Schrödinger and Dirac. At its heart quantum chemistry is concerned with finding the

eigenfunctions and eigenvalues of the time independent Schrödinger equation1-3:

ˆi i iH EΨ = Ψ (2.1.1)

where H is the molecular Hamiltonian operator, Ψi is the total wavefunction of the i-th

electronic state and Ei is the corresponding energy eigenvalue of the system of interest.

Evaluation of the total energy of a system is, of course, of great value; in addition, knowledge

of the wavefunction enables one to predict many other important properties of the atom,

molecule or solid.

In this work, as in the majority of quantum chemical calculations to date, the non-relativistic

Hamiltonian operator has been used:

ˆ ˆ ˆ ˆ ˆ ˆN e NN ee NeH T T V V V= + + + + (2.1.2)

where TN and Te are the kinetic energy operators for nuclei and electrons respectively:

2

1

1 1ˆ2

N

N II I

TM=

= − ∇∑ (2.1.3)

2

1

1ˆ2

n

e ii

T=

= − ∇∑ (2.1.4)

and VNN , Vee and VNe are the Coulombic potential energy operators representing the inter-

nuclear and inter-electron repulsions and the attraction between nuclei and electrons:

ˆ| |

NI J

NNI J I J

Z ZV

<

=−∑

R R(2.1.5)


10

1ˆ| |

n

eei j i j

V<

=−∑r r

(2.1.6)

1 1

ˆ| |

N nI

NeI i I i

ZV

= =

= −−∑∑

R r(2.1.7)

In the above equations (and throughout this thesis unless otherwise noted) uppercase letters

have been used to denote coordinates and indices relating to nuclei and lowercase for those

relating to electrons. Thus N is the total number of nuclei, n is the total number of electrons,

MI , ZI and RI are the mass, charge and position vector of the I-th nucleus and ri is the

position vector of the i-th electron. Atomic units have been used here and throughout this

work unless indicated otherwise.

Unfortunately, analytic solutions of the Schrödinger equation exist only for the simplest

systems which contain no more than two interacting particles. Real systems, that is, atoms,

molecules and solids, contain many interacting electrons and nuclei and thus approximations

must be made to allow solutions to be found. A basic aspect of quantum chemistry involves

the development of approximate yet accurate and efficient methods for calculating

wavefunctions and energy eigenvalues. The following sections describe in detail the

necessary approximations and the various resulting quantum chemical methodologies.

2.1.1 The Born-Oppenheimer Approximation

The first of these simplifications (for molecules and solids) is the Born-Oppenheimer

approximation4,5. It is based upon the understanding that, as electrons have much lower

masses than nuclei (by at least three orders of magnitude), they move much more quickly and

as such, to a good approximation, the electrons can be regarded as being able to respond

instantaneously to a change in nuclear geometry. The nuclear and electronic motions are thus

said to be “decoupled”. The total wavefunction of a given electronic state can therefore be

separated into two components: one which describes the nuclear motion, ( ){ }KΘ R (where

each ( )KΘ R represent a ro-vibrational state of the molecule), and one which describes the

motion of the electrons, ψ R rb g , for a given nuclear configuration, R:

( ) ( ) ( ), RψΨ = Θ ×R r R r (2.1.8)


11

The electronic Schrödinger equation is thus constructed and solved for a fixed nuclear

configuration using the Born-Oppenheimer (clamped nuclei) Hamiltonian:

( ) ( )ˆBO R R RH Eψ ψ=r r (2.1.9)

( ) ( ) ( ) ( )ˆ ˆ ˆ ˆe NN Ne ee R R RT V V V Eψ ψ + + + = R R r r (2.1.10)

resulting in the total electronic wavefunction, ψ R rb g , and the energy, ER , of the system for a

given nuclear configuration. The energies, ERl q , for all possible nuclear configurations form

a potential energy surface for the molecule. The nuclear (ro-vibrational) wavefunctions,

( ){ }KΘ R , can in turn be obtained by solving the nuclear Schrödinger equation:

( ) ( )N R K K KT E ε + Θ = Θ R R (2.1.11)

Errors due to the Born-Oppenheimer approximation are generally small and relatively

unimportant in chemical applications except in systems where the electronic states are

degenerate or near degenerate. In such cases the electronic states are coupled by the nuclear

motion and the wavefunction needs to be expressed as

( ) ( ) ( ),

, ,mK m Km K

c ψΨ = ×Θ∑R r R r R (2.1.12)

where the summation is over the ro-vibrational states and the electronic states which are

(near) degenerate. Such situations were not encountered in this work.


12

2.2 Ab Initio Quantum Chemistry

2.2.1 Many-Electron Wavefunctions

The electronic wavefunction, ψ R rb g , introduced above must describe the motion of all of the

electrons in the system simultaneously; it is therefore a many-electron wavefunction. In

general many-electron, viz. n-electron, wavefunctions are constructed as linear superpositions

of n-electron basis functions (called configuration state functions or CSF’s)6:

( ) ( )i ki kk

aψ φ=∑r r (2.2.1)

where ψ i rb g is the electronic wavefunction of i-th electronic state of the system (at a

particular geometry), φ k rb gm r are the configuration state functions and akil q are numerical

coefficients which can be optimised, as will be described below, so as to obtain as accurate a

description of the electronic wavefunctions of the system as possible (within the confines of

the finite basis expansion approach).

In the majority of applications the n-electron CSF’s are constructed as antisymmetrised

products of one-electron wavefunctions; these are generally atomic or molecular orbitals.

CSF’s are often defined as linear combinations of these products, such that a given CSF is

spin and symmetry adapted.

The atomic or molecular orbitals are, in turn, constructed from sets of linearly independent

one-electron basis functions:

1

m

i pi pp

cϕ χ=

=∑ (2.2.2)

In most modern applications, these basis functions, { }pχ , are atom-centred Gaussian type

functions. The coefficients of the basis functions are also optimised in order to give the best

possible description of the atomic or molecular orbitals.


13

More detailed descriptions of the formulation and optimisation of one- and many-electron

wavefunctions are presented in later sections.

2.2.1.1 The Independent Particle Model

Just as the nuclear and electronic motions are separated using the Born-Oppenheimer

approximation, the motions of the different electrons in a many-electron wavefunction can

also be separated. Thus, to a first approximation, an n-electron CSF, ( )1 2 3, , ,...,R nφ r r r r , is

expressed as a product of one electron spin orbitals:

( ) ( ) ( ) ( ) ( )

( )

1 2 3 1 1 2 2 3 3

1

, , ... ...R n n n

n

i ii

φ ϕ ϕ ϕ ϕ

ϕ=

=

=∏

r r r r r r r r

r(2.2.3)

where ϕ i irb g is the spin orbital of the i-th electron with position vector ri .

This is called the independent particle model. It is the original model used by Hartree7 in his

pioneering work on atoms and is potentially exact for systems of non-interacting particles.

Electronic wavefunctions formed as products of individual electron spin orbitals are therefore

known as Hartree products.

Most systems of interest, however, contain particles (electrons and nuclei) which do interact

with each other; in these cases the independent particle model assumes that each electron

moves independently of every other in the field of the nuclei and the average field of all the

other electrons. While it is immediately clear that this is a much more severe approximation

than the Born-Oppenheimer one (neglecting, most significantly, the fact that the total

wavefunction must be antisymmetric and also not accounting for the effects of dynamic

electron correlation, that is, the fact that individual electrons avoid each other), it allows for

significant simplification of the problem of interest. Errors introduced with this approximation

can, however, be corrected for at a later stage as described in Sections 2.2.1.3 and 2.2.3.


14

In practice, as noted earlier, accurate spin orbitals, { }iϕ , are obtained by constructing linear

combinations of m atom-centred Gaussian type basis functions, χ p , with coefficients, cpi :

1

m

i pi pp

cϕ χ=

=∑ (2.2.4)

In order to allow for adequate flexibility in the description of the orbitals, m must be

significantly larger than the number of occupied orbitals in the system. This immediately

introduces linearly independent virtual (unoccupied) orbitals (in addition to the occupied

ones). Further details on the construction of basis sets are given in Section 2.4.

2.2.1.2 Antisymmetry

Electrons are indistinguishable particles and, as such, the properties of the system should be

invariant to the interchange of the coordinates of any two electrons. In particular, the

probability density, φ rb g 2 , must remain unchanged.

As electrons are fermions (and therefore obey Fermi-Dirac statistics), the many-electron

wavefunctions must also be antisymmetric with respect to this interchange of electron

coordinates. Applying the permutation operator, Pij , to an n-electron wavefunction,φ rb g ,should, therefore, result in a change in sign:

( ) ( )( )

1 2 1 2

1 2

ˆ , , , , , , , , , , , , , ,

, , , , , , ,

ij i j n j i n

i j n

P φ φ

φ

=

= −

r r r r r r r r r r

r r r r r(2.2.5)

The Hartree products described above are clearly not antisymmetric. They can be made so,

however, by the application of an antisymmetriser, A , defined by:

( )1ˆ ˆ1!

p

P

A Pn

= −∑ (2.2.6)


15

Here the sum is over all possible permutation operators, P , for n identical particles (including

the identity); p is the parity of the relevant permutation.

The application of the antisymmetriser to a Hartree product results in a determinant:

( ) ( ) ( ) ( ) ( )( )( ) ( ) ( ) ( )( ) ( ) ( ) ( )( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

1 1 2 2 3 3

1 1 1 2 1 3 1

2 1 2 2 2 3 2

3 1 3 2 3 3 3

1 2 3

ˆ ...

1

!

n n

n

n

n

n n n n n

A

n

φ ϕ ϕ ϕ ϕ

ϕ ϕ ϕ ϕϕ ϕ ϕ ϕϕ ϕ ϕ ϕ

ϕ ϕ ϕ ϕ

=

=

r r r r r

r r r r

r r r r

r r r r

r r r r

(2.2.7)

If the orbitals, { }iϕ , are orthonormal, the factor 1

n! ensures that such an antisymmetrised

product will be normalised, thus forming an orthonormal set of basis functions.

Such antisymmetrised electronic wavefunctions are generally referred to as Slater

determinants after J. C. Slater who was instrumental in their development.8

2.2.1.3 Configuration Interaction Wavefunctions

The configuration state functions introduced earlier are usually either single Slater

determinants or linear combinations thereof. The set of all possible Slater determinants

(constructed by considering all possible arrangements of the electrons amongst the available

spin orbitals) therefore forms a set of n-electron basis functions for the total electronic

wavefunction of the system of interest, ψ rb g . If the set of one-electron basis functions (and

thus the set of atomic or molecular spin orbitals) is complete (that is, infinite), the resulting set

of Slater determinants (CSF’s) also forms a complete n-electron basis set for ψ rb g . The exact

n-electron wavefunction can therefore be formulated as:

( ) ( )i ki kk

aψ φ=∑r r (2.2.8)


16

This is called the Configuration Interaction (CI) expansion of the wavefunction.6 In practice,

of course, the set of one-electron basis functions is finite and incomplete and thus the

configuration interaction expansion is also finite and can only yield an approximation to the

true total wavefunction. Even with a finite one-electron basis set, however, the full set of

CSF’s for a molecular system may still contain far too many Slater determinants for such

calculations to be computationally feasible. In most applications, therefore, only a subset of

these configurations is used.

For most molecules, especially near their equilibrium geometries, the wavefunction is

dominated by a single CSF. In such cases the Schrödinger equation is first solved subject to

the approximation that the wavefunction consists of only this determinant. This gives both a

reference state wavefunction and a convenient set of optimised one-electron orbitals, { }iϕ ,

which can be used in the construction of other CSF’s. While such single determinant

wavefunctions do not account for the effects of electron correlation (as the independent

particle model has been applied), extending them by the inclusion of additional terms in the

configuration interaction expansion can correct for this deficiency.

Finding solutions of the Schrödinger equation therefore involves finding both the best set of

coefficients for the CSF’s, { }ka , and the optimal set of orbital coefficients, { }pic . These

coefficients can be obtained by the use of the Variation Principle (described in the next

section) or, specifically in the case of the CSF coefficients, by Perturbation Theory9,10.

Sections 2.2.2 and 2.2.3 describe in more detail a range of approaches to this problem.

2.2.1.4 The Variation Principle

Given an approximate wavefunction for a system, the corresponding total energy is, by

definition, the expectation value of the Hamiltonian operator:

[ ] HE

ψ ψψ

ψ ψ= (2.2.9)


17

The Variation Principle (theorem)11 states that if the energy is stationary with respect to any

arbitrary variation, δψ , in the wavefunction, i.e.,

0Eδ = (2.2.10)

then the wavefunction is an eigenfunction of the Hamiltonian:

H Eψ ψ= (2.2.11)

and the lowest eigenvalue, 0E , is an upper bound to the true ground state energy of the

system, ε 0 :

0 0E ε≥ (2.2.12)

Moreover, according to McDonald’s theorem12, the higher eigenvalues, { }iE , are upper

bounds to the corresponding excited state energies, { }iε .

The variational flexibility of most approximate wavefunctions is provided by the orbital and

CI coefficients { }pic and { }ka . Variation of these coefficients can be thought of as mixing or

rotation between occupied and virtual orbitals (for cpi ’s) or among the CSF’s (for ak ’s). A

“variational” wavefunction, giving the lowest possible energy, is therefore stable under such

mixings or rotations.


18

2.2.2 Hartree-Fock Self Consistent Field Theory13,14

As noted earlier, in most typical applications the wavefunction is relatively well described by

a single CSF; the first problem is, therefore, to find the energy and wavefunction of this single

determinantal reference state. This is readily achieved by the application of the Variation

Principle in order to determine the optimal one-electron occupied orbitals for this

wavefunction. This leads to Hartree-Fock Self Consistent Field Theory (HF-SCF).

For a single determinantal wavefunction, φ, the expectation value of the Hamiltonian is given

by:

HE

φ φφ φ

= (2.2.13)

where the (Born-Oppenheimer) Hamiltonian, H , is expressed in terms of one- and two-

electron contributions, as well as nuclear repulsion:

0ˆ ˆˆ ˆi ij

i i j

H h h g<

= + +∑ ∑ (2.2.14)

Here h0 is the internuclear repulsion term:

0ˆ

| |I J I J

NNI J I JI J IJ

Z Z Z Zh V

< <

= = =−∑ ∑

R R R(2.2.15)

hi is a one-electron term which contains both the kinetic energy of electron i and its

Coulombic potential energy in the field of the nuclei:

21ˆ2

Ii i

I iI

Zh = − ∇ −∑

r(2.2.16)


19

and gij is a typical inter-electron repulsion term:

1ˆij

ij

g =r

(2.2.17)

Thus, when expectation values are taken, the h0 term is simply a constant and, with the

application of the Slater-Condon rules15, the expectation value of the one-electron terms

simplifies to:

( )1

1

ˆn

i ii

E hϕ ϕ=

=∑ (2.2.18)

where ϕ il q are the occupied spin orbitals and h is a typical one-electron Hamiltonian

operator.

The expectation value of the two-electron terms is thus:

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ){ }21 2 12 1 2 1 2 12 1 2ˆ ˆ

n

i j i j i j j ii j

n

i j i ji j

E g gϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ

ϕϕ ϕϕ

<

<

= −

=

∑

∑

r r r r r r r r

(2.2.19)

where the summations are over all occupied spin orbitals.

The two terms that make up E 2b g are known as the Coulomb and exchange integrals

respectively. Their joint contribution is conveniently written using the notation: ij ij , where

i stands for the spin orbitals ϕ i , etc. The total electron-electron repulsion energy can also be

rewritten in terms of one-electron Coulomb and exchange operators ( Ji and Ki for each

occupied spin orbital, ϕ i ). These operators are defined through their action on an arbitrary

one-electron function f r1b g :

( ) ( ) ( ) ( )*

2 21 2 1

12

ˆ i iiJ f d f

r

ϕ ϕτ

= ∫

r rr r (2.2.20)


20

( ) ( ) ( ) ( )*

2 21 2 1

12

ˆ ii i

fK f d

r

ϕτ ϕ

= ∫

r rr r (2.2.21)

Thus E 2b g can be rewritten as:

( )2

,

1 ˆ ˆ2

1 ˆ ˆ2

n

i j j ii j

n

i ii

E J K

J K

ϕ ϕ

ϕ ϕ

= −

= −

∑

∑(2.2.22)

where J and K are the total (n-electron) Coulomb and exchange operators:

ˆ ˆn

ii

J J=∑ (2.2.23)

ˆ ˆn

ii

K K=∑ (2.2.24)

The total energy can therefore be written as:

1ˆ ˆ ˆ2

n n

i i i i NNi i

E h J K Vϕ ϕ ϕ ϕ= + − +∑ ∑ (2.2.25)

As noted in Section 2.2.1.4, when the occupied orbitals are fully optimised for a particular

system the energy is stationary with respect to mixing between the occupied orbitals, φ il q ,and the unoccupied (virtual) orbitals, φ al q . The derivative of the energy with respect to this

mixing is given by the Brillouin matrix elements16:

ˆ ai

ai

aiEH

Xφ φ∂ =

∂(2.2.26)

where φ ia represents a singly substituted determinant obtained by the replacement of an

occupied spin orbital, φ i , by an unoccupied orbital, φ a .


21

Application of the Slater-Condon rules leads to:

ˆ ˆ ˆ

ˆ

i a i aai

i

ai

a

Eh J K

X

F

ϕ ϕ ϕ ϕ

ϕ ϕ

∂ = + −∂

=(2.2.27)

Thus the condition for stationary energy is

ˆ 0 ,i aF i aϕ ϕ = ∀ (2.2.28)

where the (one-electron) Fock operator, F , is defined as

ˆˆ ˆ ˆF h J K= + − (2.2.29)

Thus the Brillouin matrix elements vanish for self consistent solutions of the Fock eigenvalue

equations:

ˆi i iF iϕ ε ϕ= ∀ (2.2.30)

where ε il q represents the individual orbital energies. The orbitals which satisfy these

equations (and thus give a stationary energy for the system) are called the canonical Hartree-

Fock SCF orbitals.

It should be noted that the total (n-electron) Fock operator is not equivalent to the

Hamiltonian operator:

( )

( ) ( ) ( )

ˆ ˆ

ˆ ˆ ˆ

Tot ii

i i ii

F F

h J K

=

= + −

∑

∑

r

r r r(2.2.31)


22

while

( )( )

1ˆˆ ˆ ˆ2

1ˆ ˆ ˆ2Tot

H h J K

F J K

= + −

= − −(2.2.32)

Thus the sum of occupied orbital energies, ε ii∑ , differs from the total electronic energy, E,

since the electron-electron repulsion terms are counted twice in the former.

In practice the Hartree-Fock SCF orbitals are found by solving the matrix eigenvalue

equations:

=Fc Scεεεε (2.2.33)

where F is the Fock matrix with elements:

ˆij i jF Fχ χ= (2.2.34)

c is the matrix of eigenvectors which determine the SCF orbitals:

= cϕ χϕ χϕ χϕ χ (2.2.35)

and S is the overlap matrix with elements:

ij i jS χ χ= (2.2.36)

The matrix eigenvalue equations (2.2.33) are generally known as the Roothaan-Hall17,18

equations.


23

2.2.2.1 The Self Consistent Field (SCF) Procedure

Since the Fock operator actually depends on its eigenvectors, { }iϕ (through the construction

of the Coulomb and exchange operators), the Roothaan-Hall equations must be solved using

an iterative procedure.

In most implementations of HF-SCF theory this firstly involves making a guess of the

coefficient matrix, c. This is done by either simply orthogonalising the atomic orbital basis,

by diagonalising the one-electron part of the Hamiltonian:

=hc Scεεεε (2.2.37)

or by utilising a semi-empirical method such as INDO19 or extended Hückel theory20.

Secondly the Fock matrix is constructed and then diagonalised by solving the Roothaan-Hall

equations. This is most easily done if a unitary transformation is performed in order to

orthonormalise the original basis set (so that the overlap matrix becomes the identity). The

standard approach is to use the Löwdin orthogonalisation method21 where the transformation

is made using the 1 2−S matrix:

1 2 1 2 1 2 1 2 1 2 1 2− − − −=S FS S c S SS S cεεεε (2.2.38)

which yields:

=Fc cεεεε (2.2.39)

where:

1 2 1 2− −=F S FS (2.2.40)

1 2=c S c (2.2.41)

and the eigenvalues, εεεε, are (hopefully) a more accurate estimate of the true orbital energies. In

the simplest implementation of SCF optimisation the orbitals obtained in a given

diagonalisation step are used to construct a new Fock matrix, thus allowing a new set of


24

orbitals to be generated. This process can be iterated until the coefficient matrix is unchanged

from one iteration to the next (to within a specified threshold). The orbitals are then said to be

“self consistent”. In most applications damping and convergence accelerating techniques must

be used to ensure reasonably rapid convergence to the final optimised orbitals.22-24

The choice of occupied orbitals (for the construction of F ) is a key aspect of the SCF

procedure. The application of the Aufbau Principle is often adequate, but in more complex

situations a predetermined occupancy may need to be enforced so that the calculations

converge to the state of interest.22

At convergence the total energy of the system is thus given by:

1

2orbi j

E E ij ij≠

= − ∑ (2.2.42)

where Eorb is the total orbital energy:

orb ii

E ε=∑ (2.2.43)

2.2.2.2 Spin Unrestricted Hartree-Fock Theory25

SCF theory formulated in terms of atomic or molecular spin orbitals as described above is

known as (Spin) Unrestricted Hartree-Fock Theory. In practice this gives a Fock matrix, F,

which is block diagonal with respect to the α and β spin orbitals (ϕα and ϕβ or ϕ and ϕ

respectively). The Fock operator, F , can therefore be split into α and β components:

( ) ( )ˆˆ ˆ ˆF h J Kα α= + − (2.2.44)

( ) ( )ˆˆ ˆ ˆF h J Kβ β= + − (2.2.45)


25

The non-zero matrix elements of J , K αb g and K βb g are:

( ) ( )

î j i k j k i k j k

k k

Jα β

α α β βϕ ϕ ϕϕ ϕ ϕ ϕϕ ϕ ϕ= +∑ ∑ (2.2.46)

where ϕ i and ϕ j are (spin) orbitals of the same spin (that is, both α or both β),

( )

( )

î j i k k j

k

Kα

αα α α α α αϕ ϕ ϕ ϕ ϕ ϕ=∑ (2.2.47)

( )

( )

î j i k k j

k

Kβ

ββ β β β β βϕ ϕ ϕ ϕ ϕ ϕ=∑ (2.2.48)

Given that each spin orbital is a product of spatial and spin components; that is,

( ) ( )i iαϕ ϕ α= r σσσσ (2.2.49)

( ) ( )i iβϕ ϕ β= r σσσσ (2.2.50)

where ϕ i rb g is now a spatial orbital and α σσσσb g and β σσσσb g are spin functions with σσσσ

representing the “spin coordinate”, the total wavefunction can be written as an

antisymmetrised product of an n-electron spatial function, θ, and an n-electron spin function,

Θ:

( )( )( )( ) ( )

1 2 3 4 5

1 2 3 4 5

1 2 1 2

ˆ

ˆ

ˆ , , , ,

A

A

A

ψ ϕϕ ϕ ϕ ϕ

ϕ ϕ ϕ ϕ ϕ αβαβα

θ

=

= = Θ r r ó ó

(2.2.51)

Such a wavefunction will always be an eigenfunction of the Sz spin operator as each spin

function is itself an eigenfunction, by definition. The wavefunction may not, however, be an

eigenfunction of the S 2 total spin operator; this is because the total spin function, Θ, is not an

eigenfunction of S 2 but rather a linear combination of several spin eigenfunctions which have

different eigenvalues.


26

Nevertheless, because of its simplicity, the UHF procedure is widely used, especially for open

shell systems. It is capable of providing a qualitatively correct description of bond

dissociation; UHF potential energy surfaces may, however, contain unphysical bifurcation

regions. In the region of equilibrium geometries UHF generally performs well, however

attention must be paid to the expectation value of S 2 . In this work it was found that in most

cases S 2 is close to the desired eigenvalue of S S +1b g (S = ½, 1, 1½ … for open shell

doublet, triplet, quartet, etc. systems) but occasionally the deviation is significant due to

mixing with states with higher spin (spin contamination). It is possible to obtain a more pure

spin state by projection whereby the most serious contaminants are annihilated to give a

Projected Unrestricted Hartree-Fock (PUHF) wavefunction26 which is an (approximate)

eigenfunction of S 2 . Unfortunately in some cases this method is inadequate and the resulting

wavefunction is still significantly contaminated. A better (although more computationally

expensive) alternative is to use a Restricted Hartree-Fock formalism (Section 2.2.2.4) or

Multiconfigurational SCF theory (Section 2.2.3.1).

2.2.2.3 Spin Restricted Closed Shell Hartree-Fock Theory (RHF)

Most stable molecules have singlet ground states, corresponding to closed shell

configurations; that is, each spatial orbital is occupied by a pair of electrons with opposite

spins. The Hartree-Fock wavefunction can thus be written as:

( )1 1 2 2 2 2ˆ ... n nAφ ϕ ϕϕ ϕ ϕ ϕ= (2.2.52)

Such wavefunctions are automatically eigenfunctions of S 2 . In addition, the alpha and beta

Fock matrices are identical (as the wavefunction and energy are clearly invariant under spin

interchange). This means that only one of F αb g and F βb g needs to be evaluated and thus the

computational effort involved is approximately halved. Moreover, closed shell RHF

calculations generally converge faster than their UHF counterparts. For most singlet state

molecules in the neighbourhood of their equilibrium geometries there is, in fact, no distinct

UHF solution; that is, UHF calculations converge to the RHF wave function.


27

2.2.2.4 Spin Restricted Open Shell Hartree-Fock Theory27

As noted earlier, UHF theory can sometimes yield wavefunctions with considerable spin

contamination. In order to avoid this, Restricted Open Shell Hartree-Fock Theory (ROHF)

can be applied; this method has been developed so as to ensure that the resulting

wavefunction is an eigenfunction of S 2 .

ROHF theory involves partitioning the orbital space into a subset, D, which contains doubly

occupied orbitals, a subset, P, which contains orbitals which are allowed to be partially

occupied and a subset, V, which are unoccupied (virtual). When the orbitals are optimised

under the SCF procedure, mixing between all three subsets needs to be considered. These

three types of mixing (D/P, D/V and P/V) give rise to three different Fock operators between

orbitals of different subsets; when the orbitals are fully optimised the energy will be stable

with respect to all possible mixings between the subsets. This condition is known as the

generalised Brillouin theorem22; it corresponds to the appropriate off-diagonal matrix

elements of the Fock operators being zero.

Computationally this method is significantly more expensive than the UHF or RHF

procedures, largely because ROHF wavefunctions are often difficult to converge. In this

work, therefore, it is generally only applied when an earlier UHF calculation has indicated the

need for a restricted formalism. While ROHF theory is most readily applied to high spin open

shell states, it can be generalised to cover more complex situations such as open shell singlet

or state averaged systems.


28

2.2.3 Electron Correlation

The term “electron correlation” is generally used to describe all effects which are not

accounted for by Hartree-Fock theory. This definition was originally proposed by Löwdin28,

who also introduced the concept of the correlation energy, Ecorr , defined by the equation:

corr exact HFE E E= − (2.2.53)

Here Eexact is the exact non-relativistic energy of the system of interest and EHF is the

Hartree-Fock energy. In practice Eexact is not known and must be approximated as described

below.

There are two major phenomena that contribute to the correlation energy. Non-dynamical

correlation is the term used for near-degeneracy effects which are not resolved at the Hartree-

Fock level. This usually only occurs in systems for which the highest energy (formally)

occupied orbitals are close in energy to the (formally) unoccupied orbitals, resulting in several

near-degenerate configurations. In such situations the wavefunction will not be dominated by

a single configuration (determinant), and multiconfigurational methods such as MCSCF (see

below) must be applied to obtain a good reference state.

While non-dynamical correlation only occurs in special situations, dynamical correlation

needs to be considered for all systems. As mentioned earlier, dynamical correlation describes

the fact that individual electrons avoid each other. Although the use of a single determinant

wavefunction in conjunction with the independent particle model (as for Hartree-Fock SCF

Theory) has neglected this effect, it can be corrected for by the inclusion of additional

determinants in the wavefunction. Several methods of varying complexity and accuracy have

been proposed in order to account for the dynamical correlation effects; these include the

configuration interaction method, Møller-Plesset perturbation theory and coupled cluster

theory.

Accounting for the correlation of each pair of electrons is naturally quite expensive

computationally. In many practical applications, therefore, it is only the correlation of the

valence electrons which is explicitly considered while the core electrons are left uncorrelated


29

or “frozen” (the frozen core approximation). The effects of the correlation of the core

electrons do need to be considered, however, when high accuracy is required.

2.2.3.1 Multiconfigurational SCF Theory (MCSCF)29-31

In Hartree-Fock theory the wavefunction is defined as a single Slater determinant, φ. While in

many situations such a wavefunction provides an acceptable reference state for more

extensive (correlated) calculations, it is inadequate when there are degeneracies or near

degeneracies in the valence molecular orbitals. This situation arises particularly for bond

breaking reactions, where the occupied and unoccupied orbitals converge in energy as the

bond is stretched. In such situations there is a corresponding near-degeneracy amongst the

configurations and therefore all near-degenerate Slater determinants, φ kl q , need to be

included in the wavefunction in order to properly describe the system:

k kk

aψ φ= ∑ (2.2.54)

where { }ka are the variational coefficients and the summation is over the subset of

configurations which are expected to make a significant contribution to the wavefunction.

Thus in MCSCF theory both the configuration interaction coefficients, { }ka , and the

molecular orbital coefficients, { }pic , are simultaneously optimised.

The complete active space SCF (CASSCF) method30,32 provides a well defined procedure for

choosing n-electron configurations in a MCSCF wavefunction. As in ROHF, the orbitals are

split into three subsets (spaces):

ϕ ϕ ϕ ϕ ϕ ϕ1 1 1i i i a i a i a v

inactive active virtual+ + + + + +

where the i inactive orbitals are defined as being doubly occupied, the v virtual orbitals as

unoccupied while the a active orbitals have partial occupancy. The relevant configurations are

then constructed by considering every possible way (with correct spin and spatial symmetry)

of distributing the n i- 2 active electrons amongst the a active orbitals.


30

A second order Newton-Raphson type procedure33 (or an approximate version thereof) is then

applied to determine the CI and orbital coefficients such that the generalised Brillouin

theorem is satisfied. In other words, on convergence the energy is invariant to rotations

between the inactive, active and virtual orbitals.

2.2.3.2 Configuration Interaction (CI)6

As outlined in Section 2.2.1 the full many-electron wavefunction for a system can be

expressed in terms of the configuration interaction expansion (Equation (2.2.1)). This CI

expansion involves all possible determinants which can be constructed by considering every

possible arrangement of the available electrons amongst all the linearly independent

molecular orbitals that can be formed from the one particle basis set. For many systems,

however, the many-electron wavefunction, ψ, is dominated by a single determinant, ψ 0 ; in

such cases all other configurations can be thought of as a correction, χ, to this reference

wavefunction. This correction then accounts for electron correlation.

0ψ ψ χ= + (2.2.55)

Application of the Hamiltonian operator followed by projection onto the Hartree-Fock

reference state gives:

0 0ˆE E Hψ χ= + (2.2.56)

where E is the total non-relativistic energy of the system and E0 is the Hartree-Fock reference

energy. Thus, according to the definition in Equation (2.2.53), the correlation energy is simply

given by:

0ˆ

corrE Hψ χ= (2.2.57)

This is known as the correlation energy formula.


31

The correction, χ, can be constructed in a systematic way by generating configurations which

correspond to the substitution of 1, 2, …, n occupied spin orbitals in the reference determinant

by unoccupied spin orbitals:

0, ,

a a ab abi i ij ij

i a i ja b

a aψ ψ φ φ<<

= + + +∑ ∑ (2.2.58)

where φ ia indicates a determinant obtained by single substitutions (i substituted by a), etc. and

{ }aia , { }ab

ija , ... are the CI coefficients which will be determined either variationally or by

perturbation theory. The orbitals φ ia , φ ij

ab , … are often referred to as singly, doubly, etc.

excited configurations (that is, the electron in orbital i has been excited into orbital a, etc.). As

the one-electron, viz. molecular orbital (MO), basis has already been optimised in the SCF

determination of the Hartree-Fock reference state, the CI coefficients might be expected to

show rapid convergence. Unfortunately this is not the case in practice; while the individual

coefficients of higher than double excitations do systematically decrease in magnitude, their

collective energetic contributions converge slowly with the order of the excitation. This is

associated with the difficult problem of resolving the electron cusp using wavefunctions that

do not explicitly depend on inter-electron coordinates.34

Although the full CI expansion formally has up to n-fold excitation terms (where n is the

number of electrons in the system), it can be shown that when Ecorr is evaluated by the

correlation energy formula it is only the double excitation terms which contribute. This is

because in the orthonormal SCF MO basis the Brillouin condition (Equation (2.2.28)) applies

and, according to the Slater-Condon rules, terms with higher than double excitations have

zero Hamiltonian matrix elements with the reference state, 0ψ . Thus

0,

,

ˆ ab abcorr ij ij

i ja b

abij

i ja b

E H a

ij ab a

ψ φ<<

<<

=

=

∑

∑(2.2.59)

Unfortunately, before Ecorr can be calculated via this method the coefficients { }abija must be

known. In the Full Configuration Interaction (full-CI) method the calculation of { }abija


32

involves the application of the variational principle to solve the appropriate matrix eigenvalue

equations (Equation (2.1.9)) for the full configuration interaction expansion of the

wavefunction. This is straightforward in principle but in practice the number of

configurations, and thus the computational cost of calculations, rises rapidly with the number

of electrons and the size of the MO basis. The computations can be made more efficient by

the consideration of spatial and spin symmetry and the application of the Direct CI

approach35,36 (with the Davidson diagonalisation method37). Nevertheless, full-CI calculations

are still only feasible for small molecules with up to ~ 10 electrons and modest basis sets (up

to about double zeta plus polarisation functions quality).

It is therefore common practice to truncate the CI expansion at the double excitation terms,

neglecting triple and higher excitations. While this reduces the size of the problem so that it

becomes computationally feasible, the resulting solutions are not size extensive, that is, they

do not scale correctly with the number of electrons in the system. This is a serious problem,

especially in the context of computing molecular binding energies and intermolecular forces.

A useful, although very approximate, way to correct for size extensivity is via the Davidson

correction:38

( )201Dav corrE E a= − (2.2.60)

or via39

( )20

20

1Dav corr

aE E

a

−= (2.2.61)

where 0a is the coefficient of the reference state in the normalised CI expansion. While the

variational CI method is important as background theory for other methods such as Møller-

Plesset perturbation theory and Coupled Cluster theory, it has not been used extensively in

this thesis due to the lack of size extensivity.

CI can also be extended to multireference wavefunctions, where the reference state is

typically a CASSCF wave function. This results in a method of very high accuracy but also

high cost. While the multireference CI (MRCI) method40-45 is one of the most accurate pure

ab initio techniques it has not been employed in this work.


33

2.2.3.3 Møller-Plesset Perturbation Theory (MPPT)9,10

MPPT involves the use of perturbation theory to determine the coefficients in the CI

expansion. It is based upon the assumption that the effects of dynamical correlation can be

regarded as a perturbation, V , to the all-electron Fock operator, F , (described in Section

2.2.2). The Hamiltonian is formally partitioned:

ˆ ˆ ˆH F V= + (2.2.62)

where V is known as the fluctuation operator.

Starting with the Hartree-Fock wave function as the unperturbed state, the application of

Rayleigh-Schrödinger perturbation theory yields the perturbative corrections to the

wavefunction ψ 1b g , ψ 2b g , ψ 3b g , etc.; these are constructed from the single, double, triple, etc.

excitations as specified in the configuration interaction expansion of the wavefunction

(Equation (2.2.58)). The perturbation corrections to the energy, E 1b g , E 2b g , E 3b g … (to first,

second, third, ... order) and the corresponding contributions to the coefficients ({ }aia , { }ab

ija ,

etc.) in the CI expansion can thus be determined.

As the first order energy correction is simply the expectation value of the perturbing

fluctuation operator with respect to the Hartree-Fock reference state, perturbation theory to

first order in the energy yields the original Hartree-Fock energy.

The second order energy correction, E 2b g , in the basis of the occupied (i, j, …) and

unoccupied (a, b, …) spin orbitals is found to be:

( )

2

2,,

1

4 i j i j a ba b

ij abE

ε ε ε ε=

+ − −∑ (2.2.63)

where ε i , ε j , … are the Hartree-Fock orbital energies.


34

E 2b g is known as the MP2 correlation energy. Møller-Plesset perturbation theory to second

order (in the energy) is widely used as it is computationally inexpensive and thus allows

correlated calculations to be performed for relatively large molecules.

Perturbation theory up to fourth order in the energy (MP4) is also commonly used; this

requires knowledge of the second order correction to the wavefunction, ψ 2b g , which has

contributions from single, double, triple and quadruple excitations. Accounting for triple

excitations has been found to be more difficult (and expensive) than accounting for the

quadruples so they are often neglected, giving MP4(SDQ) theory. (MP4 theory with triple

excitations included is denoted MP4(SDTQ).) It has been observed that the additional

accuracy obtainable by including higher order terms in the perturbation expansion comes at a

high additional computational expense; it is therefore more practical to use configuration

interaction or coupled cluster methods when higher accuracy is required.

Møller-Plesset perturbation theory can be applied within the framework of both single- and

multi-determinant reference states. The most successful implementation of the latter is the

complete active space second order perturbation theory (CASPT2) method of Andersson et

al.46,47 Being based on a CASSCF reference state, CASPT2 accounts for both dynamical and

non-dynamical correlation. As the formalism is significantly more complex than for single

determinant perturbation theory (due to the more complex form of the reference state) the

computational effort and cost are also greater.

2.2.3.4 Coupled Cluster Theory (CC)48-52

Coupled cluster theory represents a seemingly different approach to the electron correlation

problem from that of configuration interaction; much of this difference is, however, semantic.

A coupled cluster wavefunction is formulated in terms of the cluster operator, T :

ˆ

0eTψ ψ= (2.2.64)


35

where T is constructed from one body, two body, three body, etc. cluster terms, T1 , T2 , T3 ,

… which represent single, double, triple, etc. excitation operators:

1 2 3ˆ ˆ ˆ ˆ ...T T T T= + + + (2.2.65)

where

1,

ˆ ˆ âi a i

i a

T t a a+= ∑ (2.2.66)

2 ˆ ˆ ˆ âbij b j a i

i ja b

T t a a a a+ +

<<

= ∑ (2.2.67)

etc.

These equations have been written using the formalism of second quantisation53 where { }âa+

are creation operators which generate an electron in spin orbital a and { }îa are annihilation

operators which remove an electron from orbital i. Together they represent the excitation of

an electron from orbital i to orbital a.

The cluster amplitudes, { }ait , { }ab

ijt , etc., are simply numerical coefficients for each term.

The asymptotic expansion of the eT operator yields:

ˆ 2 3

2 31 2 1 3 1 2 1

1 2 3

1 1ˆ ˆ ê 1 ...2 6

1 1ˆ ˆ ˆ ˆ ˆ ˆ ˆ1 ...2 6

ˆ ˆ ˆ1 ...

T T T T

T T T T TT T

c c c

= + + + +

= + + + + + + + = + + + +

(2.2.68)

where c1 , c2 , c3 , etc. are one-, two-, three-, … body clusters each representing the excitation

of 1, 2, 3, … electrons from occupied to virtual spin orbitals.


36

This means that the coefficients for the double, triple, etc. excitations of CI are now expressed

in terms of one-, two-, three-, … body cluster amplitudes:

ab ab a bij ij i ja t t t= + (2.2.69)

1

6abc abc a bc a b cijk ijk i jk i j ka t t t t t t= + + (2.2.70)

As in CI, implementation of the coupled cluster method with up to n-fold excitation operators

is not feasible computationally and in practice the cluster operator is truncated after double

excitations. Thus

1 2ˆ ˆ

0

2 21 2 1 1 2 1 2

03 2 4

1 2 1

e

1 1ˆ ˆ ˆ ˆ ˆ ˆ ˆ12 2

1 1 1ˆ ˆ ˆ ...6 2 24

T T

T T T TT T T

T T T

ψ ψ

ψ

+=

+ + + + + =

+ + + +

(2.2.71)

As this expansion shows, triple, quadruple and higher excitations are accounted for as

products of single and double excitations, or, in the language of many body perturbation

theory, all connected diagrams are implicitly present.

The coupled cluster wavefunction cannot be calculated using standard eigenvalue methods

because the function is not linear in the cluster amplitudes, { }ait , { }ab

ijt , etc. Instead the

wavefunctions are obtained iteratively by solving the Schrödinger equation in the subspace of

the configurations used, that is, the reference state and the single and double excitations. The

equations which need to be solved are therefore:

ˆ

0 0ˆ eTH Eψ ψ = (2.2.72)

ˆ

0ˆ ea T a

i iH t Eφ ψ = (2.2.73)

ˆ

0ˆ eab T ab

ij ijH t Eφ ψ = (2.2.74)

where E is the coupled cluster energy.


37

This approach is known as the coupled cluster with singles and doubles method, CCSD.

Although the coupled cluster wavefunction is size extensive, the solution of Equations

(2.2.72) - (2.2.74) does not yield an upper bound to the true energy.

It is also possible to truncate the cluster expansion after the T3 term, thus including the three

body clusters (that is, connected components of the triple excitations) and resulting in the

CCSDT method. The added computational cost, however, is at present too high to allow this

method to be used routinely. An alternative is to use perturbation theory to approximate the

contribution of the connected triple excitations using the coupled cluster wavefunction as the

unperturbed reference state:54

( ) ( )( ) ( )

1 2 0 2 0ˆ ˆ ˆ ˆ ˆ1 1abc abc

ijk ijk

triplesi j k i j k a b ca b c

T T H T HE

ψ φ ψ φ

ε ε ε ε ε ε< << <

+ + +∆ =

+ + − + +∑ (2.2.75)


CCSD with perturbative triples is denoted CCSD(T); it is currently the most commonly used

method for generating highly accurate molecular energies and has been used extensively in

this thesis.

Coupled cluster theory as described above is based on a single reference determinant. The

accuracy and reliability of the results are strongly dependant on the validity of the assumption

that the reference state is dominant in the coupled cluster expansion. To determine if this

condition is satisfied the τ 1 diagnostic has been introduced.55 The quantity τ 1 is defined by:

11

nτ =

t(2.2.76)

where t1 is the vector of single excitation amplitudes and n is the number of correlated

electrons. Based on extensive computational experience it has been suggested that if τ 1 is

larger than 0.02 then non-dynamical correlation effects are potentially important and CCSD

may be unreliable. The inclusion of the perturbative triples correction has been shown to


38

reduce these problems, however, with reliable energies having been obtained when τ 1 is as

large as 0.04.56

2.2.3.5 Quadratic Configuration Interaction (QCI)57

QCI can be viewed either as an extension of the configuration interaction methods or an

approximation to coupled cluster theory where only the terms which are required to ensure

size extensivity are retained. For quadratic configuration interaction theory with single and

double excitations (QCISD) the equations which need to be solved are:

( )0 1 2 0ˆ ˆ ˆ1H T T Eψ ψ+ + = (2.2.77)

( )1 2 1 2 0ˆ ˆ ˆ ˆ ˆ1a a

i iH T T TT t Eφ ψ+ + + = (2.2.78)

21 2 2 0

1ˆ ˆ ˆ ˆ12

ab abij ijH T T T t Eφ ψ + + + =

(2.2.79)

The effects of perturbative triples can also be included for QCISD theory:

( ) ( )( ) ( )

1 2 0 2 0ˆ ˆ ˆ ˆ ˆ1 2 1abc abc

ijk ijk

triplesi j k i j k a b ca b c

T T H T HE

ψ φ ψ φ

ε ε ε ε ε ε< << <

+ + +∆ =

+ + − + +∑ (2.2.80)


The resulting QCISD(T) theory is significantly more accurate than MP4 and is, in general, a

good approximation to CCSD(T). Like standard coupled cluster theory, QCI is based on a

single reference expansion; a Q1 diagnostic (analogous to the τ 1 diagnostic) has been

introduced to test the dominance of this reference state.582


39

2.3 Density Functional Theory

Density functional theory (DFT) is an entirely different approach to computational quantum

chemistry from the wavefunction methods described in Section 2.2. It involves expressing the

energy of a system as a functional of the electron density, ρ, rather than of a wavefunction, ψ.

This is based on the proof of Hohenberg and Kohn59 that “There exists a universal functional

of the density, F ρ rb g , independent of v rb g [the external potential due to the nuclei], such

that the expression E v d F= +z r r r rb g b g b gρ ρ has as its minimum the correct ground state

energy associated with v rb g .” Density Functional Theory is thus formally an exact theory

given that the mathematical form of this universal functional is known. Unfortunately, in

practice it is not known, nor can it be precisely determined or systematically improved.

Approximate functionals have therefore been proposed, often on the basis of fits which give

the correct results for certain well characterised systems. Density functional theory is,

therefore, a semi-empirical theory. It is important to note, however, that as DFT is based upon

the actual electron density, both dynamical and non-dynamical correlation are implicitly

accounted for in DFT calculations.

2.3.1 The Kohn-Sham Equations60

The density functional energy can be written as:

[ ] [ ] [ ] [ ]Ne eeE T V Vρ ρ ρ ρ= + + (2.3.1)

where T ρ is the kinetic energy and VNe ρ and Vee ρ are the nucleus-electron and electron-

electron interaction energies.

While VNe ρ (as indicated above) is simply given by:

[ ] ( ) ( )NeV v dρ ρ= ∫ r r r (2.3.2)


40

the forms of T ρ and Vee ρ for systems containing interacting electrons are unknown; these

functionals must therefore be approximated. A starting point for this is found in Hartree-Fock

theory where it is recognised that the electron-electron interaction contains Coulomb and

exchange terms and that the Coulomb component is given by:

[ ] ( ) ( )1 2 1 212

1 1

2J d d

rρ ρ ρ= ∫ ∫ r r r r (2.3.3)

In addition, while the form of the kinetic energy functional is unknown for systems with

interacting electrons, when the electrons do not interact the kinetic energy, Ts ρ , and the

density ρ rb g are given by:

[ ] 21

2

n

s i ii

T ρ φ φ= − ∇∑ (2.3.4)

( ) ( ) 2n

ii

ρ φ=∑r r (2.3.5)

Kohn and Sham therefore proposed that the exact density for a system of interacting particles

should also be specified in terms of the spin orbitals, ( ){ }iφ r , (as in Equation (2.3.5)) and that

the energy should be partitioned as:

[ ] [ ] [ ] [ ] [ ] [ ]( ) [ ] [ ]( )[ ] [ ] [ ] [ ]

s Ne s ee

s Ne xc

E T V J T T V J

T V J E

ρ ρ ρ ρ ρ ρ ρ ρ

ρ ρ ρ ρ

= + + + − + −

= + + +(2.3.6)

where Exc ρ is the exchange-correlation energy which accounts for all the effects in the

molecule neglected by the earlier Hartree type approximations. It has an associated exchange-

correlation potential, vxc rb g :

( ) ( )( )

xc

xc

Ev

δ ρδρ

=r

rr

(2.3.7)


41

The Kohn-Sham equations are therefore formulated in terms of the Kohn-Sham orbitals,

( ){ }iφ r :

( ) ( ) ( ) ( ) ( )2 '1'

2 ' xc i i iv d vρ

φ ε φ − ∇ + + + = −

∫r

r r r r rr r

(2.3.8)

These equations are analogous to the Fock equations (2.2.30) where the exchange operator,

K , has been replaced by the exchange correlation potential, vxc rb g . The accuracy and

reliability of density functional theory as formulated above depends, therefore, on the

accuracy of the exchange-correlation functional, Exc ρ .

2.3.2 The Local Density Approximation (LDA)

A simple formulation of this exchange-correlation functional is obtained by the study of a

convenient model system, namely the uniform electron gas in the presence of a uniform

continuum of positive charge. Here VNe ρ and J ρ sum to zero and the total energy is

simply:

[ ] [ ] [ ]s xcE T Eρ ρ ρ= + (2.3.9)

[ ] [ ] [ ]s x cT E Eρ ρ ρ= + + (2.3.10)

where Exc ρ has been separated into an exchange part, Ex ρ , and a correlation part, Ec ρ .

By using a plane wave basis set with periodic boundary conditions it has been shown that60-62

[ ] ( )5 3

s FT C dρ ρ= ∫ r r (2.3.11)

[ ] ( )4 3

x xE C dρ ρ= − ∫ r r (2.3.12)


42

where the constants are:

C

C

F

x

= =

= =−

3

103 2 8712

3

43 0 7386

2 2 3

1 1 3

π

π

c h

c h

.

.

When the α and β spin densities are different, the following more general results are obtained:

( ) ( )5 3 5 32 3, 2s FT C dα β α βρ ρ ρ ρ = + ∫ r r r (2.3.13)

( ) ( )4 3 4 31 3, 2x xE C dα β α βρ ρ ρ ρ = − + ∫ r r r (2.3.14)

The exchange energy generated via this approach is called the Dirac-Slater62 exchange

although it was, in fact, first developed by Bloch63.

Finally, the correlation functional, Ec ρ , has been formulated by Vosko, Wilk and Nusair64.

This work was based on quantum Monte-Carlo simulations of the uniform electron gas

performed by Ceperley and Alder65 for a range of electron densities. The functional is

designed to ensure that E ρ as defined in Equation (2.3.10) reproduces the quantum Monte-

Carlo results; it is known as the VWN correlation functional.

This formulation of the exchange and correlation functionals is called the Local Density

Approximation. When applied to atoms and molecules via the Kohn-Sham equations it has

been found that the LDA approach is not particularly useful for quantum chemical

applications, having an accuracy which is comparable with that of Hartree-Fock SCF theory.

To improve this situation various corrections to the LDA have been introduced.


43

2.3.3 Corrections to the LDA

The most significant of these is a correction to the exchange energy (or more specifically to

its potential) introduced by Becke66 in 1988. This correction term introduces non-locality

(shell structure) to the description of the system via a dependence on the gradient, ∇ρ . The

correction is given as

( )2

1 3

1

4 3

1 6 sinhBx

x

x x

x

ε βρβ

ρρ

−= −

+

∇=

(2.3.15)

where β is an adjustable parameter determined so that ε εxDirac

xB+ correctly reproduces the

exchange energy for six noble gas atoms; the resulting value for β is 0.0042.

The correlation functional as determined for the uniform electron gas is similarly inadequate

for an accurate description of real molecules. Utilising the Colle and Salvetti67 formula for the

correlation energy for the helium atom, Lee, Yang and Parr68 (with further contributions by

Miehlich, Savin, Stoll and Preuss69) derived a functional for the correlation energy of closed

shell systems:

[ ]

( )

2 22 8 3 21 3

1 3

11 31 3

1 31 3

1 3

5 7 11

1 12 72 24

exp

1

1

c FE a d ab C dd

c

d

dc

d

ρρ ωρ ρ ρ δ ρ ρρ

ρω ρ

ρρδ ρρ

−

−−

−

−−

−

= − − + ∇ − − ∇ + −

=+

= ++

∫ ∫r r

(2.3.16)

where a = 0.04918, b = 0.132, c = 0.2533 and d = 0.349 are the empirical parameters which

were determined by Colle and Salvetti for the helium atom. The presence of these parameters

along with the β in the Becke exchange correction implies that density functional theory is a

semi-empirical computational method. Ec ρ as defined in Equation (2.3.16) is known as the

LYP correlation functional. The presence of gradient terms in this functional, in addition to it

being derived on the basis of a two-electron wavefunction, means that LYP, like the Becke


44

exchange correction, is non-local. Consequently, it is much more realistic than the VWN

correlation functional.

In general, functionals (such as the Becke correction and LYP) which can be expressed in

terms of ρ and ∇ρ :

[ ] ( )' '

, , , ,xcE F dα β αα ββ αβ

σσ σ σ

ρ ρ ρ ζ ζ ζ

ζ ρ ρ

=

= ∇ ⋅∇∫ r

(2.3.17)

are referred to as Generalised Gradient Approximation (GGA) functionals.

While many other functionals have also been developed70-72, the LYP correlation functional

along with Becke’s correction to the exchange are currently the most commonly used.

Further improvements to Exc ρ have come with the introduction of adiabatic connection

functionals73. Instead of simply using the exchange functional given by E ExDirac

xB+ , Becke74

proposed that some “exact exchange” as obtained by a Hartree-Fock calculation ( ExHF )

should be included. His three parameter B3LYP functional75 is defined as:

( ) ( )1 1Dirac HF B VWN LYPxc x x x c cE AE A E B E C E CE= + − + ∆ + − + (2.3.18)

where A, B and C are semi-empirical parameters chosen to reproduce the exchange-

correlation energy of the 31 species in the G1 molecule set. The optimum values are

A = 0.80, B = 0.72, C = 0.81. Functionals which include both Hartree-Fock and density

functional exchange are called hybrid functionals. Of these, B3LYP is currently regarded to

be the most reliable for routine use; all DFT calculations in this thesis have used the B3LYP

functional.


45

2.3.4 Implementation of DFT

While the exchange and correlation functionals described above (such as B3LYP) allow DFT

to give good descriptions of molecular energies, geometries and related properties, their

forms, in particular the presence of fractional powers of the density, mean that the integrals

involved cannot be calculated analytically. This necessitates the use of numerical quadrature

with a three dimensional grid of points spanning the space of the molecule. Full details of the

implementation of such schemes can be found in References 70-73. It is important to note that

when such numerical procedures are employed for quantum chemical calculations of energies

and their gradients the grids used must be sufficiently fine grained to guarantee adequate

precision in the quantities of interest. 76-79


46

2.4 Basis Sets

As noted earlier, the one-particle bases used for the construction of many-electron molecular

wavefunctions consist of atomic spin orbital functions. Since the formation of a molecule

results in relatively small changes in the atomic wavefunctions, these atom centred functions

provide a suitable (and easy to obtain) basis for the description of molecular wavefunctions.

Atomic orbitals are generally expressed as products of radial R rb g and angular Ylm θ φ,b gfunctions:

( ) ( ) ( ), , ,lmr R r Yχ θ φ θ φ= (2.4.1)

where r, θ and φ are the radial and angular coordinates (in a spherical polar coordinate

representation).

The angular functions, Ylm , are normalised spherical harmonics given by

( ) ( )( ) ( )( ) ( )

1

22 i

!2 1, 1 cos e

4 !

m m m mlm l

l mlY P

l mφθ φ θ

π− −+= −

+ (2.4.2)

where l and m are the angular momentum and magnetic quantum numbers and Plm cosθb g are

associated Legendre polynomials80.

The radial nature of the one-electron atomic orbitals is, naturally, critically dependant on the

Coulombic forces between the electron and the nucleus. This means that a wavefunction

would be expected to have a singularity (cusp) at the nucleus and to decay exponentially at

large values of r. It therefore seems natural to choose radial functions of the form:

( ) en rR r Cr α−= (2.4.3)

where n is an integer and C and α are constants. Atomic orbitals of this form are called Slater

type orbitals (STO’s).81 While such orbitals give the best physical description of the


47

wavefunction, difficulties associated with the calculation of multicentre electron repulsion

integrals using STO’s make them impractical for use for anything other than small linear

molecules.

2.4.1 Gaussian Type Orbitals82

A significant simplification is made by the introduction of Gaussian type orbitals (GTO’s) of

the form:

( ) 2

en rR r Cr α−= (2.4.4)

The advantage of GTO’s is that the evaluation of the necessary integrals is so much simpler

and faster than for STO’s that it is rarely the limiting step in any practical computational

study. This is a direct consequence of the Gaussian product theorem83 which states that a

product of two Gaussian functions on different centres gives a new Gaussian centred at a new

position in space. This property allows the “difficult” 3 and 4 centre repulsion integrals to be

simplified to integrals involving only two centres.

The use of Gaussian functions does have disadvantages, however. In particular, they no

longer have a cusp at r = 0 and decay too quickly as r → ∞ . This problem can be corrected

for by the formation of contracted Gaussian type orbitals (CGTO’s); that is, by defining

orbitals as combinations (sums) of several “primitive” Gaussians (Equation (2.4.4)) with a

range of exponents, α. Thus, tighter (higher exponent) functions are employed to describe the

nuclear regions while more diffuse (lower exponent) functions will describe the valence and

outer regions as r → ∞ . CGTO’s can be expressed in the form:

( )CGTO GTOq qp p p

p

Cχ χ α= ∑ (2.4.5)

where the exponents, α p , and the contraction coefficients, qpC , are usually determined on the

basis of atomic SCF or CI calculations.


48

2.4.2 Construction of Contracted Gaussian Basis Sets

In order for a set of basis functions for a particular element to be applicable to calculations

involving molecules, solids and ions as well as free atoms it is necessary for it to be

sufficiently flexible to be able to both describe the changes in electron density involved with

formation of more complex systems and resolve the effects of dynamical electron correlation.

A minimal basis set for an element contains one CGTO for each atomic orbital in every fully

or partially occupied shell. Although this should in principle give a good description of the

atom (or a more complex system of which it is a part), it cannot, in fact, adequately describe

the changes in the orbitals due to bonding (such as contraction and polarisation), nor can it

account for electron correlation. The orbital contraction effects can be corrected for by using

two or more CGTO’s (rather than just one) to describe each atomic orbital; this leads to

double-ζ (DZ), triple-ζ (TZ), etc. basis sets. In order to obtain a perfect description of a

system ideally an infinite-ζ basis set would be needed, however in reality a compromise must

be made between the accuracy required and the time and computational resources available.

Currently sextuple-ζ (6Z) are the largest basis sets in common usage and then only for small

molecules, such as first row di- and tri-atomics. As the orbital contraction effects occur

largely in the valence orbitals, it is often the case that for second or higher row elements only

the minimal number of CGTO’s are used for the core orbitals while the valence orbitals are

augmented to double-, triple- and higher-ζ quality; such basis sets are called split valence.

The formation of bonds between two or more atoms is, of course, accompanied by a

polarisation of the atomic orbitals. It is therefore necessary to include polarisation functions in

the basis sets to account for these effects. This involves augmenting the basis set with

functions of successively higher angular momentum (l); for example, p and d functions are

added to polarise s functions; d and f functions are added to polarise p functions, etc.

Diffuse functions, that is, functions with lower exponents than those found in the standard set,

may also be added to a basis in order to account for longer range electronic effects. They are

necessary for obtaining satisfactory descriptions of anions and Rydberg states and in

situations where the outer regions of the density are of importance, for example in studies of

polarisabilities and weak interactions such as hydrogen bonding and van der Waals forces.


49

The basis sets used in this thesis fall into two categories: (1) the Gaussian type basis functions

of Pople and coworkers84-90; and (2) the correlation consistent basis sets developed by

Dunning et al.91-95 Both are split valence basis sets.

2.4.3 Pople’s Gaussian Basis Sets

The standard nomenclature for these basis sets is typified by, for example:

6-31+G(2df,p)

This notation indicates that the core orbitals have a minimal description, each being

constructed from 6 primitive GTO’s, while the valence orbitals are double zeta, one CGTO

being constructed from 3 primitives and the other being uncontracted (1). The “+” indicates

that diffuse functions have been included while “2df,p” specifies that two d and one f

polarisation functions have been added to the non-hydrogen (or He) atoms and one p

polarisation function has been included for hydrogen (and helium).

2.4.4 Correlation Consistent Basis Sets

The correlation consistent (cc) basis sets, cc-pVxZ,91,94 have been constructed to form a

sequence in which, as the cardinal number of the basis set, x, (and hence the basis set size)

increases, the improvement in the description of electron correlation is systematic and

predictable. This intention was inspired by the work of Almlöf, Taylor and co-workers96,97

who observed that, when constructing atomic natural orbital (ANO) basis sets, the

introduction of functions corresponding to the same principal quantum number made similar

contributions to the correlation energy.

The application of this principle is most readily understood by an example. The smallest basis

sets in the correlation consistent sequence, cc-pVDZ, are of double-ζ (DZ) quality; for first

row atoms they have the composition [3s, 2p, 1d]. In order to improve these basis sets to

triple-ζ (TZ) quality all functions corresponding to n = 4 must be included, that is, an

additional s, p and d function as well as an f function, [1s, 1p, 1d, 1f]. The cc-pVTZ basis sets


50

therefore have the composition [4s, 3p, 2d, 1f]. Similarly, a further [1s, 1p, 1d, 1f, 1g] must

be added to give the cc-pVQZ basis sets, resulting in [5s, 4p, 3d, 2f, 1g].

The additional functions are chosen so as to maximise their contribution to the electron

correlation. This means that significant improvements in the description of the correlation

energy are seen as the basis set increases from DZ to TZ to QZ to 5Z and so on. In addition,

the exponents have been carefully chosen so as to minimise the number of primitives in the

basis sets (in comparison with ANO’s) while still achieving the same correlation energy.

The major advantage of such systematic improvements in the treatment of electron correlation

is that the energies from a sequence of correlation consistent calculations can be fitted to

smooth monotonic functions and hence extrapolated to a hypothetical complete basis set limit.

In conjunction with accurate theories such as CCSD(T), this extrapolation allows for the

calculation of highly accurate atomisation energies and heats of formation at relatively low

computational cost.

The inclusion of diffuse functions in correlation consistent basis sets is indicated by the prefix

“aug-” (or even “d-aug-” or “t-aug-” to indicate two or three sets of diffuse functions).93 The

cc-pCVxZ basis sets98 (correlation consistent polarised core-valence x zeta) have also been

used in this thesis; these basis sets are based on their cc-pVxZ analogues but have additional

tight correlating functions added in order to describe the correlation of core electrons and

between core and valence electrons.

2.4.5 Basis Set Superposition Error

A consequence of using finite (and hence incomplete) atom-centred basis sets in calculations

of interaction energies (including covalent bonding, hydrogen bonding and van der Waals

interactions) is the presence of basis set superposition error. Briefly stated, this is the

phenomenon whereby, given an interacting system AB, the moiety A can be stabilised by the

nearby presence of the basis functions belonging to moiety B (in addition to any true

interaction between A and B) and vice versa. This is because these additional basis functions

compensate for the incompleteness of A’s own basis, thus improving the description of A and


51

lowering its energy. Thus the system is not only stabilised by any true interaction between A

and B but also by this superposition effect.

An estimate of the magnitude of this effect (and hence a possible correction for it) can be

obtained via the counterpoise method of Boys and Bernardi.99 This involves calculating the

energy of each moiety (atom or fragment) both with its own basis functions, E A , EB , and in

the presence of the basis functions of the entire system E A B , E A B . The counterpoise

corrections for A and B then given by:

[ ]CPA AA BE E E∆ = − (2.4.6)

[ ]CPB BA BE E E∆ = − (2.4.7)

The sum of these counterpoise corrections, ∆ ∆E EACP

BCP+ , therefore represents the total

correction to the interaction energy and thus the counterpoise corrected interaction energy is

given by:

corrected CP CPAB A B AB A BE E E E E E∆ = + − + ∆ + ∆ (2.4.8)

It should be noted that E A B and E A B are evaluated at the geometry optimised for AB, that

is, the geometry used to calculate E AB . If A and/or B are molecular fragments, these

geometries may be different from their equilibrium geometries (those used to calculate E A

and EB ); this may be a potential source of inaccuracy in ∆ ∆E EACP

BCP+ . This further

highlights the approximate nature of the counterpoise correction.


52

2.5 Derivatives of the Energy100

The calculation of derivatives of the energy of a molecular system with respect to

perturbations of the system is essential for determining molecular properties. For instance,

first derivatives with respect to nuclear displacements yield the forces on the nuclei and allow

the identification of stationary points on the molecular potential energy surface, such as

minimum energy structures. Derivatives with respect to an applied electrostatic field yield the

dipole moment, polarisability and hyperpolarisability of the molecule and derivatives with

respect to an applied magnetic field give the magnetisability and the magnetic shielding

tensors which, along with spin-spin coupling tensors, are essential for the quantitative

prediction of NMR spectra.

Derivatives can be calculated numerically or analytically. While numerical derivatives are

conceptually simpler (simply involving the calculation of energies at different values of the

perturbation parameter followed by polynomial fitting to obtain derivatives), they are

generally more resource intensive and less accurate than analytic methods for all but the

simplest systems. This discussion will therefore focus on the calculation of analytic

derivatives.

2.5.1 Analytic Energy Derivatives

The first derivative of the energy with respect to an arbitrary perturbation parameter, λ, is

given by:

ˆˆ2

dE HH

d

ψψ ψ ψλ λ λ

∂ ∂= +∂ ∂

(2.5.1)

where it is assumed that H is Hermitian and ψ is real.

Now, the wavefunction can be affected by the perturbation through both the CI and MO

coefficients, { }ka and { }pic , collectively labelled C, and through the basis functions, { }pχ .


53

Thus,

ψ ψ ψλ λ λ

∂ ∂ ∂ ∂ ∂= +∂ ∂ ∂ ∂ ∂

C

C

χχχχχχχχ

(2.5.2)

For all perturbations apart from those which change the nuclear configuration itself the basis

functions are independent of the perturbation:

0λ

∂ =∂χχχχ

(2.5.3)

Perturbations of the molecular geometry therefore form a special case which will be dealt

with separately in Section 2.5.2. For all other perturbations, however, Equation (2.5.1)

simplifies to:

ˆˆ2

dE HH

d

ψψ ψ ψλ λ λ

∂ ∂ ∂= +∂ ∂ ∂

C

C(2.5.4)

Clearly for fully variational wavefunctions, such as HF, MCSCF and full-CI, the second term

(known as the first order response of the wavefunction to the perturbation) vanishes, as the

wavefunction has been fully optimised with respect to all coefficients, C. Such a

wavefunction therefore obeys the Hellmann-Feynmann theorem:

ˆdE H

dψ ψ

λ λ∂=∂

(2.5.5)

that is, the derivative is given by the expectation value of the perturbation to the Hamiltonian,

V :

ˆdEV

dψ ψ

λ= (2.5.6)

where H H V= +0 λ .


54

Most wavefunctions in common use, such as Møller-Plessett, (truncated) CI or coupled

cluster wavefunctions, are not fully variational, however, and the non-Hellmann-Feynman

term (the second term in Equation (2.5.4)) must also be considered. The calculation of the

response of the coefficient matrix to the perturbation, ∂∂C

λ, is a demanding task, however it

can be avoided using Lagrange’s method of undetermined multipliers101-103.

For example: for a CI wavefunction, which is non-variational with respect to the MO

coefficients, the Lagrange function is

HFCI CI

EL E

∂= +∂c

κκκκ (2.5.7)

Choosing the Lagrange multipliers, κκκκ, so that:

2

2ˆ ˆ ˆ2 2 0CI CI HF HF HF

CI HF

LH H H

ψ ψ ψ ψψ ψ ∂ ∂ ∂ ∂ ∂= + + =

∂ ∂ ∂ ∂ ∂ c c c c cκκκκ (2.5.8)

allows the derivative in Equation (2.5.4) to be simplified to

ˆ ˆCI CI HF

CI CI HF

E L H Hψψ ψ ψλ λ λ λ

∂ ∂ ∂∂ ∂= = +∂ ∂ ∂ ∂ ∂c

κκκκ (2.5.9)

where the dependence of the coefficients on the perturbation is no longer required. This

treatment is readily generalised for the calculation of analytic second derivatives.

2.5.2 Geometric Derivatives104

As noted above, when derivatives are taken with respect to geometric parameters the

wavefunction depends on the perturbation through both the coefficients, C, and the basis

functions, χχχχ. This is because, as the nuclei move, the atom-centred basis functions move with

them. It is thus easiest to derive the equations for the derivatives when the energy is expressed

in terms of these basis functions; for example, for a Hartree-Fock wavefunction:


55

, , , ,

1

2p q pq p r q s pq rs NNp q p q r s

E h D D D Vχ χ χ χ χ χ= + +∑ ∑ (2.5.10)

where Dpqn s are the elements of the density matrix

occn

pq pi qii

D c c= ∑ (2.5.11)

The first derivative of the Hartree-Fock energy is thus given by

, , , ,

1

2p q pq p r q s pq rs pq p q NNp q p q r s pq

E h D D D W Vχ χ χ χ χ χ χ χ′ ′ ′′ ′= + − +∑ ∑ ∑ (2.5.12)

where the primed notation is used to denote the derivative with respect to the change in

geometry and

1

occn

pq pi i qii

W c cε=

= ∑ (2.5.13)

where ε il q are the orbital energies.

Thus, as for derivatives with respect to other parameters, geometric first derivatives do not

require the calculation of derivatives of the density (coefficient) matrix. It is, however, now

necessary to calculate the derivatives of the one- and two-electron integrals, χ χp q′ and

χ χ χ χp r q s′ ; the calculation of the derivatives of these two-electron integrals represents

the most resource intensive aspect of the calculation of gradients.


56

Second as well as some higher geometric derivatives have also been derived. The second

derivative of the SCF energy is given by:

, , , ,

, , , ,

1

2p q pq p r q s pq rsp q p q r s

pq p q pq p qpq pq

p q pq p r q s pq rs NNp q p q r s

E h D D D

W W

h D D D V

χ χ χ χ χ χ

χ χ χ χ

χ χ χ χ χ χ

′′ ′′′′ = +

′′ ′′− −

′ ′′ ′ ′′+ + +

∑ ∑

∑ ∑

∑ ∑

(2.5.14)

where now the first derivative of the density matrix is required. This can be calculated using

the Coupled Perturbed Hartree-Fock (CPHF) method105. The theory and implementation of

this method will not be described here, however an excellent summary can be found in

Jensen’s book106.


57

2.6 Molecular Properties

2.6.1 Geometry Optimisation107

A molecule with N atoms has 3N−6 internal degrees of freedom (3N−5 if linear) in a

Cartesian coordinate system. These correspond to three degrees of freedom for each of the N

atoms less the three degrees of freedom associated with translations of the (rigid) molecule

and the three (or two) degrees of freedom corresponding to molecular rotation. The potential

energy surface (PES) of the molecule, E Rb g , is therefore a function of these 3N−6 (3N−5)

internal distortions of the molecule.

In most chemical applications one is interested in energies and other properties of molecules

at their equilibrium geometries, which represent minima on this PES, and at transition state

geometries, which correspond to first order saddle points. A local minimum on the PES is

characterised by the energy gradient, F , being zero with respect to all geometric parameters:

( )0i

i

EF i

R

∂= = ∀

∂R

(2.6.1)

Furthermore, it is also required that the Hessian, H, be positive definite; that is, have all its

eigenvalues greater than zero. H is the second derivative matrix with matrix elements:

( )2

iji j

EH

R R

∂=

∂ ∂R

(2.6.2)

For a transition state (a first order saddle point) the gradient is also zero, while the Hessian

has one negative eigenvalue. This corresponds to a geometry where the energy is a minimum

with respect to all geometric parameters except one, the reaction coordinate, for which it is at

a maximum.

In order to successfully find an equilibrium structure or transition state on the potential energy

surface it is necessary to start with a molecular configuration, R0 , which is in the


58

neighbourhood of the appropriate local minimum or saddle point geometry, Re . The PES,

E Rb g , can then be expanded as a Taylor series around R0 :

( ) ( ) ( ) ( ) ( ) ( )( )2

0 0 00

,

1

2i i i i j ji i ji i j

E EE E R R R R R R

R R R

∂ ∂= + − + − − +

∂ ∂ ∂∑ ∑R RR R (2.6.3)

( )0

1

2E + += + ∆ + ∆ ∆ +R R F R H R (2.6.4)

where ∆R R Ri i i= − 0 .

The geometry corresponding to the minimum of the above quadratic expression (Equation

(2.6.3)) is obtained by solving

( ) ( ) ( ) ( )0 0

20 0i i

ji i i j

E E ER R

R R R R= =

∂ ∂ ∂= + − =

∂ ∂ ∂ ∂∑R R R R

R R R(2.6.5)

that is,

+ ∆ = 0F H R (2.6.6)

where the gradient vector, F, and Hessian, H, are evaluated at R0 .

The solution for the required change in geometry is therefore given by

1−∆ = −R H F (2.6.7)

The truncation of the Taylor expansion at second order means that the PES has been

approximated by a parabolic surface with the same gradient and curvature as the PES at R0

(see Figure 2.6.1). Correcting R0 by ∆R moves the geometry to the stationary point of this

quadratic surface, R1 , which, if the starting geometry is within the “local” region of the

stationary point sought, will be closer to Re . This process is repeated at the new geometry

thus found until the elements of the gradient vector are below some preset convergence

threshold, at which point the geometry is said to be converged and the equilibrium geometry


59

has been found. This iterative process is known as the Newton-Raphson method; it represents

a second order local model, since in a given search it aims to find the closest stationary point.

Figure 2.6.1 Newton-Raphson steps (in one dimension) for optimising geometries.

In practice calculating the Hessian in each step is quite expensive and, if possible, it is

avoided. This can be done by making a reasonable initial guess of the diagonal elements on

the basis of computed force constants and using the gradient information to improve the

approximate Hessian during the optimisation procedure. While this process works well for

equilibrium geometries, the “local” region is generally much smaller for transition state

structures and thus much more accurate Hessians are required. If the starting geometry is

close enough to Re it is sufficient to only calculate the Hessian fully in the first Newton-

Raphson step; for more difficult cases, however, it may be necessary to recompute it at every

step.

While gradients and Hessians are initially calculated with respect to the Cartesian coordinates

of the atoms, it is usually more convenient for the purposes of geometry optimisation to

perform a conversion such that they are expressed in terms of the 3N−6 (or 3N−5) internal

coordinates of the molecule. This approach also allows experimental or empirical force

constants to be more readily utilised for the construction of approximate Hessians.

R0R2 R1Re

Energy

First approximationto the PES

Secondapproximation

to the PES

Thirdapproximation

to the PES


60

2.6.1.1 Partial Geometry Optimisation

Sometimes it is desirable to perform a geometry optimisation where various constraints have

been applied. These constraints are particularly useful when mapping potential energy

surfaces where one geometric parameter is systematically varied while the others are allowed

to relax in response. In such situations the Hessian needs to be calculated with respect to the

molecular internal coordinates. The Lagrange method described earlier (Section 2.5.1) can be

applied in order to obtain derivatives with the constraints embedded in them; these can then

be used to aid in the location of critical points as described above.

2.6.2 Normal Mode Analysis

By definition the Hessian matrix is the matrix of force constants. When expressed in terms of

internal coordinates, its elements are the harmonic force constants for the 3N−6 (3N−5)

internal degrees of freedom of the molecule of interest. These determine the molecule’s

harmonic vibrational frequencies. The latter, by definition, correspond to the normal

vibrational modes; these can be determined by a unitary transformation of the Hessian such

that the classical potential (V) and kinetic (T) energies of the system are in a diagonal

representation. In the Cartesian representation V and T are given by:

V += X HX (2.6.8)

1

2T += X MX (2.6.9)

where H is the Hessian matrix, M is the (diagonal) matrix of atomic masses and X is the

vector of Cartesian displacements of the atoms with time derivative, X .

The normal modes, Q, are related to X via a linear transformation:

=X AQ (2.6.10)


61

In the normal mode representation V and T are therefore given as:

V + +=Q A HAQ (2.6.11)

1

2T + += Q A MAQ (2.6.12)

Thus, if A satisfies the generalised eigenvalue equations

=HA MAΛΛΛΛ (2.6.13)

where ΛΛΛΛ is the diagonal matrix of eigenvalues, one obtains:

V +=Q QΛΛΛΛ (2.6.14)

1

2T += Q Q (2.6.15)

The normal mode frequencies are simply proportional to the square roots of the elements of

ΛΛΛΛ.

If the geometry of interest corresponds to a minimum on the PES, H is positive definite and

thus all diagonal elements of ΛΛΛΛ will be positive and all frequencies will be real. If the

geometry is a transition state or higher order saddle point, one or several of the elements of ΛΛΛΛ

will be negative and will thus return imaginary frequencies.

The total zero-point energy (ZPE) of the molecular system in the harmonic approximation can

be readily obtained from the vibrational frequencies by summing over the zero-point energies

of all modes:

1

2 ii

ZPE hν= ∑ (2.6.16)

where h is Planck’s constant.


62

In reality the harmonic approximation does not provide a true representation of the vibrational

modes since bond stretches are much better represented by Morse type potentials and bending

/ torsional modes are periodic. Nevertheless, so long as the vibrational amplitudes are small,

the harmonic approximation can be demonstrated to be valid for at least the lowest energy

vibrations. An anharmonic treatment or at least anharmonic corrections need to be applied in

situations where this harmonic approximation fails, such as in the computation of vibrational

overtones. No such treatments were, however, needed in this work.


63

2.7 Computational Strategies for ChemicalAccuracy

Ideally all calculations of molecular and atomic energies would be performed using full-CI

with an infinitely large (complete) basis set. In reality this is, of course, not possible and a

trade off must be made between the desired accuracy and the time and computational

resources available for the job.

Much of the work in this thesis involves calculating energies (and thus heats of formation) for

the determination of the thermochemistry and kinetics of reactions. For this purpose reaction

energies are needed which are accurate to within ± 1 kcal mol−1 (chemical accuracy). With

current algorithms and the levels of processing power available it is not presently possible to

achieve this accuracy for most systems of interest (particularly those involving heavy atoms

such as phosphorus) from a single set of calculations (at one particular level of theory with

one chosen basis set).

Various approximation schemes have, however, been proposed in order to attempt to quantify

the effects associated with potential improvements in the level of theory and increases in the

basis set size. Such procedures, including isodesmic / isogyric reaction schemes, Gaussian-n

methods and complete basis set (CBS) schemes, can be used to obtain reaction energies of

chemical accuracy at a reasonable computational cost.

2.7.1 Isodesmic and Isogyric Reaction Schemes

Isodesmic and isogyric reaction schemes provide a method for obtaining heats of formation of

reasonably high accuracy from relatively low level calculations. They rely on the principle

that for a given reaction a particular computational approach can be expected to have similar

deficiencies for both reactants and products (when these are chemically similar). The

deficiencies are therefore expected to cancel to an appreciable degree when the energy (or

enthalpy) of a reaction is calculated. If reliable experimental (or high level theoretical)

atomisation energies ( 0DΣ ) or heats of formation ( 0f H∆ ) are available for all the species in

the reaction other than the molecule of interest, these can be used in conjunction with the


64

computed reaction energy at relatively low levels of theory to obtain much better estimates of

0DΣ (or 0f H∆ ) than from the calculated atomisation energy alone. Usually several such

reactions are constructed so as to give a range of estimates of the atomisation energy which

can then be averaged.

In an isodesmic reaction scheme the number of each type of chemical bond is conserved

throughout the reaction. For example, a reasonable isodesmic reaction for the calculation of

the 0DΣ of CHClBr2 would be

CHClBr 2CH CH Cl 2CH Br2 4 3 3+ → +

where both reactants and products have one C-Cl bond, two C-Br bonds and nine C-H bonds.

Since the atomisation energies of CH4, CH3Cl and CH3Br are well known, an accurate

calculation of the energy of this reaction allows the prediction of the atomisation energy of

CHClBr2 with similar accuracy.

In an isogyric reaction scheme only the number of electron pairs are conserved, for example:

2HF H F2 2→ +

Isodesmic schemes are expected to provide more reliable error cancellation than isogyric ones

and this is usually borne out by experience. Often, however, it is not possible to find suitable

isodesmic schemes, particularly when dealing with inorganic systems, and in such cases

isogyric reactions become an attractive alternative. In many of the systems studied in this

thesis, however, there were insufficient experimental data for the construction of either

isodesmic or isogyric schemes. This necessitated the use of more complicated schemes

requiring the application of higher levels of theory and larger basis sets in order to obtain

reliable predictions of atomisation energies.


65

2.7.2 Gaussian-n (Gn) Methods

The Gaussian-n methods were first introduced by Pople et al.108 in 1989 with the aim of

generating atomisation energies for molecules containing first and second row elements to

within ± 2 kcal mol−1. Since then the Gaussian methods have been further refined so that the

most recent modification has resulted in a mean absolute deviation of less than 1 kcal mol−1

for the G3/99 test set of molecules.109 The general principle is to perform a calculation at a

high level of theory (QCISD(T)) with a relatively small basis set and then correct this value

for deficiencies in the basis set using less expensive, lower level theories such as MP4 and/or

MP2. Geometries and vibrational frequencies are obtained at even lower levels of theory, with

the assumption that these properties are relatively insensitive to the level of theory and basis

set size; that is to say, small inaccuracies in the geometry or frequencies will cause negligible

errors in the molecular energies in comparison with the overall accuracy of the methods. Spin

restricted (RHF) based formalisms are used for all singlet state molecules while unrestricted

Hartree-Fock methods (UHF) are employed for open shell systems. While the Gn type

methods were originally developed within a single reference formulation, Sølling et al.110

have recently formulated a multireference equivalent of G2(MP2) and G3(MP2) using

MRCI+Q and CASPT2 calculations in place of QCISD(T) and MP2. These methods have not,

however, been utilised in this thesis.

2.7.2.1 Gaussian-1 (G1) Theory

Gaussian-1 theory108 was the first of the Gaussian methods to be developed. Geometries are

optimised using MP2(Full) theory, that is, with all electrons correlated, in conjunction with

the 6-31G(d) basis set87,89. Harmonic vibrational frequencies (for the calculations of zero-

point energies and thermal corrections to the enthalpies and entropies) are generated using

HF/6-31G(d) and scaled by a factor of 0.8929 to account for known deficiencies in this

method for the calculation of frequencies.111 The effects of anharmonicity on the zero-point

energies are assumed to be accounted for by the scaling.


66

The determination of the G1 energy is based upon a QCISD(T) calculation using the

6-311G(d,p) basis set. Corrections are then made for the inclusion of diffuse functions,

∆E +b g , and additional polarisation functions, ∆E df p2 ,b g , using MP4.

( ) ( ) ( )MP4/6-311 G , MP4/6-311G ,E E d p E d p ∆ + = + − (2.7.1)

( ) ( ) ( )2 , MP4/6-311G 2 , MP4/6-311G ,E df p E df p E d p ∆ = − (2.7.2)

The assumption is that these corrections are additive although it was recognised that this is a

potential weakness of the theory. The addition of these two corrections to the QCISD(T)/

6-311G(d,p) energy effectively approximates a QCISD(T)/6-311+G(2df,p) calculation. As

even QCISD(T)/6-311+G(2df,p) does not adequately reproduce experimental atomisation

energies, a further empirical higher level correction (hlc) is introduced to correct for

deficiencies in the QCISD(T)/6-311+G(2df,p) calculation. This correction is based on the

number of alpha and beta electrons in the molecule and was constructed so that the correct

absolute energies would be obtained for H and H2. The hlc (in milli-Hartrees) is

( )hlc 0.19 5.95E n nα β∆ = − − (2.7.3)

The G1 molecular energy is thus defined as:

( ) ( ) ( ) ( ) ( )( )

0 G1 QCISD T /6-311G , 2 ,

hlc ZPE

E E d p E E df p

E

= + ∆ + + ∆ + ∆ +

(2.7.4)

G1 has been shown to be capable of an accuracy (when compared with experiment) of ± 2

kcal mol−1 or better for most molecules containing first row atoms and ± 3 kcal mol−1 for

molecules with second row elements.


67


Gaussian-2 theory was introduced by Curtiss et al.112 in 1991 to compensate for some of the

deficiencies in G1. There are three major improvements in G2 over G1: firstly a correction is

made for the assumption that the ∆E +b g and ∆E df p2 ,b g corrections are additive; a

correction is also made for the extension of the basis to 6-311+G(3df,2p); finally the higher

level correction is also refined. The first two corrections are both made at the MP2 level of

theory, resulting in the following expression for the G2 energy correction:

( ) ( ) ( )( ) ( )

G2 MP2/6-311+G 3 , 2 MP2/6-311G 2 ,

MP2/6-311+G , MP2/6-311G ,

E E df p E df p

E d p E d p

∆ = − − +

(2.7.5)

G2 can therefore be regarded as an approximation to QCISD(T)/6-311+G(3df,2p).

The G1 higher level correction is modified so as to minimise the deviation of the G2

atomisation energy from experimental values for a set of 55 molecules (where the

experimental atomisation energies are well known). The modification is

( )corrhlc 1.14 pairE n∆ = (2.7.6)

where npair is the number of valence electron pairs in the molecule and the units of

( )corrhlcE∆ are milli-Hartrees.

The resulting G2 energy is thus given by

( ) ( ) ( )( ) ( ) ( )( ) ( )

0

corr

G2 QCISD T /6-311G ,

2 , G2

hlc hlc ZPE

E E d p

E E df p E

E E

= + ∆ + + ∆ + ∆

+ ∆ + ∆ +

(2.7.7)

The mean absolute deviations of G2 atomisation energies from experiment for the molecules

in the test set was found to be 0.92 kcal mol−1 for species containing only first row elements

and 1.08 kcal mol−1 for molecules which also contain second row atoms.


68

Several modifications to G2 theory have been introduced113-120 with the aim of reducing the

computational cost of the method while still providing reasonable accuracy. The most well

known of these is G2(MP2) theory115, where the corrections for basis set expansion are made

using only MP2 rather than both MP2 and MP4. The G2(MP2) energy is therefore given by

( ) ( ) ( )( ) ( )

( ) ( )

0

corr

G2 QCISD T /6-311G ,

MP2/6-311+G 3 , 2 MP2/6-311G ,

hlc hlc ZPE

E E d p

E df p E d p

E E

= + −

+ ∆ + ∆ +

(2.7.8)

G2(MP2) theory has been found to yield an average deviation of 1.52 kcal mol−1 from

experiment for the atomisation energies of the 125 molecules in the test set.

2.7.2.2.1 G2-RAD Theory

As noted earlier, sometimes the use of the UHF formalism can result in spin contamination of

the reference state and thus the G2 method, as described above, cannot be reliably employed.

A modification of the G2 procedure, called G2-RAD, has been developed by Parkinson,

Mayer and Radom121 to deal with such systems. In this method an RCCSD(T) reference

energy is used rather than UQCISD(T) and all MPPT and HF-SCF calculations are performed

using the restricted open-shell formalism.


In 1998 Curtiss et al.122 proposed G3 theory as an improved Gaussian method for the

computation of thermochemical data. The geometry and vibrational frequencies are obtained

in the same way as for G1 and G2. The reference energy, however, is now calculated at the

QCISD(T)/6-31G(d) level of theory rather than with QCISD(T)/6-311G(d,p). The basis set

has been changed in response to criticism that the valence-triple zeta basis set is

unbalanced.123 Consequently in G3 theory the parent basis is 6-31G(d). The corrections due to

the addition of diffuse and extra polarisation functions are therefore given by:


69

( ) ( ) ( )MP4/6-31 G 4 / 6-31E E d E MP G d ∆ + = + − (2.7.9)

( ) ( ) ( )2 , MP4/6-31G 2 , MP4/6-31GE df p E df p E d ∆ = − (2.7.10)

The largest basis set used in G2 theory, namely 6-311+G(3df,2p), has also been modified,

both to improve the uniformity of the set and to provide corrections for further basis set

enlargement; core-polarisation functions were also included. The resulting basis set is termed

G3Large122; its composition is [4s, 2p] for H and He, [5s, 5p, 3d, 1f] for first row atoms, [7s,

6p, 4d, 3f] for second row atoms and [9s, 8p, 7d, 3f] for atoms of the third row124.

It must be noted that for third row non-transition elements G3 employs the new 6-31G(d)

basis sets (and their extensions with diffuse and polarisation functions) of Rassolov et al.125

(these differ from the 6-31G(d) sets in the basis set libraries of most computational chemistry

packages).126 In addition, the 3d electrons are included in the valence space of all frozen core

calculations. G3 theory has not yet been extended to include transition block elements.

As a further improvement over G2, core-core and core-valence correlation effects are also

accounted for by performing a MP2(Full)/G3Large calculation. ∆E G2b g is thus replaced by

the G3Large correction:

( ) [ ] ( )( ) ( )

G3Large MP2(Full)/G3Large MP2/6-31G 2 ,

MP2/6-31+G MP2/6-31G

E E E df p

E d E d

∆ = − − +

(2.7.11)

G2 theory has been found to perform relatively poorly in the description of ionisation

potentials and electron affinities. This has been largely corrected through modification of the

higher level correction term, in particular by using different formulae for atoms and

molecules. The hlc now takes the form:

( ) ( )atomshlc 6.219 1.185E n n nβ α β∆ = − − − (2.7.12)

( ) ( )moleculeshlc 6.386 2.977E n n nβ α β∆ = − − − (2.7.13)


70

In addition the effects of spin-orbit (SO) coupling corrections for the atoms are also included

in G3 theory; thus the final G3 energy is given by:

( ) ( ) ( ) ( ) ( ) ( )( ) ( )

0 G3 QCISD T /6-31G 2 , G3Large

hlc SO ZPE

E E d E E df p E

E E

= + ∆ + + ∆ + ∆ + ∆ + ∆ +

(2.7.14)

The test set for evaluating the performance of Gaussian-n theories was also extended to

include 299 energies (atomisation energies, ionisation potentials, electron affinities and proton

affinities).109 With this new test set, G2 theory now has a mean absolute deviation (MAD) of

1.48 kcal mol−1 while for G3 theory the MAD is only 1.02 kcal mol−1.

Modifications of G3 in the spirit of G2(MP2), and some other modifications such as scaling

of energies, changes in geometries and the use of coupled cluster theory rather than

QCISD(T), have also been introduced.127-132

2.7.2.3.1 G3-RAD Theory

As for G2, a G3-RAD procedure has been developed (by Henry, Parkinson and Radom133) to

describe open shell systems, particularly those which suffer from spin contamination. As for

its G2 counterpart, the G3-RAD procedure employs an RCCSD(T) reference energy and

ROMPn corrections. It uses B3LYP/6-31G(d), however, to generate geometries and

vibrational frequencies (the latter scaled by 0.9806) and the MPPT calculations are performed

using all Cartesian components of the d and f polarisation functions (6 and 10 respectively).

The higher level correction has also been reoptimised for this method and now takes the form:

( ) ( )atomshlc 6.561 1.341E n n nβ α β∆ = − − − (2.7.15)

( ) ( )moleculeshlc 6.884 2.747E n n nβ α β∆ = − − − (2.7.16)


71

2.7.2.4 Gaussian-3X (G3X) Theory

The most recent addition to the Gaussian-n family of theories is G3X (due to Curtiss,

Redfern, Raghavachari and Pople134). This was introduced specifically to correct for

deficiencies in G3 theory when describing molecules containing second row elements. In

G3X theory the geometries and vibrational frequencies are determined using density

functional theory, specifically the B3LYP functional, and the larger 6-31G(2df,p) basis set.

(Frequencies are scaled by 0.9854.) The effects of adding g functions to the basis sets for the

second row elements are also included through the introduction of the G3XLarge basis set

(formed by simply adding a g function to G3Large). This correction is applied at the SCF

level:

( ) [ ] [ ]G3XLarge HF/G3XLarge HF/G3LargeE E E∆ = − (2.7.17)

Thus the correlation effects of the g functions are not taken into account. Finally, the higher

level correction has also been reoptimised, giving A = 6.783 mEh, B = 3.083 mEh, C = 6.877

mEh and D = 1.152 mEh.

The test set of molecules has also been increased, now including 376 reaction energies

(including atomisation energies, ionisation energies and proton affinities). For this set G3 has

a MAD of 1.07 kcal mol−1 while G3X shows a small improvement with a MAD of 0.95 kcal

mol−1. While most first row molecules are hardly affected by the replacement of G3 by G3X

theory, the description of second row molecules is appreciably improved.

2.7.2.5 G3X2 Theory

Finally, Haworth and Bacskay135 have observed that G3X theory still shows systematic

deficiencies in the description of molecules containing second row atoms, particularly

phosphorus. We have therefore proposed an extension to G3X, denoted G3X2, in which the

G3XLarge correction (Equation (2.7.17)) is applied at the MP2(Full) level, thus recovering

additional correlation energy. This is equivalent to performing a G3 calculation using the

G3XLarge basis set rather than G3Large (using the B3LYP/6-31G(2df,p) geometry and

vibrational frequencies). Counterpoise corrections for BSSE in the core-valence correlation of


72

second row atoms are also included at the MP2/G3XLarge level of theory. While G3X2

shows improvement over G3X for a small set of phosphorus containing species, further

testing, particularly for molecules containing other second row atoms, is required before this

method can be recommended for general use.

2.7.3 Complete Basis Set Methods

The complete basis set methods currently represent the highest level of theoretical treatment

available for the reliable calculation of heats of formation and related properties. They employ

coupled cluster theory in conjunction with the correlation consistent basis sets.

As noted earlier, the cc-pVxZ basis sets have been constructed so that the incremental energy

lowering due to the addition of correlating functions follows well defined trends. This means

that the energies obtained in a sequence of correlation consistent calculations can be fitted to a

smooth function of x and extrapolated to a theoretical complete basis set (CBS) limit, that is,

x = ∞ .

A number of extrapolation schemes have been proposed for this purpose over the last 10

years; the most commonly used are the mixed exponential/Gaussian extrapolation of Feller136

(“mix”, Equation (2.7.18)), the Schwartz type extrapolations137 (“ lmax ”, Equation (2.7.19) and

“ n n− −+4 6 ”, Equation (2.7.20)) and the “ x−3 ” scheme of Helgaker et al.138 (Equation

(2.7.21)).

( ) ( ) ( )( )2exp 1 exp 1E x A B x C x= + − + − − (2.7.18)

( ) ( ) 4

max 1 2E x A B l−= + + (2.7.19)

( ) ( ) ( )4 6

max max1 2 1 2E x A B l C l− −= + + + + (2.7.20)

( ) 3E x A Bx−= + (2.7.21)


73

In these equations x is the cardinal number of the basis set (that is, 3 for TZ, 4 for QZ and 5

for 5Z), lmax is the highest angular momentum quantum number in the basis, and A, B and C

are fitted parameters. As a result of recent work by several groups, the x−3 scheme is emerging

as the most reliable and trusted of these extrapolations.138,139 This follows from the

observation that the error in the description of the correlation energy is roughly inversely

proportional to the number of basis functions, and the number of basis functions in the

correlation consistent sets scales as x3 .139

Usually calculations using up to (at least) the cc-pV5Z basis sets are used for the

extrapolations. Diffuse functions are also often included for electronegative atoms such as

oxygen. Furthermore, it has been found that the x−3 extrapolation scheme gives the best

results when only the energies corresponding to the largest and second largest values of x are

used in the fit.140

As CCSD(T) has emerged as the most accurate and reliable correlated method, giving an

excellent approximation to full-CI, it is the theory of choice for the CBS methods.

Furthermore, since SCF energies converge more rapidly than correlation energies, in order to

achieve the highest level of accuracy the SCF and correlation energies can be extrapolated

separately (although this is not, as a rule, part of the standard procedure).

The effects of correlation of core electrons and between core and valence electrons are usually

computed using significantly smaller basis sets than those used for the extrapolations; the

cc-pCVTZ basis sets (with augmentation if required) are often the largest which can be

employed for this purpose, although for very small molecules cc-pCVQZ may also be used.

The core-valence (CV) correlation correction is calculated as the difference in molecular

energies when all electrons are correlated and when only valence correlation is accounted for.

It is assumed that CV and valence only correlation energies are additive, so a computed CV

correction is simply added to the extrapolated valence correlated energy. While it is preferable

to compute the CV correlation correction using CCSD(T), lower levels of theory such as MP2

may also be used.

Although relativistic effects tend to be small for molecules which can be treated by CBS

methods (usually only first row elements), they are often large enough to be significant.


74

Scalar relativistic effects are usually included via the computation of the Darwin and mass-

velocity terms141,142; in our work these have been calculated using Finite Perturbation Theory

at the CASPT2 or CASSCF levels of theory. Spin orbit effects are also included in the

calculation of thermochemistry where appropriate (usually only for atoms).

As noted above, CBS methods are currently the most accurate methods available for quantum

chemical calculations of thermochemistry. Due to the use of the highly correlated CCSD(T)

method along with large basis sets, CBS methods are significantly more computationally

expensive than the Gaussian-n schemes; they are, however, also capable of delivering

significantly higher accuracy143-145, to within ± 0.2 to 2.0 kcal mol−1 for heats of formation

from atomisation energies, depending on the size of the molecule. CBS theory has therefore

successfully achieved an aim that has been a holy grail of computational chemistry, that is, the

reliable prediction of reaction energies to chemical accuracy.


75

2.8 Thermochemistry

The calculation of theoretical heats of formation is essential for many of the applications of

quantum chemistry, in particular for aiding in the interpretation of experimental results and

for the prediction of reaction kinetics. Unfortunately, however, this requires the computation

of the reaction enthalpy for the formation of a molecule relative to the standard states of its

constituent elements; in many cases these standard states are liquids or solids for which direct

calculation of the energy is not feasible. Given that accurate experimental values are available

for the enthalpies of formation of free atoms, a practical alternative is to use these in

conjunction with a theoretical prediction of the atomisation energy, 0DΣ , to predict the heat

of formation for the molecule of interest. Thus, given an atomisation energy at 0K,

0 atom moleculeatoms

D E E= −∑ ∑ (2.8.1)

(where the total molecular energy includes the zero-point vibrational energy), Hess’ law can

be applied to obtain the 00f H∆ :

( ) ( )0 00 0 0f f

atoms

H molecule H atom D∆ = ∆ −∑ ∑ (2.8.2)

Heats of formation at other temperatures ( 0f TH∆ ) as well as entropies ( ST

0 ) and Gibbs free

energies of formation ( 0f TG∆ ) can then be calculated using the standard methods of statistical

mechanics.

2.8.1 Partition Functions146,147

The first step in determining the thermal contributions to the enthalpies and entropies of a

molecule is to determine its partition function, q; this is a measure of the number of states

accessible to the molecule (translational, rotational, vibrational and electronic) at a particular

temperature.


76

Given the energies, Ei , of the available quantum states of a molecule, q is defined as:

1

e iEi

i

q g β∞

−

=

= ∑ (2.8.3)

where gi is the degeneracy of the i-th state and

1

Bk Tβ = (2.8.4)

where kB is Boltzmann’s constant and T is the temperature of interest. The summation in

Equation (2.8.3) is over all possible quantum states of the system.

It is assumed that the translational (T), rotational (R), vibrational (V) and electronic (E) modes

of the system can be separated, thus allowing the energy of each level, Ei , to be separated

into T, R, V and E contributions:

T R V Ei i i i iE E E E E= + + + (2.8.5)

While the translational modes are truly independent from the rest, the separations of the other

modes are based on approximations, in particular the Born-Oppenheimer approximation4

described in Section 2.1.1 for electronic and (ro-)vibrational motion and the Rigid Rotor

Approximation148 (which assumes that the geometry of the molecule does not change as it

rotates) for vibrational and rotational modes. Within these approximations, the total molecular

partition function can therefore be factorised into translational, rotational, vibrational and

electronic contributions:

T R V Eq q q q q= (2.8.6)

The translational partition function is given by:

3

1 2

2

T Vq

hm

βπ

=Λ

Λ =

(2.8.7)


77

where h is Planck’s constant, m is the mass of the molecule, and V is the volume available to

it; for a gas phase system this is the molar volume at the specified temperature and pressure

(usually determined by the ideal gas equation).

The formulation for rotational partition functions depends on whether or not the molecule is

linear. For linear molecules

R Bk Tq

hcBσ= (2.8.8)

and for non-linear molecules

3 1

2 21R Bk Tq

hc ABC

πσ =

(2.8.9)

where σ is the rotational symmetry number of the molecule, c is the speed of light and A, B

and C are the rotational constants.

The vibrational partition function in the harmonic approximation is

1

1 e i

Vhc

i

q β ν−=−∏ (2.8.10)

where ~ν i are the harmonic vibrational frequencies (expressed as wavenumbers) and the

product is taken over all (3N−6 or 3N−5) vibrational modes (excluding the reaction coordinate

for transition states).

For the electronic partition function it is usually assumed that there will be no thermal

excitation into higher electronic states so that the partition function, q E , is simply given by

the degeneracy of the appropriate electronic state.


78

2.8.2 Thermodynamic Properties149

The thermal contributions to thermodynamic properties such as enthalpy, entropy, free

energy, heat capacity, etc. are all derived from the molecular partition functions.

For a system of N molecules the internal energy (relative to internal energy at 0K) is given by

0 00

lnT

V

qU U N

β ∂− = − ∂

(2.8.11)

where the derivative is taken at constant volume.

The enthalpy is therefore

( )( )

0 0 0 00 0

0 00

T T

T B

H H U U p V

U U Nk T

− = − + ∆

= − +(2.8.12)

The entropy of the system is given by

( )0 000 ln

T

B

U US Nk q

T

−= + (2.8.13)

so that the change in Gibbs free energy is

( ) ( )0 0 0 0 00 0

ln

T T

B B

G G H H TS

Nk T Nk T q

− = − −

= −(2.8.14)

The Gibbs free energy change for a reaction is, of course, related to the equilibrium constant

for the reaction:

0 lnr B eqG Nk T K∆ = − (2.8.15)


79

2.9 Kinetics

2.9.1 Transition State Theory (TST)150-152

The construction of partition functions is also essential for the calculation of kinetic rate

parameters. The central principle of transition state theory (TST), or activated complex

theory, is that there is a critical point on the reaction path that connects reactants and products

called the transition state, TS‡. Once this point has been reached the formation of products is

inevitable, that is, the molecule can no longer relax to reform the reactants. For most

molecular potential energy surfaces this transition state is identified as a first order saddle

point corresponding to a maximum with respect to the reaction coordinate; that is, the

minimum energy pathway between reactants and products. This is shown schematically in

Figure 2.9.1, where the barrier height (with zero point energy included) is defined as the

critical energy, ∆E ‡ , of the reaction.

Figure 2.9.1 A schematic potential energy surface.

The statistical derivation of rate coefficients is based upon several assumptions. In addition to

the condition that all molecules which reach the transition state must go on to form products

(as noted above), it is also necessary to assume that the Born-Oppenheimer approximation4 is

valid and that both reactants and molecules at the transition state geometry are distributed

Reactants

TS‡

Products

∆E‡

∆rE

Reaction Coordinate

Energy


80

among their states according to the Maxwell-Boltzmann law (even in the absence of an

equilibrium between reactants and products). It is also assumed that motion along the reaction

coordinate in the transition state can be regarded as a translation rather than a vibration (hence

it is also left out of the calculation of the vibrational partition function as noted earlier).

The rate coefficient of a given reaction (in the high pressure limit) at a particular temperature,

k T∞ b g , has thus been derived as

( )‡ ‡

expB

ii

k T q Ek T

h q kT∞ −∆= ∏

(2.9.1)

where q‡ is the (canonical) partition function of the transition state and the qi are the

partition functions for each of the reactants.

To facilitate comparison with experimental data it is useful to re-express rate coefficients in

Arrhenius153 or modified Arrhenius form:

( ) exp a

B

Ek T A

k T∞

= −

(2.9.2)

( ) expn a

B

Ek T AT

k T∞

= −

(2.9.3)

where A, Ea and (in the modified Arrhenius fit) n are the fitted parameters. A and Ea are

known as the pre-exponential (or simply A) factor and the activation energy respectively.


81

2.9.2 Variational Transition State Theory (VTST)154-156

Many reactions, such as simple bond fissions, do not have a barrier as shown in Figure 2.9.1,

and thus a transition state cannot be identified by locating a first order saddle point. In such

cases a more general definition of the transition state must be used where it is defined as the

point corresponding to the maximum value of the free energy along the reaction coordinate;

this is equivalent to locating the minimum value of the rate coefficient. This approach is

known as variational transition state theory. When the reaction has a barrier, this minimum in

the rate coefficient effectively coincides with the top of the barrier. For barrierless reactions,

however, calculations of energies and vibrational frequencies need to be performed at several

points along the reaction coordinate and rate coefficients calculated at each of these points in

order to determine the true (minimum) reaction rate and the associated geometry. This, of

course, has a much higher computational cost than for reactions with barriers; in addition,

such dissociation reactions often have significant multiconfigurational character so MCSCF

or density functional methods must be used.

2.9.3 RRKM Theory157

RRKM theory, developed by Rice, Ramsperger, Kassel and Marcus158-162, is a reliable and

widely used formalism for predicting the rates of unimolecular decomposition and

recombination reactions. The mechanism for such reactions consists of initial activation of the

reactant molecule via collisions with a bath gas, M, followed by the actual reaction via a

(variational) transition state to form products:

*A M A M+ ↔ + (2.9.4)

( )* ‡k EA A P→ → (2.9.5)

This collisional activation introduces a pressure dependence to the reaction rate. There are

three basic regimes: a high pressure limit where the rate of collisions is sufficiently high that

Equation (2.9.5) becomes the rate limiting step and the overall rate coefficient is pressure

independent; a low pressure limit where collisions are sufficiently rare that Equation (2.9.4) is

the rate limiting step and the rate constant is proportional to the pressure; and a “fall-off”


82

region which connects the two. In order to accurately predict the rate of a unimolecular

reaction it is therefore necessary to consider the rates of both processes.

Canonical transition state theory is incapable of correctly describing the low pressure and fall-

off regimes and thus RRKM was introduced. The development of RRKM theory rests on two

major assumptions: that energy gained by collisional activation is rapidly randomised through

all degrees of freedom of the reactant (the so called “ergodicity” assumption); and that all

molecules which cross the transition state will go on to form products (the transition state

theory assumption as stated earlier). The (microcanonical) RRKM rate coefficient, k Eb g , isthen given by:

( )( )

( )( )( )

0

‡

0

‡

1

1

E E

E dE

k Eh E

W E

h E

ρ

ρ

ρ

−

+ +

=

=

∫(2.9.6)

where h is Planck’s constant, E is the energy of the system (with the understanding that most

of this will be associated with those modes which have the potential to result in reaction), E0

is the critical energy, ρ Eb g and ρ‡ E+b g are the densities of states of the reactant, A* , and the

transition state, A‡ , respectively and W E‡b g is the total number of states of the TS‡ with

energy less than E. It can be shown that, in the limit of high (∞ ) pressure, averaging k Ea fover the Boltzmann distribution yields the canonical transition state formula (Equation

(2.9.1)).

In the fall-off and low pressure regions the rate coefficient for the reaction also depends

strongly on the rate of collisional energy transfer, R E E, ′b g . This is related to the probability

of energy transfer, P E E, ′b g ; that is, the probability that a reactant with energy ′E will end

up with energy E:

( ) [ ] ( )( )

,,

M R E EP E E

Eω′

′ =′

(2.9.7)


83

where M is the number concentration of the bath gas and ω ′Eb g is the total collision

frequency.

The first moment of R E E, ′b g with respect to energy is the mean energy transferred per

collision, ∆E , while the second moment is the mean-square energy transferred per collision,

∆E 2 ; the moments are formally defined by:

( ) ( ),nn

o

E E E P E E dE∞

′ ′∆ = −∫ (2.9.8)

It has been found that the pressure dependence of the overall rate constant is fairly insensitive

to the functional form of R E E, ′b g , instead depending largely on the moments alone. This

means that it is usually sufficient to use the simple exponential-down and Gaussian forms for

R E E, ′b g :

( ) ( )1

, expE E

R E En E α

′− ′ = − ′ (2.9.9)

and

( ) ( )2

1, exp

E ER E E

n E α ′ − ′ = − ′

(2.9.10)

respectively, where n E ′b g is a normalisation factor to ensure the correct overall collision

frequency and α is a constant chosen to yield reasonable values for the moments, that is, the

values obtained from accurate trajectory calculations of collisional energy transfer.


84

The overall thermal unimolecular rate constant, kuni , can finally be obtained by solving the

master equation

( ) [ ] ( ) ( ) ( ) ( ) ( ) ( )0

, ,unik g E M R E E g E R E E g E dE k E g E∞

′ ′ ′ ′ − = − − ∫ (2.9.11)

where g Eb g is the population of reactant molecules with energy E. In practice, kuni for a

given temperature and pressure is usually calculated at a range of energies and averaged.

Recombination reaction rates can also be obtained by RRKM using the rate of the reverse

(unimolecular decomposition) reaction and the principle of microscopic reversibility.


85

2.10 Population Analysis

Quantum chemical calculations, as described in Sections 2.2 and 2.3, yield total molecular

wavefunctions and/or total molecular probability densities. From a chemical point of view,

however, it is often useful to describe a molecule in terms of electrons associated with

individual nuclei and with covalent bonds. A range of schemes have been proposed for

extracting such localised information (atomic and bond populations) from the delocalised one-

particle density function; of these, the most commonly used is the simple Mulliken method.163

Assumptions and definitions vary widely amongst the various population analysis models,

thus different methods are often found to give significantly different results. One of the more

respected methods, however, is the Roby-Davidson population analysis164-166, which has been

used in this thesis.

The Roby-Davidson procedure involves partitioning the electron density of a molecule,

ABC…, into populations (numbers of electrons) which can be associated with each atom, with

pairs of atoms, with triples of atoms, and so on. This is achieved by applying appropriate

projection operators to the total density as obtained from quantum chemical calculations.

These projection operators, PA , PB , PC , …, PAB , PAC , PBC , …, PABC , … are constructed

from the atomic orbitals of the molecule so as to span the space of individual atoms, A, B, C,

…, pairs of atoms, AB, AC, BC, …, triples of atoms, ABC, … etc. Thus

( )1

,

ˆA

A

P Sµ νµνµ ν

ϕ ϕ−

∈

= ∑ (2.10.1)

( )1

,B

B

P Sµ νµνµ ν

ϕ ϕ−

∈

= ∑ (2.10.2)

( )1

, ,

ˆAB

A B

P Sµ νµνµ ν

ϕ ϕ−

∈

= ∑ (2.10.3)

etc.

where µ and ν run over the (non-orthogonal) spin orbitals of the relevant atoms and S −1c hµν

is

an element of the inverse of the overlap matrix, S.


86

As PA is constructed from the spin orbitals of atom A, it can act on the total (one-electron)

density, D, to project out the density associated with atom A; likewise for PB , while PAB ,

being built from the spin orbitals of both atoms A and B, projects out the density associated

with the pair of atoms. The occupation numbers, nA , nB , nAB , etc. (that is, the number of

electrons associated with each atom, pair of atoms, etc.) are therefore given by

( )( )1

, ,

TrA A

A

n

D S S Sλσ σµ νλµνλ σ µ ν

−

∈

=

= ∑ ∑DP

(2.10.4)

( )TrB Bn = DP (2.10.5)

( )TrAB ABn = DP (2.10.6)

etc.

Once the occupation numbers are known, the degree of electron sharing between pairs or

multiplets of atoms can also be quantified by defining shared electron numbers, σ AB , σ AC ,

σ BC , σ ABC , etc. as

AB A B ABn n nσ = + − (2.10.7)

ABC A B C AB AC BC ABCn n n n n n nσ = + + − − − + (2.10.8)

The degree of electron sharing between two atoms can be considered as a measure of the

covalent bonding between them. While single and double covalent bonds have been found to

have shared electron numbers of approximately 1 and 2 respectively, it is important to

calibrate the shared electron numbers for any A-B bond before using them to investigate and

interpret the bonding in new molecules. For example, the S-O bond in SO only has a

population of 1.47 even though it is formally a double bond.


87

The partial charge on each atom, qA , qB , etc. is the difference between its nuclear charge and

the electron density associated with that atom; the latter is found by equally partitioning any

shared electron density.

( ) ( )( )

1 1

2 3A A AB ABCB A C B A

q n σ σ≠ ≠ ≠

= − −∑ ∑ (2.10.9)

One of the most important issues affecting the reliability of the Roby-Davidson population

analysis is the choice of the spin orbital basis used in the construction of the projection

operators. Ahlrichs and coworkers166,167 have proposed the use of a minimal set of modified

atomic orbitals (MAO’s), ϕϕϕϕ , consisting of individual atom centred minimal basis sets, ϕϕϕϕ A ,

ϕϕϕϕ B , etc. These MAO’s are constructed by firstly partitioning the molecular density matrix, D,

(expressed in terms of the original atomic orbitals, χχχχ ) into diagonal blocks associated with

each atom, DA , DB , etc.:

(2.10.10)

Each block is then diagonalised in order to give the MAO’s, ϕϕϕϕ , for the associated atom; for

example,

A A A A A A A

A

+ +==

U D U U S U d

d(2.10.11)

A B C …

A

B

C

…

χ1

χ2

χ3 …

…

DA

DB

DC

χ3

χ2

χ1

D =


88

and hence

A A A= Uϕ χϕ χϕ χϕ χ (2.10.12)

Only a minimal number of these MAO’s on each atom are used to construct the projection

operators; that is, only the MAO’s with the highest occupation numbers (eigenvalues) are

included. This means that the set of projection operators may not, in fact, fully span the space

of the molecule and some fraction of the total charge is left unassigned. This unassigned

charge, ε, is defined as:

( )n Trε = − DP (2.10.13)

where n is the total number of electrons in the system and P is the total projection operator for

all atoms in the molecule:

...A B C= + + +P P P P (2.10.14)

In practice ε is usually very small (less than 0.05) and can be safely neglected. A large

unassigned charge (0.2 or larger) is, however, believed to be indicative of hypervalency in the

molecule.168 The concept of the unshared population, uA , associated with an atom has been

introduced in an effort to quantify this hypervalency:

( )TrA AB Cu n ′ ′= − DP (2.10.15)

where AB C′ ′P is a projection operator defined in terms of the minimal MAO basis for atom A

but the non-minimal (full) MAO sets for all other atoms.


89

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145. B. Rusic, D. Feller, D. A. Dixon, K. A. Peterson, L. B. Harding, R. L. Asher and A. F.

Wagner, J. Phys. Chem. A, 2001, 105, 1.

146. P. W. Atkins, Physical Chemistry, 5th ed.; Oxford University Press: Oxford, UK,

1994, 694.

147. D. A. McQuarrie, Statistical Mechanics; Harper & Row: New York, 1973, 129.


1994, 556.


1994, 692.

150. J. I. Steinfeld, J. S. Francisco and W. L. Hase, Chemical Kinetics and Dynamics;

Prentice Hall: Engelwood Cliffs, NJ, 1989, 308.

151. M. G. Evans and M. Polanyi, Trans. Faraday Soc., 1935, 31, 875.

152. H. Eyring, J. Chem. Phys., 1935, 3, 107.

153. S. Arrhenius, Z. Physik. Chem., 1889, 4, 226.

154. E. Pollack, in Theory of Chemical Reaction Dynamics, M. Baer, Ed.; Vol. 3; CRC

Press, Inc.: Boca Raton, Florida, 1985, p. 128.

155. W. L. Hase, S. L. Mondro, R. J. Duchovic and D. M. Hirst, J. Am. Chem. Soc., 1987,

109, 2916.

156. D. G. Truhlar and B. C. Garrett, Acc. Chem. Res., 1980, 13, 440.

157. R. G. Gilbert and S. C. Smith, Theory of Unimolecular and Recombination Reactions;

Blackwell Scientific Publications: Oxford, UK, 1990.

158. O. K. Rice and H. C. Ramsperger, J. Am. Chem. Soc., 1928, 50, 617.

159. L. S. Kassel, J. Phys. Chem., 1928, 32, 1065.

160. R. A. Marcus, J. Chem. Phys., 1952, 20, 359.

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162. H. M. Rosenstock, M. B. Wallenstein, A. L. Wahrhaftig and H. Eyring, Proc. Natl.

Acad. Sci. USA, 1952, 38, 667.

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164. E. R. Davidson, J. Chem. Phys., 1967, 46, 3320.

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167. C. Ehrhardt and R. Ahlrichs, Theor. Chim. Acta, 1985, 68, 231.

168. D. W. J. Cruickshank and E. J. Avaramides, Phil. Trans. R. Soc. Lond. A, 1982, 304,

533.

3 Thermochemistry of Fluorocarbons

Chapter 3

Thermochemistry of

Fluorocarbons

Chapter 3. Fluorocarbons

97

3.1 Introduction

With the recent international restrictions on the production and deployment of chloro- and

bromo-fluorocarbons (CFC, BFC), much effort is currently devoted to the search for suitable

ozone-friendly replacements. An important use of halons, such as trifluorobromomethane

(CF3Br), has been as fire suppressants. Unfortunately, the bromine atoms that are so efficient

in extinguishing flames, by removing hydrogen radicals, are also efficient catalysts of the

ozone reduction process. Indeed, the ozone depletion potential of CF3Br is an order of

magnitude greater than that of most CFC’s.1 Fluorocarbons and hydro-fluorocarbons have

been identified as promising candidates for fire-suppressants2 and considerable effort is being

devoted to their study. This has resulted in the generation of extensive thermochemical and

kinetic databases.3,4 Unlike CFC’s and BFC’s, fluorinated hydrocarbons have zero ozone

depleting potential, although they are potential greenhouse gases. Fluorocarbons are also

widely used as lubricants, blowing and sterilising agents, anaesthetics, propellants,

refrigerants, and in the preparation of semiconductors.

There is considerable current interest in 2-H-heptafluoropropane (CF3CHFCF3, FM-200) as a

potential fire retardant.5,6 Unlike bromine, fluorine forms much stronger bonds and thus

fluorine atoms are not recycled in the flame, as one fluorine radical will terminate just one

hydrogen radical. Hence, with 7 fluorines per molecule, it is not surprising that flame tests

have shown CF3CHFCF3 to be a very effective fire retardant7. The pyrolysis kinetics of

CF3CHFCF3 at 1200-1500 K has been the subject of a recent shock tube and kinetic modelling

study by Hynes et al.8 The dominant initiation pathways were identified as HF elimination and

CC bond fission:

3 3 3 6CF CHFCF C F + HF→ (3.1)

and

3 3 3 3CF CHFCF CF CHF + CF→ (3.2)


98

The most important subsequent reactions are

(1) the decomposition of the CF3CHF radical

3 2CF CHF CF =CHF + F→ (3.3)

(2) the abstraction of H from the parent molecule

CF3CHFCF3 + F → CF3CFCF3 + HF

followed by the decomposition reactions

3 3 3 3CF CFCF CF CF: + CF→ (3.4)

3 2 2 2 2CF CF: CF =CF CF + CF→ → (3.5)

and the secondary reaction

2 2 2 3 6CF =CF + CF cyclo-C F→ (3.6)

and

(3) the radical recombination reactions

3 2 3 2CF CHF + CHF CF CHFCHF→ (3.7)

and

3 3 3 3CF + CF CF CF→ (3.8)

As the thermochemistry of a number of species participating in the above reactions had been

poorly characterised at the time, ab initio quantum chemical calculations were carried out

concurrently with the modelling studies, generating heats of formation for most of the


99

intermediates in reactions (3.1) to (3.8). These calculations constitute the major part of this

chapter.

In subsequent work Hynes et al.9 studied the kinetics of high-temperature oxidation of C3F6 by

O(3P), where the initial step is the addition of an oxygen atom across the double bond of C3F6:

3 2 3 2CF CF=CF + O CF CFCF O→ (3.9)

The resulting triplet biradical could:

(1) simply decompose to the triplet CF3CF: and CF2O,

(2) undergo a 1,2 F-atom shift and decompose to form CF3CF2 + CFO and

(3) lose fluorine, to yield CF3CFCF=O + F.

Some of these reactions were also studied using ab initio techniques and the thermochemical

information generated was subsequently used in the kinetic modelling studies of Hynes et al.9

The most recent work in this area has been the shock tube kinetic study10 of the high

temperature reaction of H atoms with hexafluoropropene (C3F6) over the temperature range of

1250-1550 K, in an effort to understand the role C3F6 plays in a flame (given that it is a

pyrolysis product of 2-H-heptafluoropropane). Addition of H across the double bond yields

CF3CHFCF2 or CF3CFCHF2 which can then decompose by CC bond scissions to yield CF3 +

CHFCF2, CF3CHF + CF2, etc., or after F loss, CF2CHF + CF2. Again, ab initio calculations

were carried out to compute, in particular, the heat of formation of the hexafluoropropyl

radicals.

The work presented in this chapter, including the computation of heats of formation of C1, C2

and C3 halons (closed shell singlets, radicals and carbenes), therefore complements and

extends the thermochemical database representing ~ 30 years of experimental work by

numerous scientists as well as ab initio theoretical work principally by Westmoreland,

Zachariah and co-workers2,3,11-13 and by Francisco and co-workers14-17 over the last 7 years. Of


100

particular significance is the work of Smith18 who computed the heats of formation of all the

halons of importance in the kinetic modelling studies using the approximate Gaussian-2

technique: G2(MP2)19.

An important advance in the computation of thermochemistry, particularly of fluorine containing

molecules, was made with the introduction of Gaussian-3 (G3) theory20. G3 has been

demonstrated to be significantly more accurate than Gaussian-2 (G2)21 as well as being

computationally cheaper. This work, therefore, comprises of the recalculation, using G3, of the

heats of formation previously obtained by Smith, as well as a number of other C1 and C2 halons

of general interest, in particular those included in the set of molecules studied by Zachariah et

al.3 by the bond additivity corrected MP4 (BAC-MP4) method22-24.

The accuracy of these computations was maximised, where possible, by calculating the heats of

formation of the species of interest via suitable isodesmic reactions, that is, utilising G3 heats of

reaction in conjunction with accepted literature values for all other species in the reactions.

While this approach is generally more accurate than using computed atomisation energies, its

accuracy is also limited by the reliability of the available literature data. Two approximate

schemes that derive from G3 are also presented which reduce the computational cost of G3 and

therefore allow heats of formation for larger molecules (i.e., those with more than 6 heavy

atoms) to be obtained on modest workstations.

3.2 Theory and Computational Methods

Recent advances in computational quantum chemistry have made the ab initio calculation of

heats of formation via the computation of atomisation energies a realistic endeavour. As noted

in Section 2.7.2, the Gaussian methods, G221 and more recently G320, developed by Pople and

co-workers, achieve this via accurate estimates of the atomic and molecular energies in a near-

complete one-particle basis (and by incorporating an empirical “higher level” correction term).

This is done by correcting the energy obtained in a quadratic configuration interaction

(QCISD(T)) calculation in a small split valence + polarisation functions basis (6-311G(d,p) or

6-31G(d)) by MP4 and MP2 estimates of the changes in the energy with systematic


101

enlargement of the basis sets. Alternatively, as pioneered by Martin25,26, Dixon and Feller27,28

as well as others29,30, the same high level of theory (most commonly the coupled cluster

(CCSD(T)) method) is employed in successively larger correlation consistent basis

computations, such that the computed energies can be confidently extrapolated to an

effectively complete basis limit (see Section 2.7.3). Using computed atomisation energies at 0

K in conjunction with experimental heats of formation of the elements in their atomic states,

the heats of formation of the molecules at 0 K and hence at 298 K are readily obtained, as

discussed in detail by Curtiss et al.31, by calculating also the appropriate thermal contributions

to the atomic and molecular enthalpies.

Direct use of atomisation energies for the computation of heats of formation of chemical

accuracy (usually understood to be ~ 1 kcal mol−1) requires, of course, that level of accuracy

in the computed atomisation energies. For small molecules this is achievable. For example,

for the Gaussian data set of 299 molecules, on average, the G2 and G3 atomisation energies

have been found to be within 1.48 and 1.02 kcal mol−1 of experiment respectively.20 More

recently, Martin and Oliveira25, using a range of extrapolation schemes for CCSD(T) energies,

demonstrated an even higher level of accuracy of ± 0.24 kcal mol−1 in the computation of

heats of formation of some 30 small first and second row molecules.

The major part of the work reported in this chapter was carried out using the G3 level of theory.

Unfortunately, given the current limitations of our computing resources, it was not practicable to

carry out G3 calculations for molecules with more than 6 heavy atoms. To treat larger molecules

two approximations to G3 are proposed in the spirit of the G3(MP2)32 and G2(MP2)19; these, in

our view, retain the major advantages of G3 while offering considerable reductions in

computational cost. To develop and justify the proposed approximations we write the

(vibrationless) equilibrium G3 energy (at the MP2/6-31G(d) geometry), E0 G3b g , as

( ) ( ) ( ) ( ) ( ) ( )( ) ( )

0 G3 QCISD T /6 31G 2 , G3Large

hlc SO

E E d E E df p E

E E

= − + ∆ + + ∆ + ∆ + ∆ + ∆

(3.10)

where ∆E +b g , ∆E df p2 ,b g , ∆E G3Largeb g , ∆E SOb g and ∆E hlcb g are corrections for diffuse,


102

higher polarisation and larger basis set effects (along with core-valence correlation and non-

additivity in the latter), spin-orbit coupling effects and the so-called “higher level” corrections

respectively, as defined by Curtiss et al.20 In G3, ∆E +b g and ∆E df p2 ,b g are evaluated at the

MP4(SDTQ) level, while ∆E G3Largeb g consists of MP2 energies, including MP2(Full)/

G3large. As the most expensive step in a G3 calculation is the MP4(SDTQ)/6-31G( 2 ,df p )

computation of the energy in ∆E df p2 ,b g , which is dominated by the evaluation of the triple

excitations’ contribution, an obvious and reasonable approximation to G3 is to calculate the

contributions to ∆E df p2 ,b g or even both ∆E +b g and ∆E df p2 ,b g at a lower level, such as

MP4(SDQ), MP3 or even MP2. Thus we define the G3(MP4SDQ) approximation as

( )( ) ( ) ( ) ( )( ) ( )( ) ( )

0 MP4SDQ

MP4SDQ

G3 MP4SDQ QCISD T /6 31G

2 , G3Large

hlc SO

E E d E

E df p E

E E

= − + ∆ + + ∆ + ∆

+ ∆ + ∆

(3.11)

The MP2 alternative then trivially results in the G3[MP2(Full)] approximation

( )( ) ( ) ( )( ) ( ) ( )

( ) ( )

0 G3 MP2 Full QCISD T /6 31G

MP2 Full /G3Large MP2 Full /6 31G

hlc SO

E E d

E E d

E E

= − + − −

+ ∆ + ∆

(3.12)

The G3[MP2(Full)] method is, of course, closely related to the G3(MP2)32 method. In the latter

the MP2 correction does not include core-valence correlation and thus the G3large basis is

reduced to the smaller G3MP2large set.

The proposed G3[MP2(Full)] and G3(MP4SDQ) methods can be further improved by

optimisation of the hlc terms, as done for G3(MP2). As discussed in the following section, in

order to minimise the deviation between the G3 and G3(MP4SDQ) or G3[MP2(Full)] heats of

formation for the molecules studied in this work, we propose an adjustment to the hlc terms of

the atoms only, namely to C and D, in the expression

( ) ( )atomshlcE Cn D n nβ α β∆ = − − − (3.13)


103

For G3(MP4SDQ) these are (in mEh) C = 5.708, D = 0.922, while for G3[MP2(Full)] C = 6.461,

D = 0.979. The equilibrium geometries and vibrational frequencies in all these approximate G3

schemes are identical to those defined by G3.

In order to complement the G3 (and approximate G3) calculations described above, Dr

George Bacskay completed a more extensive computational study on a group of small

molecules (namely the closed shell HCCH, HCCF and FCCF acetylenes and the HCC, FCC

and formyloxyl (HCOO) radicals), in which their heats of formation were computed by a

complete basis set extrapolation technique, as recommended by Dixon and Feller27. This work

is an integral part of our collaborative project, and as such it is included here for

completeness. It is particularly relevant as it demonstrates the reliability of G3 results in cases

where they differ significantly from experimentally determined values.

In this work the equilibrium geometries were optimised at the CCSD(T)/cc-pVTZ level. The

zero point vibrational energies of these molecules (with the exception of HCOO) were

calculated at the MP2/cc-pVTZ level and scaled by a factor of 0.96, as in previous work by

this group33. (A very similar factor, 0.9646, was proposed by Pople et al.34 for the scaling of

zero point energies obtained at the MP2(Full)/6-31G(d) level.) The open shell coupled cluster

and MP2 calculations were carried out using the restricted formalisms, viz. RCCSD(T) and

ROMP2. As discussed in a subsequent section, the HCOO frequencies were taken from the

published work of Rauk et al.35

The electronic energies of the molecules and their constituent atoms were computed at the

valence (R)CCSD(T) level using the sequence of (diffuse function) augmented correlation

consistent basis sets aug-cc-pVxZ, x = 2 (D), 3 (T), 4 (Q)36-38. The resulting energies E xb gwere then fitted to a mixed exponential/Gaussian function

( ) ( ) ( )( )2exp 1 exp 1E x A B x C x= + − + − − (3.14)

and to the asymptotic formula


104

( ) ( ) 4

max 1 2E x A B l−= + + (3.15)

where A, B and C are (fitted) constants and lmax is the highest angular momentum quantum

number in the basis set. The constant A thus represents the complete basis set (CBS) limit to

the valence (R)CCSD(T) energy ( x → ∞ ). Using the notation of Dixon and Feller27, the

resulting extrapolated energies are denoted CBS(aDTQ/mix) and CBS(aTQ/lmax), indicating

the extrapolation technique and the sequence of basis sets used. (Note that the lmax type fit

utilises only the (augmented) triple and quadruple zeta basis sets.) The extrapolated energies

were then corrected for core-valence correlation (CV corr), using the cc-pCVQZ basis39,40, by

computing the difference between the all-electron (R)CCSD(T)/cc-pCVQZ and valence

(R)CCSD(T)/cc-pVQZ energies. The energies were further corrected for scalar relativistic

effects, by computing, using first order perturbation theory, the Darwin and mass-velocity

contributions41,42. As in previous work by this group on the heats of formation of

halocarbenes33, these relativistic corrections were computed at the complete active space 2nd

order perturbation theory (CASPT2) level of theory43,44 with full valence complete active

space self-consistent field (CASSCF)45,46 reference states, using the G3large basis.

After combining the molecular electronic and zero point vibrational energies and correcting

the computed atomic energies for spin-orbit coupling, the atomisation energies at 0 K, 0DΣ ,

and hence the heats of formation at 0 K were computed. By adding to 00Hf∆ the appropriate

enthalpy differences ( H H2980

00− ), for which accurate experimental values are available in the

case of the elements and which can be readily calculated for the molecule of interest from the

rotational constants and vibrational frequencies (as described in Section 2.8), the heats of

formation at 298 K are obtained, as discussed in detail by Curtiss et al.31

All Gaussian-3 and related calculations were carried out using the Gaussian98 programs47.

The (R)CCSD(T) and ROMP2 computations of the CBS studies were performed using the

MOLPRO48-50 , CADPAC651 and ACES252 programs, while MOLCAS453 was used to carry

out the CASPT2 relativistic correction calculations. All computations were performed on

DEC alpha 600/5/333 and COMPAQ XP1000/500 workstations of the Theoretical Chemistry

group at the University of Sydney.


105

3.3 Results and Discussion

3.3.1 Heats of Formation from G3 and Related Atomisation

Energies

The G3 energies (including zero point vibrational contributions) for the C1 and C2 halons are

listed in Table 3.1 and Table 3.2 respectively along with the heats of formation at 298 K

obtained from atomisation energies computed at the G3, G3[MP2(Full)] and G3(MP4SDQ)

levels of theory; the G2 results of Berry12,54 and Curtiss21 and the G2(MP2) results of Smith18 are

also included for comparison. The appropriate atomic data used in the computation of the

molecular atomisation energies and heats of formation are given in Appendix 1.1.

As the geometries of the majority of these molecules, calculated at the SCF/6-31G(d) level, were

published by Zachariah et al.3, the MP2(Full)/6-31G(d) geometries obtained in this work are not

included in this thesis. However, as all rotational constants and vibrational frequencies are given

in Appendices 1.2 – 1.4, any additional thermochemical data could be readily generated.

Table 3.1 and Table 3.2 also contain current literature data, that is, experimental values and/or

the results of accurate, high level ab initio computations. In the majority of cases the G3 heats of

formation agree with the literature values to within ~ 1 kcal mol−1, once allowance is made for

the quoted uncertainties in the latter. In some instances, however, larger discrepancies (in excess

of 2 kcal mol−1) are noted, e.g. for CF2O, CFO, CF2CF2, CF3O and HCOO. The first three of

these molecules were recently the subject of an extensive theoretical study by Dixon, Feller and

Sandrone27,28 who concluded that the heats of formation of these molecules at 0 K are −145.2 ±

0.8, −44.1 ± 0.5 and −159.8 ± 1.5 kcal mol−1. These differ from the accepted experimental

estimates by up to 6 kcal mol−1, but are consistent with the G3 predictions. The theoretical value

for tetrafluoroethylene has been recently confirmed by the high level computations of

Bauschlicher and Ricca55, who obtained ∆ f H2980 = −160.5 ± 1.5 kcal mol−1. The remaining

problem cases, including CF3O, will be discussed in the next section on isodesmic calculations.

For a number of systems no errors are reported in the literature cited. For these the estimated

errors of Zachariah et al.3 have therefore been quoted.

Chapter 3. F

luorocarbons

106

Table 3.1 C1 fluoro hydrocarbons: G3 energies and computed and literature heats of formation (in kcal mol−1 unless indicated otherwise).

Molecule E0(G3)/Eh ∆ f H2980 ∆ f H298

0 ∆ f H2980 ∆ f H298

0 ∆ f H2980 ∆ f H298

0Diff

G3 G3[MP2(Full)] G3(MP4SDQ) G2a G2(MP2) Literatureb G3−Lit

CH4 −40.45762 −18.1 −17.5 −18.7 −18.6 −18.1 −17.90±0.08 c −0.2

CH3F −139.64964 −56.9 −56.3 −57.3 −58.3 −58.6 −55.6±2.0 d −1.3

CH2F2 −238.86227 −108.4 −107.9 −108.8 −110.8 −111.6 −108.1±0.4 d −0.3

CHF3 −338.08656 −167.1 −166.9 −167.6 −170.9 −171.8 −166.7±0.6 d −0.4

CF4 −437.30780 −223.9 −223.7 −224.7 −228.6 −230.1 −223.0±0.4 d −0.9

−223.1±1.1 e −0.8

CH3 −39.79329 34.0 34.5 33.4 35.1 35.6 35.1±0.1 f −1.1

CH2F −138.98968 −7.7 −7.4 −8.3 −7.9 −7.8±2.0 g 0.1

CHF2 −238.20132 −58.6 −58.6 −59.2 −60.6 −59.2±2.0 g 0.6

CF3 −337.41737 −112.2 −112.3 −112.9 −114.7 −115.8 −112.8 h −0.6

−112.5±1.0 i −0.5

CH2 (1A1) −39.10301 101.9 102.2 101.2 101.4 101.7 101.7±0.7 j 0.2

102.6±1.0 k −0.7

CHF −138.34011 34.8 34.7 34.4 31.7 32.6 34.2±3.0 l 0.6

35.1±1.0 k −0.3

CF2 −237.60041 −46.6 −47.2 −47.0 −48.2 −50.7 −44.6 m −2.0

−44.0 l −2.6

−45.9±0.3 i −0.7

106

Chapter 3. F

luorocarbons

107

Table 3.1 continued


0 ∆ f H2980 ∆ f H298

0 ∆ f H2980 ∆ f H298

0Diff


CH −38.45831 141.1 141.3 140.4 141.9 142.2 142.0±0.1 c −0.9

CF −137.72111 58.0 57.6 57.4 57.0 59.4±0.3 i −1.4

CH2O −114.43106 −26.6 −26.5 −26.3 −27.9 −26.5 −26.0±1.5 n,o −0.6

CHFO −213.66577 −92.0 −92.0 −91.7 −93.0 −90.0±3.6 c,o −2.0

−91.6±1.7 p 0.4

CF2O −312.88194 −145.7 −145.7 −145.5 −148.6 −147.8 −152.7±0.4 c 7.0

−145.9±0.8 q 0.2

CHO −113.79156 9.7 9.4 10.0 9.3 10.8 9.96±0.20 f −0.3

CFO −213.00549 −42.7 −43.0 −42.4 −43.0 −38.5±1.7 r −4.9

−44.0±0.5 q 1.3

CH3OH −115.62921 −48.1 −47.3 −47.8 −49.4 −47.8 −48.1±0.1 s,o 0.0

CH2FOH −214.84531 −101.9 −101.1 −101.4 −102.9

CHF2OH −314.07127 −161.6 −161.0 −161.2 −163.9

CF3OH −413.29243 −218.3 −217.7 −218.1 −222.1 −217.7±2.0 p −4.8

CH3OF −214.71751 −21.5 −20.1 −21.1 −21.9 −17.3±3.0 t,o −4.2

CH2FOF −313.92729 −71.2 −69.9 −70.7 −73.1

CHF2OF −413.14211 −123.9 −122.7 −123.6 −126.4

CF3OF −512.35912 −178.0 −176.8 −177.8 −183.0 −173.0±2.0 p 4.8

107

Chapter 3. F

luorocarbons

108

Table 3.1 continued


0 ∆ f H2980 ∆ f H298

0 ∆ f H2980 ∆ f H298

0Diff


CH3O −114.96272 4.9 5.6 4.3 7.0 4.1±0.2 u 0.8

CH2FO −214.17891 −48.9 −48.4 −49.3 −48.3

CHF2O −313.38786 −98.0 −97.6 −98.6 −99.0

CF3O −412.60361 −151.2 −150.9 −151.9 −153.8 −149.2±2.0 p 5.5

CH2OH −114.97710 −3.9 −3.4 −3.7 −3.8 −2.1 −4.08±0.8 f 0.2

CHFOH −214.18595 −53.0 −52.6 −52.7 −54.9

CF2OH −313.40523 −108.7 −108.3 −108.4 −110.0

CH2OF −214.05996 −26.1 −27.3 −26.5 −27.2

CHFOF −313.25679 −15.3 −14.2 −15.3 −18.0

CH3OOH −190.72485 −30.1 −28.7 −29.0 −28.9 −33.2 v 3.1

−31.3±2.0 n,o 1.2

CF3OOH −488.37663 −193.1 −191.8 −192.2 −194.9

CH3OO −190.09001 2.9 4.1 3.0 5.7 2.2 v 0.7

CF3OO −487.73047 −152.9 −151.9 −153.0 −154.7 −144.0±3.0 t,o −8.9

HCOOH −189.65671 −90.6 −90.2 −89.4 −92.5 −85.6 −90.5±0.1 n,o −0.1

FCOOH −288.87711 −146.9 −146.6 −145.8 −145.4

108

Chapter 3. F

luorocarbons

109

Table 3.1 continued


0 ∆ f H2980 ∆ f H298

0 ∆ f H2980 ∆ f H298

0Diff


HCOO (2A1) −188.98028 −31.1 −30.3 −29.7 −26.8 −37.7±3.0 w 5.6

−29.3±1.0 x −2.8

FCOO (2B2) −288.19901 −86.5 −86.1 −85.5 −85.5

CH2OHOH −190.82596 −93.9 −92.8 −92.6 −92.7 −93.5±2.0 y −0.4

CF2OHOH −389.27646 −212.3 −211.3 −211.1 −213.7

OCH2OH −190.15797 −39.9 −39.0 −39.5 −37.0

OCF2OH −388.59024 −146.9 −146.2 −146.7 −139.3

a G2 results from Refs 12, 21 and 54.b Experimental value unless otherwise indicated by

italics and footnotes.c Ref. 56. 56

d Ref. 57. 57

e CCSD(T)/CBS computations, Ref. 55. 55

f Ref. 58.58

g Ref. 59.59

h Ref. 60.60

i CCSD(T)/CBS computations, Ref. 28, with

thermal corrections from this work.j Ref. 61.61

k CCSD(T)/CBS computations, Ref. 33.l Ref. 62. 62

m Ref. 63.63

n Ref. 64.64

o Error as given in Ref. 3.p Ref. 65.65

q CCSD(T)/CBS computations, Ref. 27,

with thermal corrections from this work.r Ref. 66.66

s Ref. 67. 67

t Ref. 68. 68

u Ref. 69. 69

v Ref. 70.70

w Ref. 71. 71

x Ref. 72.72

y Ref. 73.73

109

Chapter 3. F

luorocarbons

110

Table 3.2 C2 fluoro hydrocarbons: G3 energies and computed and literature heats of formation (in kcal mol−1 unless indicated otherwise).


0 ∆ f H2980 ∆ f H298

0 ∆ f H2980 ∆ f H298

0Diff

G3 G3[MP2(Full)] G3(MP4SDQ) G2a G2MP2 Literatureb G3−Lit

CH3CH3 −79.72340 −20.4 −19.9 −20.9 −20.6 −19.9 −20.1±1.0 c,d −0.3

CH3CH2F −178.92623 −65.7 −65.2 −66.1 −71.2 −66.8 −66.1±1.0 e 0.4

CH2FCH2F −278.12348 −107.3 −106.9 −107.6 −109.9 −110.9 −103.7±2.8 f −3.6

CH3CHF2 −278.14559 −121.3 −120.9 −121.6 −123.9 −123.9 −119.7±1.5 g −1.6

CHF2CH2F −377.33990 −161.1 −160.7 −161.3 −164.2 −165.3 −158.9±1.0 h −2.2

CH3CF3 −377.37214 −181.3 −181.0 −181.7 −184.5 −185.3 −178.2±0.4 g −3.1

CHF2CHF2 −476.55281 −212.5 −212.2 −212.7 −216.7 −216.9 −209.8±4.2 f −2.7

CH2FCF3 −476.56312 −219.0 −218.7 −219.3 −223.3 −224.7 −214.1±2.0 g −4.9

CHF2CF3 −268.2 −268.8 −273.9 −264.0±1.1 g

CF3CF3 −323.8 −324.5 −330.7 −320.9±1.5 g

CH3CH2 −79.06400 28.7 29.0 28.0 29.9 30.7 28.3±1.0 c,d 0.4

CH2FCH2 −178.26370 −14.7 −14.4 −15.2 −14.4 −14.2±2.0 i −0.5

CH3CHF −178.26902 −18.2 −18.0 −18.7 −18.0 −16.8±2.0 i 1.4

CH2FCHF −277.46342 −58.1 −58.0 −58.6 −59.6 −57.0±3.0 f −1.1

CHF2CH2 −277.47984 −68.3 −68.0 −68.7 −69.5 −68.3±3.6 f 0.0

CH3CF2 −277.48538 −71.9 −71.9 −72.5 −73.5 −72.3±2.0 j 0.4

CH2FCF2 −376.67696 −110.0 −110.0 −110.5 −113.3 −107.5±3.6 f −2.5

CHF2CHF −376.67800 −110.6 −110.6 −111.1 −113.6 −109.0±3.6 f −1.6

CF3CH2 −376.70469 −127.3 −127.1 −127.7 −129.8 −123.6±1.0 j −3.7

CF3CHF −475.90075 −168.3 −168.3 −168.8 −172.7 −162.7±2.3 k −5.6

CHF2CF2 −475.88795 −160.3 −160.4 −160.7 −165.0 −158.9±4.5 f −1.4

CF3CF2 −216.5 −216.9 −213.0±1.0 j

110

Chapter 3. F

luorocarbons

111

Table 3.2 continued


0 ∆ f H2980 ∆ f H298

0 ∆ f H2980 ∆ f H298

0Diff


CH2CH2 −78.50742 12.3 11.7 12.1 12.8 13.2 12.54±0.07 l −0.3

CH2CHF −177.71256 −34.4 −35.1 −34.5 −34.9 −35.0 −33.5±0.6 m −0.9

CHFCHF−Z −276.90631 −73.8 −74.6 −73.9 −76.0 −71.0±2.4 n −2.8

CHFCHF−E −276.90730 −74.5 −75.3 −74.7 −76.9 −70.0±2.4 n −4.5

CH2CF2 −276.92299 −84.5 −85.2 −84.6 −86.4 −80.4±1.0 m −4.1

CHFCF2 −376.11080 −120.1 −120.9 −120.2 −123.7 −117.4±2.2 m −2.7

CF2CF2 −475.30917 −162.3 −163.2 −162.6 −165.6 −167.5 −157.4±0.7 l −4.9

−160.6±1.5 o −1.7

−160.5±1.5 p −1.8

CH3CH −78.38810 87.5 87.4 87.0 87.7

CH2FCH −177.59324 40.7 40.6 40.5 39.6

CHF2CH −276.79333 q −12.1 q

CF3CH −376.01256 −58.2 −58.2 −58.5 −63.4

CH3CF −177.62985 17.9 17.5 17.6 16.2

CH2FCF −276.82347 −21.7 −22.1 −21.9 −25.2

CHF2CF −376.02767 −67.7 −68.1 −67.9 −73.0

CF3CF −475.24834 −124.0 −124.4 −124.2 −131.0

CH2CH −77.83307 70.5 70.0 70.2 72.7 73.4 71.6±0.8 r −1.1

CHFCH−Z −177.03040 28.7 28.1 28.4 29.8

CHFCH−E −177.03102 28.3 27.7 27.9 29.4

CH2CF −177.03465 26.0 25.3 25.6 26.9

CHFCF−Z −276.22409 −10.8 −11.6 −11.2 −11.7

CHFCF−E −276.22357 −10.4 −11.1 −10.8 −11.2

111

Chapter 3. F

luorocarbons

112

Table 3.2 continued


0 ∆ f H2980 ∆ f H298

0 ∆ f H2980 ∆ f H298

0Diff


CF2CH −276.23666 −18.7 −19.4 −19.1 −19.2

CF2CF −375.42392 −54.0 −54.8 −54.4 −56.4 −45.9±2.0 s −8.1

CH3C −77.75331 120.6 120.4 120.0 122.0

CH2FC −176.95053 78.8 78.6 78.4 78.6

CHF2C −276.14927 36.1 35.8 35.6 34.8

CF3C −375.36705 −18.4 −18.7 −18.9 −22.0

HCCH −77.27596 54.9 53.6 55.3 55.8 56.3 54.2±0.2 c,d 0.7

HCCF −176.45463 24.8 23.4 25.0 24.9 30.0±5.3 l −5.2

FCCF −275.62524 −0.0 −1.6 −0.1 −1.1 5.0±5.0 l −5.0

CH2C −77.20691 98.5 97.4 98.3 99.3

CHFC −176.38031 71.5 70.4 71.3 70.4

CF2C −275.57646 30.4 29.2 30.2 27.5

CCH −76.56469 136.3 135.1 136.1 138.7 139.4 135.0±1.0 t 1.3

CCF −175.73867 109.3 107.8 108.9 110.5 110.0±5.3 f −0.7

CH2CO −152.50687 −12.1 −13.2 −11.5 −12.1 −10.4 −11.4±0.4 u −0.7

CHFCO −251.68018 −38.8 −39.9 −38.2 −38.9

CF2CO −350.86874 −75.0 −76.0 −74.4 −76.5

CHCO −151.84066 40.9 39.8 41.2 43.4 41.9±2.0 r −1.0

CFCO −251.00583 19.3 18.1 19.6 24.1

CH3CHO −153.71480 −39.8 −39.8 −39.4 −41.0 −39.1 −39.7±0.1 v,d −0.1

CH2FCHO −252.90987 −80.2 −80.2 −79.7 −80.1

CHF2CHO −352.12071 −130.4 −130.4 −129.8 −132.8

CF3CHO −451.34093 −186.5 −186.5 −186.0 −190.3

112

Chapter 3. F

luorocarbons

113

Table 3.2 continued


0 ∆ f H2980 ∆ f H298

0 ∆ f H2980 ∆ f H298

0Diff


CH3CFO −252.95069 −105.8 −105.8 −105.4 −107.7 −106.2

CH2FCFO −352.14170 −143.6 −143.7 −143.1 −144.7

CHF2CFO −451.34766 −190.7 −190.8 −190.2 −194.6

CF3CFO −550.56716 −246.3 −246.3 −245.8 −251.4

CH3CO −153.07373 −2.5 −2.7 −2.1 −2.8 −0.9 −2.4±0.3 r −0.1

CH2FCO −252.26707 −41.9 −42.1 −41.5 −41.7

CHF2CO −351.47501 −90.3 −90.5 −89.8 −92.1

CF3CO −450.69399 −145.6 −145.9 −145.2 −148.9

a G2 results from Refs 12, 21 and 54.b Experimental value unless otherwise indicated by italics and footnotes.c Ref. 74. 74

d Error as given in Ref. 3.e Ref. 75.75

f BACMP4, Ref. 3. g Ref. 76.76

h Ref. 77.77

i Ref. 78.78 j Ref. 63. k Ref. 79. 79

l Ref. 56.m Ref. 60.n Ref. 80. 80

o CCSD(T)/CBS computations, Ref. 28, with thermal corrections from Ref. 55.p CCSD(T)/CBS computations, Ref. 55.q Computed at HF/6-31G(d) geometry, see text. r Ref. 58. s Ref. 81. 81

t Ref. 59.u Ref. 82. 82

v Ref. 64.

113


114

On the whole the G3[MP2(Full)] and G3(MP4SDQ) results are in reasonable agreement with

those obtained by the application of G3. The average absolute deviations of G3[MP2(Full)] and

G3(MP4SDQ) from G3 are ~ 0.5 and 0.4 kcal mol−1 respectively, the largest deviation being 1.6

kcal mol−1 in the case of FCCF. The deviations are significantly larger when the G2(MP2) and

G3 results are compared, up to 6 kcal mol−1. However, as discussed in the next section, the

consistency between the computed heats of formation is much improved once isodesmic

reaction schemes are used.

It is noted that no equilibrium structure was found at the MP2(Full)/6-31G(d) level for the

CHF2CH carbene. The MP2 as well as B3LYP/6-31G(d) density functional optimisations

converge to difluoro ethylene, CF2CH2. These results suggest that CHF2CH may not exist as a

distinct molecule. Nevertheless, to give an estimate of the energy of this probably metastable

carbene, its G3 heat of formation was computed at the HF/6-31G(d) geometry, as at that level of

theory there is a local minimum on the potential surface for CHF2CH.

Comparison of the G3 heats of formation with the BAC-MP4 values for the C1 and C2 halons

studied by Zachariah et al.3 suggests remarkably good agreement on the average, the mean

absolute deviation between the two sets being just 1.6 kcal mol−1. While the agreement is mostly

excellent (~ 1 kcal mol−1 or better), for a number of molecules, e.g. FCCF, CCH, CH and

CH2FOF, substantial disagreement (~ 5 kcal mol−1) has been noted.

3.3.2 Heats of Formation from G3 and related Isodesmic Reaction

Enthalpies

The calculation of accurate atomisation energies and hence heats of formation is a stringent and

demanding test of the quantum chemical methodology since the molecules of interest and their

constituent atoms need to be described in an accurate and balanced manner. It has long been

recognised, however, that the computation of isodesmic reaction energies, where the number of

bond types is conserved, is much less demanding with respect to the resolution of electron

correlation. Therefore reasonably accurate predictions of heats of formation are possible by

utilising isodesmic schemes, even at relatively low levels of theory. The success of such an


115

approach, however, crucially depends on the availability of accurate thermochemical data for

molecules that are chemically similar to those under study, that is, molecules with the same type

of bonds. Given the demonstrated accuracy of G3 theory in the calculation of atomisation

energies, major improvements are not expected in the heats of formation when these are

recalculated from suitable isodesmic reaction energies. It is expected, however, that there will be

a higher level of consistency between the three methods used, viz. G3, G3[MP2(Full)] and

G3(MP4SDQ) (and with the G2(MP2)-ID values of Smith), than observed in the data in Table

3.1 and 3.2. The application of isodesmic schemes to the heats of formation obtained from

atomisation energies can therefore also be regarded as a test of the consistency of the

calculations and their results.

There are relatively few bond types among the molecules in this study (C-H, C-F, C-C, C=C,

C-O, C=O, etc.) but, as may be noted on inspection of the data in Table 3.1 and 3.2, the number

of molecules with accurate (≤ 1 kcal mol−1) experimental or computed heats of formation is

quite small; this means that not all bond types are represented by the selected set: CH4, CF4,

CH3, CH2, CF2, CF2O, CFO, CH3OH, CH3O, C2H6, C2H4 and C2H2. Nevertheless, using these

12 molecules it is possible to construct isodesmic reactions for the majority of the molecules

studied in this work, as demonstrated by the results summarised in Table 3.3. For example, the

heats of formation of all hydrofluoroethanes can be obtained from the computed heats of the

reactions

2 6 4 2 6 4C H CF C H F CH4 4x x

x x−+ → + (3.16)

and experimental enthalpies of formation of C2H6, CH4 and CF4. As discussed by Berry et al.12,

such use of isodesmic reactions is equivalent to applying a bond additivity correction to the

heat of formation of the molecule of interest, C2H6−xFx in the current example. Such a bond

additivity corrected enthalpy of formation is then written

0 0298 298 CC CH CF(BAC) (calc) (6 )f fH H x x∆ = ∆ − ∆ − − ∆ − ∆ (3.17)

where ∆ f H2980 ( )calc is the enthalpy of formation of C2H6−xFx calculated from its atomisation

energy. The bond correction parameters ∆CC , ∆CH and ∆CF are obtained by comparison of


116

∆ f H2980 ( )calc and ∆ f H298

0 ( )expt of the reference molecules C2H6, CH4 and CF4, for example:

0 01CH 298 4 298 44 (CH ,calc) (CH ,expt)f fH H ∆ = ∆ − ∆ (3.18)

As expected, the G3, G3[MP2(Full)], G3(MP4SDQ) and G2(MP2) heats of formation obtained

from the corresponding isodesmic reaction enthalpies, as listed in Table 3.3, are in much closer

agreement than those obtained from atomisation energies. The differences are generally no

greater than 0.3 kcal mol−1. Clearly, considerable error cancellation occurs when we compute

the heats of isodesmic reactions. It is worth noting also that all empirical hlc contributions to the

Gaussian-2, -3, etc. energies completely cancel when one computes isogyric or isodesmic

reaction energies. Nevertheless, the differences between the G3 heats of formation when

obtained from atomisation energies and isodesmic reaction enthalpies are moderately small: ~

0.9 kcal mol−1 on the average and no larger than 1.6 kcal mol−1. This is of course expected, given

that the heats of formation of the above 12 reference molecules are quite accurately predicted

from the G3 atomisation energies. On the other hand, in the case of certain heats of formation,

such as CF3O, CH2FCF3 and CF3CHF, where initially large discrepancies (~ 5 kcal mol−1)

between the G3 and literature values of were noted (see Tables 3.1 and 3.2), the application of

isodesmic (viz. bond additivity) corrections, does not significantly improve the situation. We

believe that in the case of such molecules the precision in the literature values is considerably

less than implied by the quoted errors.

Table 3.4 summarises the heats of formation for the C3 systems that were of direct interest in

the kinetic modelling studies of Hynes et al.10 The various schemes yield very consistent results

in that the isodesmic heats of formation are, with one exception, within 0.2 kcal mol−1 of each

other and up to ~ 3 kcal mol−1 higher than when obtained from atomisation energies. The

variations are largest for hexafluoropropene and the hexafluoropropyl radical. Utilising the

isodesmic results, it is estimated that the heats of formation of these two species as −276.2 ± 2

and −266.4 ± 3 kcal mol−1 respectively, on the basis of the spread of computed values and the

expected intrinsic accuracy of the G3 method. Given the good agreement between the computed

(isodesmic) and experimental heats of formation for propene, n-propyl and hexafluoropropene,

the computed value for hexafluoropropyl, −266.4 ± 3 kcal mol−1, is expected to be similarly

reliable.

Chapter 3.

Fluorocarbons

Chapter 3. F

luorocarbons

117

Table 3.3 C1 and C2 fluoro hydrocarbons: Computed heats of formation via isodesmic (ID) reactions of selected species (in kcal mol−1).

Molecule Eqn G3 Diffb G3[MP2(Full)] G3(MP4SDQ) G2(MP2) Literaturec Diff

ID ID − AE ID ID ID G3(ID) − Lit

CH3F 1 −56.5 0.4 −56.4 −56.3 −56.7 −55.6±2.0 −0.9

CH2F2 1 −107.8 0.6 −107.7 −107.5 −108.0 −108.1±0.4 0.2

CHF3 1 −166.4 0.7 −166.5 −166.1 −166.5 −166.7±0.6 0.4

CH2F 2 −6.5 1.2 −6.7 −6.3 −6.8 −7.8±2.0 1.3

CHF2 2 −57.2 1.4 −57.5 −57.0 −57.7 −59.2±2.0 2.0

CF3 2 −110.7 1.5 −111.0 −110.6 −111.3 −112.5±1.0 d 1.8

CHF 3 35.7 0.9 35.4 36.0 35.2 35.1±1.0 e 0.6

CF2 3 −45.5 0.9 −46.2 −45.1 −46.3 −45.9±0.3 d 1.2

CHF 4 35.1 0.3 35.5 35.6 35.7 35.1±1.0 e 0.0

CH2O 5 −27.2 −0.6 −27.3 −27.2 −28.0 −26.0 −1.2

CHFO 5 −92.4 −0.4 −92.5 −92.3 −92.8 −90.0 −2.4

CHO 6 8.0 −1.7 7.9 8.0 7.9 10.4±2.0 −0.9

CH2FOH 7 −101.7 0.2 −101.6 −101.5 −101.5

CHF2OH 7 −161.2 0.4 −161.2 −161.0 −160.8

CF3OH 7 −217.8 0.5 −217.7 −217.7 −217.2 −213.5 −4.3

CH2FO 8 −49.6 −0.7 −49.7 −49.4 −49.6

CHF2O 8 −98.6 −0.6 −98.7 −98.5 −98.6

CF3O 8 −151.6 −0.4 −151.7 −151.5 −151.7

117

Chapter 3.

Fluorocarbons

Chapter 3. F

luorocarbons

118

Table 3.3 continued



CH3CH2F 9 −65.2 0.5 −65.1 −65.1 −65.3 −62.9±0.4 −2.3

CH2FCH2F 9 −106.6 0.7 −106.5 −106.3 −107.7 −103.7±2.8 −2.9

CH3CHF2 9 −120.6 0.7 −120.5 −120.3 −120.7 −119.7±1.5 −1.0

CHF2CH2F 9 −160.2 0.9 −160.1 −159.8 −160.4 −158.9±1.0 −1.4

CH3CF3 9 −180.5 0.8 −180.4 −180.2 −180.4 −178.2±0.4 −2.3

CHF2CHF2 9 −211.5 1.0 −211.3 −211.0 −210.2 −209.8±4.2 −1.7

CH2FCF3 9 −218.0 1.0 −217.8 −217.6 −218.0 −214.1±1.0 −3.9

CHF2CF3 9 −267.0 −266.9 −264.0±1.1 1.3

CF3CF3 9 −322.3 −322.3 −320.9±1.5 −2.1

CH3CH2 10 29.8 1.1 29.7 29.8 29.7 28.3 1.5

CH2FCH2 10 −13.4 1.3 −13.5 −13.2 −13.6 −11.40±0.24 −2.0

CH3CHF 10 −16.9 1.3 −17.1 −16.7 −17.3 −18.2±1.4 1.3

CH2FCHF 10 −56.6 1.5 −56.7 −56.3 −57.1 −57.0±3.0 0.4

CHF2CH2 10 −67.0 1.3 −66.9 −66.8 −67.2 −68.3±3.6 1.3

CH3CF2 10 −70.6 1.3 −70.8 −70.6 −71.0 −72.3±2.0 1.8

CH2FCF2 10 −108.5 1.5 −108.6 −108.4 −109.1 −107.5±3.6 −1.0

CHF2CHF 10 −109.1 1.5 −109.2 −109.0 −109.4 −109.0±3.6 −0.1

CF3CH2 10 −125.8 1.5 −125.7 −125.6 −125.8 −123.6±1.0 −2.3

CF3CHF 10 −166.7 1.6 −166.7 −166.5 −167.0 −162.7±2.3 −4.0

CHF2CF2 10 −158.7 1.6 −158.8 −158.4 −159.3 −158.9±4.5 0.2

CF3CF2 10 −214.6 −214.3 −213.0±1.0

118

Chapter 3.

Fluorocarbons

Chapter 3. F

luorocarbons

119

Table 3.3 continued



CH2CHF 11 −34.0 0.4 −34.0 −33.8 −34.0 −33.5±0.6 −0.5

CHFCHF−Z 11 −73.2 0.6 −73.2 −73.0 −73.3 −71.0±2.4 −2.2

CHFCHF−E 11 −73.9 0.6 −73.9 −73.8 −74.1 −70.0±2.4 −3.9

CH2CF2 11 −83.9 0.6 −83.8 −83.7 −83.7 −80.4±1.0 −3.5

CHFCF2 11 −119.3 0.8 −119.2 −119.1 −119.3 −117.4±2.2 −2.0

CF2CF2 11 −161.3 1.0 −161.2 −161.2 −160.6 −160.5 −0.8

CH2CH 12 71.4 0.9 71.6 71.3 71.7 71.6±0.8 −0.2

CHFCH−Z 12 29.8 1.1 30.0 29.7 30.0

CHFCH−E 12 29.4 1.1 29.6 29.2 29.6

CH2CF 12 27.1 1.1 27.2 26.9 26.9

CHFCF−Z 12 −9.6 1.2 −9.5 −9.7 −9.9

CHFCF−E 12 −9.2 1.2 −9.0 −9.3 −9.4

CF2CH 12 −17.5 1.2 −17.3 −17.6 −17.4

CF2CF 12 −52.6 1.4 −52.4 −52.6 −52.9 −45.9±2.0 −6.7

CH3CH 13 88.3 0.8 88.0 88.4 88.2

CH2FCH 13 41.7 1.0 41.5 42.1 41.9

CHF2CH 13 −10.9 1.2

CF3CH 13 −56.9 1.3 −56.8 −56.4 −57.7

CH3CF 14 18.5 0.6 18.7 18.5 18.8

CH2FCF 14 −20.9 0.8 −20.6 −20.8 −20.9

CHF2CF 14 −66.7 1.0 −66.3 −66.6 −66.9

CF3CF 14 −122.8 1.2 −122.3 −122.6 −123.9

119

Chapter 3.

Fluorocarbons

Chapter 3. F

luorocarbons

120

Table 3.3 continued



HCCF 15 24.3 −0.5 24.3 24.1 24.6 30.0±5.3 −5.7

FCCF 15 −0.4 −0.4 −0.5 −0.8 0.3 5.0±5.0 −5.4

CCH 16 136.2 −0.1 135.6 136.2 136.3 135.0±1.0 1.2

CCF 16 109.4 0.1 108.6 109.4 109.2 110.0±5.3 −0.6

a Isodesmic Reactions:

(1) 1 4− xb gCH4 + x4 CF4 → CH4xFx

(2) CH3 + x4 CF4 → x

4 CH4 + CH3xFx

(3) CH2 + x4 CF4 → CH2xFx + x

4 CH4

(4) 12 CH2 + 1

2 CF2 → CHF

(5) CF2O + x4 CH4 → CHxF2xO + x

4 CF4

(6) CFO + 14 CH4 → CHO + 1

4 CF4

(7) CH3OH + x4 CF4 → x

4 CH4 + CH3xFxOH

(8) CH3O + x4 CF4 → x

4 CH4 + CH3xFxO

(9) C2H6 + x4 CF4 → x

4 CH4 + C2H6xFx

(10) C2H6 + CH3 + x4 CF4 → 1 4+ xb gCH4 + C2H5xFx

(11) C2H4 + x4 CF4 → x

4 CH4 + C2H4xFx


(13) C2H6 + CH2 + x4 CF4 → 1 4+ xb gCH4 + CH3xFxCH

(14) C2H6 + CF2 + x−14 CF4 → x+3

4b gCH4 + CH3xFxCF

(15) C2H2 + x4 CF4 → x

4 CH4 + C2H2xFx


b Difference between G3 heats of formation obtained via isodesmic (ID)

reactions and atomisation energies (AE).c Experimental value as in Tables 3.1 and 3.2, unless otherwise indicated.d CCSD(T)/CBS computations, Ref. 28, with thermal corrections from this

work.e CCSD(T)/CBS computations, Ref. 33.

120

Chapter 3.

Fluorocarbons

Chapter 3. F

luorocarbons

121

Table 3.4 C3 fluoro hydrocarbons: G3 energies and computed values of heats of formation from atomisation energies (AE) and isodesmic

reactions (ID) as specified (in kcal mol−1 unless indicated otherwise).


0 ∆ f H2980 ∆ f H298

0

G3 G3[(MP2(Full)] G3(MP4SDQ) Experiment

AE ID AE ID AE ID

CH3CHCH2 −117.78219 4.7 5.0 3.8 4.8 4.6 5.0 4.88b

CH3CH2CH2 −118.33332 24.5 25.7 24.6 25.5 23.8 25.4 23.9 ± 0.5c

CF3CFCF2 −713.01767 −277.6 −276.2 −278.3 −275.6 −277.5 −275.7 −275.3 ± 1.1d

CF3CHFCF2 −269.1 −266.5 −269.3 −266.3

a Isodesmic Reactions:

CH3CH3 + CH2CH2 → CH3CHCH2 + CH4

2CH3CH3 + CH3 → CH3CH2CH2 + 2CH4

CH3CH3 + CH2CH2 + 32 CF4 → CF3CFCF2 + 5

2 CH4

2CH3CH3 + CH3 + 32 CF4 → CF3CHFCF2 + 7

2 CH4

b Ref. 83.83

c Ref. 84.84

d Ref. 85.85

121


122

3.3.3 Comparison of G2 and G3 Methods: Analysis of Atomisation

Energies of Fluoromethanes

As the results of the previous sections clearly indicate, the G3 method is superior to G2 and

G2(MP2) in the prediction of heats of formation of fluoro hydrocarbons from the computed

atomisation energies. In an effort to gain some understanding of the reasons for this an analysis

of the G2 and G3 energetics for the fluoromethanes CH4, CH3F, CH2F2, CHF3 and CF4 was

carried out, where the individual contributions to the composite G2 and G3 atomisation energies

were compared.

Using the decomposition scheme of Equation (3.10), Table 3.5 lists the G2 and G3

atomisation energies (AE) obtained by the appropriate QCISD(T) calculations, followed by the

MP4 and MP2 corrections (for basis incompleteness) and the zero point corrections. Up to

this point the differences between G2 and G3 are due to the different “parent” bases:

6-311G(d,p) for G2 and 6-31G(d) for G3, and the different “large” bases: 6-311+G(3df,2p) for

G2 and the G3large set for G3. Note that thus far all correlated energies, including the

MP2/(G3large), are valence only and thus the sum of these contributions is denoted AE(valence).

The core valence correlation (CV) corrections to the G3 energies are listed separately, along with

the empirical hlc terms and the spin-orbit coupling corrections that are implicit in G3 and,

finally, the resulting total atomisation energies at 0 K.

The trends displayed by the data in Table 3.5 are interesting and informative. The largest

corrections to the QCI values of the atomisation energies (apart from ZPE) are the MP4/(2df,p)

terms. While these are relatively constant in the G3 calculations, ranging from 23.3 to 26.5 kcal

mol−1, in the case of G2 they vary from 5.7 to 30.1 kcal mol−1. In contrast with these, the

MP4/(+) corrections are more significant for G3, especially in CHF3 and CF4. These trends point

to some basic differences between G2 and G3 in the quality of the respective QCI energies and

the relative importance of the MP4/(+) and MP4/(2df,p) corrections in the two schemes. As a

further illustration of this point, Figure 3.1 shows a plot of the QCI atomisation energies,

corrected by the MP4/(+) and zero-point contributions, against the G3 total atomisation energies.

The resulting QCISD(T) + MP4/(+) + ZPE energies, as obtained in the G2 and G3 calculations,

correlate linearly with the benchmark G3 (total) atomisation energies but the two slopes are very

Chapter 3. F

luorocarbons

123

Table 3.5 Comparison of G2 and G3 methods: Analysis of atomisation energies (kcal mol−1) of CH4, CH3F, CH2F2, CHF3 and CF4.

CH4 CH3F CH2F2

G2 G3 G3 − G2 G2 G3 G3 − G2 G2 G3 G3 − G2

AE [QCISD(T)] 401.6 382.9 −18.6 397.9 385.4 −12.5 408.2 402.9 −5.3

∆AE [MP4/(+)] −0.4 −1.3 −0.9 1.9 1.3 −0.6 1.6 −0.1 −1.7

∆AE [MP4/(2df,p)] 5.7 26.5 20.7 10.9 24.9 14.0 16.7 23.8 7.1

∆AE [MP2/(G3large)] 4.3 2.7 −1.6 4.2 1.7 −2.5 4.2 1.5 −2.7

∆AE [ZPE] −26.8 −26.8 0.0 −23.8 −23.8 0.0 −20.2 −20.2 0.0

AE (valence)a 384.5 384.0 −0.5 391.1 389.5 −1.5 410.5 407.9 −2.6

∆AE [CV] 0.0 1.1 1.1 0.0 1.2 1.2 0.0 1.4 1.4

∆AE [hlc] 8.7 7.7 −1.0 8.7 8.0 −0.7 8.7 8.3 −0.4

∆AE [Spin Orbit] 0.0 −0.1 −0.1 0.0 −0.5 −0.5 0.0 −0.9 −0.9

AE (total)b 393.2 392.8 −0.4 399.8 398.3 −1.5 419.2 416.7 −2.5

123

Chapter 3. F

luorocarbons

124

Table 3.5 continued

CHF3 CF4

G2 G3 G3 − G2 G2 G3 G3 − G2

AE [QCISD(T)] 425.8 429.5 3.7 441.4 454.5 13.1

∆AE [MP4/(+)] −0.2 −5.1 −5.0 −2.8 −12.3 −9.5

∆AE [MP4/(2df,p)] 23.2 23.3 0.1 30.1 23.5 −6.6

∆AE [MP2/(G3large)] 4.2 1.6 −2.6 4.3 2.1 −2.2

∆AE [ZPE] −15.8 −15.8 0.0 −10.7 −10.7 0.0

AE (valence)a 437.4 433.5 −3.8 462.4 457.2 −5.2

∆AE [CV] 0.0 1.7 1.7 0.0 1.9 1.9

∆AE [hlc] 8.7 8.6 −0.1 8.7 8.9 0.2

∆AE [Spin Orbit] 0.0 −1.2 −1.2 0.0 −1.6 −1.6

AE (total)b 446.1 442.5 −3.5 471.1 466.4 −4.7

a AE (valence) = AE [QCISD(T)] + ∆AE [MP4/(+) + MP4/(2df,p) + MP2/(G3large) + ZPE]

b AE (total) = AE (valence) + ∆AE [CV + hlc + Spin Orbit ]

124


125

Figure 3.1 Comparison of G2 and G3 atomisation energies of fluoromethanes: Correlation of

the QCISD(T) + MP4/(+) + ZPE components with the G3 total atomisation energies.

different: 1.03 for G3 and 0.74 for G2. Thus, even at this base level of theory, viz. QCISD(T) +

MP4/(+), the G3 values of these scale significantly better with the number of fluorines than the

corresponding G2 values. This, of course, is also reflected in the large variation in the

MP4/(2df,p) corrections in the case of G2, as remarked above. This behaviour points to some

imbalance in the QCI component of the G2 atomisation energies due to inadequacies of the

6-311G(d,p) basis.

Core-valence correlation increases the G3 atomisation energies by 1.1 to 1.9 kcal mol−1, while

spin-orbit coupling corrections change them by −0.1 to −1.6 kcal mol−1, resulting in net changes

of 0.3 to 1.0 kcal mol−1. The G2 and G3 hlc contributions to the atomisation energies differ by

1.0 kcal mol−1 at most, but such that they reduce the differences due to core-valence correlation

and spin-orbit coupling. Thus, effectively, the differences between the total G2 and G3

atomisation energies are almost fully reproduced by the valence calculations alone.

In summary, the shortcomings of G2 when applied to the above molecules are traced to

inadequacies in the 6-311G(d,p) basis. These problems were briefly discussed by Curtiss et al. in

their first paper on G320, although not actually quantified or analysed as in our work.

390 400 410 420 430 440 450 460 470

350

360

370

380

390

400

410

420

430

440 CF4

CHF3

CH2F2

CH3FCH4

G3 (slope = 1.03)

G2 (slope = 0.74)

AE

[Q

CIS

D(T

) +

MP

4/(

+)

+ Z

PE

] /k

cal m

ol-1

AE [G3 (total)] /kcal mol-1


126

3.3.4 Heats of Formation by Complete Basis Set Coupled Cluster

Calculations

HCCH, HCCF, FCCF, CCH, CCF and HCOO were selected for further study, carried out by

Dr George Bacskay, whereby their heats of formation were calculated using the coupled

cluster RCCSD(T) method and large basis sets, allowing the sequences of atomic and

molecular energies to be extrapolated to the hypothetical complete basis set (CBS) limit.

These small molecules were chosen for further study partly because the experimental heats of

formation of several of these (HCCF, FCCF, CCF) are poorly characterised, with estimated

errors of ~ 5 kcal mol−1 in the literature values. The BAC-MP4 heats of formation for HCCF,

FCCF and CCH are also at significant variance with the G3 values. The formyloxyl (HCOO)

radical is an unusual system in that it has several low-lying electronic states. An excellent

summary of the theoretical literature on this interesting molecule is provided in a relatively

recent paper by Rauk et al.35, who also report the results of an extensive CASPT2 and multi-

reference CI (MRCI) study of formyloxyl. Rauk et al. were unable to conclude unequivocally

whether the ground state is 2A1 or 2B2, since the order of the two states (separated by no more

than 2.2 kcal mol−1) was found to be dependent on the method of calculation, although the

broken symmetry 2A′ state consistently appeared to be an excited state. According to G3 the

ground state is 2A1, but the G3 prediction of ∆ f H2980 = −32.1 kcal mol−1 could be regarded as

being equally consistent with the two conflicting literature values −37.7 ± 3.0 and −29.3 ± 1.0

kcal mol−1. Consequently, formyloxyl represents an interesting and challenging application for a

coupled cluster CBS study.

As indicated in Section 3.2, the CBS energies of the above molecules and their constituent

atoms were obtained by extrapolating the sequence of valence correlated (R)CCSD(T)

energies computed using the aug-cc-pVxZ (x = D, T, Q) basis sets, followed by corrections for

core-valence correlation, scalar relativistic effects and zero point vibrational contributions.

The latter were computed at the (RO)MP2/cc-pVTZ level of theory, except in the case of

HCOO, for which the CASPT2 harmonic frequencies of Rauk et al.35, all scaled by 0.96, were

utilised. Table 3.6 contains a representative part of the raw data, namely the total valence

CCSD(T) energies of the molecules obtained in the aug-cc-pVQZ basis, along with the

Chapter 3. F

luorocarbons

127

Table 3.6 Computed and extrapolated CCSD(T) energies, core-valence correlation corrections, zero point vibrational energies, thermal

corrections to enthalpies at 298 K and relativistic corrections (in Eh unless otherwise indicated).

CCSD(T) CCSD(T) CCSD(T) CV corra ZPE 298H relE b

aug-cc-pVQZ CBS(mix) CBS(lmax) /kcal mol−1 /kcal mol−1

C2H2 −77.21098 −77.22119 −77.22182 −0.11010 16.08 18.50 −0.02956

CFCH −176.35179 −176.37717 −176.37800 −0.17499 12.21 14.93 −0.11622

C2F2 −275.48392 −275.52446 −275.52548 −0.23992 8.09 11.34 −0.20291

C2H - 2Σ −76.48915 −76.49876 −76.49922 −0.10949 8.61 10.94 −0.02957

C2F - 2Σ −175.62574 −175.65050 −175.65120 −0.17442 5.12 8.04 −0.11722

HCOO - 2A1 −188.87336 −188.90056 −188.90120 −0.17690 10.07 12.70 −0.11897

HCOO - 2B2 −188.87499 −188.90196 −188.90261 −0.17673 11.75 14.27 −0.11885

HCOO - 2A′ −188.87209 −188.89898 −188.89963 −0.17679 12.03 14.60 −0.11888

H −0.49995 −0.50000 −0.50000 0.00000 1.48 0.00000

C −37.78660 −37.78940 −37.78950 −0.05317 1.48 −0.01501

O −74.99484 −75.00401 −75.00424 −0.06065 1.48 −0.05230

F −99.65266 −99.66690 −99.66710 −0.06463 1.48 −0.08699

a Core-valence correlation from cc-pCVQZ calculations.b Scalar relativistic correction from CASPT2/G3large calculations.

127


128

corresponding extrapolated values and the core-valence correlation corrections, zero point

vibrational energies, thermal corrections to the enthalpies and scalar relativistic corrections.

The resulting atomisation energies at 0 K are given in Table 3.7. Although the effect of the

extrapolation on the total molecular energies is ~ 10 - 25 kcal mol−1 in comparison with those

obtained at the CCSD(T)/aug-cc-pVQZ level of theory, the effect on the atomisation energies

is a modest 3 - 4 kcal mol−1. The mix and lmax methods yield comparable results, so the CBS

atomisation energies were defined as the average of the two sets of extrapolated values. Core-

valence correlation further increases the atomisation energies by ~ 2 kcal mol−1. The scalar

relativistic corrections to the atomisation energies are generally quite small, the largest

correction being just −0.7 kcal mol−1 (for FCCF).

The heats of formation at 0 and 298 K, that were computed from the atomisation energies are

summarised in Table 3.8 along with the corresponding G3 as well as G2 values. The

agreement between the CBS and G3 results is excellent for all molecules, except HCOO,

where the deviation is 2 kcal mol−1. The agreement between the G2 and CBS heats of

formation is generally less good, the maximum difference being 2.8 kcal mol−1. In line with

previous work of this quality, the CBS heats of formation are expected to be accurate to

within 1 kcal mol−1, although this may prove to be a conservative estimate. In the case of

acetylene, where the heat of formation is known accurately, the CBS prediction is in excellent

agreement with experiment. For CCH the theoretical results agree well with the experimental

value of McMillen and Golden59. Given the high level of disagreement when the former are

compared with the current JANAF value56 of 114.0 ± 6.9 kcal mol−1, we must conclude that

the JANAF value is seriously in error. For CFCH, C2F2 and C2F the theoretical predictions,

while consistent with the available experimental estimates, are expected to be more reliable

than the latter. The overall agreement between the CBS and G3 results further supports the

reliability of G3 in predicting heats of formation.

For formyloxyl the 2A1 state is found to be the ground state, with the 2B2 and 2A′ states being

just 1.0 and 3.1 kcal mol−1 higher in energy at 0 K. This ordering is largely the result of the

zero point energies, since in the absence of zero point correction the 2B2 would be predicted to

be the ground state. The resulting heat of formation of HCOO (2A1) at 0 K, viz. −29.4 kcal

mol−1, is in excellent agreement with the experimental value of −28.6 ± 0.7 kcal mol−1


129

Table 3.7 Atomisation energiesa ΣD0 at 0 K computed at various levels of theory (kcal mol−1).

CCSD(T) CCSD(T) CCSD(T) CCSD(T) CCSD(T)

aug-cc-pVQZ CBS(mix) CBS(lmax) CBSb CBSb

+ CV corr + CV corr + relE c

C2H2 384.02 386.85 387.12 389.34 389.04

CFCH 380.03 383.48 383.75 386.13 385.63

C2F2 370.85 374.91 375.18 377.75 377.06

C2H 252.25 254.74 254.91 256.79 256.50

C2F 245.25 248.35 248.53 250.60 250.72

HCOO - 2A1 364.63 367.87 367.92 369.41 369.01

HCOO - 2B2 363.98 367.07 367.12 368.51 368.02

HCOO - 2A′ 361.34 364.92 364.97 366.40 365.94

a Using atomic energies corrected for spin-orbit contributions (from Ref. 20).b Average of CBS(aDTQ/mix) and CBS(aTQ/lmax) results.c Scalar relativistic corrections from CASPT2/G3large calculations.

Table 3.8 Computed heats of formation at 0 and 298 K (kcal mol−1).

∆ f H00 ∆ f H298

0

CBSa G2 G3 CBSa G2 G3 Experiment

C2H2 - 1Σg 54.2 56.0 55.1 54.0 55.8 54.9 54.2±0.2 b

CFCH - 1Σ 24.4 25.0 24.8 24.6 25.0 24.8 30.3±5.3 c

C2F2 - 1Σg −0.2 −0.7 −0.5 0.5 −0.2 0.0 −5.5±5.0

c

C2H - 2Σ 135.1 137.8 135.4 135.9 138.7 136.3 135.0±1.0 d

114.0±6.9 c

C2F - 2Σ 107.7 109.4 108.0 109.1 110.7 109.3 110.0±5.3 e

HCOO - 2A1 −29.4 −31.2 −30.4 −30.1 −31.9 −31.1 −29.3±0.7f

−37.7±3.0g

HCOO - 2B2 −28.4 −29.9 −28.6 −29.3 −30.7 −29.4

HCOO - 2A′ −26.3 −27.9 −27.0 −27.1 −28.7 −27.8

a Average of CBS(aDTQ/mix) and CBS(aTQ/lmax)

results and including scalar relativistic

corrections.b Ref. 74.c Ref. 56.

d Ref. 59.e Estimated from bond dissociation energies, Ref. 3.

f Based on ∆ f H00 = 28.6 ± 0.7 kcal mol−1 from Ref.

72 with thermal corrections from this work.g Ref. 71.


130

reported by Langford et al.72, who used H (Rydberg) atom photofragment translational

spectroscopy to deduce the OH bond dissociation energy of formic acid and hence heat of

formation of formyloxyl. It is worth noting that using G2(MP2) in conjunction with several

isodesmic reactions Yu et al.86 deduced a value of −30.3 ± 0.7 kcal mol−1 for ∆ f H2980 which is

clearly in very good agreement with the current CBS prediction as well as experiment.

3.4 Conclusion

Using the G3 and related methodologies the heats of formation of ~ 120 C1 and C2

hydrofluorocarbons and oxidised hydrofluorocarbons, including a number of C2 carbenes,

were computed. For most molecules studied in this work the G3 heats of formation are in

good agreement with the available experimental data, attesting to the capability and reliability

of G3. Indeed, there is growing evidence, in the form of accurate ab initio values, that where

the discrepancy between G3 and experiment is in excess of 2 kcal mol−1, it may well signal

inaccuracies in the latter. While for most molecules the G3 predictions agree well with those

of the BAC-MP4 method, there are also sizeable discrepancies. Given the apparent robustness

of G3 and its relative ease of application as displayed in this chapter, it is highly

recommended for use in the computation of thermochemical data. The application of suitable

isodesmic reaction schemes, as expected, has the potential to improve the accuracy and

consistency of the predictions, especially when using approximate forms of G3, such as

G3[MP2(Full)] and G3(MP4SDQ). Using this approach, the heat of formation of the

hexafluoropropyl radical, an important intermediate in the high temperature reaction of H

atoms with hexafluoropropene, was computed and subsequently used in the kinetic model

describing the pyrolysis of 2-H-heptafluoropropane.10 In addition to the G3 and related

applications, the heats of formation of the fluoroacetylenes (HCCF and C2F2 as well as C2H2)

and the C2H, C2F and formyloxyl radicals were computed using the coupled cluster method,

with extrapolations to the CBS limit. The computed heats of formation are believed to be

accurate to within 1 kcal mol−1, providing useful and reliable data for HCCF, C2F2 and C2F,

while in the case of formyloxyl it strongly supports the experimental value of 29.3 ± 0.7 kcal

mol−1 of Langford et al.72 These results provide additional support for our confidence in the

reliability of the G3 method.


131

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Serrano-Andrés, P. E. M. Siegbahn and P.-O. Widmark, MOLCAS Version 4, Lund

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59. D. F. McMillen and D. M. Golden, Annu. Rev. Phys. Chem., 1982, 33, 493.

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62. J. C. Poutsma, J. A. Paulino and R. R. Squires, J. Phys. Chem., 1997, 101, 5327.

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64. D. L. Baulch, R. A. Cox, R. F. Hampson, Jr., J. A. Kerr, J. Troe and R. T. Watson, J.

Phys. Chem. Ref. Data, 1984, 13, 1259.

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4 The Role of Phosphorus Compounds in the H + OH Recombination Reaction

Chapter 4

The Role of Phosphorus

Compounds in the

H + OH Recombination

Reaction

Chapter 4. Phosphorus in the H + OH Reaction

137

4.1 Introduction

The recombination reaction of H and OH radicals to form water is a key exothermic reaction

in a range of combustion processes, particularly in flames and in the combustion of hydrogen

fuel in the presence of oxygen. As pointed out by Twarowski,1-4 this last reaction is of special

importance in supersonic aircraft engines, where recombination must be as fast as possible in

order to maximise engine efficiency. Twarowski carried out a number of experimental studies

of the H + OH recombination reaction in the presence of phosphine and concluded the

reaction is catalysed by the oxidation products of PH3, namely PO2, HOPO and HOPO2. Two

possible reaction sequences were proposed to account for this catalysis:

2H + PO (+ M) HOPO (+ M)→ (4.1.a)

2 2H + HOPO H + PO→ (4.1.b)

2 2OH + H H O + H→ (4.1.c)

and

2 2OH + PO (+ M) HOPO (+ M)→ (4.2.a)

2 2 2H + HOPO H O + PO→ (4.2.b)

The net result of both proposed catalytic cycles is the recombination reaction of interest:

(4.1)(4.2)

2H + OH H O→ (4.3)

The full reaction model put forward by Twarowski3 consists of 175 reactions involving 24

species, 17 of which contain P atoms. A serious limitation of the model has been the lack of

reliable rate and thermochemical data; the rate coefficients of 162 of these reactions were

estimated by Benson’s rules.5

This chapter describes the resolution of this problem through the computation of accurate

theoretical heats of formation for the phosphorus containing molecular species in

Twarowski’s model3 and the subsequent re-evaluation of the rate coefficients for the critical

reactions that make up the proposed catalytic cycles (4.1) and (4.2) given above. The


138

thermochemical calculations were performed using Gaussian-2 (G2)6, Gaussian-3 (G3)7, and

Gaussian-3X (G3X)8 theories as well as other ab initio quantum chemical methods. The

applicability and reliability of the Gaussian methods to phosphorus containing molecules such

as PO2, HOPO and HOPO2 are tested by comparing the results obtained with those from other

studies using alternative methods such as the coupled cluster method.

Given the documented catalytic properties of these simple phosphorus containing molecules it

is highly likely that these, as well as various organic derivatives, would be efficient fire

retardants. The work reported here was motivated to a large extent by this idea and represents

the initial steps of a study that focuses on the investigation of the flame suppression

mechanisms by organophosphorus compounds, using both computational and experimental

techniques. Achieving an improved level of understanding of the mechanisms of flame

suppression by these species has the potential to provide guidance for the development of

efficient and environmentally friendly fire retardants. Indeed, it has been shown recently that

dimethyl methylphosphonate (DMMP) or trimethylphosphate (TMP) when added to flames

retard the flame velocity with an efficiency comparable with that of CF3Br.9 This agent, prior

to the Montreal protocol, was in widespread use as a fire suppressant especially for aircraft

engines and in military applications. A further potential bonus of phosphorus compounds is

that they can be added to polymer blends whereby they can act as condensed phase inhibitors

promoting the formation of chars to inhibit the burning of plastics; upon vaporisation they

also act as vapour phase inhibitors.10

A further recent interest in phosphorus flame chemistry has arisen from the need to destroy

toxic chemical waste and chemical warfare agents such as sarin.11 Recent investigations12,13 of

incineration of these agents have involved a study of the combustion of organophosphorus

compounds, including DMMP and TMP as models for sarin. Korobeinichev et al.12 measured

the concentrations of PO, PO2, HOPO and HOPO2 in the burnt gases of premixed low

pressure flames of H2 / O2 / Ar doped with DMMP. Using kinetic modelling, they optimised

rate coefficients for key phosphorus flame gas reactions to their measured profiles. They

concluded that organophosphorus additives can actually promote low pressure H2 / O2 / Ar

flames. In a subsequent study13 they found, however, that TMP inhibits atmospheric CH4 / O2

/ Ar and H2 / O2 / Ar flames. Macdonald et al.14 found that the inhibiting ability of DMMP in

non-premixed CH4 / O2 / Ar flames diminishes with increasing adiabatic flame temperature.

The ability to model these inhibition and promotion characteristics of organophosphorus


139

additives to flames is very dependent on the availability of reliable rate coefficients for key

phosphorus flame reactions over a wide range of pressures and temperatures.


The heats of formations of approximately 30 molecules were obtained by quantum chemical

computations of energies and enthalpies, utilising the Gaussian-n methods (G2, G3 and G3X)

of Pople and co-workers6-8. In these approaches, as described in Section 2.7.2, the energies of

the atomic and molecular species of interest are obtained via quadratic configuration

interaction (QCISD(T)) calculations in small valence triple and double zeta + polarisation

functions bases (6-311G(d,p) and 6-31G(d)) respectively, which are then corrected by MP4,

MP2 and SCF estimates of the energy changes with systematic enlargement of the basis sets

and by empirical higher level corrections. Open shell systems are treated by the unrestricted

versions of the above approaches. The heats of formation at 298 K are obtained from the

computed G2, G3 and G3X values of the atomisation energies (at 0 K) in conjunction with

experimental heats of formation of the elements in their atomic states and thermal corrections,

as discussed in detail by Curtiss et al.15 For some radical species (particularly PO), a variant

of the G3(RAD) approach of Radom and co-workers16-18 was also used, where the

unrestricted Hartree-Fock (UHF) and Møller-Plesset (UMP) computations are replaced by

their restricted open shell ROHF and ROMP analogues, while the (unrestricted) quadratic CI

(QCISD(T)) method19 implicit in G2 and G3 is replaced by the restricted singles and doubles

coupled cluster method with perturbative correction for triples (RCCSD(T))20,21.

The rates of the reactions in Twarowski’s reaction sequence were determined using transition

state theory (TST)22 (Section 2.9.1). In the case of reactions with well defined transition states

(that is, first order saddle points) the G3 method was used, whereby the saddle points were

located at the appropriate SCF and MP2 levels of theory. Using the appropriate energies and

partition functions, the limiting high pressure rate coefficient k∞ at any given temperature T is

then obtained from:

( )‡ ‡

expB

i Bi

k T q Ek T

h q k T∞

−∆= ∏

(4.4)


140

where ∆E ‡ is the critical energy of reaction, Bk is the Boltzmann constant, h is Planck’s

constant, ‡q is the partition function of the transition state and qil qare the partition functions

of the reactants. The latter are obtained readily from the computed rotational constants and

vibrational frequencies, using the standard formulas of statistical mechanics, with the rigid

rotor and harmonic vibrations approximations as described in Section 2.8.1.23

The computed rate coefficients, over a range of temperatures, could then be fitted to the

standard Arrhenius form

( ) exp a

B

Ek T A

k T∞

= −

(4.5)

(where Ea is the activation energy and A is the Arrhenius pre-exponential factor). Rate

coefficients were generated at 1000, 1250, 1500, 1750 and 2000 K. These temperatures gave a

sensible range over which rate coefficients could be fitted and spans the temperatures studied

by Twarowski1-4 and Korobeinichev13. Alternatively, the rate coefficients could be fitted to a

modified Arrhenius equation:

( ) expn a

B

Ek T AT

k T∞

= −

(4.6)

As a number of the reactions studied are barrierless recombinations, with no saddle-point to

define the transition state, variational transition state theory (VTST, Section 2.9.2)24-26 was

used to calculate the appropriate rate coefficients. These calculations were carried out by

evaluating the rate coefficients disk of the (reverse) dissociation reactions along the intrinsic

reaction coordinate and thus locating the geometry (at a given temperature) where the rate

coefficient is a minimum27,28. As such transition states generally correspond to molecules near

a bond dissociation limit, the VTST geometries were located via complete active space SCF

(CASSCF) calculations29,30, using Dunning’s correlation consistent cc-pVDZ basis sets31,32.

The active spaces in these CASSCF calculations (on H + PO2 and OH + PO2 transition states)

typically correspond to 8 - 18 active electrons in 6 - 13 active orbitals. The critical energy of

the dissociation reaction is then obtained by correcting the dissociation energy obtained by G3

or other high level theory, by the computed energy difference between the dissociated system


141

and the transition state that had been determined at the CASSCF/cc-pVDZ level of theory. In

the case of reaction (4.2.a) the energy difference was recalculated using complete active space

second order perturbation (CASPT2)33,34 theory in conjunction with Dunning’s cc-pVTZ

basis. (Use of a higher level of theory and larger basis set, as exemplified by the CASPT2/cc-

pVTZ approach, is expected to yield more reliable energies than the lower level CASSCF/cc-

pVDZ method that was employed for the determination of geometries and frequencies.) The

rate coefficient assk for the association reaction is then obtained from

ass

disc

kK

k= (4.7)

where cK is the equilibrium constant for the association (recombination) reaction; it is readily

calculated at any temperature from the appropriate Gibbs free energy of reaction, 0rG∆ .

The pressure dependence of the dissociation rate coefficients in a bath gas of N2 was

determined via the RRKM model (Section 2.9.3) in the weak collision approximation, using

an average collisional energy transfer parameter, α, of 400 cm−1, at pressures ranging from 1

to 104 torr and temperatures 1000 - 2000 K. In order to obtain the rate coefficients in a

convenient form for use by combustion modellers the RRKM rate coefficients were then

expressed in terms of a Troe fit35, as defined by the equations

( )00

01

k kkF k k

k k k∞

∞∞ ∞

= +

(4.8)

( )( )

( )( )

0 2

0

0

loglog

log1

log

centFF k k

k k c

N d k k c

∞

∞

∞

= +

+ − +

(4.9)

where

( )0.4 0.67 log centc F= − − (4.10)

( )0.75 1.27 log centN F= − (4.11)

0.14d = (4.12)


142

and

**

*** *(1 )exp exp expcent

T T TF a a

T T T

= − − + − + − (4.13)

0k is the pseudo-first order limiting low pressure rate coefficient, that is, equal to the

bimolecular rate coefficient multiplied by the bath gas concentration. k∞ is the high pressure

rate coefficient calculated by transition state theory. a, *T , **T , ***T are fitted parameters

expressing the variation with temperature of the pressure-dependant rate coefficients.

All G2, G3 and G3X calculations were carried out using the Gaussian98 programs36. The

ROMP and RCCSD(T) computations were performed using ACESII37, while DALTON38 and

MOLCAS439 were used for the CASSCF geometry optimisations and CASPT2 energy

calculations respectively. The CHEMRATE40 programs were employed for the RRKM

calculations. All computations were performed on DEC alpha 600/5/333 and COMPAQ

XP1000/500 workstations of the Theoretical Chemistry group at the University of Sydney.


4.3.1 G2, G3, and G3X Thermochemistry

The energies and heats of formation of the 24 species involved in Twarowski’s reaction

schemes1-4, calculated using the G2, G3 and G3X methods, are reported in Table 4.1. As will

be discussed later, the standard G3 and G3X results for PO (based on spin unrestricted

calculations) are regarded as unreliable. We recommend instead the G3(RAD) and

G3X(RAD) values, which are also listed in Table 4.1. Where available, experimental and/or

other theoretical heats of formation41-45 are also listed for comparison. The computed

equilibrium geometries, rotational constants and vibrational frequencies are listed in

Appendix 2.1. A number of species, such as P2O2, have several geometric isomers as well as

low lying excited electronic states, giving rise to potential uncertainties as to the nature of the

ground electronic state. In such cases all possible isomers as well as a range of electronic

Chapter 4. P

hosphorus in the H +

OH

Reaction

143

Table 4.1 Total energies and heats of formation computed at the G2, G3 and G3X levels of theory.

Species E0(0 K) /Eh ∆ f H2980 /kcal mol−1

G2 G3 G3X G2 G3 G3X Literature a

H −0.50000 −0.49959 −0.50097 52.1030 ± 0.0014b

O −74.98203 −75.02957 −75.03224 59.553 ± 0.024b

P −340.81821 −341.11502 −341.11699 75.62 ± 0.24b

H2 - 1Σg (D∞h) −1.16636 −1.16738 −1.16721 −1.1 −0.5 −0.4 0.0

O2 - 3Σg (D∞h) −150.14821 −150.24821 −150.25248 2.4 1.1 0.0 0.0

P2 - 1Σg (D∞h) −681.81931 −682.41600 −682.41907 35.6 35.5 34.3 34.3 ± 0.5

b

P4 - 1A1 (Td) −1363.71963 −1364.91465 −1364.92008 19.6 18.2 16.2 14.1 ± 0.05

b

OH - 2Π (C∞v) −75.64391 −75.69490 −75.69607 9.1 8.4 8.4 9.32 ± 0.29b

8.83 ± 0.09c

H2O - 1A1 (C2v) −76.33205 −76.38204 −76.38323 −58.1 −57.5 −57.5 −57.798 ± 0.010b

HO2 - 2A′ (Cs) −150.72792 −150.82689 −150.82950 3.3 3.3 3.2 0.5 ± 2.0

b

PH - 3Σ (C∞v) −341.42844 −341.73033 −341.73131 57.7 56.0 55.7 60.6 ± 8.0d

PH2 - 2B1 (C2v) −342.04913 −342.34974 −342.35096 32.9 32.6 32.2 26 ± 23

e

PH3 - 1A′ (Cs) −342.67900 −342.97851 −342.98004 2.0 3.1 2.4 1.3 ± 0.4

b

PO - 2Σ (C∞v) −416.02430 −416.37332 −416.38022 −6.4 −7.6 −10.8 −5.6 ± 1.0b

−7.1f −7.7

f −6.8 ± 1.9g

−7.8h

PO2 - 2A1 (C2v) −491.19500 −491.59301 −491.59877 −66.4 −67.5 −69.2 −66.6 ± 2.6

g

−69.1f −70.2

f −70.3h

PO3 - 2A2″ (D3h) −566.32361 −566.77130 −566.78381 −100.1 −101.7 −106.3 −107.5

h

PO3 - 2B2 (C2v) −566.77102 −101.4

PPO - 1Σ (C∞v) −756.94248 −757.58837 −757.59417 5.8 5.5 3.3

P2O - 1A1 (C2v) −756.93141 −757.57613 12.7 13.1

143

Chapter 4. P


OH

Reaction

144

Table 4.1 continued

Species E0(0 K) /Eh ∆ f H2980 /kcal mol−1

G2 G3 G3X G2 G3 G3X Literature a

P2O2 - planar - 1Ag (D2h) −832.12205 −832.81642 −832.82515 −59.8 −59.8 −63.4

P2O2 - butterfly – 1A (C1) −832.10890 −832.80295 −51.8 −51.6

P2O2 - cis - 3A″ (Cs) −832.10775 −832.80583 −50.7 −53.0

P2O2 - trans - 3A″ (Cs) −832.10457 −832.80259 −48.5 −50.8

P2O3 - gauche - 1A (C2) −907.34228 −908.08682 −908.09849 −150.3 −151.1 −155.3

HPO - 1A′ (Cs) −416.62879 −416.97614 −416.98030 −21.1 −20.5 −22.0 −22.6h

POH - 3A″ (Cs) −416.59568 −416.95018 −0.2 −4.2

HPOH - trans - 2A″ (Cs) −417.21175 −417.56145 −417.56447 −22.2 −22.5 −23.4

HPOH - cis - 2A″ (Cs) −417.21070 −417.56045 −21.4 −21.7

H3PO - 1A1 (C3v) −417.83301 −418.18166 −418.18604 −47.9 −46.9 −48.6

H2POH - trans - 1A′ (Cs) −417.83334 −418.18206 −418.18558 −47.8 −46.8 −48.0

H2POH - cis - 1A′ (Cs) −417.83292 −418.18168 −47.6 −46.7

HOPO - cis - 1A′ (Cs) −491.84249 −492.23992 −492.24608 −108.1 −108.3 −110.3 −110.6 ± 3i

−112.4h

HOPO - trans - 1A′ (Cs) −491.83848 −492.23577 −105.6 −105.7

HPO2 - 1A1 (C2v) −491.82318 −492.22032 −96.1 −96.1

HOPO2 - planar - 1A′ (Cs) −567.00756 −567.45385 −567.46232 −164.7 −164.8 −167.4 −168.8 ± 4i

−171.4h

a Experimental values unless otherwise indicated

by italics and footnotes.b Ref. 41.c Ref. 42.

d Semiempirical estimate, Ref. 41.e Estimate, Ref. 41. f Computed by G3(RAD) and G3X(RAD) type

procedures.

g Ref. 43.h RCCSD(T)/CBS computations, Ref. 44.i Ref. 45.

144


145

states were explicitly considered in the calculations. The results in Table 4.1 and Appendix

2.1 pertain to the electronic ground states thus located for the lowest energy isomers.

Comparison with the available experimental data suggests that for most systems the

difference between theory and experiment is ~ 2 kcal mol−1 or less. For some molecules,

however, notably P4 and HOPO2, the difference between the G3 result and experiment can be

up to ~ 4 kcal mol−1. The G3X results, as expected, are generally superior to those obtained by

G3. However, quite large discrepancies are noted when G3 and G3X heats of formations are

compared with those calculated by Bauschlicher.44 The latter were obtained by extrapolation

of (R)CCSD(T) energies, obtained with the cc-pVxZ (for P and H) and aug-cc-pVxZ (for O)

basis sets (x = T, Q, 5), to the complete basis set (CBS) limit, and include core-valence

correlation, scalar relativistic and spin-orbit corrections. Given the discrepancies between the

G3X and Bauschlicher’s CBS heats of formation, Bauschlicher’s recommended values for

PO2, HOPO and HOPO2 have been used in the computation of rate coefficients and reaction

enthalpies. The demonstrated weakness of the Gaussian-n approach for some of these systems

is further analysed in the next section.

HOPO and HOPO2, which are particularly important species in Twarowski’s reaction

scheme1-4, possess low frequency torsional modes. Treating these as harmonic oscillators in

the calculation of partition functions might be expected to affect the accuracy of the computed

thermal contributions. The reliability of the harmonic model for these cases was checked by

computing full (360°) torsional potentials by a series of MP2/6-31G(d) calculations (at 5 - 10

degree intervals) and hence the corresponding energy eigenvalue spectra and partition

functions. For HOPO the torsional partition function was found to be 5.05 at 2000 K (the

highest temperature of interest), indicating that effectively only the fifth torsional energy level

was available to the molecule. The energy of this level was 6.08 kcal mol−1 above the

minimum in the potential, which corresponds to being 4.42 kcal mol−1 below the height of the

barrier to rotation. Consequently, no rotation would be expected to occur in this molecule.

Similarly, for HOPO2 the torsional partition function at 2000 K was computed to be 7.74

indicating occupancy of the seventh torsional level, which has an energy of 4.45 kcal mol−1,

and which is again significantly lower than the rotational barrier of 7.68 kcal mol−1. The

hindered rotor correction therefore would result in insignificant changes in the molecular

partition functions. Thus use of the all-vibration model for the computation of thermal

corrections has been validated.


146

The reaction enthalpies for the 175 reactions that make up Twarowski’s scheme1-4, as

obtained from the G3 and G3X heats of formation, are listed in Appendix 2.2.

4.3.2 Reliability of G3, G3X and Related Methods

As noted above, our G3 heats of formation for a number of phosphorus containing molecules,

especially P4, PO2, PO3 and HOPO2, differ by up to ~ 6 kcal mol−1 from experiment or the

computed values of Bauschlicher44. Errors of this magnitude were found for several other

non-hydrogen systems such as SF6 and PF5,46 although these problems appear to have been

overcome by the recent introduction of G3X8. Consequently, testing G3X on some of the

problem molecules encountered in this work is particularly relevant.

G3X differs from G3 in three major respects: in the calculations of geometries and vibrational

frequencies (at B3LYP/6-31G(2df,p) level in G3X) and in the inclusion of an SCF energy

correction for basis set expansion (to G3XLarge in the G3X method). In addition the higher-

level correction parameters have been revised.8 As a small modification to G3X, it is

proposed that the G3XLarge basis set expansion correction be applied at the MP2(Full) level,

(equivalently the MP2(Full)/G3Large computation in G3 theory is replaced with

MP2(Full)/G3Xlarge). This modified G3X technique is denoted G3X2.

The computed G3X and G3X2 heats of formation are given in Table 4.2, where they are also

compared with the G3 values and Bauschlicher’s CBS results44. As the latter contain scalar

relativistic corrections, the same corrections are applied to the G3, G3X and G3X2 results, so

as to make the comparisons more meaningful. As discussed below, in the case of PO the

G3(RAD), G3X(RAD) and G3X2(RAD) results are preferred over their standard

(unrestricted) counterparts. With that proviso, it is noted that going from G3 to G3X and to

G3X2 does yield significant improvements. In the case of PO3 much of the improvement can

be traced to the lower zero point energies at the B3LYP level in comparison with the SCF

values which are used in G3. The (scaled) UB3LYP/6-31G(2df, p) and UHF/6-31G(d) zero

point energies of PO3 are 5.65 and 8.20 kcal mol−1 respectively; the 2.6 kcal mol−1 difference

is thus responsible for 55% of the improvement in the heat of formation of PO3 that occurs


147

Table 4.2 Phosphorus oxides and acids: Comparison of computed heats of formation (∆ f H2980

/kcal mol−1)a.

G3 G3X G3X2 RCCSD(T)/CBSb

PO −7.3 (−6.8)c −10.5 (−7.4)c −11.9 (−8.8)c −7.8

PO2 −66.7 (−69.1)c −68.4 (−70.2)c −70.6 (−72.2)c −70.3

PO3 −100.5 −105.2 −108.3 −107.5

HPO −19.9 −21.6 −23.6 −22.6

HOPO −107.5 −109.5 −111.6 −112.4

HOPO2 −163.2 −165.8 −168.8 −171.4

a All heats of formation corrected for scalar relativistic effects, as in Ref. 44.b Ref. 44.c Computed by G3(RAD), G3X(RAD) and G3X2(RAD) type procedures.

when G3 is replaced by G3X. (In the other systems the differences between the G3X and G3

zero point energies are less than 0.2 kcal mol−1.) Significant further improvements are

obtained, however, with the introduction of the G3X2 method. The heats of formation for

PO2, PO3, HPO and HOPO are lowered by a further 2 - 3 kcal mol−1 so that the G3X2 values

are generally within 1 kcal mol−1 of Bauschlicher’s CBS results44. In the case of HOPO2,

where Bauschlicher judges the reliability of the CBS heat of formation as ± 2 kcal mol−1, the

discrepancy between the G3X2 and the CBS values at 2.6 kcal mol−1 can still be regarded as

acceptable.

4.3.2.1 PO and G3(RAD) Procedures

The problem with PO, where the improvements in the level of theory appear to destroy the

initial agreement between G3 and CBS, was traced to the presence of spin contamination in

the UHF based calculations that leads to a quite bizarre bond distance dependence. Figure 4.1

shows the distance dependence of the UHF, UMP2, UMP4 and UQCISD(T) energies, all

computed with the 6-31G(d) basis, along with the UMP2/G3Large energy. In the region of

about 1.47 - 1.53 Å the UHF energy appears to flatten out, remaining too low, due to a

noticeable increase in spin contamination, as indicated by the expectation value of the total


148

-5

0

5

10

15

20

1.35 1.40 1.45 1.50 1.55 1.60

R (Å)

E (

kca

l mo

l -1)

HF/6-31G(d)

MP2/6-31G(d)

MP4(SDTQ)/6-31G(d)

QCISD(T)/6-31G(d)

E(MP2(Full)/G3Large)

Figure 4.1 Potential energy curves of PO obtained at unrestricted Hartree-Fock, MP2, MP4

and QCISD(T) levels of theory (relative to respective equilibrium values).

spin operator 2S which increases from 0.76 to 1.16 over the above range of distances. There

appears to be a discontinuity at a distance slightly greater than 1.53 Å, so that at 1.54 Å the

value of 2S has fallen to 0.77 with a corresponding jump in the energy. The UMP2 and

UMP4 energies, with much larger discontinuities, further amplify the un-physical behaviour

of the UHF wave function and energy. The UQCISD(T) energy shows normal behaviour,

demonstrating the robustness of the underlying coupled cluster expansion of the wave

function to spin contamination in the reference state. Interestingly, when larger, extended

basis sets are used, for example G3Large, there is only a small blip in the MP2 energy instead

of the ~ 20 kcal mol−1 jump shown by the MP2/6-31G(d) energies.

The UHF and UMP2 energies are contrasted with the restricted open shell ROHF and

ROMP2 energies in Figure 4.2. The latter behave in a perfectly sensible manner, with the

ROHF energies being near-identical to the UHF energies outside the 1.47 - 1.53 Å region, but

the former smoothly bridge the gap where UHF is discontinuous. In Figure 4.3 the behaviour


149

-5

0

5

10

15

20

1.35 1.40 1.45 1.50 1.55 1.60R (Å)

E (

kca

l mo

l -1)

HF/6-31G(d)

MP2/6-31G(d)

ROHF/6-31G(d)

ROMP2/6-31G(d)

Figure 4.2 Potential energy curves of PO obtained at restricted and unrestricted Hartree-Fock

and MP2 levels of theory (relative to respective minimum values).

-2

0

2

4

6

8

10

1.35 1.40 1.45 1.50 1.55 1.60R (Å)

E (

kca

l mo

l -1)

QCISD(T)/6-31G(d)

RCCSD(T)/6-31G(d)

B3LYP/6-31G(d)

Figure 4.3 Potential energy curves of PO obtained at unrestricted QCISD(T), B3LYP and

restricted CCSD(T) levels of theory (relative to respective minimum values).


150

of the high level theories RCCSD(T), UQCISD(T) and B3LYP are compared. Overall, the

UQCISD(T) energies closely match the RCCSD(T) values, although a small blip in the

former can now be clearly seen at 1.54 Å. The behaviour of the UB3LYP energies is

completely sensible and predicts an equilibrium PO distance of ~ 1.50 Å, which is ~ 0.015 Å

lower than the RCCSD(T) value.

In light of the above findings the unusual behaviour of the G3 and G3X results is easy to

rationalise. The G3 energies were obtained at the UMP2/6-31G(d) bond distance of 1.4715

Å, that is, just outside the problem region identified above, and thus a seemingly sensible

value for the heat of formation was obtained. The G3X and G3X2 calculations, at the

UB3LYP/6-31G(2df,p) bond distance of 1.4988 Å, are affected by spin contamination. Note,

however, that the UMP2/6-31G(d) value for the bond distance cannot be accepted as reliable,

as the corresponding minimum is largely an artefact of the non physical behaviour of the

UMP2 energy. In contrast with other (well-behaved) P and O containing molecules, where the

MP2 PO distances are typically ~ 0.02 Å longer than the B3LYP values, in PO the UMP2

distance is actually ~ 0.03 Å shorter than the UB3LYP value. Therefore, the seemingly

sensible heat of formation of PO at the G3 level is fortuitous.

4.3.2.2 Comparison with QCISD(T,Full)

As a further test of the G3, G3X and G3X2 methods, the atomisation energies of PO, PO2,

PO3, HPO, HOPO and HOPO2, obtained at the above levels of theory are summarised in

Table 4.3, along with the atomisation energies computed at the QCISD(T,Full) level of theory

using both the G3Large and G3XLarge basis sets, with the higher level correction terms

implicit in the G3 techniques included. Such a comparison of G3 and QCI results is relevant

since the Gaussian methods aim to produce reliable estimates of the QCISD(T,Full) energies

in the largest basis sets used, viz. G3Large and G3XLarge, by applying a series of lower level

quantum chemical methods in conjunction with a range of basis sets. In the case of the PO

and PO2 radicals the standard UHF based results are compared with those obtained by the

appropriate ROHF based techniques. The latter are analogous to Radom’s G3(RAD) and

related methods16-18. Finally, comparisons are also made with Bauschlicher’s (R)CCSD(T)

results44. In all cases G3 reproduces the QCISD(T,Full)/G3Large atomisation energies to

within ~ 1 kcal mol−1, but mostly better. Similarly, G3X2 reproduces the QCISD(T,Full)/

Chapter 4. P


OH

Reaction

151

Table 4.3 Phosphorus oxides and acids: Comparison of computed atomisation energies (in kcal mol−1).

Atomisation Energy a

PO PO2 PO3 HPO HOPO HOPO2

Method/Reference State UHF ROHF b UHF ROHF b UHF

G3 143.54 143.05 264.15 263.06 360.79 211.92 362.60 480.76

UQCISD(T,Full)/G3largec 143.69 142.36 265.00 263.11 361.69 211.87 362.37 480.84

G3X 146.72 143.66 265.79 264.13 363.39 213.23 364.64 483.51

G3X2 148.10 144.98 268.07 266.15 366.52 215.20 366.71 486.56

UQCISD(T,Full)/G3Xlarged 145.86 144.49 269.43 266.81 368.00 215.50 366.87 487.37

RCCSD(T)/TZ+CVe 135.16 253.10 346.14 204.35 352.11 466.70

RCCSD(T)/QZ+CVe 140.27 261.72 357.74 210.37 361.39 479.97

RCCSD(T)/5Z+CVe 142.12 264.83 361.89 212.39 364.43 na

RCCSD(T)/CBS+CVe 144.05 268.09 366.25 214.51 367.61 489.65

a Not including zero point energy.b All ROHF based calculations performed at UB3LYP/6-31G(2df,p) geometries and use RCCSD(T) in place of QCISD(T) where appropriate.c Including G3 higher level correction.d Including G3X higher level correction.e Including core-valence correlation corrections, Ref. 44.

151


152

G3XLarge atomisation energies to within ~ 1.5 kcal mol−1, except in the case of PO, where

the discrepancy is ~ 2.3 kcal mol−1, for the UHF based calculations. The consistency of the

ROHF based results is significantly better. Comparison with Bauschlicher’s results

demonstrates that the G3X2 method is capable of yielding chemically accurate atomisation

energies and hence heats of formation for this class of difficult molecules. The worst

discrepancy between G3X2 and the CBS results occurs for HOPO2, where the difference is

3.1 kcal mol−1, although at the QCISD(T,Full)/G3XLarge level it is reduced to 2.3 kcal mol−1.

In light of the encouraging performance of the G3X2 method for the above six molecules a

larger systematic evaluation of G3X2 has been undertaken, applying it to a larger number of

phosphorus containing molecules, including all those in Table 4.1. These results are reported

in Chapter 5.

4.4 Kinetic Parameters

A primary aim of this chapter is the calculation of rate coefficients for the PO2 + H and PO2 +

OH recombination reactions (4.1.a) and (4.2.a), and for the subsequent abstraction reactions,

(4.1.b), (4.1.c) and (4.2.b). The rates of these reactions that make up the catalytic cycles are

compared with the (experimental) rates of the (uncatalysed) H + OH recombination reaction

(4.3). The computed geometries of the transition states of these reactions are shown in Figure

4.4. The corresponding heats of formation at 298 K, rotational constants and vibrational

frequencies are summarised in Table 4.4. The geometries of the transition states of reactions

(4.1.b), (4.1.c) and (4.2.b) correspond to well-defined first order saddle points. However, the

recombination reactions (4.1.a) and (4.2.a) are barrierless and thus require a VTST treatment

to locate the geometries of the appropriate transition states at a given temperature, as outlined

in the Sections 2.9.2 and 4.2. The variational transition states were determined at five

temperatures in the range 1000 - 2000 K. The parameters in Figure 4.4 and Table 4.4

pertaining to the transition states of reactions (4.1.a) and (4.2.a) were obtained at 1000 K. The

full set of geometries, rotational constants and vibrational frequencies are given in

Appendices 2.3 and 2.4.


153

Figure 4.4 Geometries of transition states: Variational transition state geometries at 1000 K

for reactions (4.1.a) and (4.2.a) obtained at CASSCF/cc-pVDZ level of theory. All others are

saddle points computed at MP2/6-31G(d) level.

Given the barrierless nature of the recombination reactions (4.1.a) and (4.2.a), the dissociation

rates of HOPO and HOPO2 were initially computed using RRKM at a number of temperatures

in the range 1000 - 2000 K, at pressures 1 - 104 Torr, for a bath gas of N2. The heats of

reactions involving phosphorus containing species were computed using Bauschlicher’s CBS

heats of formation44 (at 298 K) for PO2, HOPO and HOPO2 and experimental literature values

for all other species41-43,45, with the appropriate thermal corrections also taken from JANAF

tabulations41 or computed on the basis of the B3LYP geometries and frequencies44. The rate

coefficients were then fitted to the Troe equations (4.8) - (4.13)35. Figure 4.5 – 4.8 display the

individual rate coefficients and the resulting Troe fits of these. Clearly, the quality of the fits

is generally very good.

O

P O

H

O

P O

H

H

H

O

H

H

P

O

O

O

H

P

O2

O3

O1

H1

H2

2.501

1.497

1.464129.0° 95.6°

0.858

1.310

1.564

1.491

178.9°

122.2°121.4°

1.482 1.481

2.927

0.961

112.0°

1.761

0.979 1.293

0.841

98.8° 167.3°

129.4°

118.7°

1.483 1.487

0.989 1.285

135.7°

112.3°

107.2°

135.3°

Planar

H1-O3-P-O1 = 21.3°H2-O3-P-O1 = -106.8°Planar

PlanarPlanar

1a: H + PO2 → HOPO

2b: H + HOPO2 → H2O + PO22a: OH + PO2 → HOPO2

1c: OH + H2 → H2O +H1b: H + HOPO → H2 + PO2

Chapter 4. P


OH

Reaction

154

Table 4.4 Computed heats of formation, vibrational frequencies and rotational constants of transition states.a

Reaction ∆ f H2980

Rotational Constants Vibrational Frequencies

/kcal mol−1 /cm−1 /cm−1

1a: H + PO2 → HOPO −20.3 1.2566 0.2936 0.2380 340i 82 152 406 1022 1329

1b: HOPO + H → PO2 + H2 −45.1 1.0019 0.2722 0.2140 2668i 224 270 469 774 787

899 1276 1477

1c: H2 + OH → H2O + H 12.9 18.5846 2.9867 2.5732 2813i 628 675 1283 1443 3602

2a: PO2 +OH → HOPO2 −66.9 0.2919 0.1124 0.0812 200i 86 126 164 426 470

1087 1212 3982

2b: HOPO2 + H → PO2 + H2O −90.2 0.2708 0.2481 0.13213665i

244 279 327 438 530

632 738 1115 1238 1405 3530

a Computed at 1000 K for variational transition states (4.1.a) and (4.2.a).

154


155

-10

-8

-6

-4

-2

0

2

4

6

0.5 0.6 0.7 0.8 0.9 1103/T (K)

log( k

)1 Torr

10 Torr

100 Torr

532 Torr

760 Torr

1000 Torr

10000 Torr

High Pressure Limit

Figure 4.5 Arrhenius plots of RRKM rate constants for HOPO → PO2 + H reaction at a

range of pressures. (Symbols = computed rate constants, Lines = Troe fits.)

-10

-8

-6

-4

-2

0

2

4

0.5 0.6 0.7 0.8 0.9 1103/T (K)

log( k

)

1 Torr

10 Torr

100 Torr

532 Torr

760 Torr

1000 Torr

10000 Torr

High Pressure Limit

Figure 4.6 Arrhenius plots of RRKM rate constants for HOPO2 → PO2 + OH reaction at a

range of pressures. (Symbols = computed rate constants, Lines = Troe fits.)


156

-6

-5

-4

-3

-2

-1

0

-1.0 0.0 1.0 2.0 3.0 4.0

log(p /Torr)

log( k

/ k∞)

1000K

1250K

1500K

1750K

2000K

-6

Figure 4.7 Pressure dependence of RRKM rate constants for HOPO → PO2 + H reaction at a

range of temperatures. (Symbols = computed rate constants, Lines = Troe fits.)

-5

-4

-3

-2

-1

0

-1 0 1 2 3 4

log(p /Torr)

log( k

/ k∞)

1000K

1250K

1500K

1750K

2000K

Figure 4.8 Pressure dependence of RRKM rate constants for HOPO2 → PO2 + OH reaction at

a range of temperatures. (Symbols = computed rate constants, Lines = Troe fits.)


157

The calculated rate coefficients of all the reactions studied in this chapter, in the form of Troe,

Arrhenius or modified Arrhenius fits, are summarised in Table 4.5, in the temperature range

1000 - 2000 K. Further, pertinent computational details of the individual reactions are

discussed below. The rate coefficients of the association (that is, recombination) reactions

were calculated by utilising the appropriate equilibrium constants, as given by Equation (4.7).

The calculated Gibbs free energies of reactions and reaction enthalpies at a number of

temperatures are given in Table 4.6.

HOPO → → → → H + PO2. Initially, the potential energy surfaces of both the cis and trans isomers

were studied in the region of dissociation at the CASSCF/cc-pVDZ level of theory. In the

case of the trans isomer a saddle point was found at an O…H distance of 2.54 Å. This

transition state was ~ 7 kcal mol−1 higher in energy than the dissociation products H + PO2. In

contrast with such a 7 kcal mol−1 barrier to the recombination reaction to give the trans

isomer, no barrier could be found for the dissociation of the lower energy cis isomer. This

indicates that the latter mechanism represents the preferred reaction channel. The geometries

and frequencies of the variational transition states were located at CASSCF/cc-pVDZ level of

theory. As the potential energy surface is very flat in the critical region, the transition state

geometries at the various temperatures show little variation, as the data in Appendices 2.3

and 2.4 indicate. The energies of the transition states relative to the dissociated products were

obtained from CASSCF/cc-pVDZ calculations.

H + HOPO →→→→ H2 + PO2 and OH + H2 →→→→ H2O + H. The transition state geometries

(corresponding to first order saddle points) and critical energies were computed at the G3

level of theory.

HOPO2 →→→→ OH + PO2. The geometries of the variational transition states were located at the

CASSCF/cc-pVDZ level of theory. The energies of the transition states relative to the

dissociated products were calculated at the CASPT2/cc-pVTZ level. As before, the enthalpy

of dissociation was determined using Bauschlicher’s CBS heats of formation (at 298 K) for

PO2 and HOPO2 and experimental values for OH.

Chapter 4. P


OH

Reaction

158

Table 4.5 Computed rate coefficients: Troea, Arrheniusb and modified Arrheniusb fit parameters.

High pressure limit Low pressure limit

Arrhenius Modified Arrhenius Arrhenius Modified Arrhenius

log A Ea log A n Ea log A Ea log A n Ea

HOPO → PO2 + H 15.34 94.9 13.57 0.50 93.6 16.57 82.1 34.97 −5.13 96.1

PO2 + H → HOPO 10.06 1.29 −1.5 31.46 −4.33 1.02

HOPO + H → H2 + PO2 c 14.28 16.1 7.33 1.94 10.8

H2 + OH → H2O + H c 14.33 9.9 5.92 2.34 3.5

HOPO2 → HO + PO2 15.56 105.2 24.44 −2.48 112.0 18.41 91.5 57.26 −10.83 121.0

HO + PO2 → HOPO2 14.19 −0.24 0.0 47.01 −8.59 9.0

HOPO2 + H → PO2 + H2O c 14.10 26.7 8.74 1.49 22.6

Troe Fit

a 1−a T* T** T***

HOPO → PO2 + H 1.00 0.00 950.72 4797.20 −115.31

PO2 + H → HOPO

HOPO + H → H2 + PO2 c

H2 + OH → H2O + H c

HOPO2 → HO + PO2 1.00 −2.22×10−6 640.64 3973.10 −202.75

HO + PO2 → HOPO2

HOPO2 + H → PO2 + H2O c

a See equations (8) - (13); T*, T**, T*** in K.b See equations (5) and (6); Ea in kcal mol−1, A in s−1 or cm3 mol−1 s−1.

c These reactions are not pressure dependent.

158


159

Table 4.6 Computed enthalpies and Gibbs free energies of reaction (in kcal mol−1) at a range

of temperatures.

H + PO2 → HOPO OH + PO2 → HOPO2

T /K 0Tr H∆ 0

TrG∆ 0Tr H∆ 0

TrG∆298.15 −94.2 −86.6 −110.4 −99.1

1000 −95.4 −67.4 −110.0 −72.4

1250 −95.6 −60.4 −109.5 −63.1

1500 −95.7 −53.4 −108.9 −53.9

1750 −95.7 −46.3 −108.3 −44.7

2000 −95.7 −39.3 −107.6 −35.7

H + HOPO2 →→→→ H2O + PO2. The search for a transition state for this reaction initially yielded

a minimum corresponding to a weakly bound PO2…H2O dimer. This dimer had a binding

energy of 4.8 kcal mol−1 after applying the counterpoise correction for basis set superposition

effects. However, from the point of view of this work the existence of such a dimer is of

academic interest only as it would not be expected to be stable at ~ 2000 K. The transition

state for OH abstraction by H is characterised by a long OH bond and a slightly elongated PO

bond. The geometry and barrier height were located at the G3 level of theory.

Finally, the calculated rate coefficients at the temperatures of 1000 and 2000 K and a pressure

of 532 torr are summarised in Table 4.7 where they are compared with Twarowski’s

estimated values (which were obtained by the application of Benson’s rules), the modelling

values of Korobeinichev et al.13 and experiment.47

At 2000 K the rate coefficients in the two reaction schemes considered are comparable in

magnitude, suggesting that both routes are important, in qualitative agreement with

Twarowski’s conclusions. However, in an absolute sense, the rate coefficients obtained in this

work are significantly lower than Twarowski’s, especially at 1000 K, although order of

magnitude differences can exist even at the higher temperature. Our values agree somewhat

better with the modelled values of Korobeinichev et al.,13 but there remain significant

differences. Under the conditions studied by Twarowski (0.7 atm pressure and 1970 K) the

rate coefficient for (4.1.a) is substantially into falloff (approximately 1,000 fold lower than the

high pressure limit), whereas the coefficient for (4.2.a) is significantly less so (only ~ 60 fold

Chapter 4. P


OH

Reaction

160

Table 4.7 Comparison of computed and experimental rate coefficients at 532 Torr pressure at 1000 and 2000 K (in cm3mol−1s−1).

Reaction T (K) k (This work) k (Twarowskia) k (Korobeinichev et al.b) k (expt.)

1a: PO2 + H → HOPO 1000 7.03×1012 4.42×1012 c4.55×1013 c

2000 2.78×1011 2.87×1011 c6.51×1012 c

1b: HOPO + H → H2 + PO2 1000 6.08×1010 3.15×1013 7.7×1011

2000 3.53×1012 3.16×1013 7.8×1011

1c: H2 + OH → H2O + H 1000 1.52×1012 6.28×1013 1.26×1012 d

2000 1.88×1013 6.30×1013 8.33×1012 d

2a: HO + PO2 → HOPO2 1000 7.40×1012 1.74×1014 c1.71×1013 c

2000 3.68×1011 5.62×1012 c1.89×1012 c

2b: HOPO2 + H → PO2 + H2O 1000 1.93×1081 3.16×1013 1.55×1091

2000 1.60×1011 3.16×1013 3.13×1010

3: H + OH → H2O 1000 3.56×1091 c7.12×1012 c,e

2000 1.17×1091 c8.91×1091 c,e

a Ref. 3.b Ref. 13.c Calculated as k[M].d Fit to all NIST data, Ref. 48.e Ref. 47.

160


161

lower than the high pressure limit). Thus, whilst it might be acceptable to use the termolecular

rate coefficient for (4.1.a) in modelling, in light of its proximity to the limiting low pressure

value, the appropriate falloff value of the rate coefficient should be used for (4.2.a). For the

H2 + OH abstraction reaction (4.1.c) the computed G3 rate coefficients agree with the

observed values48 to within a factor of approximately two or better, depending on the

temperature.

4.5 Conclusion

Using ab initio quantum chemical and RRKM techniques, theoretical rate coefficients were

obtained for the H + PO2 (4.1.a) and OH + PO2 (4.2.a) recombination reactions and for the

subsequent H + HOPO (4.1.b) and H + HOPO2 (4.2.b) abstraction reactions, which, along

with the OH + H2 abstraction reaction (4.1.c) constitute the catalytic pathway for the H + OH

recombination reaction (4.3), as formulated by Twarowski1-4 and also by Korobeinichev et

al.13 The computed rate coefficients for (4.1.c) agree well with experiment (within 20% at

1000 K and almost within a factor of 2 at 2000 K), while for the other reactions the rate

coefficients are consistent with the modelled values of Korobeinichev et al.,13 although for

several key reactions ((4.1.b), (4.2.a), (4.2.b)) they are substantially lower than Twarowski’s

values. Whilst we utilised Bauschlicher’s44 recent thermochemical data in the derivation of

our rate coefficients, we note that reasonable accuracy could be achieved by the application of

the G3X and G3X2 methods to the computation of the heats of formation of the phosphorus

containing species. Using the G2, G3 and G3X methods we also computed the

thermochemistry of Twarowski’s reaction model which includes 17 phosphorus containing

molecules.


162

4.6 References





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14. M. A. MacDonald, F. C. Gouldin and E. M. Fisher, Combustion and Flame, 2001,

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23. D. A. McQuarrie, Statistical Mechanics; Harper & Row: New York, 1973, 129.

24. E. Pollack, in Theory of Chemical Reaction Dynamics. M. Baer, Ed.; Vol. 3; CRC

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27. G. B. Bacskay, M. Martoprawiro and J. C. Mackie, Chem. Phys. Lett., 1999, 300, 321.

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B. Stefanov, G. Lui, A. Liashenko, P. Piskorz, I. Komaromi, R. Gomperts, R. L.

Martin, D. J. Fox, T. Keith, M. A. Al-Laham, C. Y. Peng, A. Nanayakkara, C.

Gonzalez, M. Challacombe, P. M. W. Gill, B. G. Johnson, W. Chen, M. W. Wong, J.

L. Andres, M. Head-Gordon, E. S. Replogle and J. A. Pople, Gaussian 98 (Revision

A.7), Gaussian, Inc.: Pittsburgh, PA, 1998.


164





included are VMOL (J. Almlöf and P. R. Taylor); VPROPS (P. Taylor); ABACUS (T.

Helgekar, H. J. Aa. Jensen, P. Jørgensen, J. Olsen and P. R. Taylor).

38. "DALTON, an ab initio electronic structure program, Release 1.0 (1997)" written by

T. Helgaker, H. J. Aa. Jensen, P. Jørgensen, J. Olsen, K. Ruud, H. Ågren, T.

Andersen, K. L. Bak, V. Bakken, O. Christiansen, P. Dahle, E. K. Dalskov, T.

Enevoldsen, B. Fernandez, H. Heiberg, H. Hettema, D. Jonsson, S. Kirpekar, R.

Kobayashi, H. Koch, K. V. Mikkelsen, P. Norman, M. J. Packer, T. Saue, P. R. Taylor

and O. Vahtras.



Serrano-Andrés, P. E. M. Siegbahn and P.-O. Widmark, MOLCAS (Version 4), Lund

University: Lund, Sweden, 1997.

40. V. Mokrushin, V. Bedanov, W. Tsang, M. Zachariah and V. Knyazev, Chemrate

(1.10), National Institute of Standards and Technology: Gaithersburg, MD, USA,

1999.

41. M. W. Chase, Jr., J. Phys. Chem. Ref. Data, 1998, Monograph 9, 1.



43. J. Drowart, C. E. Myers, R. Szwarc, A. Vander Auwera-Mahieu and O. M. Uy, J.

Chem. Soc., Faraday Trans. 2, 1972, 68, 1749.

44. C. W. Bauschlicher, Jr., J. Phys. Chem. A, 1999, 103, 11126.

45. D. L. Hildenbrand and K. H. Lau, J. Chem. Phys., 1994, 100, 8373.


112, 7374.

47. D. L. Baulch, C. J. Cobos, R. A. Cox, C. Esser, P. Frank, T. Just, J. A. Kerr, M. J.

Pilling, J. Troe, R. W. Walker and J. Warnatz, J. Phys. Chem. Ref. Data, 1992, 21,

411.

48. W. G. Mallard, F. Westley, J. T. Herron, R. G. Hampson and D. H. Frizzell, NIST

Chemical Kinetic Database: Version 2Q98. 1998, National Institute of Standards and

Technology, Gaithersburg, MD.

5 Accurate Thermochemistry of Phosphorus Compounds

Chapter 5

Accurate

Thermochemistry

of Phosphorus

Compounds

Chapter 5. Accurate Phosphorus Thermochemistry

166

5.1 Introduction

During the last decade a considerable degree of interest has developed in phosphorus

containing molecules as potential catalysts, with the ability to accelerate a range of radical

recombination reactions.1-7 Such catalytic activity has implications in a number of fields, such

as flame suppression, engine fuel efficiency and nerve gas disposal. Unfortunately, the

development of reliable kinetic models for these processes has been hampered by a lack of

accurate thermochemical and kinetic data. The work reported in this chapter aims to aid in

remedying this problem by undertaking the computation of accurate heats of formation for a

number of small phosphorus-containing molecules using current techniques of computational

quantum chemistry.

As in other work8-17 concerned with the ab initio computation of thermochemistry, the

approach here has been to compute the atomisation energies and hence heats of formation of

the molecules of interest by the application of the CCSD(T) coupled cluster method in

conjunction with correlation consistent basis sets (aug)-cc-pVxZ (x = T, Q, 5), followed by

extrapolation to the complete basis set (CBS) limit, and inclusion of corrections for core-

valence correlation, spin-orbit coupling and scalar relativistic effects. Such calculations have

already been reported by Bauschlicher15 for PO, PO2, PO3, HPO, HOPO and HOPO2. In

addition to these phosphorus oxides and acids, heats of formation for P2, P4, PH, PH2, PH3,

P2H2, P2H4, P2O, P2O2, HPOH, H2POH, and H3PO have been calculated and are reported here.

Regarding the computed CCSD(T)/CBS thermochemistry as a benchmark, the second aspect

of this work is a critical test of Gaussian-3 type methods, in particular G318, G3X19 and

G3X220, when applied to the above set of molecules. As these techniques are considerably

less computer resource intensive than the CCSD(T)/CBS approach, they would have a wider

range of applicability, especially for larger systems. However, given that the original test set

of molecules that was used to calibrate the G3 and G3X methods contains comparatively few

phosphorus containing species (due largely to the lack of reliable experimental information),

an assessment of the capability and reliability of these methods for a larger number of

phosphorus containing molecules is clearly desirable.


167

The heats of formation of the above phosphorus compounds, as calculated by G3 and G3X,

have been reported in the previous chapter (and also published20), although the main focus of

that work was the computation of rate coefficients for the H + PO2 and OH + PO2

recombination reactions and for the H + HOPO → H2 + PO2, OH + H2 → H + H2O, and H +

HOPO2 → H2 + PO2 abstraction reactions, which constitute a catalytic pathway for the H +

OH recombination reaction. Comparison of the G3 and G3X heats of formation with the

CCSD(T)/CBS values of Bauschlicher for the six phosphorus oxides and acids PO, PO2, …

HOPO2 revealed that the G3 and G3X heats of formation were in error by up to ~ 9 and 6 kcal

mol−1 respectively, given that the CCSD(T)/CBS values are believed to be essentially

chemically accurate, with estimated errors of ~ 1 kcal mol−1. The performance of G3X2 was

found to be superior to G3X, with the maximum deviation from the CCSD(T)/CBS values

being ~ 3 kcal mol−1.) Extending such comparisons to a larger set of molecules, as undertaken

in this study, is seen as definitely warranted.

This work, therefore, has two primary aims: firstly to generate benchmark atomisation

energies (AE) and heats of formation for a larger selection of small phosphorus containing

molecules and secondly to use these results to assess the reliability of the G3 type methods for

these systems.

5.2 Theory And Computational Methods

The application of coupled cluster theory in conjunction with correlation consistent basis sets

and extrapolation to the CBS limit has become well established as a reliable approach for

generating accurate heats of formation from atomisation energies.8-17 This is the scheme that

was employed by Bauschlicher15 to calculate thermochemical data for the phosphorus oxides

and acids. The approach here for the generation of benchmark heats of formation is

effectively the same as Bauschlicher’s.

The geometries and vibrational frequencies in this work were obtained using density

functional theory, employing the B3LYP exchange-correlation functional21-24 and the

6-31G(2df,p) basis set, as prescribed by G3X theory (with a scale factor of 0.9854 for the

computation of zero point energies). This basis set is somewhat different from those used by


168

Bauschlicher, viz. 6-31+G(2df) for geometries and 6-31G(d) for frequencies. The single point

energy calculations at the B3LYP optimised equilibrium geometries were performed using

coupled cluster theory with single and double excitations, with the inclusion of triples by

perturbation theory (CCSD(T))25,26. The open shell reference state orbitals and energies were

generated by restricted Hartree-Fock (RHF) theory, while the subsequent coupled cluster

energies were obtained by the restricted RCCSD(T) method.

The basis sets employed in the sequence of valence-correlated coupled cluster calculations

(where the 1s electrons on O and 1s, 2s, 2p electrons on P are left uncorrelated) are the

correlation consistent valence-polarised (cc-pV) triple-ζ (TZ), quadruple-ζ (QZ) and

pentuple-ζ (5Z) basis sets developed by Dunning et al.27-29 As recommended by

Bauschlicher15, an additional tight d function was added to each of the phosphorus basis sets

with an exponent three times that of the largest d function in the original set (that is, 1.956,

3.11 and 8.00 for the TZ, QZ and 5Z sets respectively). Diffuse functions were also included

in the oxygen basis sets; that is, for O atoms the aug-cc-pVxZ basis sets27,28 are used. The

resulting basis sets for molecules with P, O and H atoms are denoted (aug-)cc-pVxZ+d.

The three valence (R)CCSD(T) single point energies were then extrapolated to the CBS limit

({T,Q,5}). All four extrapolation schemes mentioned in Section 2.7.3 were employed; that is,

the mixed exponential/Gaussian extrapolation of Feller30 (“mix”, Equation (5.1)), the

Schwartz type extrapolations31 (“ maxl ”, Equation (5.2) and “ 4 6n n− −+ ”, Equation (5.3)) and

the “ 3x− ” scheme of Helgaker et al.16 (Equation (5.4)).

( ) ( ) ( )( )2exp 1 exp 1E x A B x C x= + − + − − (5.1)

( ) ( ) 4

max 1 2E x A B l−= + + (5.2)

( ) ( ) ( )4 6

max max1 2 1 2E x A B l C l− −= + + + + (5.3)

( ) 3E x A Bx−= + (5.4)


169

The core-valence (and core-core) correlation energies (CV) were computed via (R)CCSD(T)

calculations using both the correlation consistent polarised core valence triple-ζ (cc-pCVTZ)

basis sets of Dunning et al.27,29,32 (aug-cc-pCVTZ for oxygen) as well as the core-valence

basis sets proposed by Bauschlicher15. The CV contributions to the atomisation energies were

corrected for basis set superposition effects (BSSE) by application of the counterpoise (CP)

method of Boys and Bernardi33. The CP treatment was restricted to the phosphorus atoms of

any given molecule; that is, CP corrections to the CV component of the energies, denoted

CP(CV), were computed only for the P atoms.

The spin-orbit coupling energy corrections for the atomic species were taken from the

tabulation of Curtiss et al.18 The only molecule in the set which would be expected show

appreciable spin-orbit coupling is PO; in this case the value calculated by Bauschlicher15 was

used. Scalar relativistic corrections, viz. the Darwin and mass-velocity contributions34,35, were

calculated using finite field perturbation theory, utilising the complete active space second

order perturbation theory (CASPT2) method36,37 with complete active space self-consistent

field (CASSCF)38,39 reference states, and the G3Large basis sets of Curtiss et al.18

The Gaussian methods, G140, G241, G318 and their variants42-46, have been designed to

approximate a molecular energy that would result from the application of a high level of

quantum chemical theory in conjunction with a large basis set by a series of lower level

calculations and/or smaller basis sets. This, of course, results in significant savings of

computer resources. The G3 and G3X methods have already been described in detail in

Section 2.7.2. In brief, in G3 the aim is to effectively reproduce the electronic energy of a

QCISD(T, Full) calculation obtained with the G3Large basis set.

[ ] [ ][ ] [ ]{ }[ ] [ ]{ }[ ] [ ]

[ ] [ ]

0 G3 QCISD(T)/6-31G( )

MP4/6-31 G( ) MP4/6-31G( ) ( correction)

MP4/6-31G(2 , ) MP4/6-31G( ) (2 , correction)

MP2(Full)/G3Large MP2/6-31G(2 , )(G3La

MP2 6-31 G MP2 6-31G

E E d

E d E d

E df p E d df p

E E df p

E d E d

=

+ + − +

+ −

− + − / + ( ) + / ( )

ZPE SO hlc

rge correction)

E E E+ ∆ + ∆ + ∆

(5.5)


170

G3X theory is a derivative of G3, designed primarily so as to improve agreement between

theoretical and experimental heats of formation for molecules which contain second row

atoms (Na-Ar). This is achieved by using improved geometries and vibrational frequencies

and by the inclusion of a so-called G3XLarge correction at the SCF level using the G3XLarge

basis set:

[ ] [ ] [ ] [ ]{ }0 0G3X G3 HF/G3XLarge HF/G3Large

G3XLarge correction

E E E E= + − (5.6)

(with the understanding that the equilibrium geometry and zero point vibrational correction

are obtained by B3LYP/6-31G(2df,p) calculations).

The recently proposed G3X2 method (Section 2.7.2.5, Chapter 4) differs from G3X only in

that the G3XLarge correction is now evaluated using MP2(Full):

[ ] [ ] [ ] [ ]{ }0 0G3X2 G3 MP2(Full)/G3XLarge MP2(Full)/G3LargeE E E E= + − (5.7)

The G3X2 approach can also be regarded as a G3 calculation (with the geometry and zero

point energy correction determined at the B3LYP/6-31G(2df,p) level) where the G3Large

basis has been replaced by the G3XLarge set:

[ ] [ ][ ] [ ]{ }[ ] [ ]{ }[ ] [ ]

[ ] [ ]

0 G3X2 QCISD(T)/6-31G( )



MP2(Full)/G3XLarge MP2/6-31G(2 , )(G

MP2 6-31 G MP2 6-31G

E E d

E d E d

E df p E d df p

E E df p

E d E d

=

+ + − +

+ −

− + − / + ( ) + / ( )

ZPE SO hlc

3X2 correction)

E E E+ ∆ + ∆ + ∆

(5.8)

G3X2 represents an approximation to a QCISD(T,Full)/G3XLarge computation and, as it

accounts for a greater degree of electron correlation energy than G3X, it is reasonable to

expect that it would be superior to G3X in an absolute sense, that is, yield closer agreement

with experiment. Indeed, as remarked in the introduction to this chapter (Section 5.1), we


171

found that for the six phosphorus oxides and acids PO, PO2, … HOPO2 studied by

Bauschlicher the G3X2 heats of formation were in significantly better agreement with

Bauschlicher’s CCSD(T)/CBS values than those obtained by G3X.

A further suggested refinement of G3X2 which is introduced in this work is to correct the CV

component of the energy for BSSE, which can be quite large for phosphorus containing

molecules. The analogous corrections for BSSE effects within the valence space are assumed

to be taken care of by the higher level corrections. In G3X2 the latter are defined the same

way as in G3X, that is, in terms of the valence electrons of a given molecules, according to

Equations (2.7.12) and (2.7.13). While ultimately the parameters A, B, C and D would need to

be reoptimised for G3X2, no such optimisation has been undertaken here, as there are too few

molecules in our data set for such results to have wider applicability. Therefore, the hlc

parameters of G3X, given in Section 2.7.2.4, have been adopted for the current

implementation of G3X2.

The standard Gaussian theories treat open shell systems (atoms and molecules) by the

unrestricted methods: UHF, UMP2, UMP4 and UQCISD(T). An alternative approach is the

Gn(RAD) type approach of Radom and co-workers47-49, where the UHF and UMP

computations are replaced by their restricted open shell ROHF and ROMP analogues and

RCCSD(T) is used instead of the UQCISD(T) method implicit in G2 and G3. In this work

both the standard (unrestricted) formulation of G3X2 as well as the (restricted) G3X2(RAD)

procedure have been used for all atoms as well as several open shell molecules.

The B3LYP geometries and vibrational frequencies as well as the G3X2 energies were

generated using the Gaussian98 suite of programs50. The (R)CCSD(T) and ROMP

calculations were performed with MOLPRO26,51,52 and ACESII53 respectively. The CASSCF

and CASPT2 calculations of the scalar relativistic corrections were carried out using

MOLCAS54. The computations were performed on DEC alpha 600/5/333 and COMPAQ

XP100/500 workstations of the Theoretical Chemistry group at the University of Sydney and

on the COMPAQ AlphaServer SC system of the Australian Partnership for Advanced

Computing National Facility at the National Supercomputing Centre, ANU, Canberra.


172


5.3.1 CCSD(T) Benchmark Calculations

The geometric parameters, rotational constants and vibrational frequencies of the species

studied in this investigation can be found in Appendix 2.1.

Table 5.1 contains the absolute energies of the molecules of interest and their component

atoms, as obtained at the highest level of (valence) correlated theory, namely (R)CCSD(T)/

(aug-)cc-pV5Z+d, along with the {T,Q,5} 3x− extrapolated energies, CV correlation

corrections, CP corrections for BSSE in the latter, zero point vibrational energies, thermal

corrections to the enthalpies and the scalar relativistic energy corrections.

5.3.1.1 Testing the B3LYP Geometry

As a test of the reliability of the B3LYP functional for the computation of molecular

geometries, P4 was taken as test case and its equilibrium geometry re-computed using

CCSD(T). The results are summarised in Figure 5.1 which shows the computed molecular

energy as a function of the P-P distance in tetrahedral P4. The B3LYP and valence correlated

CCSD(T) calculations yield effectively the same geometry, but a shorter bond length is

obtained when the CCSD(T) calculations include CV correlation. (The force constants from

these three calculations are comparable.) These findings are interesting, since the DFT

calculations do include CV correlation. In the context of the current work, however, the

discrepancy in the B3LYP geometry is negligible, as the resulting error in the molecular

energy is only ~ 0.04 kcal mol−1. As in this work P4 is probably the most difficult molecule to

describe accurately, it is expected that any inaccuracies in the molecular energies of other

species, resulting from errors in the choice of equilibrium geometries will be similarly

negligible.

Chapter 5. A

ccurate Phosphorus T

hermochem

istry

173

Table 5.1 Total CCSD(T) energies, core-valence (CV) correlation corrections, counterpoise (CP) corrections to CV, zero point energies,

thermal corrections to enthalpies and scalar relativistic corrections (in Eh unless otherwise indicated).

CCSD(T)

(aug-)cc-pV5Z+d

CCSD(T)

CBS( 3x− )CV corr1 a CV corr2 b

CP(CVcorr2) c

/kcal mol−1

ZPE/kcal mol−1

298 0H H−/kcal mol−1 relE

P2 - 1Σg (D∞h) −−681.84172 −−681.85074 −0.65340 −0.67441 0.20 −1.13 2.12 −1.62549

P2 - 1Σg (D∞h)

d −−681.84172 −−681.85025 −0.65340 −0.67441 0.20 −1.13 2.12 −1.62549

P2 - 1Σg (D∞h)

e −−681.84189 −−681.85080 −0.65340 −0.67441 0.20 −1.13 2.12 −1.62549

P4 - 1A1 (Td) −1363.77256 −1363.79264 −1.30798 −1.34968 0.83 −3.84 3.40 −3.25105

P4 - 1A1 (Td)

d −1363.77256 −1363.79168 −1.30798 −1.34968 0.83 −3.84 3.40 −3.25105

P4 - 1A1 (Td)

e −1363.77274 −1363.79233 −1.30798 −1.34968 0.83 −3.84 3.40 −3.25105

PH - 3Σ (C∞v) −−341.44587 −−341.44963 −0.32637 −0.33675 0.08 −3.29 2.07 −0.81280

PH2 - 2B1 (C2v) −−342.07416 −−342.07892 −0.32660 −0.33697 0.16 −8.26 2.38 −0.81249

PH3 - 1A′ (Cs) −−342.71361 −−342.71915 −0.32685 −0.33721 0.26 14.79 2.42 −0.81222

P2H2 - 1Ag (Cs) −−683.02997 −−683.04004 −0.65348 −0.67431 0.35 10.90 2.63 −1.62437

P2H4 - 1Ag (Cs) −−684.24539 −−684.25625 −0.65369 −0.67442 0.51 21.52 3.30 −1.62460

PO - 2Σ (C∞v) −−416.05555 −−416.06728 −0.38065 −0.39187 0.23 −1.77 2.08 −0.86497

PO2 - 2A1 (C2v) −−491.25104 −−491.27104 −0.43455 −0.44653 0.46 −3.94 2.60 −0.91602

PO3 - 2A2″ (D3h) −−566.40722 −−566.43531 −0.48881 −0.50135 0.59 −5.65 3.40 −0.96742

P2O - 1Σ (C∞v) −−756.99067 −−757.00777 −0.70762 −0.72935 0.54 −3.40 2.84 −1.67676

P2O2 - 1Ag (D2h) −−832.19200 −−832.21693 −0.76187 −0.78362 0.61 −5.62 2.92 −1.72910

173

Chapter 5. A


hermochem

istry

174

Table 5.1. continued

CCSD(T)

(aug-)cc-pV5Z+d

CCSD(T)

CBS( 3x− )CV corr1 a CV corr2 b

CP(CVcorr2) c

/kcal mol−1

ZPE/kcal mol−1

298 0H H−/kcal mol−1 relE

HPO - 1A′ (Cs) −−416.66743 −−416.67983 −0.38068 −0.39180 0.29 −6.09 2.40 −0.86449

HPOH - 2A″ (Cs) −−417.26372 −−417.27646 −0.38084 −0.39176 0.25 13.39 2.63 −0.86372

H2POH - 1A′ (Cs) −−417.89452 −−417.90796 −0.38095 −0.39183 0.33 19.43 2.81 −0.86396

H3PO - 1A1 (C3v) −−417.89612 −−417.91002 −0.38055 −0.39161 0.45 19.07 2.51 −0.86326

HOPO - 1A′ (Cs) −−491.90932 −−491.92982 −0.43510 −0.44683 0.38 10.95 2.77 −0.91634

HOPO2 - 1A′ (Cs) −−567.13090 −0.48901 −0.50149 0.59 14.36 3.13 −0.96707

H −−−−0.49999 −−−−0.50005 −0.00000 −0.00000 1.01 −0.00000

O −−−75.00041 −−−75.00653 −0.05354 −0.05414 1.04 −0.05230

P −−340.82973 −−340.83229 −0.32617 −0.33656 1.28 −0.81296

P d −−340.82973 −−340.83208 −0.32617 −0.33656 1.28 −0.81296

P e −−340.82980 −−340.83235 −0.32617 −0.33656 1.28 −0.81296

a Calculated using Dunning’s cc-pCVnZ basis sets, Refs 27, 29, 32.b Calculated using Bauschlicher’s core-valence basis sets, Ref. 15.c Counterpoise correction for BSSE on P atoms in CV corr2.d Extrapolation process includes the cc-pV6Z+d energy.e Calculated using cc-pCV(n+d)Z basis sets, Ref. 56.

174


175

Figure 5.1 P4: Comparison of computed equilibrium geometries and potential energy surfaces

with respect to the symmetric stretch.

5.3.1.2 Atomisation Energies and Extrapolation Schemes

It is generally recognised that atomisation energies provide a far more convenient tool than

absolute energies for investigating the merits of the various extrapolation schemes and the

relative magnitudes of the corrections involved in the calculations of thermochemistry.

Therefore a range of these, as obtained with all three basis sets used for valence correlated

calculations (triple-, quadruple- and pentuple-ζ) and all four extrapolation methods, along

with the CV and scalar relativistic corrections are listed in Table 5.2. We note here that in the

case of HOPO2 the RCCSD(T)/(aug-)cc-pV5Z+d calculation proved too demanding for our

computing resources and for this molecule therefore, as in Bauschlicher’s work, the

extrapolations (for HOPO2 and its atoms) are based on the triple- and quadruple-ζ energies

alone ({T,Q}). (This necessarily means that only the maxl and 3x− extrapolation schemes

could be implemented for HOPO2.) Bauschlicher expressed some concern about this in his

paper, stating that the two-point {T,Q} extrapolation is likely to result in an overestimation of

the atomisation energy by 0.85 kcal mol−1.

-0.02

0.00

0.02

0.04

0.06

0.08

0.10

2.198 2.200 2.202 2.204 2.206 2.208 2.210 2.212 2.214 2.216

R (Å)

E (

kca

l mo

l-1)

CCSD(T)/cc-pVTZCCSD(T)/cc-pCVTZB3LYP/6-31G(2df,p)

Chapter 5. A


hermochem

istry

176

Table 5.2 Atomisation energies at 0K (including zero point energy and spin-orbit corrections; all energies in kcal mol−1).

CCSD(T)(aug-)

cc-pVTZ+d

CCSD(T)(aug-)

cc-pVQZ+d

CCSD(T)(aug-)

cc-pV5Z+d

CCSD(T)CBS(mix)

CCSD(T)

CBS( maxl )

CCSD(T)CBS

( 4 6n n− −+ )

CCSD(T)

CBS( 3x− )

CCSD(T)

CBS( 3x− ) + CV corr1 a

CCSD(T)

CBS( 3x− )+ CV corr2 b

CCSD(T)

CBS( 3x− )+ CV corr2+ CP(CV)

CCSD(T)

CBS( 3x− )+ CV corr2

+ CP(CV) + Erel

P2 104.4 110.9 113.2 114.6 114.9 115.1 115.7 116.3 116.5 116.3 116.0P2

c104.4 110.9 113.2 114.7 115.0 115.3 115.6 116.3 116.4 116.2 116.0

P2 d

104.6 111.0 113.3 114.6 114.9 115.2 115.6 116.3 116.4 116.2 116.0P4 257.8 274.5 280.8 284.5 284.9 286.1 287.0 289.1 289.1 288.3 287.8P4

c257.8 274.5 280.8 284.6 285.2 286.3 286.9 289.0 289.1 288.3 287.8

P4 d

258.6 274.6 280.8 284.4 284.6 286.5 286.6 288.7 288.8 288.0 287.5PH −67.2 −69.1 −69.6 −69.9 −70.1 −70.0 −70.3 −70.4 −70.4 −70.3 −70.2PH2 140.7 144.2 145.1 145.7 146.0 145.8 146.4 146.7 146.7 146.5 146.2PH3 220.2 224.8 226.1 226.9 227.3 227.0 227.9 228.3 228.3 228.0 227.6P2H2 210.9 219.0 221.6 223.1 223.7 223.6 224.6 225.4 225.4 225.0 224.1P2H4 334.1 343.3 346.2 347.8 348.5 348.3 349.6 350.5 350.5 349.9 349.1PO 132.6 137.7 139.5 140.6 140.8 141.0 141.5 142.1 142.2 142.0 141.8PO2 247.5 256.3 259.5 261.4 261.6 262.1 262.7 263.5 263.8 263.3 262.4PO3 338.8 350.9 355.3 357.9 358.3 358.9 359.8 361.0 361.2 360.7 359.1P2O 190.7 200.5 204.0 206.0 206.4 206.7 207.6 208.7 208.9 208.4 207.5P2O2 310.9 323.4 327.6 330.0 330.8 330.8 332.3 333.9 333.7 333.1 332.2

176

Chapter 5. A


hermochem

istry

177

Table 5.2 continued

CCSD(T)(aug-)

cc-pVTZ+d

CCSD(T)(aug-)

cc-pVQZ+d

CCSD(T)(aug-)

cc-pV5Z+d

CCSD(T)CBS(mix)

CCSD(T)

CBS( maxl )

CCSD(T)CBS

( 4 6n n− −+ )

CCSD(T)

CBS( 3x− )

CCSD(T)

CBS( 3x− ) + CV corr1 a

CCSD(T)

CBS( 3x− )+ CV corr2 b

CCSD(T)

CBS( 3x− )+ CV corr2+ CP(CV)

CCSD(T)

CBS( 3x− )+ CV corr2

+ CP(CV) + Erel

HPO 197.1 203.2 205.3 206.6 206.9 207.0 207.6 208.2 208.3 208.0 207.5HPOH 250.1 256.6 258.5 259.6 260.2 259.8 261.0 261.7 261.6 261.4 260.4H2POH 324.7 332.3 334.5 335.8 336.5 336.1 337.4 338.2 338.1 337.8 337.0H3PO 324.6 333.1 335.9 337.5 338.0 338.0 339.1 339.6 339.6 339.2 337.9HOPO 339.2 348.6 351.8 353.6 354.2 354.3 355.3 356.5 356.6 356.2 355.4HOPO2 449.8 463.2 470.6 473.0 474.4 474.6 474.0 472.4

a Calculated using Dunning’s cc-pCVTZ basis sets, Refs 27, 29, 32.b Calculated using Bauschlicher’s core-valence basis sets, Ref. 15.c Extrapolation includes the cc-pV6Z+d atomisation energy: AE(P2) = 114.2 kcal mol−1, AE(P4) = 283.2 kcal mol−1.d Calculated using the cc-pV(n+d)Z basis sets, Ref. 56.

177


178

On inspection of the results in Table 5.2 it can be seen that the atomisation energies increase

by approximately 6 to 10 kcal mol−1 as the basis set is enlarged from triple- to quadruple-ζ

and then by another 2 to 4 kcal mol−1 when going to the pentuple-ζ basis set. These

corrections scale roughly with molecular size, viz. number of electrons, as expected,

becoming as large as 16.7 kcal mol−1 (TZ to QZ) and 6.4 kcal mol−1 (QZ to 5Z) for the largest

molecule in the set, P4. Some degree of scatter is found among the atomisation energies given

by the various extrapolation schemes. The 3x− scheme consistently yields the largest values,

increasing the atomisation energy by roughly the same amount as the QZ to 5Z gap, while the

mixed exponential/Gaussian extrapolation gave the lowest results, only increasing the AE by

half of this amount. As a result, the scatter is also found to increase with molecular size; in

general it is less than 2 kcal mol−1, but it rises to ~ 2.4 kcal mol−1 for P2O2 and P4. As the

uncertainties in the computed AE’s are also expected to increase with the number of electrons,

the observed scatter in the extrapolated results is expected to provide a reasonable measure of

the expected errors in the AE’s. Based on this scatter, therefore, our estimated uncertainties

are ± 1 kcal mol−1 for PH, PH2, PH3, P2, PO and HPO, ± 1.5 kcal mol−1 for P2H2, PO2, P2O,

HPOH, H2POH, H3PO and HOPO, ± 2 kcal mol−1 for P2H4 and PO3 and ± 2.5 kcal mol−1 for

P4 and P2O2. While for HOPO2 the uncertainty cannot be estimated this way, it is comparable

in size to PO3, and thus the same uncertainty is assigned to both.

As a test of the extrapolations, for P2 and P4 we computed the next energies in the sequence,

that is, with the cc-pV6Z+d basis (the additional d exponent was 12.9024) and re-fitted these

according to Equations (5.1) - (5.4). As can be seen from the results in Table 5.2, the 6Z

calculations add 1.0 kcal mol−1 to the 5Z atomisation energy for P2 and 2.4 kcal mol−1 for P4.

However, no appreciable difference can be discerned between the {T,Q,5} and {T,Q,5,6}

extrapolated atomisation energies or the scatter among each set of energies.

The fits to the calculated atomisation energies of P4 (including the 6Z result) are shown in

graphical form in Figure 5.2. The plots nicely illustrate the asymptotic behaviour of the

various equations utilised in the fits and extrapolations. The lmax and mix procedures suggest

rapid convergence to the hypothetical CBS limit, while convergence (to a higher CBS limit) is

slower for the 3x− and 4 6n n− −+ approaches. While the latter two procedures, along with mix,

provide fits of comparable quality to the computed points, the lmax fit is found to be

significantly poorer. Clearly, more energy points would be required before a definitive choice


179

between the mix and 3x− (or 4 6n n− −+ ) could be made on the basis of quality of fit.

Nevertheless, in light of the accumulated compelling numerical evidence from other

workers16,17 concerning the accuracy and reliability of the 3x− procedure, we too have

adopted it in this work as our preferred extrapolation technique.

Figure 5.2 P4: Comparison of fits to CCSD(T)/cc-pVnZ+d atomisation energies.

After the completion of this work it was suggested by Klopper and Radom55 that, in the

context of the 3x− extrapolation scheme, better quality results could be obtained by fitting

only the final two energies in a given sequence, rather than by the inclusion of all x ≥ 3 points,

as has been done in this work. On testing this procedure for the current set of molecules it was

found that the 3x− extrapolation of the QZ and 5Z energies ({Q,5}) yields atomisation

energies which agree with those obtained by the three-point fits (as listed in Table 5.2) to

within 0.3 kcal mol−1 for all systems except P4, PH3, P2H4, P2O2, HPOH, and H2POH, where

the differences between the two- and three-point fits are 0.5, −0.4, −0.5, −0.4, −0.5, and −0.6

kcal mol−1 respectively. In the case of P4, however, the two-point fit of the 5Z and 6Z energies

({5,6}) yields an extrapolated atomisation energy which is 0.4 kcal mol−1 lower than obtained

by the three-point {T,Q,5} and four-point {T,Q,5,6} fits. Such fluctuation may be interpreted

as a measure of the degree of convergence and is thus indicative of the expected uncertainty

in the atomisation energy of P4. This uncertainty, we believe, is adequately accommodated by

the proposed conservative error estimate of ± 2.5 kcal mol−1. In light of these results, given

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

260

265

270

275

280

285

290

295

Data mix l

max

n-4+n

-6

x-3

Ato

mis

atio

n E

ner

gy

/kca

l mo

l-1

x (=lmax

)


180

the relatively small differences between the two- and three-point fits relative to the proposed

uncertainties, the heats of formation are calculated from the latter, that is, the atomisation

energies in Table 5.2.

Recently, Dunning et al.56 formulated a new set of (re-optimised) correlation consistent basis

sets for second row atoms, denoted cc-pV(x+d)Z, which also include an additional tight d

function in the basis. By way of comparison, we have carried out a set of calculations with

this new set for P2 and P4. The resulting atomisation energies are given in Table 5.2. Clearly,

there is very little difference (no more than 0.4 kcal mol−1) between the extrapolated energies

generated by this new cc-pV(x+d)Z and the “standard” cc-pVxZ+d basis sets. Since P4 is

effectively the largest of our systems of interest, such basis set effects are expected to be even

less pronounced for the other molecules. This is clearly borne out for P2.

5.3.1.3 Core-Valence Correlation, BSSE and Scalar Relativistic Effects

A legitimate question in the context of atomisation energy calculations concerns the presence

and importance of basis set superposition error (BSSE). As the basis set is systematically

expanded the magnitude of any BSSE will be reduced, so one may expect the extrapolated

energy to be essentially free of superposition errors. As a test of this hypothesis, we applied

the appropriate CP corrections to the energies of P4 at the CCSD(T)/cc-pVxZ+d levels of

theory (x = T, Q, 5) and extrapolated these to the CBS limit, as described above. The resulting

CP corrections to the extrapolated atomisation energies are −0.46, −0.40, 0.19, and 0.0006, as

obtained by the mix, maxl , 3x− and 4 6n n− −+ procedures respectively. Thus, the 4 6n n− −+

extrapolation scheme can be seen to display near-ideal behaviour, as the “extrapolated” BSSE

is effectively zero. Given that the actual CP correction at the CCSD(T)/cc-pV5Z+d levels of

theory is only −0.67 kcal mol−1, an extrapolated correction of ~ −0.5 kcal mol−1, obtained by

the mix and maxl methods is regarded as unrealistic. The positive correction of ~ 0.2 kcal

mol−1, corresponding to the 3x− approach, while non physical, is probably a reflection of the

fact that the CP method itself is approximate. On the whole, in the light of these results, we

believe that the 3x− and 4 6n n− −+ extrapolated atomisation energies can be accepted as being

essentially free of BSSE artefacts.


181

The CV contributions to the AE’s are relatively small. Moreover, in the absence of CP

corrections, both basis sets used to compute CV effects, namely Dunning’s cc-pCVTZ (CV

corr1) and Bauschlicher’s core-valence sets (CV corr2), yield essentially identical results. The

CP corrections are, however, significantly smaller when using Bauschlicher’s basis, by up to

0.5 kcal mol−1 (in the case of P4), suggesting that it is a better balanced basis in the context of

CV calculations. Therefore, the heats of formation reported in the next section are based on

AE’s generated via the 3x− extrapolation and CV corrections obtained via Bauschlicher’s

basis set, with the inclusion of CP corrections for BSSE.

The CV contribution to the AE is largest for P4, as might be expected, where the CP corrected

and uncorrected (CV corr2) values are 1.3 and 2.1 kcal mol−1 respectively. As indicated in the

previous section, such corrections were applied only to the phosphorus atoms in each

molecule, as corrections to the oxygen atoms are expected to be an order of magnitude

smaller. This has been verified by computing the oxygen CP corrections in PO3 and PPO,

which resulted in BSSE estimates of 0.01 kcal mol−1 for each oxygen atom (in both

molecules), compared with 0.59 kcal mol−1 for the phosphorus atom in PO3 or with 0.54 and

0.28 kcal mol−1 for the two P atoms in PPO. Accepting the CP corrected CV values as the

more reliable, we see that the CV contribution to AE’s is generally ~ 1 kcal mol−1 or less.

It is also clear from the data in Table 5.2 that both CV and scalar relativistic corrections are

relatively minor in comparison with the effects of extrapolation. In most cases the scalar

relativistic corrections are less than ~ 1 kcal mol−1. In addition, there is some cancellation

between these two contributions, so that the net effects of CV and scalar relativistic

corrections to the AE’s are often below ~ 0.5 kcal mol−1.

The above observations concerning the relative importance of CV corrections in the

computation of atomisation energies and heats of formation, however, are in disagreement

with the findings of Persson et al.57, who reported a value of ~ 6 kcal mol−1 as the CV

correction to the energy of the P4 → 2P2 reaction. This result was obtained by MP2

computations in a [6s,5p,4d,3f,2g,1h] atomic natural orbital (ANO) basis. Subsequent

investigations, however, revealed that the exponent range of the polarisation functions used in

the construction of the ANO’s was not adequate to describe the 2s2p correlation reliably.58

For a P atom this ANO basis resolves only ~ 60% of the CV correlation energy obtained with


182

the cc-pCVTZ basis, while with regard to the CV component of the energy of the P4 → 2P2

reaction the results of subsequent counterpoise calculations are indicative of substantial

superposition errors.

5.3.2 G3, G3X and G3X2 Calculations

The computed atomisation energies obtained by application of the G3, G3X, G3X2 and

G3X2(RAD) methods are displayed in Table 5.3, along with the (R)CCSD(T)/CBS

benchmark values for ready comparison. As discussed in Section 5.2, G3X2 represents an

approximation to the QCISD(T,Full)/G3XLarge method (provided the latter also includes the

G3X higher level correction). It is therefore instructive to compare the G3X2 atomisation

energies with those obtained by QCISD(T,Full)/G3XLarge calculations. Such QCI results for

P4, PH, P2H2, P2O, HPO and HOPO are thus also included in Table 5.3. These results are to

be compared with the benchmark energies, both with and without the scalar relativistic

corrections, since there seems to be some doubt as to whether G3 type results should be

compared with benchmarks that include scalar relativistic corrections. The study by Kedziora

et al.59 concluded that in G3 such corrections are in fact compensated for by the higher level

corrections and that their explicit inclusion in the G3 methodology leads to slightly worse

agreement with experiment (even when the hlc is reoptimised).

In the previous chapter it was shown that it was necessary to employ restricted open shell

methods when performing G3n type calculations for the PO radical since large spin

contamination occurred in some of the UHF calculations. While no comparable spin

contamination could be discerned for the other radical species studied here (atoms and

molecules), we investigated whether the use of the G3n(RAD) procedure would be superior to

the standard method. As the results in Table 5.3 demonstrate, there is little difference between

the standard G3X2 and G3X2(RAD) generated atomisation energies. Therefore, the routine

use of Gn(RAD) procedures for open shell atoms and molecules is unwarranted, unless the

unrestricted formalism is unusable due to spin contamination. Note, however, that in the case

of PO, where the UHF based formalisms are inapplicable, all the Gn results quoted in Table

5.3 are actually Gn(RAD) values.


183

Table 5.3 Comparison of atomisation energies at 0K from CCSD(T)/CBS benchmark and

G3n calculations (including zero point energy and spin-orbit corrections; all energies in kcal

mol−1).

CCSD(T)CBS a

CCSD(T)CBS b

G3 G3X G3X2G3X2

+ CP(CV)G3X2(RAD)c QCI d

P2 116.0 116.3 114.9 116.1 120.4 119.2 120.3 120.5P4 287.8 288.3 281.7 283.7 296.6 292.7 296.2 293.6PH −70.2 −70.3 −70.8 −71.1 −71.9 −71.8 −72.6 −73.0PH2 146.2 146.5 145.1 145.6 147.2 146.9 147.6PH3 227.6 228.0 225.3 226.0 228.3 227.8 228.2P2H2 224.1 225.0 221.5 222.7 227.6 226.4 228.2P2H4 349.1 349.9 344.3 345.8 351.0 349.7PO c 141.8 142.0 141.3 141.9 143.2 142.4 143.2PO2 262.4 263.3 260.2 261.9 264.1 262.5 262.2PO3 359.1 360.7 352.6 357.7 360.9 358.6 360.5P2O 207.5 208.4 203.6 205.8 211.2 208.9 209.9 213.0P2O2 332.2 333.1 327.3 330.5 334.8 332.6 334.5HPO 207.5 208.0 205.6 207.1 209.1 208.2 208.8 209.2HPOH 260.4 261.4 258.5 259.3 261.1 260.3 260.7H2POH 337.0 337.8 333.6 334.7 337.1 336.2 336.6H3PO 337.9 339.2 333.4 335.0 337.8 336.7 337.0HOPO 355.4 356.2 351.7 353.7 355.8 354.4 355.2 355.5HOPO2 472.4 474.0 466.5 469.1 472.2 469.9 471.2

a Including scalar relativistic corrections.b Not including scalar relativistic corrections.c Calculated by G3n(RAD) methods.d QCISD(T,Full)/G3XLarge calculation including G3X higher level correction.

When compared with the benchmark atomisation energies, it is clear, as already noted by

Curtiss et al., that G3 performs relatively poorly for molecules containing second row atoms.

The G3 atomisation energies are found to be consistently too low, by over 6 kcal mol−1 in the

worst case (HOPO2). It should be noted, however, that in the case PO3 much of the 6.5 kcal

mol−1 difference between the G3 and benchmark results can be traced to the large difference

between the respective zero point energies (computed at the UHF/6-31G(d) and UB3LYP/

6-31G(2df,p) levels of theory respectively), as discussed in Chapter 4. For the molecules of

this study, the root-mean-square (rms) deviation between the G3 and benchmark results is 3.8


184

kcal mol−1 when the latter include scalar relativistic correction. The deviation is 4.6 kcal mol−1

in the absence of such corrections.

In most cases the G3X atomisation energies are found to be appreciably closer to the

benchmark results than those from G3 calculations. The superior performance of G3X is

partly due to the introduction of the G3XLarge correction, which increases the atomisation

energies by typically ~ 0.5 kcal mol−1, as indicated by the data in Table 5.4. The use of DFT

optimised geometries instead of MP2 is found to yield significantly lower total energies for all

phosphorus oxides and acids (and P4) and hence higher atomisation energies, by ~ 0.5 kcal

mol−1 on the average. The differences in G3 and G3X zero point energies, with the exception

of PO3, are quite small at ~ 0.2 kcal mol−1. However, as the data in Table 5.4 shows, the hlc’s

contribution to the atomisation energies are quite large and the small differences between the

G3X and G3 hlc parameters are responsible for an additional ~ 0.5 kcal mol−1 difference in

the resulting atomisation energies. For the phosphorus compounds of this work the combined

effect of the changes to G3, that define G3X, is that the rms deviations between the G3X and

benchmark results (with and without scalar relativistic corrections) are significantly lower at

2.8 and 2.0 kcal mol−1 respectively. However, the atomisation energies are still

underestimated by up to ~ 4 kcal mol−1.

At first sight, the G3X2 calculations do not seem to represent a significant improvement over

G3X. Whereas G3X generally underestimates the AE’s, G3X2 overestimates them by a

comparable amount, due to much larger G3XLarge corrections (Table 5.4) which in G3X2

are evaluated at the MP2 level. However, with one exception (P4), the G3X2 atomisation

energies agree with the QCISD(T,Full)/G3XLarge values at least to within ± 2 kcal mol−1.

Four of the seven molecules (P2, P4, P2H2, P2O) for which we carried out these large QCI

calculations were chosen specifically because of the large deviations of their G3X2

atomisation energies from the benchmarks. The overall good agreement between the G3X2

and QCI results suggests that with the exception of “pathological” cases, such as P4, the G3X2

approach is capable of providing a good approximation to a QCISD(T,Full)/G3XLarge

calculation. Note that G3X represents a lower level of electron correlation treatment, since it

is an approximation to QCISD(T,Full) with the smaller G3Large basis, where the energetic

effects of basis set extension to G3XLarge is obtained at the SCF level. In spite of this, G3X2

actually appears to be inferior to G3X when comparing the rms deviations from the


185

Table 5.4 Various components of the G3X and G3X2 atomisation energies at 0K (in kcal

mol−1).

SCFG3XLargecorrection

MP2(Full)G3XLargecorrection

CP(CV)correction of

G3X2G3X hlc

P2 0.2 −4.5 −1.1 −8.3

P4 0.8 13.7 −3.8 16.6

PH 0.0 −0.9 −0.2 −5.2

PH2 0.1 −1.7 −0.3 −6.8

PH3 0.1 −2.4 −0.4 −8.4

P2H2 0.3 −5.2 −1.2 11.1

P2H4 0.3 −5.5 −1.3 13.9

PO a 0.4 −1.8 −0.8 −6.7

PO2 0.7 −3.0 −1.6 −9.3

PO3 1.1 −4.2 −2.3 12.0

P2O 0.6 −6.0 −2.2 11.0

P2O2 1.9 −6.2 −2.2 13.7

HPO 0.4 −2.4 −0.9 −8.3

HPOH 0.5 −2.2 −0.7 −9.5

H2POH 0.5 −2.9 −0.9 11.1

H3PO 0.5 −3.3 −1.1 11.1

HOPO 0.8 −2.8 −1.4 10.9

HOPO2 1.1 −4.2 −2.3 13.6

a Calculated by G3n(RAD) methods.

benchmark. In the case of G3X2 these are 2.9 and 2.6 kcal mol−1 respectively, when the

benchmarks do and do not include scalar relativistic corrections. The apparent inferiority to

G3X is, however, largely due to the inclusion of P4, an extreme outlier, for which G3X2

predicts the atomisation energy to be up to ~ 9 kcal mol−1 higher than the benchmark. For this

molecule the failure of the G3X2 procedure is partly due to a 3 kcal mol−1 overestimation of

the QCISD(T,Full)/G3XLarge atomisation energy, which itself is a further 5 kcal mol−1

higher than the benchmark. For several other molecules, however, due to the fortuitous

cancellation of errors, the G3X2 results agree slightly better with the benchmarks than those

obtained by QCI. P4 is clearly a special case, and as such, it will be the subject of a more


186

detailed discussion later in this chapter. At this point, however, it is reasonable to re-evaluate

the rms errors when P4 is removed from the test set. These are now 3.6 and 4.5 kcal mol−1 for

G3; 1.8 and 2.7 kcal mol−1 for G3X and 2.1 and 1.7 kcal mol−1 for G3X2, the two sets of

values corresponding to the inclusion and exclusion of scalar relativistic corrections in the

benchmark results respectively. Thus, G3X2 represents a modest improvement upon G3X

theory for this set of molecules, when compared with benchmark values without scalar

relativistic corrections.

As remarked already, a legitimate modification of G3X and G3X2 would be to consider

corrections to the CV component of the atomisation energies for BSSE. The computed

corrections to the G3X2 energies, that is, at the MP2/G3XLarge level of theory, are given in

Table 5.4. They range from −3.8 to −0.2 kcal mol−1 and reduce the rms deviation from the

benchmark values (with and without scalar relativistic corrections) to 1.8 and 2.0 kcal mol−1

respectively. Omitting P4 from the set results in further reductions in the above deviations to

1.5 and 1.8 kcal mol−1. The analogous corrections to G3X are somewhat smaller but as they

are negative their impact would be to degrade the agreement between the G3X and

benchmark values. The introduction of these proposed modifications of course necessitates

changes to the higher level corrections. As the current set of molecules is too small and not

sufficiently representative, no modifications to the hlc are proposed at this stage. Further work

is under way, however, to produce G3X2 atomisation energies for all the molecules in the

G3/99 test set which contain second row atoms. This will allow re-optimisation of the hlc in

the presence of BSSE corrections as well as a fuller investigation of the need (or otherwise) to

correct the G3X2 energies for scalar relativistic effects. With the current (G3X) choice of hlc

the fluctuation in the deviations of both G3X and G3X2 results is such that, in the rms sense,

a comparable level of agreement is achieved with the two sets of benchmark values. It is

possible, however, that re-optimisation of the hlc on the basis of G3X2 data will change this

apparent insensitivity to relativistic corrections as well as the magnitude of the above rms

deviations.

The deviations of the G3X, G3X2 and QCISD(T,Full)/G3XLarge atomisation energies from

the benchmark results (including scalar relativistic corrections) are summarised graphically in

Figure 5.3. It can be clearly seen that in most cases G3X2 is a good approximation to

QCISD(T,Full)/G3XLarge. Where the difference between G3X2 and QCI is largest, both are


187

seen to show relatively poor agreement with the corresponding benchmark result, although

corrections to the CV components for BSSE (in both G3X2 and QCI) do improve the

agreement. This occurs for P4, as noted above, and also for PPO. Both are somewhat unusual

molecules. Tetrahedral P4, with 60º bond angles, is a very “strained” molecule, while the

bonding in PPO is best described in terms of a P-P+ triple bond and a P+-O− semipolar bond,

as suggested by the results of a Roby-Davidson population analysis60-62. Thus it should not be

too surprising that such molecules would be difficult to describe accurately.

Figure 5.3 Deviations of G3X, G3X2 and QCISD(T,Full)/G3XLarge atomisation energies

from CCSD(T)/CBS benchmark values (with scalar relativistic corrections).

As we expect that, in general, the G3X2 results would have an uncertainty of approximately ±

2 kcal mol−1, it is noteworthy that all the molecules which fall outside this range (P2, P4, P2H2

and PPO) have multiple or strained P-P bonds. As P2H4 (with a P-P single bond) and P2O2

(with two phosphorus atoms but no P-P bond) are both described adequately by G3X2, it

appears that this method could be expected to provide a satisfactory description of molecules

with second row atoms but care should be taken when these atoms have formal multiple

bonds with each other. As remarked earlier, G3X produces atomisation energies that are

consistently lower than the benchmark values, while in most cases the G3X2 results are on the

-6.0

-4.0

-2.0

0.0

2.0

4.0

6.0

8.0

10.0

Devi

atio

n /

kcal m

ol -1

G3X

G3X2

G3X2 + CP

QCISD(T)

P2

P4

PH

PH2

PH3

P2H2

P2H4

PO

PO2

PO3

P2O

P2O2 HPOH

H2POH

H3PO

HOPO

HPO

HOPO2


188

high side. There is only one molecule, HOPO2, for which the G3X2 atomisation energy is

significantly below the benchmark result; however, as noted in the previous section, the latter

may be 0.85 kcal mol−1 too high due to the absence of the CCSD(T)/(aug-)cc-pV5Z+d energy

in extrapolations.

5.3.2.1 Analysis of Molecules for Which G3n Methods Perform Poorly

5.3.2.1.1 P4

As noted above, in the case of P4 there is significant deviation between the G3X2 atomisation

energies and those resulting from QCISD(T,Full)/G3XLarge calculations. The discrepancy

suggests at least a partial failure of the underlying assumption of G3 type theories for P4; that

is, that as the basis set is extended, the increased degree of correlation can be adequately

approximated by lower levels of theory than QCISD(T). In order to identify the actual source

of this problem, the individual G3X and G3X2 corrections, namely (+), (2df,p), G3Large,

G3XLarge and G3X2, as defined in Equations (5.5) to (5.8), computed using MP4, MP2 (and

SCF in the case of G3XLarge), are compared with the corresponding corrections evaluated by

QCISD(T). The results for the phosphorus atom and P4 are given in Table 5.5. In the case of

P, the agreement between the corrections obtained by QCISD(T) and the relevant lower levels

of theory is clearly very good, as may be expected. The largest discrepancy is 0.7 kcal mol−1

(in the MP2 determination of the G3Large correction) but in the case of G3X2 this is largely

cancelled by the negative discrepancy in the G3XLarge term. Somewhat fortuitously, thus,

G3X2 represents an excellent approximation to the QCI energy in the case of P. The situation

is quite different for P4, where MP4 overestimates the magnitude of the (2df,p) correction by

3.6 kcal mol−1. The trends and deviations in the other corrections, specifically the G3Large

and G3XLarge corrections, are similar to those observed for the phosphorus atom, provided

we allow for the presence of four atoms in the molecule. Therefore, the major deviation

between the G3X2 and QCI atomisation energies of P4 results from the poor performance of

MP4 in the prediction of the (2df,p) correction, as shown by the analysis of the atomisation

energy in Table 5.5. The total error in the G3Large and G3XLarge terms as obtained by MP2

is −1.1 kcal mol−1. Thus again there is some fortuitous error cancellation, resulting in a total

discrepancy of 3.0 kcal mol−1 in the G3X2 atomisation energy of P4.

Chapter 5. A


hermochem

istry

189

Table 5.5 Basis set enlargement corrections to total G3X and G3X2 energies of P4 and P and to the atomisation energy of P4 at

various levels of theory (in kcal mol−1).

P4 PContribution to

Atomisation Energy

Correction Level of TheoryAbsolutecorrection

Deviation fromQCISD(T) a

Absolutecorrection


Absolutecorrection


+ MP4 −−−4.3 −0.2 −−−0.6 −0.0 −2.0 −−0.2

QCISD(T) −−−4.1 −−−0.6 −1.8

2df,p MP4 −118.8 −3.6 −−16.3 −0.1 53.5 −−3.8

QCISD(T) −115.2 −−16.4 49.7

G3Large MP2(Full) −797.6 −3.6 −198.3 −0.7 −4.4 −−0.7

MP4(Full) −803.3 −2.1 −199.1 −0.0 −7.0 −−1.9

QCISD(T,Full) −801.2 −199.0 −5.1

G3XLarge SCF −−−0.8 18.4 −−−0.0 −1.3 −0.8 −13.2

MP2(Full) −−20.8 −1.6 −−−1.8 −0.5 13.7 −−0.4

MP4(Full) −−20.0 −0.8 −−−1.3 −0.0 14.9 −−0.9

QCISD(T,Full) −−19.2 −−−1.3 14.0

G3X2 MP2(Full) −818.4 2.0 −200.1 −0.2 18.1 −−1.1

MP4(Full) −823.3 −2.9 −200.3 −0.0 21.9 −−2.8

QCISD(T,Full) −820.5 −200.3 19.2

a QCISD(T,Full) for G3Large, G3XLarge and G3X2.

189


190

5.3.2.1.2 P2O, P2, P2H2

Analysis of G3X and G3X2 results, as applied to P4, was also carried out for P2O, P2, P2H2

and HOPO. The various corrections to the atomisation energies and their deviation from the

QCI values for these four molecules, as well as P4 for ready comparison, are listed in Table

5.6. We recall that for P4 and PPO significant discrepancy in the G3X2 atomisation energy

has been noted in comparison with QCISD(T,Full)/G3XLarge while in the case of P2 and

P2H2 there were no major discrepancies however the results were in poor agreement with the

benchmark calculations. In contrast to these species, good agreement was found between the

G3X2, QCISD(T) and CCSD(T)/CBS results for HOPO. For this set of molecules, the (+)

correction is found to be consistently the smallest (less than 5 kcal mol−1 but becoming larger

as oxygen atoms are added); deviations of MP4 from QCI are found to be small (< 1 kcal

mol−1) and positive. The (2df,p) correction, on the other hand, is by far the largest (up to about

50 kcal mol−1 for P4). The deviations between the MP4 and QCI corrections are again found

to be small and positive for P2, P2H2 and HOPO; for PPO the deviation is also small although

negative, but for P4, as mentioned earlier, it is significantly larger at +3.8 kcal mol−1. The

G3Large, G3XLarge and G3X2 corrections are of intermediate magnitude (4 - 10 kcal mol−1

for G3Large, 3 - 15 kcal mol−1 for correlated G3XLarge corrections and 10 - 20 kcal mol−1 for

G3X2). We note that in general the MP2 results show slightly better agreement with

QCISD(T) (0.5 - 1.5 kcal mol−1 deviation for G3X2) than MP4 (1.0 - 2.8 kcal mol−1). In

addition, the correction terms obtained by MP2 are slightly lower than at the QCI level

(giving a negative deviation), whereas the MP4 differences are all positive. This leads to

significant cancellation of errors when the (+) and (2df,p) (obtained by MP4) and G3X2 (by

MP2) corrections are added, resulting in good agreement between the final QCISD(T) and

G3X2 atomisation energies for P2, P2H2 and HOPO. As noted above, in the case of P4, the

(2df,p) correction is too large to be compensated for by the G3X2 correction, while for PPO

all errors, with the exception of that in the (+) term, tend to reinforce each other, leading to the

observed lack of agreement between G3X2 and QCISD(T,Full)/G3XLarge.

Chapter 5. A


hermochem

istry

191

Table 5.6 Basis set enlargement corrections to the atomisation energies of P4, P2O, P2, P2H2 and HOPO evaluated at various levels of theory

(in kcal mol−1).

P4 P2O P2

Correction Level of Theory CorrectionDeviation from

QCISD(T) aCorrection


CorrectionDeviation from

QCISD(T) a

+ MP4 2.0 0.2 3.2 0.1 0.6 0.1

QCISD(T) 1.8 3.1 0.5

2df,p MP4 53.5 3.8 29.9 −0.5 13.0 0.3

QCISD(T) 49.7 30.3 12.7

G3Large MP2(Full) 4.4 −0.7 6.9 −1.1 4.4 −0.4

MP4(Full) 7.0 1.9 9.4 1.4 5.7 0.9

QCISD(T,Full) 5.1 8.0 4.8

G3XLarge SCF 0.8 −13.2 0.6 −5.8 0.2 −4.3

MP2(Full) 13.7 −0.4 6.0 −0.4 4.5 −0.1

MP4(Full) 14.9 0.9 6.8 0.4 4.8 0.3

QCISD(T,Full) 14.0 6.4 4.5

G3X2 MP2(Full) 18.1 −1.1 12.9 −1.4 8.9 −0.5

MP4(Full) 21.9 2.8 16.2 1.9 10.5 1.2

QCISD(T,Full) 19.2 14.3 9.4

191

Chapter 5. A


hermochem

istry

192


P2H2 HOPO

Correction Level of Theory CorrectionDeviation from

QCISD(T) aCorrection


+ MP4 0.8 0.1 4.3 0.9

QCISD(T) 0.8 3.4

2df,p MP4 28.6 0.6 32.7 0.6

QCISD(T) 27.9 32.1

G3Large MP2(Full) 5.0 −0.9 8.5 −0.8

MP4(Full) 6.5 0.6 10.3 1.0

QCISD(T,Full) 5.9 9.3

G3XLarge SCF 0.3 −5.3 0.8 −2.4

MP2(Full) 5.2 −0.4 2.8 −0.4

MP4(Full) 5.8 0.3 3.4 0.2


G3X2 MP2(Full) 10.2 −1.3 11.3 −1.2

MP4(Full) 12.4 0.9 13.7 1.1


a QCISD(T,Full) for G3Large, G3XLarge and G3X2.

192


193

5.3.3 Enthalpies of Formation

The heats of formation (at 0 and 298 K) of the molecules studied in this work, generated from

the computed atomisation energies (Table 5.3), are listed Table 5.7, along with experimental

and theoretical values from the chemical literature for comparison. For a number of molecules

there has been an absence of experimental and/or theoretical values; for these the current

calculations provide reliable heats of formation. For several others, where the imprecision in

the literature values is quite large, we are able to offer improved data. It is gratifying,

however, that in all cases the benchmark results agree with the accepted literature values

within the respective error margins, even for P4.

The information in Table 5.7 also allows one to assess the performance of the G3, G3X and

G3X2 methods in the context of thermochemistry. For most molecules, given the large

experimental errors, there appears to be reasonable agreement between the available

experimental heats of formation and those from any of the G3, G3X or G3X2 calculations,

with the obvious exception of P4. The differences are more evident if the comparisons are

made with the benchmark values, in which case the G3X and G3X2 results are clearly

superior to those of G3. As discussed already in the context of AE’s, in general G3X2 appears

to be more accurate than G3X, especially if corrections for BSSE in the CV contributions are

included in the calculations.

Chapter 5. A


hermochem

istry

194

Table 5.7 Enthalpies of formation at 0 and 298K (in kcal mol−1).

Heat of Formation (0K) Heat of Formation (298K)

CCSD(T)CBS a

G3 G3X G3X2G3X2

+ CP(CV)QCI b

CCSD(T)CBS a

G3 G3X G3X2G3X2

+ CP(CV)QCI b Literature c

P2 34.8 35.9 34.7 30.5 31.6 30.4 34.4 ± 1.0 35.5 34.3 30.0 31.2 29.9 34.3 ± 0.5 d

P4 13.9 20.0 18.0 5.1 8.9 8.1 12.1 ± 2.5 18.2 16.2 3.4 7.2 6.4 14.1 ± 0.05 d

PH 56.8 56.2 55.9 55.1 55.2 56.6 ± 1.0 56.0 55.7 54.9 55.0 60.6 ± 8.0 e

57.4 ± 0.6 f

PH2 32.4 33.5 33.1 31.5 31.8 31.5 ± 1.0 32.6 32.2 30.6 30.8 26 ± 23 g

33.1 ± 0.6 f

PH3 2.8 5.0 4.3 2.1 2.5 0.9 ± 1.0 3.1 2.4 0.2 0.6 1.3 ± 0.4 d

P2H2 30.0 32.6 31.4 26.5 27.7 25.9 28.1 ± 1.5 30.6 29.4 24.6 25.8 23.9

P2H4 8.2 13.1 11.6 6.4 7.7 4.9 ± 2.0 9.8 8.3 3.1 4.4 5.0 ± 1.0 h

PO i −7.4 −6.9 −7.5 −8.8 −8.0 −7.6 ± 1.0 −7.1 −7.7 −9.0 −8.2 −5.6 ± 1.0 d

−6.8 ± 1.9 j

−7.8 k

PO2 −69.0 −66.8 −68.5 −70.7 −69.1 −69.7 ± 1.5 −67.5 −69.2 −71.5 −69.9 −66.2 ± 3 j

−70.3 k

PO3 −106.7 −100.2 −105.3 −108.5 −106.2 −107.7 ± 2.0 −101.7 −106.3 −109.5 −107.2 −107.5 k

P2O 2.4 6.2 4.0 −1.4 0.9 −3.1 1.6 ± 1.5 5.5 3.3 −2.1 0.1 −3.9

P2O2 −63.4 −58.5 −61.7 −66.0 −63.8 −65.1 ± 2.5 −59.8 −63.4 −67.7 −65.5

194

Chapter 5. A


hermochem

istry

195

Table 5.7 continued

Heat of Formation (0K) Heat of Formation (298K)

CCSD(T)CBS a

G3 G3X G3X2G3X2

+ CP(CV)QCI b

CCSD(T)CBS a

G3 G3X G3X2G3X2

+ CP(CV)QCI b Literature c

HPO −21.5 −19.6 −21.1 −23.1 −22.2 −22.4 ± 1.0 −20.5 −22.0 −24.0 −23.1 −22.6 k

HPOH −22.7 −20.9 −21.7 −23.4 −22.7 −24.4 ± 1.5 −22.5 −23.4 −25.1 −24.4

H2POH −47.7 −44.3 −45.4 −47.8 −46.9 −50.2 ± 1.5 −46.8 −48.0 −50.3 −49.5

H3PO −48.6 −44.1 −45.7 −48.5 −47.4 −51.5 ± 1.5 −46.9 −48.6 −51.4 −50.2

HOPO −110.4 −106.7 −108.7 −110.7 −109.7 −110.4 −112.0 ± 1.5 −108.3 −110.3 −112.3 −111.3 −112.0 −110.6 ± 3 l

−112.4 k

HOPO2 −168.3 −162.5 −165.1 −168.2 −165.9 −170.6 ± 2.0 −164.8 −167.4 −170.5 −168.1 −168.8 ± 4 l

−171.4 k

a Including scalar relativistic corrections.b QCISD(T,Full)/G3XLarge calculation including G3X higher level correction.c Experimental values unless otherwise indicated by italics and footnotes.d Ref. 63.e Semiempirical estimate, Ref. 63.f Ref. 64.

g Estimate, Ref. 63.h Ref. 65.i Calculated by G3n(RAD) methods.j Ref. 66.k RCCSD(T)/CBS computations, Ref. 15.l Ref. 67.

195


196

5.4 Conclusion

Using the (R)CCSD(T) quantum chemical method in conjunction with correlation consistent

basis sets, accurate heats of formation have been obtained for a series of small phosphorus

containing molecules. These are regarded as convenient and useful benchmark values,

especially in light of the paucity of accurate experimental values. The computed atomisation

energies and hence heats of formation include complete basis estimates of valence correlation

contributions, core-valence contributions which are corrected for basis set superposition and

scalar relativistic corrections. The equilibrium geometries and vibrational frequencies were

obtained by density functional theory. The resulting benchmark heats of formation are in good

agreement with the available experimental and other high level quantum chemical data.

Utilising the calculated benchmark values, we were able to carry out a critical study of the

accuracy and reliability of three G3n type procedures, G3, G3X and G3X2, for phosphorus

containing molecules. We have found that in general the G3X and G3X2 results are of

comparable accuracy, both reproducing the benchmark heats of formation, on the average, to

within ± 2 kcal mol−1. The relative accuracy of G3X2 improves, however, on the introduction

of BSSE corrections to the core-valence correlation contributions. The problem cases for the

G3n methods appear to be molecules with unusual P-P bonding, such as P2 and P4. 63 64-66 67


197

5.5 References





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7. M. A. MacDonald, F. C. Gouldin and E. M. Fisher, Combustion and Flame, 2001,

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17. T. Helgaker, P. Jørgensen and J. Olsen, Molecular Electronic-Structure Theory; John

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6 The Role of the NNH +O Reaction in the Production of NO in Flames

Chapter 6

The Role of the

NNH + O Reaction

in the Production of

NO in Flames

Chapter 6: NNH + O in NO Production

202

The work described in this chapter involves a thermochemical and kinetic study of the

importance of the NNH + O reaction in the production of NO in flames. This work comprises

of two halves: the quantum chemical investigation of the potential energy surface (including

the characterisation of all stationary points) followed by the calculation of the rates of all

possible reactions and the kinetic modelling of various flame systems. The latter half of the

work was performed by Associate Professor John Mackie.

6.1 Introduction

The presence of nitric oxide in the atmosphere is largely the consequence of combustion

processes. As a result, many governments have recently placed stringent limits on the levels

of emission of NO from combustion facilities. It is important in the development of new

combustion technology that accurate chemical kinetic models of the combustion process be

developed to predict the concentrations of NO produced.

Nitric oxide is produced in combustion via four main pathways.1 The first of these is the well

known Zel’dovich or thermal route, initiated by the reaction

2N + O N + NO (6.1)

The prompt-NO route is initiated by

2CH + N HCN + N (6.2)

A third route involves the formation of the intermediate, N2O, and, for fuels containing fuel-

bound nitrogen, there is a fuel-NO route.

Recently, Bozzelli and Dean2 have discovered an additional pathway involving the

intermediate NNH radical, formed by the reaction of H with molecular nitrogen:

2N + H NNH (6.3)


203

The subsequent reaction of NNH with atomic oxygen then yields the products NO and NH:

NNH + O NO + NH→ (6.4)

Using the QRRK technique, Bozzelli and Dean2 estimated the rate coefficient for this reaction

to be 134 7 10k = × cm3 mol−1 s−1 at 2000 K. This reaction and its rate coefficient have been

subsequently incorporated into detailed chemical kinetic reaction models such as GRIMech

2.113 and its successor, GRIMech 3.04, developed to predict species profiles in the

combustion of C1 and C2 hydrocarbons.

There have been no direct experimental studies of the reactions of NNH + O → products. In a

study of NO profiles in laminar premixed flames of H2 / O2 / N2, however, Hayhurst and

Hutchinson5 observed enhanced production rates of NO in the burnt gases that could not be

explained on the basis of the Zel’dovich mechanism alone. They attributed their faster

observed rates to the operation of the NNH + O pathway and, from a steady state analysis of

their experimental NO profiles, arrived at a value of 4 ,3Pk K = ( )91.4 10 exp 2760 T× − cm3

mol−1 s−1 over the temperature range of 1800 – 2500 K; here 4k is the rate coefficient of

Reaction (6.4) and ,3PK is the equilibrium constant of Reaction (6.3). From estimates of the

NNH thermochemistry, they evaluated the equilibrium constant and arrived at a value of

144 1 10k = × cm3 mol−1 s−1, constant within a factor of 2.5, over the temperature range of 1800

– 2500 K. This result is comparable with the rate coefficient originally estimated by Bozzelli

and Dean2.

Recently, experimental and modelling studies of NO profiles in premixed flames6 and in

completely stirred reactors7 have concluded, however, that in circumstances where the NNH +

O route is likely to be important (that is, where the thermal and prompt NO routes are

unimportant), models which employ the above rate coefficient for Reaction (6.4) lead to

overprediction of NO production.

The current study is therefore motivated to make a detailed investigation of the NNH + O

reaction potential energy surface using current techniques of computational quantum

chemistry. Such a study will allow the calculation of reliable thermochemical and kinetic data


204

which can subsequently be applied to the modelling of combustion reactions in the presence

of nitrogen.

Following Bozzelli and Dean, we have included

2NNH + O N O + H→ (6.5)

and

2NNH + O N + OH→ (6.6)

in our system of reactions (in addition to Reactions (6.3) and (6.4)) as potentially competing

channels to the production of NO in Reaction (6.4).

Reactions (6.4) to (6.6) are expected to take place via a common ONNH intermediate, which

decomposes to yield NO + NH, N2O + H or N2 + OH. While no studies of the O + NNH

association have previously been reported, the three decomposition reactions have been

extensively investigated in the chemical literature over the past 18 years8-17. It has been found

that ONNH, a planar molecule, is stable in both cis and trans forms. Although the latter is

generally predicted to be the more stable10-13, it has been concluded that decomposition to

N2O + H and N2 + OH actually occurs via the cis isomer.12,13

Several other reactions are also considered in this work, in particular

NNH O products+ → → (6.7)

where the intermediate, ONHN, is an isomer of ONNH which can yield HNO + N, N2O + H

and N2 + OH as decomposition products.

N

N

HO


205

The isomerisation reactions between cis- and trans-ONNH and between trans-ONNH and

ONHN are alternative channels which have also been studied, along with the direct

abstraction reaction

2NNH O N OH+ → + (6.8)

The stability of the NNH intermediate plays a crucial role in determining the relative

importance of channels (6.4), (6.5) and (6.6) in the chemistry of nitrogen in flames, in

particular the reaction flux which passes through channel (6.4) to generate NO. High level

quantum chemical calculations by Walch and Partridge18 and by Gu et al.19 have predicted the

exothermicity of Reaction (6.3) to be between 3.8 and 4.3 kcal mol−1, with a barrier height of

10.0 - 11.3 kcal mol−1. As modelling studies are extremely sensitive to the stability of NNH,

in the present work we have also undertaken the computation of the heats of formation of both

NNH and the transition state leading to its formation (NN-H) at the highest currently

achievable level of theory available to us, viz. complete basis estimates of the CCSD(T)

(coupled cluster theory with single, double and perturbative triple excitations) energetics

based on extrapolation of aug-cc-pVxZ results with x = 5, 6.

This work therefore involves an extensive investigation of the N2OH potential energy surface

using high level quantum chemical methods followed by the computation of the rate

parameters for all NNH + O → products channels. Finally, in conjunction with the data set of

the GRIMech 3.0 model, we use our revised thermochemistry and rate data to model the lean

combustion of CO / H2 / air and CH4 / air in completely stirred reactors.


6.2.1 Quantum Chemical Calculations of Thermochemistry

Heats of formation and other thermochemical data were computed via two different

approaches: the Gaussian-3X (G3X) method of Curtiss et al.20 and a CCSD(T)/CBS type

scheme, utilising coupled cluster theory with single, double and (perturbative) triple

excitations (CCSD(T)21,22) in conjunction with the correlation consistent basis sets23-25, aug-


206

cc-pVQZ and aug-cc-pV5Z, and extrapolation to the hypothetical complete basis set (CBS)

limit.

In both schemes molecular geometries and vibrational frequencies for equilibrium structures

and transition states were determined by Density Functional Theory (DFT), using the B3LYP

hybrid density functional26-28 with the 6-31G(2df,p) basis set (frequencies scaled by 0.9854).

Single point energy calculations were then performed at these geometries as required for the

two different methodologies, viz. G3X and CCSD(T)/CBS. Open shell systems were treated

by unrestricted calculations in the DFT geometry optimisations (UB3LYP) and the

subsequent implementation of G3X but by restricted, RCCSD(T), methods in the

CCSD(T)/CBS computations.

In the case of the CCSD(T)/CBS{Q,5} scheme, the single point energies were extrapolated to

the complete basis limit using the 3x− extrapolation29

( ) 3E x A Bx−= + (6.9)

where x = 4, 5. In the case of a few species, e.g. NNH, more extensive calculations were

performed where the sequence of basis sets include aug-cc-pV6Z, viz. x = 6. Core-core and

core-valence correlation (CV) corrections were evaluated at the CCSD(T)/aug-cc-pCVQZ

level of theory. Scalar relativistic effects (Darwin and mass-velocity terms)30,31 were

determined by complete active space SCF (CASSCF)32,33 theory using cc-pVTZ basis sets.

Spin-orbit corrections were also included for atomic species.34

As several important reactions, including the formation of ONNH and its decomposition to

NO + NH, are barrierless, Variational Transition State Theory (VTST)35-37 was utilised to

locate and characterise the transition states at a range of temperatures between 1000 and 2500

K. As in previous work of ours38, density functional theory (B3LYP/6-31G(2df,p)) was used

to map the minimum energy path (MEP) along the potential energy surface (PES) as a

function of the reaction coordinate. The latter was approximated as the critical bond forming

or bond breaking distance; thus this critical bond distance was systematically varied while all

other geometric parameters were allowed to relax. At each such point along the reaction

coordinate the rate coefficient was calculated by the application of the canonical transition


207

state formula39 at the given temperature, thus allowing the geometry which yielded the

minimum rate to be identified as the variational transition state.

The Gaussian 98 programs40 were used to perform all DFT calculations (geometry

optimisations and PES scans) as well as the G3X calculations, while MOLPRO21,41,42 was

utilised for all (R)CCSD(T) computations. The CASSCF calculations of scalar relativistic

corrections were carried out using DALTON43 for all molecules and MOLCAS44 for atomic

species. The computations were performed on DEC alpha 600/5/333 and COMPAQ

XP100/500 workstations of the Theoretical Chemistry group at the University of Sydney and

on the COMPAQ AlphaServer SC system of the Australian Partnership for Advanced

Computing National Facility at the National Supercomputing Centre, ANU, Canberra.

6.2.2 Derivation of Rate Coefficients for Individual Reaction

Channels

As discussed above, addition of O to NNH produces three intermediates via chemical

activation. These intermediates, which represent local minima on the potential energy surface,

are trans-ONNH, cis-ONNH and ONHN. Further reaction leads to four product channels, that

is, to NO + NH, N2O + H, N2 + OH and HNO + N. All but the last of these are exothermic

processes. To derive rate coefficients for the overall reaction to the four product channels we

have separately considered the three reaction surfaces for

NNH + O ONNH productstrans - → (6.10)

NNH + O ONNH productscis - → (6.11)

and

NNH + O ONHN products→ (6.12)

We then assume that the vibrationally excited adduct (trans-ONNH, cis-ONNH or ONHN), is

formed from NNH and O at an energy, E, and will undergo the reverse reaction at an energy-


208

specific rate coefficient, ( )k E . The limiting high-pressure rate coefficient for this reverse

(dissociation) reaction, ,unik ∞ , is given by

0,

1( ) ( ) exp( / )

( )uni BEk k E E E k T dE

q Tρ

∞

∞ = −∫ (6.13)

where ( )q T is the internal partition function of the adduct calculated at the translational

temperature, T; ( )Eρ is the density of states and Bk is Boltzmann’s constant. The lower limit

for integration is the critical energy of reaction, 0E . The recombination rate coefficient in the

high pressure limit, ,reck ∞ , is obtained from ,unik ∞ by detailed balance using the equilibrium

constant ( )cK T as given by

( ),

,uni

recc

kk

K T∞

∞ = (6.14)

The overall pressure-dependent rate coefficient via each adduct to each of the product

channels is obtained from

,overall products reck f k ∞= × (6.15)

where prodsf is the fraction of reaction flux to each product channel. We have carried out an

RRKM analysis using the MultiWell suite of programs developed by Barker45 to solve the

internal master equation with densities of states calculated by an exact count method.

Collisional energy transfer parameters were taken from the work of Barker.46 Lennard-Jones

parameters have been taken from the Chemkin Collection.47 As all three adducts lead to the

four reaction channels (but with different values of productsf ), the final overall rate coefficients

for NNH + O → products were obtained by summing the contributions from the three

surfaces.


209


6.3.1 Quantum Chemistry

All potential stationary points on the NNH + O PES were investigated using B3LYP/

6-31G(2df,p) on both the A′ and A″ surfaces. The A′ species were found to be consistently

lower in energy (as found by other workers), so in general only results for this surface are

presented.

The electronic energies of the stationary points on the PES as calculated at the (valence

correlated) CCSD(T)/aug-cc-pVQZ and CCSD(T)/aug-cc-pV5Z levels of theory, along with

the extrapolated results, are presented in Table 6.1. This Table also provides details of the

CCSD(T)/aug-cc-pCVQZ core-valence correlation corrections, the scalar relativistic

corrections and the zero point energies for each molecule, and thus the total CCSD(T)/CBS

energy at 0K together with the corresponding G3X result. The thermal corrections between 0

and 298K for each molecule are also reported here. Unfortunately, it was not possible to

perform such CCSD(T)/CBS calculation on the cis to trans transition state of ONNH (ONNH

c-t TS), as the lack of symmetry in this molecule made the aug-cc-pV5Z and aug-cc-pCVQZ

calculations too large. The geometries, rotational constants and vibrational frequencies for all

equilibrium structures, transition states and variational transition states are summarised in

Appendix 3 (schematic structures for these species can also be found in Figures 6.1 to 6.3).

Table 6.2, in turn, contains the atomisation energies for each species as calculated using both

the CCSD(T)/CBS and G3X approaches, as well as the resulting heats of formation at 0 and

298 K and literature values for the latter where available. Core-valence correlation and

relativistic contributions to the atomisation energies are also included in this Table.

Examination of the latter reveals that both effects are relatively small in magnitude and that

their respective contributions cancel out to a large extent, resulting in a net contribution of

~ 0.5 kcal mol−1 or less.

According to Curtiss et al.20, the heats of formation obtained from the G3X method are

expected to have a mean absolute deviation of 1.0 kcal mol−1 from experiment. The

CCSD(T)/CBS results are expected to have a higher degree of accuracy; our conservative

Chapter 6: N

NH

+ O

in NO

Production

210

Table 6.1 Total CCSD(T) and G3X energies, core-valence (CV) correlation corrections, scalar relativistic corrections (in hE ) along with zero

point energies and thermal corrections to enthalpies (in kcal mol−1).

Valence correlated energy CV corr relE∆ ZPE Total Energyb 0 0298 0H H−

CCSD(T) CCSD(T) CCSD(T) CCSD(T) CASSCF B3LYP CCSD(T) B3LYP

aug-cc-pVQZ aug-cc-pV5Z CBS{Q,5}a aug-cc-pCVQZ cc-pVTZ 6-31G(2df,p) CBS{Q,5}G3X

6-31G(2df,p)

N 4S −54.52506 −54.52780 −54.53068 −0.05613 −0.02921 −54.61601 −54.56490 1.04

O 3P −74.99493 −75.00041 −75.00615 −0.05907 −0.05237 −75.11795 −75.03224 1.04

H 2S −0.49995 −0.49999 −0.50004 0.00000 0.00000 −0.50004 −0.50097 1.01

N2 1Σg −109.40724 −109.41551 −109.42418 −0.11344 −0.05818 3.42 −109.59034 −109.48808 2.07

NH 3Σ −55.15574 −55.15913 −55.16269 −0.05628 −0.02906 4.57 −55.24075 −55.19350 2.07

NO 2Π −129.75792 −129.76818 −129.77893 −0.11585 −0.08085 2.80 −129.97117 −129.83624 2.07

OH 2Π −75.66426 −75.67038 −75.67680 −0.05929 −0.05187 5.21 −75.77967 −75.69607 2.07

NNH 2A′ −109.90091 −109.90907 −109.91763 −0.11344 −0.05790 8.18 −110.07595 −109.97501 2.39

NNO 1Σ −184.46683 −184.48141 −184.49671 −0.17313 −0.10956 6.98 −184.76828 −184.58323 2.27

HNO 1A′ −130.34240 −130.35288 −130.36388 −0.11587 −0.08067 8.54 −130.54681 −130.41227 2.37

trans-ONNH 2A′ −185.00994 −185.02643 −185.04373 −0.17290 −0.10949 13.04 −185.30534 −185.11715 2.52

cis-ONNH 2A′ −185.00041 −185.01644 −185.03325 −0.17297 −0.10948 12.56 −185.29569 −185.10725 2.56

ONHN 2A′ −184.97089 −184.98820 −185.00636 −0.17270 −0.10952 12.52 −185.26863 −185.07844 2.53

NN-H 2A′ −109.88513 −109.89336 −109.90200 −0.11332 −0.05813 4.05 −110.06699 −109.96623 2.46

ONN-H 2A′ −184.95143 −184.96843 −184.98626 −0.17301 −0.10955 7.38 −185.25707 −185.06898 2.76

ON2-H 2A′ −184.92953 −184.94647 −184.96424 −0.17276 −0.10957 7.77 −185.23418 −185.04605 2.57

NNOHsq 2A′ −184.94179 −184.95694 −184.97283 −0.17210 −0.10985 8.91 −185.24059 −185.05381 2.50

NNOHtr 2A′ −184.92091 −184.93646 −184.95278 −0.17257 −0.10976 8.20 −185.22205 −185.03713 2.71

ONNH c-t TS 2A −185.08082

ONHN-ONNHt 2A′ −184.91787 −184.93264 −184.94813 −0.17260 −0.10956 9.02 −185.21592 −185.03186 2.63

a Extrapolated CCSD(T) energy to CBS (x = ∞) limit using x = 4, 5 data. b Including CV, rel

E∆ and ZPE corrections in CCSD(T)/CBS energies.

Chapter 6: N

NH

+ O

in NO

Production

211

Table 6.2 CCSD(T)/CBS{Q,5} and G3X atomisation energies (along with CV correlation and scalar relativistic contributions to CBS) and

heats of formation (at 0 and 298K) of reactants, products, intermediates and first order saddle points on the N2OH surface (in kcal mol−1).

Atomisation Energya 00f H∆ 0

298f H∆

CV corr relE∆ CCSD(T)b

CBSG3X

CCSD(T) b

CBSG3X

CCSD(T) b

CBSG3X Experiment

N2 0.74 −0.15 224.9 224.8 0.2 0.2 0.2 0.2 0.00

NH 0.09 −0.09 78.3 80.1 85.9 84.1 85.9 84.1 85.32 ± 0.02c

NO 0.41 −0.46 148.9 150.0 22.7 21.5 22.7 21.5 21.82 ± 0.04d

OH 0.14 −0.31 101.5 102.2 9.2 8.4 9.2 8.5 8.83 ± 0.09e

NNH 0.74 −0.33 215.8 216.0 60.9 60.7 60.2 60.0

NNO 1.13 −0.77 262.5 264.3 21.6 19.8 20.7 18.9 19.6 ± 0.1f

HNO 0.42 −0.57 196.3 197.1 26.9 26.0 26.2 25.3 25.60.60.1

+−

g

trans-ONNH 0.99 −0.82 285.7 285.0 50.0 50.7 48.4 49.1

cis-ONNH 1.03 −0.82 279.7 278.8 56.0 56.9 54.5 55.4

ONHN 0.86 −0.80 262.7 260.7 73.0 75.0 71.4 73.4

NN-H 0.67 −0.18 210.2 210.5 66.5 66.2 65.9 65.6

ONN-H 1.05 −0.78 255.4 254.8 80.3 80.9 78.9 79.6

ON2-H 0.90 −0.77 241.1 240.4 94.6 95.3 93.1 93.8

NNOHsq 0.48 −0.59 245.1 245.2 90.6 90.5 89.0 88.8

NNOHtr 0.78 −0.65 233.5 234.8 102.2 100.9 100.8 99.5

ONNH c-t TS 262.2 73.5 71.9

ONHN-ONNHt 0.80 −0.77 229.6 231.5 106.1 104.2 104.6 102.7

a Including zero-point corrections.b Including CV correlation and

scalar relativistic corrections.

c 0

0fH∆ from Ref. 48 with thermal corrections

from this work.48

d Ref. 49.49

e Ref. 50.50

f Ref. 49.g Ref. 51.51


212

estimate for the maximum uncertainty in any of the CBS heats of formation computed in this

work is ± 1.0 kcal mol−1. Comparison of the CCSD(T)/CBS and G3X heats of formation

indicates the two sets of results agree with each other and with the available experimental data

within their respective error margins.

In light of the sensitivity of the kinetic models to the stability of NNH, as noted above, a more

extensive investigation was carried out for NNH, N2 and the transition state, NN-H. The

geometries and harmonic frequencies were redetermined at the CCSD(T)/aug-cc-pVQZ level

of theory using numerical differentiation to obtain the appropriate force constants. The

CCSD(T) valence correlated energies were calculated at the revised geometries using the aug-

cc-pV5Z and aug-cc-pV6Z basis sets52,53 and extrapolated as before. CCSD(T)/aug-cc-

pCVQZ CV corrections, CASSCF/cc-pVTZ scalar relativistic corrections and atomic spin

orbit corrections were again applied. In order to account, at least in part, for the effects of

anharmonicity in the calculation of the zero-point energies and thermal corrections, the NNH

bending frequencies were scaled by a factor of 0.97, the N-N stretching frequencies by 0.988

and the N-H stretching frequencies by 0.95. (These scaling factors were chosen by

comparison of experimental harmonic and anharmonic frequencies of N2, NH3 and H2O.) The

results of these high level calculations are summarised in Table 6.3. As can be seen by

comparison with the results in Table 6.2, the application of a substantially higher level of

theoretical treatment leaves the heats of formation for N2 and NNH largely unchanged; it

does, however, reduce the barrier for the dissociation of NNH by ~ 1 kcal mol−1.

Comparison of the CCSD(T)/aug-cc-pV5Z energies at the revised CCSD(T) geometries (in

Table 6.3) with those at B3LYP geometries (in Table 6.1) demonstrates that for N2 and NNH

the small geometric changes (less than 0.01 Å in bond lengths and 0.6° in the NNH bond

angle) have a negligible effect on the energies. For the NN-H transition state, however, the

energy has been lowered by ~ 1 kcal mol−1; this is accompanied by a decrease of 0.12 Å in the

N-H bond length. The zero-point energies are effectively unchanged (differences of ~ 0.1 kcal

mol−1 or less). For N, N2 and NNH the CCSD(T)/CBS{5,6} energies are 0.54, 1.16 and 1.54

kcal mol−1 higher than the corresponding {Q,5} results. Thus, the effects of the extra degree

of theoretical complexity implicit in the aug-cc-pV6Z calculations largely cancel in the

atomisation energy computations of N2 while the value for NNH is reduced by ~ 0.4 kcal

mol−1. Interestingly, for the NN-H transition state the combined effects of geometry changes


213

Table 6.3 Summary of the energetic contributions to the CCSD(T)/CBS{5,6} extrapolated

atomisation energies (AE) and heats of formation for N2, NNH and the NN-H transition state

(in hE unless otherwise noted).

N H N2 NNH NN-H

CCSD(T)/aug-cc-pV5Z −54.52780 −0.49999 −109.41550 −109.90913 −109.89164

CCSD(T)/aug-cc-pV6Z −54.52865 −0.50000 −109.41838 −109.91167 −109.89596

CCSD(T)/CBS{5,6} −54.52982 −0.50001 −109.42233 −109.91517 −109.90200

CCSD(T)/aug-cc-pCVQZ(core + valence) −54.58205 −109.52248 −110.01611 −109.99847

CCSD(T)/aug-cc-pCVQZ(valence only) −54.52592 −109.40906 −109.90267 −109.88515

CV correction −0.05613 −0.11342 −0.11343 −0.11332

relE∆ −0.02921 −0.05818 −0.05790 −0.05811

ZPE 0.00526 0.01287 0.00644

CCSD(T)/CBS {5,6} a −54.61516 −0.50001 −109.58867 −110.07363 −110.06698

CCSD(T)/CBS {5,6} AE a

/kcal mol−1 224.87 215.43 211.26

CCSD(T)/CBS {5,6} 00f H∆ a

/kcal mol−1 0.19 61.26 65.43

0 0298 0H H− /kcal mol−1 1.04 1.01 2.07 2.39 2.44

CCSD(T)/CBS {5,6} 0298f H∆ a

/kcal mol−1 0.18 60.56 64.78

a Including CV, rel

E∆ and ZPE corrections.

and larger basis set have resulted in a CCSD(T)/CBS{5,6} energy that is essentially the same

as that obtained via CCSD(T)/CBS{Q,5}. The net results is therefore a lower atomisation

energy for NN-H; that is, the barrier to dissociation is reduced, by ~ 1 kcal mol−1. These

results are in support of our proposed uncertainty of ± 1 kcal mol−1 in our CCSD(T)/CBS

heats of formation.

In Table 6.4, the G3X and CCSD(T)/CBS energetics are compared with the earlier theoretical

work of Walch12,18, Gu19 and Durant13. As these workers reported their results as energies

relative to N2 + H (for NNH and NN-H) and NO + NH for all other species, we also present


214

Table 6.4 Energiesa of NNH and N2OH species relative to N2 + H and NO + NH respectively

computed at different levels of theory (in kcal mol−1).

CCSD(T)CBS {Q,5}

CCSD(T)CBS {5,6}

G3X G2b MR-CIcCCSD(T)/

aug-cc-pVQZd

MR-CI +Dav.e

NNH 9.1 9.4 8.8 8.6 ± 0.5 9.1

NN-H 14.7 13.6 14.3 14.4 ± 1.0 16.3

trans-ONNH −58.6 −54.9 −56.0 −51.5

cis-ONNH −52.6 −48.6 −48.9 −46.2

ONHN −35.6 −30.6

O + NNH 11.3 14.1

NNO + H −35.4 −34.2 −34.5 −31.7

N2 + OH −99.2 −96.9 −96.9

ONNH c-t TS −32.1 −30.8

ONN-H −28.3 −24.6 −25.3 −21.4

ON2-H −14.0 −10.2

NNOHsq −18.0 −15.1 −17.9 −15.4

NNOHtr −6.4 −4.6

ONHN-ONNHt −2.5 −1.3

a Including zero-point corrections.b Ref. 13.c Ref. 12.d Ref. 19 + B3LYP/6-31G(2df,p) zero point energy.e MR-CI values, including Davidson’s correction, from Ref. 18 + B3LYP/6-31G(2df,p) zero point energy.

our results in this form for easy comparison. Note also that the previous calculations on NNH

and NN-H by Walch and Partridge18 and by Gu, Xie and Schaefer19 reported only electronic

energy differences; we have therefore added our B3LYP/6-31G(2df,p) zero-point energies to

their values for consistency. The results of Gu et al.19 were obtained via CCSD(T)/

aug-cc-pVQZ calculations while Walch and Partridge utilised multireference CI (MRCI)

(including Davidson’s correction) in conjunction with the cc-pVxZ, x = D, T, Q, 5 basis sets

and extrapolation via the exponential formula:

( ) ( )expE x A B Cx= + − (6.16)


215

While there is broad agreement with respect to the stability of NNH, our best estimate of the

barrier height for its dissociation is lower that those obtained by either of the other groups.

Thus our results indicate that NNH is significantly less stable to dissociation and has a shorter

lifetime than previously predicted.

The energies of the N2OH species are compared in Table 6.4 with the G2 values of Durant13

and the MRCI results of Walch12. As may be expected, the G2 and G3X results are, in most

cases, in good agreement. Walch’s MRCI results, however, are found to be consistently

higher than those obtained by either G2, G3X or CCSD(T)/CBS. We believe that this

discrepancy is due to size extensivity problems in the MRCI approach, which in this case did

not include Davidson’s correction. Such problems are expected to be most serious in the

calculation of dissociation energies. Consequently, as noted by Durant13, in Walch’s

calculations the stabilities of the N2OH species (as well as of N2O + H) relative to NO + NH

are underestimated by ~ 3 kcal mol−1; when a correction of this size is applied to the MRCI

results the agreement with the G2 and G3X results is much improved. It is seen, however, that

the CCSD(T)/CBS{Q,5} results are generally significantly lower than their G3X counterparts,

by up to 3.7 kcal mol−1.

We believe that for the systems studied here the CCSD(T)/CBS approach, as outlined in this

work, represents potentially the highest level of size consistent treatment of electron

correlation that is currently available. Consequently, we expect our CCSD(T)/CBS results to

be more accurate than those obtained previously.

6.3.2 Potential Energy Surfaces and Reaction Paths

Schematic potential energy diagrams showing the major stationary points on the N2OH

potential energy surface corresponding to the three main reaction channels (Equations (6.10)

to (6.12)) are shown in Figures 6.1 to 6.3. The relative enthalpies (at 298K) shown in these

diagrams are CCSD(T)/CBS{Q,5} values, except for ONNH c-t TS where the G3X result has

been used. Clearly, the reaction channels producing N2O + H and N2 + OH are

thermodynamically favoured over the formation of NO + NH. On the other hand, the reaction

to form N + HNO is calculated to be endothermic by 19.4 kcal mol−1 and thus unlikely to

compete with the other more favourable channels.


216

-47.9

-65.3

-30.8

-110.4

-11.2

-71.4

-40.9

-46.9

0.00

NNH + O

NO + NH

trans -ONNH

ONN-H

NNO + H

ONNH c -t TS

cis -ONNH

N2 + OH

NNOHsq

Figure 6.1 Schematic of potential energy surfaces for the reactions of trans-ONNH. Relative

enthalpies at 298 K (in kcal mol−1) from CCSD(T)/CBS{Q,5} calculations.

-46.9

-40.9

-65.3

-30.8

-110.4

0.0

-11.2

NNH + O

NO + NH

ONN-H

NNO + H

cis -ONNH

NNOHsq

N2 + OH

Figure 6.2 Schematic of potential energy surfaces for the reactions of cis-ONNH. Relative


N

H

NO

N N

HO

O

N N

H

O

N N

H

N N

H

O

N

H

NO

N

O

N

HO

N N

H

N N

H

O

N N

H

O

O

N N

H

N N

H

O


217

19.4

-46.9-26.7

-15.20.0

-71.4

-110.4

-18.9

-48.3

NNH + O

trans -ONNH

ONHN

ONHN-ONNHt NNOHtr

N2 + OH

ON2-HNNO + H

HNO + N

Figure 6.3 Schematic of potential energy surfaces for the reactions of ONHN. Relative


As several of the reactions on this PES involve barrierless recombinations or dissociations,

variational transitions state theory was applied in order to locate and characterise the

(temperature dependent) transition states, as described in the Section 6.2.1. Energies, and thus

heats of formation, at the CCSD(T)/CBS level of theory were estimated by utilising the

B3LYP/6-31G(2df,p) estimate of the energy difference between the TS and the dissociated

adducts along with the CCSD(T)/CBS energies for the dissociated species. The resulting heats

of formation are listed in Table 6.5. Geometries, rotational constants and vibrational

frequencies for these species are given in Appendices 3.3 and 3.4.

N

N

HO

N

O

N

H

N

N

O H

N

N

HO

N

N

O H

O

N

H

N

N

N

O

H


218

Table 6.5 G3X and CCSD(T)/CBS{Q,5} total energies, atomisation energies (at 0K) and

heats of formation (at 0 and 298K) of variational transitions states (in kcal mol−1).

Reaction Temp. /K00f H∆ 0

298f H∆

G3X CBS G3X CBS

trans-ONNH → NNH + O 1000 116.1 116.3 115.1 115.3

1500 114.2 114.5 113.1 113.4

2000 112.1 112.3 110.9 111.1

2500 109.6 109.8 108.3 108.5

cis-ONNH → NNH + O 1000 116.8 117.1 115.9 116.1

1500 116.2 116.4 115.2 115.4

2000 114.9 115.1 113.9 114.1

2500 112.7 112.9 111.5 111.8

ONHN → NNH + O 1000 117.0 117.2 116.1 116.3

1500 115.5 115.7 114.5 114.8

2000 111.4 111.7 110.3 110.5

2200 107.0 107.2 105.7 105.9

trans-ONNH → NO + NH 1000 99.8 102.9 98.9 102.0

1500 98.6 101.6 97.6 100.6

2000 97.1 100.1 96.1 99.1

2400 96.7 99.7 95.6 98.6

cis-ONNH → NO + NH 1000 99.6 102.6 98.7 101.7

1500 98.2 101.3 97.3 100.3

2000 96.7 99.7 95.6 98.6

2500 96.2 99.2 95.1 98.1

ONHN → N + HNO 1000 135.5 136.4 134.2 135.0

1500 135.1 135.9 133.7 134.6

2000 134.5 135.3 133.1 133.9

2500 134.1 135.0 132.7 133.5

NNH + O → N2 + OH 1000 117.8 118.0 116.8 117.0

1500 116.9 117.2 115.8 116.1

2000 115.7 116.0 114.6 114.8

2500 115.3 115.5 114.1 114.3


219

As noted in Section 6.2.1, the MEP’s for all potential reaction channels were mapped using

B3LYP/6-31G(2df,p). The important features of each surface are discussed here:

NNH (2A′′′′) + O (3P) →→→→ cis- and trans-ONNH (2A′′′′). Both reactions, as indicated in Figures

6.1 and 6.2, are simple barrierless recombinations leading to the cis- and trans-ONNH

adducts. Variational transition states were determined for both reactions, as described in

Section 6.2.1. Figure 6.4 shows the plots of the minimum energy paths along the PES for all

three possible NNH + O recombination reactions (giving cis- and trans-ONNH as well as

ONHN). It is interesting to note that, although the cis- isomer is higher in energy than the

trans- at their respective equilibrium geometries, the cis- form is more stable over most of the

PES. In the vicinity of the minima for cis- and trans-ONNH, the cis-trans interconversion

takes place via torsion. The ONNH dihedral angle in the isomerisation transition state

structure, ONNH c-t TS, was found to be 90.9°.

-80

-70

-60

-50

-40

-30

-20

-10

0

1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0

R (N-O) /Å

E /

kca

l mo

l -1

cis-ONNH

trans-ONNH

ONHN

Figure 6.4 B3LYP/6-31G(2df,p) potential energy surfaces for NNH + O recombination

reactions. Energy relative to NNH + O.


220

NNH (2A′′′′) + O (3P) →→→→ ONHN (2A′′′′). This is also a barrierless recombination reaction as

shown in Figures 6.3 and 6.4. The ONHN adduct and the variational transition states leading

to it are all planar 2A′ states. As shown in Figure 6.5, however, application of the time

dependent density functional approach, viz. TD-B3LYP, at the ground state geometries

reveals that the first two excited states (2A″ and 2A′ respectively) are quite close in energy to

the ground state. Figure 6.5 also suggests that there are two avoided crossings on this PES,

one at ~ 1.45 Å (between the two excited states) and one at ~ 1.65 Å (between all three

states). This indicates that, although the ground state is planar, the equilibrium geometries of

the excited state molecules are likely to be non-planar, thus raising the symmetry constraints

and allowing direct interaction between the states. In the context of the current work,

however, these unusual features are of purely academic interest, the important point being the

existence of a direct pathway leading to ONHN on the 2A′ surface.

Figure 6.5 Ground and excited state B3LYP/6-31G(2df,p) potential energy surfaces for the

NNH + O → ONHN recombination. Energy relative to NNH + O.

-60

-50

-40

-30

-20

-10

0

10

20

1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9

R(N-O) /Å

E /

kca

l mo

l -1

ground state -

first excited state -

second excited state -

2A′

2A′

2A″


221

cis- and trans-ONNH (2A′′′′) →→→→ NO (2ΠΠΠΠ) + NH (3ΣΣΣΣ). The dissociation of both isomers was

found to occur via a common, barrierless, non-planar surface, as shown in Figures 6.1 and

6.2. The MRCI studies of Walch12 have previously established that dissociations of both cis-

and trans-ONNH on the 2A′ surface must pass over activation barriers while the analogous

2A″ surfaces are barrierless. The same features are also observed on our B3LYP/6-31G(2df,p)

PES, shown in Figure 6.6, with the surface crossings from 2A′ to 2A″ occurring at R(N-N) ≈

1.67 Å for the trans isomer and R(N-N) ≈ 1.58 Å for cis-ONNH. When the minimum energy

path for the dissociation of the trans isomer was mapped with no symmetry constraints

applied, the trans 2A′ surface was followed until the neighbourhood of the surface crossing

from which point the molecule became non-planar and the dissociation followed a surface

very similar to that of 2A″ trans-ONNH; it therefore appears that the 2A states of the non-

planar variational transition states correlate with the 2A″ surfaces of ONNH. Such states are

accessible via torsion of the molecule, as shown in Figure 6.7. Here the potential energy

surfaces for torsion of ONNH are shown for various N-N bond lengths. At low N-N

separations there is a high barrier between 2A′ cis- and trans-ONNH; as the bond stretches,

however, the cis 2A″ surface falls below the 2A′ surface so that at R(N-N) = 1.6 Å the torsion of

2A′ trans-ONNH actually results in the cis isomer on the 2A″ surface. As the N-N bond is

stretched beyond 1.65 Å, a minimum appears on the torsional PES at a dihedral angle of

~ 130°; this is consistent with the observed non-planar variational transition states. Further

stretching of R(N-N) results in the 2A″ state of trans-ONNH falling below the 2A′ state.

Although at 1.75 Å 2A″ trans-ONNH is higher in energy than the non-planar structure, the

energy difference is only ~ 0.6 kcal mol−1; this barrier reduces further as the molecule

dissociates (as seen in Figure 6.6) such that the non-planar MEP for the dissociation is

effectively on the trans 2A″ surface. In summary, therefore, we are proposing that the

dissociation reaction of ONNH occurs via 2A′ → 2A″ surface crossing which can be achieved

by the out-of-plane distortion, viz. torsion of the molecules. This is reasonable for the sort of

systems we are modelling, where the ONNH adducts are generated in highly vibrationally

excited states as a result of chemical and/or collisional activation. Surface crossing, via

vibronic coupling, predominantly involving torsion, is expected to occur readily. Note,

however, that even though there is a common MEP for both cis and trans dissociations, their

respective variational transition states differ slightly because of the differences in the ground

state energies and partition functions of the two ONNH isomers.


222

-70

-60

-50

-40

-30

-20

-10

0

1.25 1.3 1.35 1.4 1.45 1.5 1.55 1.6 1.65 1.7 1.75 1.8 1.85 1.9 1.95R (N-N) /Å

E /

kca

l mo

l -1

cis - 2A'

trans - 2A'

cis - 2A"

trans - 2A"

trans - 2A' with nosymmetry constraints

cis - 2A′

trans - 2A′

cis - 2A″

trans - 2A″

trans - 2A′

Figure 6.6 B3LYP/6-31G(2df,p) 2A′ and 2A″ PES’s for ONNH → NO + NH showing the

stretching of the N-N bond in the cis- and trans-isomers and in the non-planar reaction path

for 2A′ trans-ONNH with no symmetry constraints. (Energy relative to NO + NH.)

15

20

25

30

35

40

45

50

-20 0 20 40 60 80 100 120 140 160 180 200dihedral angle /degrees

E /

kca

l mo

l -1

1.75 1.65

1.60 1.55

1.52 1.442A′

2A′

2A′

2A′

2A′

2A′

2A′

2A′

2A′

2A″

2A″

2A″

2A″

cis -ONNH trans -ONNH

Figure 6.7 B3LYP/6-31G(2df,p) torsional potentials for ONNH at various R(N-N) (in Å)

distances. (Energy relative to 2A′ trans-ONNH.)


223

ONHN (2A′′′′) →→→→ HNO (1A′′′′) + N (4S). This reaction (Figure 6.3) is complicated by a change

in spin multiplicity from doublet to quartet as the N-N bond breaks. By mapping the

minimum energy paths on both the doublet and quartet surfaces as a function of N-N distance,

it was found that the intersection occurs at a distance of approximately 1.94 Å (Figure 6.8).

The variational transition states for the dissociation at all temperatures are found to occur on

the (barrierless) quartet surface at N-N distances of ~ 2.1 Å. We expect that the intersystem

crossing will be substantially faster than the classical dissociation, hence the rate of this

reaction was calculated using variational transition state and RRKM theory, utilising the

transition state structures identified on the quartet surface.

-80

-70

-60

-50

-40

-30

-20

-10

0

10

1.2 1.4 1.6 1.8 2 2.2

R (NN) /Å

E /

kca

l mo

l -1

doublet surface

quartet surface

Figure 6.8 B3LYP/6-31G(2df,p) PES for ONHN → HNO + N showing the doublet and

quartet surfaces. (Energy relative to HNO + N.)

cis- and trans-ONNH (2A′′′′) →→→→ N2O (1ΣΣΣΣ) + H (2S). The dissociation of the cis isomer takes

place over a barrier of 24.4 kcal mol−1 (6.1 kcal mol−1 above products) as shown in Figure

6.2. The B3LYP calculations reveal that in this transition state, labelled ONN-H, the ONN

moiety is near-linear (A(NNO) = 173°) and the N-H separation is 1.66 Å. Attempts to find an

analogous transition state on the trans surface were, however, unsuccessful as calculations


224

converged either to the cis transition state or to a second order saddle point. Mapping the

minimum energy path as a function of N-H distance for trans-ONNH revealed a monotonic

increase in the energy, significantly exceeding the barrier height to ONN-H in the region of

R(N-H) ≈ 1.50 Å; at this point, however, the NNO angle of 156° is only 24° larger than at

equilibrium. Stretching the N-H bond further results in a rapid increase in the NNO angle and

collapse onto the cis surface (Figure 6.9). Further exploration of the potential energy surface

revealed that there is a family of low energy (~ 5 – 10 kcal mol−1) trans to cis isomerisation

pathways that occur via the linearisation of the NNO moiety such that the maximum energy is

below the energy of ONN-H. In summary, therefore, both cis- and trans-ONNH dissociate to

N2O + H via a common transition state as shown in Figures 6.1 and 6.2, with the

understanding that the trans to cis isomerisation is part of the overall mechanism.

-35

-30

-25

-20

-15

-10

-5

0

5

10

15

1 1.2 1.4 1.6 1.8 2

R (N-H) /Å

E /

kca

l mo

l -1

trans-ONNH

cis-ONNH

Figure 6.9 B3LYP/6-31G(2df,p) PES for ONNH → N2O + H. (Energy relative to N2O + H.)

ONHN (2A′′′′) →→→→ N2O (1ΣΣΣΣ) + H (2S). As shown in Figure 6.3, this reaction proceeds via a

transition state (denoted ON2-H) with a critical enthalpy of 21.7 kcal mol−1 and an

exothermicity of 1.4 kcal mol−1 at 298 K.


225

cis-ONNH (2A′′′′) →→→→ N2 (1ΣΣΣΣg) + OH (2ΠΠΠΠ). This reaction (Figure 6.1 and 6.2) occurs via a

cyclic NNOH transition state (designated NNOHsq) followed by dissociation to N2 + OH.

Given the geometry of this transition state, the reaction can only proceed from the cis form of

ONNH. The computed reaction barrier is 34.5 kcal mol−1, that is, ~ 10 kcal mol−1 higher than

for the N2O + H channel. However, while ONNH → N2O + H is endothermic, the ONNH →

N2 + OH reaction is highly exothermic (by 45.1 kcal mol−1).

ONHN (2A′′′′) →→→→ N2 (1ΣΣΣΣg) + OH (2ΠΠΠΠ). This reaction proceeds by a 1,2 hydrogen shift from the

central nitrogen to the oxygen, yielding a cyclic transition state, followed by decomposition

into N2 + OH (Figure 6.3). The transition state has been labelled NNOHtr; it is 29.4 kcal

mol−1 higher in energy than ONHN and 91.4 kcal mol−1 higher than the dissociated products.

NNH (2A′′′′) + O (3P) →→→→ N2 (1ΣΣΣΣg) + OH (2ΠΠΠΠ). This reaction represents the direct abstraction of

the hydrogen of NNH by an oxygen atom. Somewhat surprisingly, no barrier was found for

this reaction. This is likely to be a consequence of the very weak (breaking) N-H bond and the

very strong interaction between the hydrogen and oxygen atoms, which is attractive at all O-H

separations.

trans-ONNH (2A′′′′) →→→→ ONHN (2A′′′′). These two intermediates can interconvert via a 1,2

hydrogen shift, although the barrier is rather high at 53.6 kcal mol−1 above trans-ONNH and

29.3 kcal mol−1 above ONHN (Figure 6.3). We also considered the possibility that this

isomerisation could occur via a 1,2 oxygen shift. Although a transition state was found for

this process using B3LYP/6-31G(2df,p), the subsequent RCCSD(T) calculations using this

geometry showed large values of the 1τ diagnostic and are therefore judged to be unreliable.

Similarly, the corresponding higher level calculations which make up G3X were also not

accepted as reliable due to the presence of significant spin contamination. As the energy of

this transition state is ~ 28 kcal mol−1 higher than the analogous H transfer transition state at

the B3LYP level of theory, this process was not investigated any further.


226

6.3.3 Kinetic Parameters

For the three reaction potential energy surfaces (viz. those shown in Figures 6.1 to 6.3)

chemical activation simulations were carried out over the temperature range of 1000 – 2600 K

and at pressures ranging from 1 to 10000 Torr using the MultiWell code. This temperature

range spans the temperatures of relevance in flame studies. For each potential energy surface

there are four barrierless reactions. These are the reverse fission reactions, adduct → NNH +

O together with cis-ONNH → NO + NH, trans-ONNH → NO + NH and ONHN → HNO +

N. For each of these reactions a variational transition state (VTS) was located at temperatures

of 1000 K, 1500 K, 2000 K and at a higher temperature (2500 K except for trans-ONNH →

NO + NH and ONHN→ O + NNH which were evaluated at 2400 K and 2200 K respectively).

The VTS evaluated at 1000 K was used in the MultiWell modelling for temperatures 1000

and 1200 K, the VTS at 1500 K used for temperatures 1400 and 1600 K, the VTS at 2000 K

for temperatures 1800, 2000 and 2200 K and the high temperature VTS for 2400 and 2600 K.

Pressure-dependent overall rate coefficients to the four product channels were calculated

using Equations (6.14) and (6.15).

No stabilisation of adducts or intermediates was found at any temperature or pressure in the

studied range. Rate coefficients derived for individual reaction channels for a pressure of 1

atm are shown in Table 6.6. The rate coefficients did not show any significant pressure

dependence between 1 and 10000 Torr and were not strongly dependent on temperature. The

summed contribution to each channel from the two surfaces can be fitted by a modified

Arrhenius expression and these rate coefficients are given in Table 6.7. From Tables 6.6 and

6.7 we see that the major product channel is to N2O + H, with all three surfaces contributing

an approximately equal amount of reaction flux. The main contribution to the N2 + OH

channel is through the intermediate ONHN, with a somewhat smaller contribution arising

from the cis-ONNH. Nearly all flux to NO + NH is through the cis- and trans- adducts. The

endothermic reaction to HNO + N is only a very minor pathway and nearly all reaction flux is

via the ONHN intermediate. A significant additional contribution to the production of N2 +

OH can also arise from the barrierless direct abstraction reaction for NNH + O.

Chapter 6: N

NH

+ O

in NO

Production

227

Table 6.6 Rate coefficients (cm3 mol−1 s−1) for NNH + O → product channels shown via adducts cis-ONNH, trans-ONNH and ONHN at 1

atm pressure.

via adduct

cis-ONNH trans-ONNH

T/K NO + NH N2O + H N2 + OH HNO + N NO + NH N2O + H N2 + OH HNO + N

1000 6.51×1012 1.64×1013 3.19×1012 9.45×1012 2.17×1013 8.36×1011

1200 7.23×1012 1.56×1013 3.09×1012 9.35×1012 1.82×1013 6.33×1011

1400 7.07×1012 1.56×1013 3.18×1012 7.65×1012 1.90×1013 8.13×1011

1600 7.88×1012 1.52×1013 3.19×1012 7.60×1012 1.66×1013 6.44×1011

1800 7.62×1012 1.69×1013 3.55×1012 7.55×1012 1.69×1013 6.07×1011

2000 8.16×1012 1.65×1013 3.50×1012 8.09×1012 1.64×1013 5.59×1011

2200 8.84×1012 1.60×1013 3.50×1012 7.45×1012 1.35×1013 4.62×1011

2400 9.40×1012 1.63×1013 3.58×1012 7.15×1012 1.32×1013 4.89×1011

2600 9.77×1012 1.57×1013 3.50×1012 7.05×1012 1.19×1013 4.50×1011 2×109

Chapter 6: N

NH

+ O

in NO

Production

228


via adduct

ONHN

T/K NO + NH N2O + H N2 + OH HNO + N

1000 3.61×1011 1.94×1013 6.55×1012

1200 2.52×1011 1.99×1013 6.13×1012

1400 3.00×1011 1.86×1013 5.88×1012

1600 5.07×1011 1.70×1013 5.89×1012 −3×109

1800 5.06×1011 2.10×1013 7.22×1012 −7×109

2000 5.40×1011 1.83×1013 6.41×1012 −6×109

2200 5.84×1011 1.60×1013 6.12×1012 12×109

2400 5.52×1011 1.63×1013 6.10×1012 20×109

2600 5.27×1011 1.43×1013 5.48×1012 16×109


229

Table 6.7 Modified Arrhenius parameters for NNH + O → products via adducts cis-ONNH,

trans-ONNH and ONHN ( ( )expnak AT E RT= − ).

Reactions A/cm3 mol−1 s−1 n Ea/cal mol−1

NNH + O → NO + NH 7.80×1010 0.642 −1830.

NNH + O → N2O + H 2.40×1016 −0.765 1540.

NNH + O → N2 + OH 2.57×1010 0.702 −2320.

NNH + O → HNO + N 6.2×10−7 4.84 0.

NNH + O → N2 + OH a 3.00×1013 0 0.

a Direct abstraction reaction via abstraction transition state.

Our branching ratios into the three principal reaction channels are very different from those

estimated by Bozzelli and Dean.2 In their QRRK analysis they only considered a single

ONNH adduct. They also did not consider the ONHN adduct which we have discovered in the

present work. The ONHN well is considerably shallower than that for cis- or trans-ONNH

and significant flux flows through the ONHN adduct both to N2O + H and to N2 + OH (but

not to NO + NH).

6.3.4 Comparison with Experiment

As mentioned in the Introduction (Section 6.1), Hayhurst and Hutchinson5 reported a value

for 4 ,3Pk K from which they then estimated a rate coefficient, 4k , for reaction to NO + NH.

Their method involved the assumption that every NH radical produced by Reaction (6.4)

rapidly reacts to yield a second NO molecule. For fuel-rich flames of CH4 / O2 / N2 and of H2

/ O2 / N2 the above assumption leads to the equation

4 ,3 42 H

[NO] 1 1

2[N ][O]P

dk K k

dt x

= ⋅ − ⋅

(6.17)

where Hx is the mole fraction of H in the burnt gas. To obtain Hx , Hayhurst and Hutchinson

measured OH and temperature profiles in the burnt gas. There are, however, very significant


230

random errors in their data, ranging from nearly 3 orders of magnitude at 2500 K to nearly an

order of magnitude at 1800 K. If we fit our computed values of 4k from the MultiWell

simulations and ,3PK derived from our thermochemical calculations we obtain

( )74 ,3 3.7 10 exp 2800Pk K T= × − cm3 mol−1 s−1. Comparing with Hayhurst and Hutchinson’s

value of ( )94 ,3 1.4 10 exp 2760Pk K T= × − cm3 mol−1 s−1 shows that our value at 2000 K is

over an order of magnitude lower than theirs but probably still within the considerable

random error in their data.

6.3.5 Kinetic Modelling

As discussed earlier, detailed chemical reaction models, such as the two formulations of

GRIMech3,4 (containing the rate coefficient for 4k estimated by Bozzelli and Dean2), have

recently been found to overestimate the level of NO produced by combustion systems. We

have chosen to use the GRIMech 3.0 model with our new NNH thermochemistry and kinetics.

The value of 4k in this model was altered to the value given in Table 6.7 and the other three

addition and decomposition reaction channels also included together with the direct

abstraction of H by O atoms. This modified mechanism has been used to model two series of

data54 from a completely stirred reactor – a fuel lean methane / air combustion and a lean

combustion of CO / H2 / air at residence times, τ, between about 3 to 4 ms and equivalence

ratios, φ, between 0.5 and 0.6 approximately. The first of these cases has been used to

benchmark the performance of GRIMech.4 Figure 6.10 compares the performance of our

modified kinetic model with that of the original GRIMech 3.0 formulation and experimental

NO profiles.

As can be seen from Figure 6.10(a) both the original formulation of GRIMech 3.0 and our

modified version with new NNH thermochemistry and kinetics reproduce the experimental

NO data from CH4 / air satisfactorily although our model gives a closer fit to experiment.

With the runs in CO / H2 / air (Figure 6.10(b)), however, significantly poorer performance

occurs when using the original GRIMech 3.0 model. Whereas our present model gives a good

fit to experiment, GRIMech 3.0 overestimates the level of NO by nearly a factor of two. To


231

ascertain the reason for this difference in performance we have carried out reaction path

analyses on both kinetic models.

1650 1700 1750 18000

5

10

15

NO

/pp

m

(a)CH4/air

0.512 ≤φ≤ 0.6223.22 ≤τ≤ 3.54 ms

1660 1680 1700 1720T/K

0

10

20

30

40

50

60

70

NO

/pp

m

(b)air/CO = 82/17.4 mol %H2 = 0.69 - 0.25 mol %

3.87 ≤τ≤ 4.03 ms

Figure 6.10 Comparison of NO profiles of combustion in a completely stirred flame reactor.

(a) CH4 / air, (b) H2 / CO / air. Filled circles: Experimental data from Ref. 54. Dashed lines:

predictions using GRIMech 3.0. Full lines: prediction using GRIMech 3.0 with NNH

thermochemistry and kinetics from the present study.


232

We have sought to quantify the contribution that each of the four reaction pathways

(Zel’dovich, Prompt-NO, N2O and NNH + O) makes to NO production by using the

following simplified procedure. In order to determine the contribution of a particular pathway,

that pathway is eliminated from the kinetic model and the modified mechanism is then run to

ascertain the effect of its omission. This is repeated in turn for each pathway. The basic

assumption in this method is that there are no cross correlations between the pathways. It has

been shown, however, that the error resulting form such neglect of cross correlation is of 5%

or less in the total contribution of all pathways when applied to NO profiles in atmospheric

opposed flow methane-air flames.55 In our present path analysis the summation error is

significantly less than 3%.

The contribution of the thermal pathway is assessed by eliminating its initiation reaction,

(6.1). The prompt-NO pathway contribution was also determined by elimination of its

initiation reaction, (6.2). To assess the contribution of the N2O intermediate pathway, it was

necessary to eliminate all reactions involving N2O from the reaction model while the NNH +

O route was quantified by elimination of its initiation reaction, (6.4). This procedure is

similar to that used previously.6, 55

Figures 6.11(a) and (b) compare the contribution of each reaction pathway for our present

model and for the original formulation of GRIMech 3.0 to methane-air studies in a completely

stirred reactor.54 The computed contributions of the N2O intermediate, thermal and prompt-

NO pathways are similar for both models. A larger contribution of the NNH + O pathway is

calculated using GRIMech 3.0. Nevertheless, because the first three pathways all make

significant contributions to the NO profile, the two models both give reasonable reproductions

of the experimental data, although GRIMech 3.0 does somewhat overestimate the NO levels.

It is with the runs in CO / H2 / air (Figure 6.11(c) and (d)), however, that the computed

contributions of the two models radically differ. The prompt-NO pathway makes no

contribution in these runs. Again, the calculated contributions of the N2O and thermal

pathways are quite similar for both models, however GRIMech 3.0 predicts that the NNH + O

pathway will make the greatest contribution to the NO profiles whereas our model, with new

NNH thermochemistry and kinetics, predicts that this pathway makes only a very small

contribution to the total. It is apparent, therefore, that by overestimating the contribution of

the NNH + O pathway GRIMech 3.0 predicts too high a level of NO in this system. In

methane-air combustion, where there are significant contributions from the prompt and


233

thermal routes, the overestimation of the NNH + O pathway is masked. Where the prompt and

thermal pathways are minimised this overestimation becomes obvious. A detailed testing of

our new thermochemistry and kinetics in modelling premixed and opposed flow flames will

be presented elsewhere.56

1650 1700 1750 18000

5

10

15

NO

/pp

m

TotalNNH + ON2O intermediatePrompt-NOThermal

(a)

1650 1700 1750 18000

5

10

15 (b)

1660 1680 1700 1720

T/K

0

10

20

30

40

50

60

70(c)

Prompt-NO

1660 1680 1700 17200

10

20

30

40

50

60

70(d)

Prompt-NO

Figure 6.11 Predictions of the contribution of individual pathways to NO formation.

(a) Present model predictions for the CH4 / air data of Figure 6.10(a).

(b) GRIMECH 3.0 predictions for the CH4 / air data of Figure 6.10(a).

(c) Present model predictions for the H2 / CO / air data of Figure 6.10(b).

(d) GRIMECH 3.0 predictions for the H2 / CO / air data of Figure 6.10(b).


234

6.4 Conclusions

Three reaction potential energy surfaces for NNH + O → products have been investigated by

ab initio quantum chemical calculations. Three adducts, namely trans-ONNH, cis-ONNH and

ONHN, have been identified through which reaction to three exothermic product channels,

NO + NH, N2 + OH, N2O + H, and one endothermic channel, HNO + N, takes place. Rate

coefficients to each reaction channel have been obtained by RRKM analysis. The rate

coefficient at 2000 K to the NO + NH channel is predicted to be approximately a factor of

four lower than had been previously estimated2 (and included in detailed reaction models such

as GRIMech 3.0)4. A new value of 0298f H∆ (NNH) = 60.6 ± 0.5 kcal mol−1 has been obtained

by CCSD(T) calculations which include extrapolation to the complete basis limit. This value,

together with the rate coefficients we have derived for the NNH + O → products reactions,

have been used to modify the GRIMech 3.0 reaction model. Using this new formulation we

could satisfactorily model NO profiles produced in a completely stirred reactor54 from both

methane / air and CO / H2 / air mixtures. Overestimation of NO profiles from the latter

mixtures by GRIMech 3.0 has been shown, by reaction path analysis, to result from too high a

rate coefficient for initiation of the NNH + O pathway. On the basis of the present work we

conclude that this pathway represents a very minor route to NO in most combustion systems.


235

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237








43. "DALTON, an ab initio electronic structure program, Release 1.0 (1997)" written by

T. Helgaker, H. J. Aa. Jensen, P. Joergensen, J. Olsen, K. Ruud, H. Aagren, T.

Andersen, K. L. Bak, V. Bakken, O. Christiansen, P. Dahle, E. K. Dalskov, T.

Enevoldsen, B. Fernandez, H. Heiberg, H. Hettema, D. Jonsson, S. Kirpekar, R.

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Combustion & The 8th Australian Flames Days, Melbourne, December 8-9, 2003.

7 The Enthalpy and Mechanism of the

Photolysis of CFClBr2

Chapter 7

The Enthalpy and

Mechanism of the

Photolysis of CFClBr2

Chapter 7. Photolysis of CFClBr2

240

The work described in this chapter represents a combined experimental and theoretical

investigation into the thermochemistry of the bromomethane CFClBr2. It is being jointly

published in Chemical Physics Letters.1 The experimental work (Section 7.2) was performed

by N. L. Owens, K. Nauta and S. H. Kable and has also been reported in the Honours thesis of

Owens2. The theoretical and experimental aspects of this work are heavily interreliant and

thus the work has been reported here in its entirety.

7.1 Introduction

The role of halons and CFC’s in the chemistry of the atmosphere has been well documented

for many decades. As a general class of molecules, halomethanes absorb ultraviolet light via a

σ*←σ transition, which results in the cleavage of the weakest C−X bond (where X is halogen

or hydrogen). The fate of the atom is very well understood and central to the depletion of

stratospheric ozone. The fate of the halomethyl radical is less well understood, but the

primary reaction is probably O2 addition leading to the formation of halogen-substituted

aldehydes and slow transport back to the troposphere.

Over the past couple of decades, several reports of UV or vacuum-UV excitation of

halomethanes, resulting in cleavage of two C–X bonds, have appeared. Bromo- and

iodomethane species seem to exhibit triple fragmentation pathways when the wavelength is

shorter than 200 nm.3-10 For wavelengths longer than 200 nm, however, only the difluoro-

species, CF2I2, CF2Br2 and CF2BrI have been reported to undergo triple fragmentation. CF2I2

undergoes a single C–I cleavage at longer wavelengths, which is believed to become two

sequential C–I cleavages and finally concerted loss of two I-atoms as the energy of the

dissociating photon is increased.10 Triple fragmentation of CF2Br2 and CF2BrI are also

thought to be stepwise processes.5-9


241

In a series of papers on carbene spectroscopy, Kable and co-workers have created carbenes by

photolysis of suitable halon precursors, for example CFCl from CFClBr2,11 and both CHF and

CFBr from CHFBr2.12 At the time, no attempt was made to elucidate the mechanism of

carbene formation. In this paper we investigate the mechanism by which CFCl is formed from

CFClBr2 (halon-1112).

Interpretation of the results is made more difficult by the absence of thermochemical data

concerning halon-1112. In fact, we could locate thermochemical data for only two

dibromomethane compounds: CF2Br2 13 and CH2Br2

14. The literature thermodynamic data

for carbenes, including CFCl, are also highly varying. The most reliable current values are

probably theoretical values obtained by Dixon et al. for CF2,15,16 CHBr and CBr2

17 and by

Sendt and Bacskay18 for CFCl. Both groups utilised the coupled cluster method with

extrapolation to the complete basis limit and expect their results to be accurate to within ± 4

kJ mol−1. The Gaussian-3 (G3) calculations of Sendt and Bacskay18 for CF2 and CFCl and the

Gaussian-2 results of Cameron and Bacskay19 for CHBr and CBr2 are in good agreement with

these coupled cluster results. More recently, the G3 procedure has been extended to be able to

describe molecules containing atoms from the third row.20 We have therefore calculated bond

energies and heats of formation for a number of molecules containing bromine atoms using

the G3 methodology. In addition to facilitating the interpretation of the experimental data, the

calculations also permit us to test the accuracy of the G3 method for larger bromine

containing molecules, especially since the largest in the G3 test set is CH3Br.

The objectives of the current work are threefold. Firstly we seek to establish whether there is a

triple fragmentation pathway for CFClBr2 following absorption of a single photon at

stratospherically relevant wavelengths. Secondly, we use these data to determine

thermochemical properties for the various species involved, including heats of formation and

bond energies. Finally, we calculate those same thermochemical properties using the new

formulation of G3 theory for third row atoms to verify both the experimental data and the

validity of the theoretical method for larger bromine-containing compounds.


242

7.2 Experimental Methods and Results

7.2.1 Methodology

Details of the experiment can be found in the previous work on CFCl of Guss et al.11 Very

briefly, helium (2 bar) was bubbled through CFClBr2 (l, 0°C) and expanded via a pulsed

nozzle into a vacuum chamber. CFClBr2 was photolysed at the nozzle orifice by a Nd:YAG

pumped OPO laser (Coherent Infinity 40-100 and OPASCAN). The output of the OPO was

varied from 480 to 560 nm and frequency doubled to yield light from 240 to 280 nm. The

ensuing CFCl fragments were probed about 10 mm downstream, approx. 6 µs after the

photolysis pulse, by an excimer pumped dye laser (Lambda Physik Lextra 200 and LPD

3001E, Exalite 398 dye). By allowing the CFCl to cool in the expansion, any effect of

different CFCl product state distributions as a function of photolysis energy is negated. The

trade-off in doing this is that the parent CFClBr2 is fairly warm (we estimate 100-200 K).

Fluorescence from CFCl was imaged onto the slits of a Spex Minimate monochromator with

5 mm slits, which acts like a 20 nm triangular bandpass filter. This filter provides no real

rotational or vibrational resolution for the CFCl fluorescence. The broadband fluorescence

was detected by an EMI 9789QB photomultiplier, the signal passed to an SRS-250 boxcar

averager and finally to an SRS-245 A/D board and a personal computer. The experiment was

timed using the internal variable delays provided by the Infinity laser.

7.2.2 Results

An absorption spectrum of CFClBr2, diluted in air, measured on a Cary 4E spectrometer, is

shown in Figure 7.1 (top). The spectrum shows negligible absorption in the actinic range (λ >

295 nm) and no sharp features, which is typical of halon and CFC species. At least two broad

overlapping features are evident, the one to the red (around 240 nm) forming a pronounced

shoulder on the stronger feature to the blue, which peaks further to the blue than the range of

the spectrometer. By analogy with other CFC’s and halons, these features are likely to be

σ*←σ transitions involving a C–Br bond.


243

240 250 260 270 2800.0

0.2

0.4

0.6

0.8

1.0

LIF

/ A

bsor

ptio

n

Wavelength (nm)

0.0

0.2

0.4

0.6

0.8

1.0 LIF signal Absorption

LIF

Sig

nal (

arb.

)

210 240 270 3000.0

0.5

1.0

1.5

2.0

2.5

Abs

orpt

ion

(arb

.)

Figure 7.1 (top) CFClBr2 absorption spectrum; (centre) fluorescence excitation spectrum of

CFClBr2; (bottom) ratio of fluorescence signal to absorption signal.


244

The photolysis of CFClBr2 is known to result in the formation of CFCl.11 Several LIF spectra

of CFCl produced in this way are shown in Figure 7.2. The origin region is shown, which is

quite weak due to poor Franck-Condon overlap with the ground state. However, it is relatively

uncongested, least affected by chlorine isotopic features, and also has well-assigned hot bands

and so we have concentrated on this region. The spectral features pertinent to this work are

the origin transition and a variety of hot-band transitions emanating from the υ2 = 1 and υ3 = 1

levels (ν2 = bend, ν3 = C–Cl stretch). The main rotational sub-branches are indicated by a

comb. The main feature is the strong rQ0 sub-branch, flanked by the rQ1 to shorter wavelength

and the pQ1 to longer wavelength. The reader is referred to previous work on CFCl

spectroscopy11 for any further details.

The mechanism of CFCl production from CFClBr2 was not explored previously. The first test

that was performed was to establish the dependence of the CFCl signal on pump laser power.

The LIF signal from the central peak of the 000 transition was monitored as the pump power

was varied randomly. The resulting dependence is shown in Figure 7.3 and indicates that the

observed signal varies linearly with laser power.

The LIF intensity from 000 transition was also monitored as the pump wavelength was

changed randomly between 240 and 275 nm as shown in Figure 7.1 (centre). The data points

in the figure arise from about 1000 laser shots and were take taken every 2 nm below 260 nm

and then every 1 nm above 260 nm. For comparison, the absorption spectrum is also plotted

on the same axes, arbitrarily normalised to the value at 240 nm. The shapes of the two spectra

are similar, but the excitation spectrum rises more rapidly towards 240 nm than does the

absorption spectrum. Additionally, the excitation spectrum has reached zero (no CFCl

observed) at 275 nm while the absorption spectrum is still about 10% of the 240 nm value

and, as shown in the top panel, continues well beyond 280 nm. The actual wavelength at

which the excitation spectrum reaches zero is rather difficult to determine in this spectrum as

the data converge asymptotically to zero over about 5 nm.

The difference between the excitation and absorption spectrum is accentuated in Figure 7.1

(bottom) by plotting the ratio of the excitation to absorption intensity. Quite clearly, relative

to the absorption spectrum, the intensity of the excitation spectrum drops consistently from

240 to 260 nm and then drops quite sharply towards 275 nm. The line is a heavily smoothed


245

395 396 397

22

03

0

1

31

1

00

0

21

1

x15

x4

240 nm

260 nm

270 nm

Wavelength (nm)

Figure 7.2 CFCl fluorescence excitation spectra in the origin region following dissociation in

CFClBr2 at 240, 260 and 270 nm.


246

0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0

LIF

sig

nal f

rom

CF

Cl (

arb

)

Pump Power (arb)

Figure 7.3 Power dependence of the CFCl signal following dissociation of CFClBr2 at 240

nm.

spline fit, which cuts the abscissa at 274 nm and we estimate the threshold for production of

CFCl from CFClBr2 to be 274 ± 2 nm, which corresponds to a photon energy of 36,500 ± 280

cm−1.

More information about the threshold can be obtained by examining the CFCl spectrum as a

function of wavelength. Three such spectra are shown in Figure 7.2, obtained following

excitation of CFClBr2 at 240, 260 and 270 nm. The 240 nm spectrum was discussed earlier

and shows features arising from population in υ = 0, υ2 = 1 and υ3 = 1, which have vibrational

energies of 0, 447 and 753 cm−1 respectively.11 Although the 112 transition is strongest, the

Franck-Condon factor for the 112 transition is about 15× larger than the 0

00 transition11 so the

population in the υ2 = 1 level is actually about an order of magnitude less than υ = 0. The 113

Franck-Condon factor is similar to 000 11 and therefore the υ3 = 1 population is also much less

than υ = 0.


247

At 260 nm the relative population in these vibrational levels has not changed much, although

the overall signal is reduced by a factor of about four. The 270 nm spectrum, however, shows

CFCl population only in the υ = 0 and υ2 = 1 levels, with the 000 transition dominant. The

overall signal is now about 15 times reduced in comparison with the 240 nm spectrum.

The data above are summarised in Figure 7.4. A photon energy of 36,360 cm−1 (275 nm) does

not lead to detectable CFCl. At 37,037 cm−1 (270 nm) CFCl in the υ = 0 and υ2 = 1 states are

produced and at 38,460 cm−1 (260 nm), υ = 0, υ2 = 1 and υ3 = 1 levels are populated. The

energy level diagram shows that all three observations are satisfied if the combined

dissociation energy for both C–Br bonds is 36,475 ± 120 cm−1. This is in good agreement with

the threshold from the excitation spectrum of Figure 7.1, which was 36,500 ± 280 cm−1.

Figure 7.4 Energy level diagram of CFClBr2 and various reaction products calculated in this

work.


248

There are two concerns at this stage about proclaiming this to be an unambiguous

experimental measurement of the threshold to triple fragmentation of CFClBr2: i) the

observation of a linear power dependence for CFCl production is a necessary but not

sufficient condition for the process to be single photon, and ii) we have not addressed the

possibility that CFCl is produced in concert with the Br2 molecule, rather than two Br atoms.

These issues could be resolved by detailed consideration of the thermochemistry of the

appropriate processes. In the absence of reliable heats of formation for CFClBr2 or CFClBr

we have carried out high level ab initio calculations of these properties, which are presented

in the next section. The theoretical and experimental data are tied together to resolve these

issues in the Discussion.

7.3 Theoretical Methods and Results

7.3.1 Methodology

The G3 procedure for molecules with third row atoms has only been developed very

recently.20 The method is essentially the same as that initially proposed for first and second

row atoms21, where the energy of a hypothetical QCISD(T) calculation with a large basis set

is approximated by performing such a calculation with a significantly smaller basis set,

followed by corrections for enlargement of the basis set, which are computed at lower levels

of theory, namely MP2 and MP4. The G3 energy can be summarised as

[ ] [ ][ ] [ ]{ }[ ] [ ]{ }[ ] [ ]

[ ] [ ]

0 G3 QCISD(T)/6-31G( )



MP2(Full)/GTLarge MP2/6-31G(2 , )(G3L

MP2 6-31 G MP2 6-31G

E E d

E d E d

E df p E d df p

E E df p

E d E d

=

+ + − +

+ −

− + − / + ( ) + / ( )

ZPE SO hlc

arge correction)

E E E+ ∆ + ∆ + ∆

(7.1)


249

where the appropriate corrections due to successive basis set enlargement are evaluated at the

MP4 and MP2 levels of theory. ZPEE∆ is the zero point vibrational energy, SOE∆ denotes

spin-orbit coupling corrections and hlcE∆ is an (empirical) higher level correction.

The geometry and zero point vibrational energy are determined at the MP2(Full)/6-31G(d)

and HF/6-31G(d) levels of theory respectively. Note that the G3 formulation for third row

atoms recommends the use of a new set of basis functions20 which are based on the 6-31G

basis of Rassolov et al.22 As for first and second row atoms, the G3 calculations on molecules

containing third row atoms are performed with six d and seven f functions, except in the case

of G3Large calculations which utilise the spherical harmonic representation for all orbitals,

viz. (5d, 7f). Furthermore, following the work of Duke and Radom23, who noted an

improvement in the accuracy of the G2 procedure when the 3d electrons of third row atoms

were unfrozen to correlation, the G3 procedure also prescribes the inclusion of the 3d orbitals

of third row atoms in the valence space.

Once G3 absolute energies have been calculated, atomisation energies and thus heats of

formation can be evaluated using the relevant (experimental) atomic data. Heats of formation

can also be determined on the basis of reaction enthalpies, preferably those of isodesmic

reactions where the number of bonds of each type are conserved in the reaction. Errors in the

theoretical description of a particular atom or bond are then expected to cancel, resulting

potentially in a relatively accurate estimate of the heat of the reaction. Application of Hess’

Law then allows the determination of the heat of formation of a given species, provided

accurate (experimental) heats of formation are available for all other species in the reaction.

Isodesmic reaction schemes have been used in this work to confirm the reliability of the heats

of formation obtained from G3 atomisation energies.

The quantum chemical computations were carried out using the Gaussian98 suite of

programs24 on DEC alpha 600/5/333 and COMPAQ XP100/500 workstations of the

Theoretical Chemistry group at the University of Sydney and on the COMPAQ AlphaServer

SC system of the Australian Partnership for Advanced Computing National Facility at the

National Supercomputing Centre, ANU, Canberra


250

7.3.2 Results

The computed G3 total energies, atomisation energies and heats of formation at 0 and 298K

are reported in Table 7.1. The absolute energies of Br2, Br and CH3Br have been reported

earlier by Curtiss et al.20 As they did not report the atomisation energies or heats of formation

of Br and CH3Br, we have generated these additional data and include them here for

completeness. The G3 results for CFCl, CH4, CH2F2, CH3Cl, CH2Cl2 and CF4 have also been

published previously18,21 but are quoted here because of their importance in the decomposition

reactions of CFClBr2 and in the isodesmic reaction schemes.

Table 7.1 Energies, atomisation energies and heats of formation (at 0 and 298K) (kJ mol−1

unless otherwise noted).

−

Species 0E /Eh AE00f H∆ a 0

298f H∆

G3 G3 G3 G3/AE a G3/ID b Experimental

CFClBr2 −5745.04026 1319.2 −187.4 −188.2 −190.6 −184 ± 5

CFClBr −3171.42502 1062.5 −42.5 −−42.9

CFCl −−597.83777 −879.2 28.9 −−29.8 31 ± 13c

Br2 −5147.10746 −190.4 33.3 −−34.6 30.91 ± 0.11d

Br −2573.51747 111.8 111.87 ± 0.12d

CH3Br −2613.41950 1500.0 −28.9 −−36.1 −34.2 ± 0.8e

−36 ± 1f

CH4 −−40.45762 1643.3 −68.0 −−75.9 −74.87d

CH2F2 −238.86226 1743.6 −445.8 −453.4 −450.66d

CH3Cl −499.91301 1552.5 −73.6 −−81.5 −83.68d

CH2Cl2 −959.37121 1469.2 −86.7 −−93.5 −95.52d

CF4 −437.30780 1951.4 −931.1 −936.8 −933.04 ± 0.7g

a Heat of formation calculated from atomisation energies.b Heat of formation calculated by isodesmic schemes.c Ref. 24.25

d Ref. 25.26

e Ref. 26.27

f Ref. 27.28

g Ref. 28.29


251

Table 7.2 lists the various isodesmic reactions which were employed to determine the heat of

formation of CFClBr2 from the G3 reaction energies, along with the resulting enthalpies of

formation. The spread of values about the mean is less than 4 kJ mol−1, suggesting chemical

accuracy in the results. The average value obtained by this approach differs by only 2.4 kJ

mol−1 from the heats of formation obtained from the G3 atomisation energies, which is well

within the expected uncertainties of the G3 calculations. This suggests that the G3 procedure

is indeed capable of producing accurate heats of formation from atomisation energies for

larger third row containing molecules and therefore the use of isodesmic schemes is not

warranted.

Table 7.2 Isodesmic reaction schemes and resulting G3 enthalpies of formation for CFClBr2

(kJ mol−1).

Reaction0298f H∆

2CFClBr2 + 4CH4 → CH2F2 + CH2Cl2 + 4CH3Br −190.5

2CFClBr2 + 5CH4 → CH2F2 + 2CH3Cl + 4CH3Br −192.2

2CFClBr2 + 3CH4 + CH2F2 → CH2Cl2 + 4CH3Br + CF4 −190.9

2CFClBr2 + 3CH4 + 2CH3Cl → 2CH2Cl2 + 4CH3Br + CH2F2 −188.9

Average for CFClBr2 −190.6

7.4 Discussion

The experimental data demonstrate conclusively that CFCl is a by-product of CFClBr2

photolysis and that the process is likely to involve the absorption of only one photon. The

threshold for CFCl production was found to be 436 ± 2 kJ mol−1 or 36,460 ± 150 cm−1. CFCl

can be formed from CFClBr2 by two different reactions:

2 2CFClBr CFCl + Brhν→ Reaction 7.1

2CFClBr CFCl + 2 Brhν→ Reaction 7.2


252

The observed appearance threshold for CFCl could correspond to either of these reactions. A

schematic of the chemical energies involved in each process in shown in Figure 7.4. The left

side of the figure shows the Br2 production, while the right hand side shows sequential or

concerted elimination of two Br atoms. To help distinguish between these two reactions we

turn to the ab initio results from above.

The G3 method has provided estimates of the heats of formation ( 0f H∆ ) for all species

involved in this work, see Table 7.1. (The heats of formation of Br and Br2 are well known.26)

These thermochemical data were used to evaluate the reaction energies for various reactions,

as shown in Table 7.3. The calculated energy required for Reaction 7.1 is 250 ± 5 kJ mol−1

and for Reaction 7.2 is 440 ± 5 kJ mol−1. The value for Reaction 7.2 is comfortably within the

mutual error limits for the theoretical and experimental estimates. This does not absolutely

discount Reaction 7.1 because as a three-centre elimination it would probably require a

barrier. However it would seem fortuitous that the barrier height is exactly the same as the

thermochemical threshold for the other channel. Therefore we have rejected the three-centre

elimination pathway as much less likely than the triple fragmentation pathway. These

conclusions are also in accord with the analogous findings for the CF2Br2 8 and CF2BrI 5

molecules.

Table 7.3 Reaction energies for possible decomposition pathways for CFClBr2 (kJ mol−1).

Since 0f H∆ is known for CFCl and Br we can use the threshold energy to determine an

experimental 00f H∆ for CFClBr2. The value so determined is −183.4 ± 5 kJ mol−1 (using the

G3 value for CFCl as the experimental value is quite uncertain). The G3 method also provides

heat capacity correction factors between 0 K and 298 K. For CFClBr2 this difference is 0.8 kJ

Reaction r E∆ (0K)

G3

0298r H∆

G3

CFClBr2 → CFCl + 2Br 439.9 441.7

CFClBr2 → CFClBr + Br 256.7 257.2

CFClBr → CFCl + Br 183.2 184.5

CFClBr2 → CFCl + Br2 249.5 252.6


253

mol−1, which provides an estimate for CFClBr2 of 0298f H∆ = −184.2 ± 5 kJ mol−1. The G3

values are in excellent agreement with this experimental heat of formation.

The triple fragmentation reactions of several halomethane species containing Br and I have

been reported previously. Most of the experiments have been carried out in the far or vacuum

ultraviolet region (λ < 200 nm), which typically excites the second absorption band (see

Figure 7.1 for this band in CFClBr2). For example CF2BrCl 3, CF2BrI 5, CH2BrI 4, CF2Br2 4-9

and CF2I2 10 have all been reported to exhibit competing chemical channels, cleaving one or

other of the C−Br or C−I bonds (if different), and for cleavage of both bonds. Chloro- and

fluoromethane species (containing no Br or I), conversely, do not show triple fragmentation

(involving C-Cl cleavage), at least down to 193 nm.

The mechanism for the triple fragmentation of halomethanes is not assured with some studies

favouring a concerted triple whammy, while others favour the formation of a hot halomethyl

radical intermediate, followed by spontaneous dissociation into the carbene. The problem is

that, although the spectra of these species appear simple, there are actually several electronic

transitions that contribute to each “peak” in the spectrum. To our knowledge, there has been

no definitive theoretical study of a bromo- or iodomethane species where one of these states

has been shown to correlate with concerted triple fragmentation. In the absence of theoretical

assistance experimentalists rely on the nuances of the atom recoil energy distributions to try

and decide between concerted and stepwise mechanisms.

In the first electronic absorption band of bromomethane species there is fairly uniform

agreement that a σ*←σ transition localised on a C-Br bond is excited, which leads to

formation of Br plus a halomethyl radical with quantum yields approaching unity. Only two

other bromomethanes have been reported to undergo triple fragmentation within this band,

namely CF2BrI 4 and CF2Br2 7,9. The work on CF2Br2 has been quite extensive over a couple

of decades and the consensus now seems to be that CF2Br2 undergoes triple fragmentation for

λ < 260 nm. At these wavelengths the primary process is still the breaking of one C−Br bond.

The resultant CF2Br radicals are born with a wide range of internal energy, some having

sufficient energy for spontaneous decomposition into CF2 + Br. The experimental data for

CFClBr2 bear a striking similarity to CF2Br2, which leads us to suspect that the mechanism of

triple fragmentation is probably the same; that is, direct loss of one Br atom, followed by

spontaneous loss of the second Br atom from a hot intermediate CFClBr radical.


254

7.5 Conclusion

In this work we have established that CFCl is formed from single photon dissociation of

CFClBr2 for wavelengths shorter than 274 nm. We attribute this to a thermochemical

threshold, and hence determine the energy required to break both C-Br bonds to be 436 ± 2 kJ

mol−1. Ab initio calculations using the G3 method confirm that two bromine atoms are the

partners in this reaction. The heat of formation of CFClBr2 is inferred from these experiments

to be 0298f H∆ = −184 ± 5 kJ mol−1, in excellent agreement with the computed G3 value of

–188 ± 5 kJ mol−1. These ab initio calculations are the first reports of dibromo species at the

G3 level. Comparison of the heat of formation by G3 calculation with that calculated as the

average value from a set of isodesmic reactions shows agreement to within 2.4 kJ mol−1,

thereby confirming the reliability of the G3 method for these species.


255

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17. D. A. Dixon, W. A. de Jong, K. A. Peterson and J. S. Francisco, J. Phys. Chem. A,

2002, 106, 4725.


19. M. R. Cameron and G. B. Bacskay, Chem. Phys., 2000, 104, 11202.

20. L. A. Curtiss, P. C. Redfern, V. Rassolov, G. Kedziora and J. A. Pople, J. Chem.

Phys., 2001, 114, 9287.


Phys., 1998, 109, 7764.

22. V. A. Rassolov, M. A. Ratner, J. A. Pople, P. C. Redfern and L. A. Curtiss, J. Comput.

Chem., 2001, 22, 976.

23. B. J. Duke and L. Radom, J. Chem. Phys., 1998, 109, 3352.




256










25. J. C. Poutsma, J. A. Paulino and R. R. Squires, J. Phys. Chem., 1997, 101, 5327.


27. K. C. Ferguson and E. Whittle, J. Chem. Soc., Faraday Trans. 1, 1972, 68, 295.

28. G. P. Adams, A. S. Carson and P. G. Laye, Trans. Faraday Soc., 1966, 62, 1447.

29. L. V. Gurvich, I. V. Veyts and C. B. Alcock, Thermodynamic Properties of Individual

Substances; CRC Press: Boca Raton, Florida, 1994.

8 The Molecular Structure and Intra- and Inter-Molecular Bonding of PSOrn

Chapter 8

The Molecular

Structure and Intra-

and Inter-Molecular

Bonding of PSOrn

Chapter 8. Structure and Bonding in PSOrn

258

8.1 Introduction

Nδ-(N′-Sulfodiaminophosphinyl)-L-ornithine (PSOrn) is the active component of

phaseolotoxin, which in turn is derived from a toxin produced by Pseudomonas syringae pv.

phaseolicola. PSOrn has been found to bind to the E. coli enzyme ornithine

transcarbamoylase (OTCase) with a dissociation constant of 1.6 × 10−12 M at 37°C, pH = 8.1

OTCase catalyses the reaction of carbamoyl phosphate with L-ornithine, forming L-citrulline

and phosphate and forms part of the urea cycle for mammals. It is also involved in the

synthesis of arginine by plants and bacteria. The binding of PSOrn to OTCase irreversibly

halts this catalysis, resulting in cell death.

The X-ray crystal structure of PSOrn within the enzyme has been determined at 1.8 Å

resolution by Langley et al.1 The resulting heavy atom backbone of this molecule is shown in

Figure 8.1. Since crystal structures refined at this resolution define only the positions of non-

hydrogen atoms, the chirality, tautomeric form and the ionisation state of the bound inhibitor

could at best be inferred from the structural data using chemical considerations. This study

aims to investigate, using the methods of computational quantum chemistry, the chemical

identity of PSOrn in both free and bound states, determine their relative stabilities and clarify

the nature of bonding both within the inhibitor and between the enzyme and inhibitor.

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O

N2

C

S

N

N1

P

O2

O3

O1

O4

C2 N3

O5

C1 C3

C4

Figure 8.1 Structure of PSOrn within OTC as determined by X-ray diffraction.1


259

Langley et al.1 proposed that PSOrn acts as a transition state analogue within the enzyme; that

is, it adopts the same conformation and forms the same type of hydrogen bonds with the

enzyme as in the suggested carbamoyl phosphate + L-ornithine transition state. They

examined the self-consistency of possible hydrogen bonding networks at the enzyme active

site and concluded that the most likely chemical form of the bound inhibitor is a doubly

ionised “imino” tautomer and that the phosphotriamide is the R enantiomer as shown in

Figure 8.2 (along with the substrates, proposed transition states and a more conventional

amino form of the inhibitor). The observed potent inhibitory activity of PSOrn could thus be

rationalised because this species is a structural mimic of the substrates in a proposed transition

state.

Figure 8.2 Imino and amino tautomers of PSOrn as transition state analogues for the OTC

catalysed reaction, as proposed by Langley et al.1

Conventional chemical wisdom suggests that free PSOrn would be more stable in the amino

form, that is, with the P-N-S nitrogen protonated. Moreover, such protonation would not

necessarily preclude hydrogen bond donation to this nitrogen, given the presence of a lone

pair on N. The current investigations were undertaken with the primary aim of elucidating the

nature of the interaction between PSOrn and some of the important enzyme residues, in

particular the hydrogen bonding of an arginine to the P-N-S nitrogen of the inhibitor. To

reduce computational costs the (CH2)3CH(NH3+)(CO2

−) side-chain of PSOrn has been

replaced by a methyl group. The relative stabilities of several tautomeric amino and imino

forms of the resulting neutral model compound are investigated by density functional theory,

NH3+

C

(CH2)3

N HH

CH2N

OO

P

O-

O-

O

NH3+

C

(CH2)3

N+ HH

CH2N

O-O

P

O-

O-

O

NH3+

C

(CH2)3

N H

PH2N

ON

S

O

O-

O

Substrates for ReactionCatalysed by OTC

Proposed TransitionState

Proposed IminoStructure for

Phaseolotoxin

CO2-H CO2

-H H CO2-

NH3+

C

(CH2)3

N H

PH2N

ONH

S

O

O-

O

Proposed AminoStructure for

Phaseolotoxin

H CO2-

-


260

followed by similar studies on adducts of these with one and two (model) arginine molecules.

In addition to providing information on the relative stabilities of the amino and imino forms of

the inhibitor in both free and bound forms, these studies also yield charge distribution data

and thus some insight into the nature of bonding within the inhibitor as well as between

inhibitor and enzyme.

8.2 Methods

In the work reported in this chapter neutral PSOrn is represented by model compound,

denoted PSO, obtained by the replacement of the (CH2)3CH(NH3+)(CO2

−) side-chain of

PSOrn by a methyl group. The structure of an amino form of PSO is shown schematically in

Figure 8.3. As indicated by the X-ray data, the side-chain is not directly involved in the

binding of PSOrn to the arginine residues. Therefore, the above simplification of the inhibitor

is not expected to significantly affect the inhibitor/arginine interactions, especially since the

CH(NH3+)(CO2

−) group, being at the end of a fully extended saturated alkyl chain, would only

marginally affect the covalent bonding pattern (and hence electron density) within the

“active” PSO moiety (either via through-space or through-bond interactions).

Figure 8.3 The structure (connectivity) of PSO, the model compound for PSOrn.

The quantum chemical calculations on PSO and the PSO/arginine adducts were carried out

using density functional theory (see Section 2.3), utilising the B3LYP exchange-correlation

hybrid functional2-4 and the 6-31G(d) basis set. Full geometry optimisations were performed

as well as constrained optimisations, where only the hydrogen coordinates were allowed to

relax while the heavy atom coordinates were constrained at the X-ray values. For a number of

CH3

NH

PH2N

ONH

S

O

OHO


261

species the energies were recalculated using the fully polarised 6-31G(d,p) basis. The

inclusion of polarisation functions on the hydrogens resulted in effectively negligible changes

in the relative energies. Due to computer resource limitations vibrational frequencies and

hence zero point energy (ZPE) corrections were not, in general, computed. On the basis of

ZPE computations on the Zw1 and Zw2 dimers and their constituent PSO and arginine

monomers, the dissociation energy of a dimer would be reduced by ~ 5 - 10 kJ mol−1, viz. up

to ~ 15%, by the inclusion of ZPE. As this work does not aim to produce energies of chemical

accuracy, the omission of ZPE is justifiable.

All calculations were carried out using the Gaussian98 programs5 on DEC alpha 600/5/333

and COMPAQ XP1000/500 workstations of the Theoretical Chemistry group at the

University of Sydney and the 64 processor SGI Origin 2400 of the Australian Centre for

Advanced Computing and Communications (ac3).


8.3.1 Free (Model) Inhibitor

All chemically reasonable tautomers of the model inhibitor PSO were considered in an effort

to locate the most stable tautomer and to quantify their relative stabilities. Free PSO, and thus

PSOrn, has the potential to form intramolecular hydrogen bonds which, in all probability, will

have a considerable effect on these stabilities. The optimised structures of five amino and

three imino tautomers of neutral PSO are shown in Figure 8.4 and Figure 8.5, along with

their relative energies. Although it may not be immediately obvious from these figures, the

geometries of the heavy atom backbones of the various tautomers are quite different,

especially in the angles. The variation is attributed, in part at least, to the effects of

intramolecular hydrogen bonding which, of course, are tautomer dependent. Constraining the

heavy atom coordinates to their X-ray values effectively eliminates most of the intramolecular

hydrogen bonds at an energy cost of ~ 400 kJ mol−1. As only part of this energy could be

reasonably attributed to the hydrogen bonds, the relaxation energy associated with

optimisation of the (covalent) bond distances and bond angles is obviously substantial. In

light of such demonstrated sensitivity of the energy to relatively small variations in the


262

Figure 8.4 Amino tautomers of PSO. (Energies relative to amino1.)

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Amino1: E = 0.0 kJ mol−1 Amino2: E = 8.9 kJ mol−1

Amino3: E = 19.1 kJ mol−1 Amino4: E = 22.8 kJ mol−1

Amino5: E = 37.5 kJ mol−1


263

Figure 8.5 Imino tautomers of PSO. (Energies relative to amino1.)

geometry, there is clearly a need for fully relaxed calculations; i.e., we cannot rely entirely on

the results of constrained computations.

According to the computed equilibrium energies listed in Figure 8.4 and 8.5 the imino

tautomers are substantially less stable than the amino forms. This was expected, as in the

former the S−N−P nitrogen would have two lone pairs of electrons and a formal negative

charge; to stabilise such a structure considerable charge delocalisation would be needed. As

will be shown later (Section 8.3.3), the sulfur and phosphorus atoms in their respective

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Imino1: E = 52.3 kJ mol−1




264

environments in PSO cannot participate in π bonding to any appreciable degree; that is, no

significant π charge delocalisation occurs. This is contrary to the implications of the Lewis

structure of the imino form of PSOrn in Figure 8.2. Hence the marked difference in stabilities

between amino and imino tautomers.

8.3.2 Bound (Model) Inhibitor

The crystal structure of PSOrn in OTCase indicates that the inhibitor is hydrogen bonded to

an arginine residue (Arg57), as shown in Figure 8.6. In particular, the distance of 2.79 Å

between the S−N−P nitrogen (N2) and the C−N−C nitrogen (NArg) of the arginine residue

suggests a strong hydrogen bond mediated N…N interaction, as noted by Langley et al.1

However, the N2…NArg interaction may be expected to be destabilising in the case of an

amino tautomer, since the near-planar arrangement of the S−N2−P and C−NArg−C groups

would imply that the N2−H and NArg−H groups would be pointing towards each other, which

would result in strong repulsion between the hydrogens.

Figure 8.6 X-ray structure of PSOrn fragment with Arg57 residue.

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2.79 Å

N2

O N

CA

S P

O

O O

C

N

NArg

3.04 Å

CA1

NA1

NA2


265

As this interaction is expected to have the most significant effect on the relative stabilities of

the tautomers, it was studied in some detail. Two approaches were used: (a) full optimisation

of the geometry (relaxed calculation) and (b) partial optimisation, where the heavy atom

backbone is constrained at the X-ray geometry and only the hydrogen positions are optimised

(unrelaxed calculation). The first approach has the advantage of yielding optimal geometries,

including hydrogen bond distances, and interaction energies. However, the strong

intramolecular hydrogen bonds in the inhibitor, as discussed in the previous section, could

considerably deform the structure, thus making comparisons between the computed

equilibrium geometries and the X-ray values effectively meaningless. (In reality the enzyme

bound PSOrn forms intermolecular hydrogen bonds to the various residues around it in

preference to intramolecular hydrogen bonds.) To simplify the calculations only the

interaction between PSO and a truncated form of the arginine residue (C2N3H7), as shown in

Figure 8.6, was studied.

Initially the range of PSO-arginine adducts that were studied were hydrogen bonded

complexes of the various (amino and imino) tautomers of PSO (as shown in Figure 8.4 and

8.5) and neutral arginine, i.e., dimers. The lowest energy dimer in this group is a complex

involving the amino1 tautomer, with binding energies computed as 27.5 and 11.3 kJ mol−1

from the relaxed and unrelaxed calculations respectively. This suggests that the N2…NArg

interaction is stabilising, although the N2…NArg distance is considerably longer than in the X-

ray structure. As can be seen from the structure in Figure 8.7, in the complex the arginine

moiety is distorted, with the NArg−H bond rotated out of the molecular plane. The interaction

of the imino1 tautomer with arginine gives rise to considerably larger binding energies: 50.9

and 26.8 kJ mol−1 from the relaxed and unrelaxed calculations respectively. Nevertheless, in

absolute terms the amino1-arginine complex is more stable by ~ 29 kJ mol−1, as indicated by

the relaxed calculations.

On extending the calculations to zwitterionic dimers, that is, complexes of deprotonated PSO

(denoted PSO−) and protonated arginine, it was found that two of these are more stable, even

in gas phase, than the dimers between neutral partners. The structures of these complexes

(denoted Zw1 and Zw2) are also shown in Figure 8.7. The PSO− moieties in both of these

dimers are imino tautomers. As can be seen from the tabulated distances in Table 8.1, the


266

Figure 8.7 Structures of the three most stable PSO…Arg dimers.

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N

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2.74 Å 2.79 Å

H N

PSO−(Imino)…ArgH+ (Zw1)Dimer

PSO−(Imino)…ArgH+ (Zw2)Dimer

PSO(Amino1)…Arg Dimer

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2.81 Å 2.79 Å

N

H


267

Table 8.1. Selected X-ray distances for enzyme bound PSOrn and corresponding computed

distances in PSO…Arg dimers and PSO…(Arg)2 trimer.a

Atom-Atom Distance /Å

Bound PSOrnX-ray

PSO…Arg(Zw1)

PSO…Arg(Zw2)

PSO…(Arg)2

Trimer

S−O 1.51 1.48 1.49 1.50

S−O1 1.34 1.63 1.50 1.52

S−O2 1.58 1.46 1.48 1.49

S−N2 1.60 1.62 1.69 1.65

N2−P 1.66 1.65 1.59 1.63

P−O3 1.48 1.49 1.49 1.52

P−N 1.72 1.71 1.69 1.71

P−N1 1.61 1.74 1.89 1.70

N1−C 1.48 1.47 1.49 1.47

N2−NArg 2.79 2.81 2.74 2.72

O−NA1 3.04 2.76 2.76 2.70

CA−NArg 1.46 1.46 1.45 1.45

NArg−CA1 1.33 1.32 1.32 1.32

CA1−NA2 1.33 1.36 1.37 1.37

CA1−NA1 1.33 1.34 1.34 1.34

O1−Na1 2.76 2.72

O3−Na2 2.73 2.69

Ca−Na 1.45 1.47

Na−Ca1 1.33 1.36

Ca1−Na1 1.32 1.34

Ca1−Na2 1.32 1.34

a Labelling of atoms as indicated in Figure 8.1.

Subscripts Arg, A, A1, A2 refer to atoms of Arg57 (See Figure 8.6).

Subscripts a, a1, a2 refer to atoms of Arg106.

fully optimised geometry of Zw1 matches the X-ray data reasonably well. Agreement

between theory and experiment is less convincing in the case of Zw2, where the P−N1 bond

distance of 1.89 Å is clearly at variance with the X-ray value of 1.61 Å. Zw2, however,

appears to be the more stable (by 13.2 kJ mol−1) of the two dimers. Compared with the lowest

energy amino1-arginine complex, Zw1 and Zw2 were computed to be 24.0 and 37.2 kJ mol−1

more stable respectively, corresponding to binding energies of 51.5 and 64.7 kJ mol−1 with


268

respect to neutral arginine and the amino1 form of PSO. The stabilities of the various dimers,

as well as of a trimer (as discussed below), are summarised in Table 8.2 as dissociation

energies to a range of neutral and charged moieties.

Table 8.2 Computed dissociation energies of PSO…Arg dimers and trimers.

∆E /kJ mol−1

PSO(Amino1) … Arg Dimer → PSO(Amino1) + Arg −−27.5

PSO(Imino1) … Arg Dimer → PSO(Amino1) + Arg −−−1.3

PSO(Imino1) … Arg Dimer → PSO(Imino1) + Arg −−50.9

PSO− (Imino) … ArgH+ Dimer (Zw1) → PSO(Amino1) + Arg −−51.5

PSO− (Imino) … ArgH+ Dimer (Zw1) → PSO− (Imino) + ArgH+ −332.8

PSO− (Imino) … ArgH+ Dimer (Zw2) → PSO(Amino1) + Arg −−64.7

PSO− (Amino) … ArgH+ Dimer (Zw3) → PSO(Amino1) + Arg −126.5

PSO2− (Imino) … (ArgH+)2 Trimer → PSO(Amino1) + 2 Arg −170.8

PSO2− (Imino) … (ArgH+)2 Trimer → PSO− … ArgH+ Dimer (Zw1) + Arg −119.3

PSO2− (Imino) … (ArgH+)2 Trimer → PSO2− + 2 ArgH+ 1284.2

The relative stabilities of PSO and PSO…Arg dimers, as obtained in constrained and relaxed

calculations are shown in Figure 8.8. The trends in the stabilities appear to be qualitatively

reproduced by the constrained optimisations, but clearly the energy differences, especially

between the PSO(Amino1)…Arg dimer and Zw1, are predicted to be considerably larger by

the constrained calculations. As remarked in the previous section, in light of the large energy

differences between the unrelaxed and relaxed structures, we regard the latter as the more

reliable.

Interactions between amino tautomers of PSO− and protonated arginine (denoted ArgH+) were

found to be repulsive, as expected. Although in the latter complexes ArgH+ did bind to PSO−,

this did not occur via N2, as would be required for a valid description of the binding of PSOrn

in the enzyme.


269

Figure 8.8 Relative energies (in kJ mol−1) of PSO tautomers and PSO…Arg dimers from

constrained and relaxed calculations.

According to the X-ray data, PSOrn interacts with two arginine residues, the second (Arg106)

effectively bridging the O1 and O3 atoms of PSOrn (see Figure 8.1). A trimer of PSO with

two arginines is clearly a more realistic model for the binding of PSOrn to the enzyme. Given

the apparent propensity of arginine to exist in protonated form, our trimer calculations were

restricted to a complex of a (doubly deprotonated) dinegative PSO (denoted PSO2−) and two

ArgH+ subunits. The computed structure of this trimer is shown in Figure 8.9. The key

interatomic distances are listed in Table 8.1. The agreement with the X-ray data is good,

given the relatively high estimated errors of ± 0.2 Å in the X-ray distances. The large binding

energy of 170.8 kJ mol−1, relative to neutral PSO and two arginines, is consistent with the

action of PSOrn as an effective inhibitor that binds irreversibly to the enzyme.

PSO (Amino1)…Arg Dimer

PSO (Imino1)…Arg Dimer

PSO− (Imino)…ArgH+ Dimer (Zw1)

PSO− (Imino)…ArgH+ Dimer (Zw2)

−390.9

52.7

−466.2

0.0

28.8 24.0

−482.5

−453.7

0.0

37.0

−506.5

−100.2

−519.713.2

75.4Relaxed

Unrelaxed

PSO PSO…Arg Dimers


270

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NNC

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2.75 Å

2.74 Å

H

Figure 8.9 Structures of PSO…(Arg)2 trimer and PSO…Arg (Zw3) dimer.

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Arg57+

Arg106+

PSO2-

H

H H

O

S N

P

O

O O C

H

H

H

H

H

H

H

H H

H

C C H

2.72 Å

2.69 Å

H

H

H

H

H

H

H H

C C

2.72 Å 2.70 Å

N

N

N N

N

N

N N


271

The very much higher stability of the trimer (to dissociation) than of the dimers Zw1 and Zw2

suggests that the interaction between a PSO− ion and ArgH+ (representing the Arg106 residue)

is actually the dominant contribution to the overall stability, rather than the interaction with

the (protonated) Arg57 residue. To test this hypothesis the structure of a third zwitterionic

dimer (Zw3) was optimised. This derives from the trimer by the removal of neutral Arg57

(See Figure 8.9). PSO− in Zw3 was chosen to be an amino tautomer. According to the

calculations Zw3 is more than twice as stable as Zw1, its dissociation energy to PSO and Arg

having been computed to be 126.0 kJ mol−1. Thus the overall binding energy of the trimer is,

to a good approximation, the sum of the individual binding energies to the two arginine

residues.

In the zwitterionic dimers, as well as in the trimer discussed above, the overall binding

between the negatively charged PSO− and ArgH+ moieties, in addition to the hydrogen

bonding, has a substantial ionic (electrostatic) contribution. This can be quantified through the

analysis of the binding energies of Zw1, Zw2 and the trimer, relative to neutral PSO and Arg

as well as relative to the ions PSO−, PSO2− and ArgH+. These results are included in Table

8.2. Thus Zw1 and Zw2 are bound by nearly 350 kJ mol−1 relative to the ions, but because of

the very different proton affinities of PSO− and Arg, the binding energies relative to the most

stable amino tautomer of PSO and Arg are nearly an order of magnitude smaller.

Furthermore, the PSO2−…(ArgH+)2 trimer is bound by 1284 kJ mol−1 relative to the ions.

8.3.3 Charge Distribution and Bonding

In Figure 8.2, following the usual convention, the Lewis structures of amino and imino

tautomers of PSOrn were drawn with several PO and SO double bonds, with the implication

that P and S are hypervalent; that is, they accommodate more than eight electrons in their

valence shells. In the case of the P=O and S=O bonds this implies utilisation of the 3d atomic

orbitals of P and S in the formation of the appropriate P-O and S-O π molecular orbitals. The

validity of such “expansion of the octet” has been strongly debated in the literature over the

past 20 years.6-9 On the basis of careful quantum chemical studies several authors have

concluded that the electronic structures of molecules with apparently hypervalent multiply

bonded second row atoms are best described by invoking semipolar bonds, where, for


272

example, the P and S atoms acquire formal charges of up to +2, which then allows these

atoms to form up to four covalent bonds with O− (or other atoms or ions)6,7,9,10. The semipolar

bonds thus have both covalent and ionic components and are comparable in strength with

double bonds, with bond lengths to match.

In light of the above observations, the amino structure of PSOrn in Figure 8.2 would be more

correctly drawn by replacing each P=O and S=O with P+− O− and S+− O− semipolar bonds

(which would result in S with a formal 2+ charge), as shown in Figure 8.10. The imino

tautomer, however, as drawn in Figure 8.2, relies on π resonance to partially delocalise the

negative charge on the PNS nitrogen due to the two lone pairs. Thus, if P and S cannot

participate in π bonding, the above mechanism for charge delocalisation cannot be invoked.

Hence our initial suspicion that amino forms of PSO would be considerably more stable than

any imino tautomer. In the case of isolated PSO these suspicions proved well-founded as the

computations located five amino type tautomers which were more stable than the lowest

energy imino tautomer (Figures 8.4 and 8.5).

Figure 8.10 Lewis structure of an amino tautomer of PSOrn with semipolar bonds.

NH3+

C

(CH2)3

N H

P+

H2NO-

NH

S++

O-

O-

O-

H CO2-


273

8.3.3.1 Population Analysis

As in previous studies which have addressed the problem of hypervalency, we have used the

Roby-Davidson (RD) method11-14 (with the B3LYP/6-31G(d) density matrix) to carry out

population analyses on free as well as bound PSO, yielding atomic charges and shared

electron numbers (σ) for pairs of atoms. The latter is interpreted as a direct measure of the

covalent character of a given bond. It must be noted, however, that shared electron numbers

are not bond orders and thus their interpretation requires calibration. This was carried out by

analysing the shared electron numbers of a range of small molecules (H2PNH2, PO(NH2)3,

H3PNH, HPNH, HPO, HOPO, H3PO, HSNH2, H2SNH, SNH, HSOH, H2SO4, H3SO, SO) and

correlating the shared electron numbers with bond lengths and bond orders, provided the latter

could be reasonably assigned, for example double bonds in S=O and HP=O, and single bonds

in H2P−NH2 and HS−NH2. Thus, as shown in Table 8.3, for PN and PO: σ = 1.0 - 1.22 are

consistent with single bonds, σ = 1.76 - 1.83 describe double bonds, while σ = 1.40 - 1.50

apply to semipolar bonds. Lower σ values describe such bonds in the case of SN and SO

linkages. Atomic charges can also be calibrated, with Nq or Oq values of ~ −0.25 to ~ −0.5

being consistent with single or double bonds and ~ −0.6 to ~ −0.75 consistent with semipolar

bonding. Pq and Sq show greater variation with the nature of their bonding partners, in

general, however, q = 0.1 to 0.5 indicate single or double bonds whereas q > 0.6 (with only

one N or O partner) indicate the presence of a semipolar bond. Bonding to additional N or O

atoms increases Pq or Sq such that S in H2SO4 (with two single and two semipolar SO bonds)

has a charge of 1.87. As noted earlier, this calibration shows that it is not possible to

distinguish between double and semipolar bonds on the basis of bond lengths (which may

have led to the original assumption that semipolar bonds were π bonds); semipolar bonds can

be identified, however, by the lower values of σ and the higher atomic charges.

The computed shared electron numbers and atomic charges for the amino1 and imino1

tautomers of PSO, along with those of the zwitterionic dimer Zw1 are listed in Table 8.4 and

Table 8.5. Almost all the bonds between the heavy atoms of PSO are described as single or

semipolar bonds. A partial double bond character has been assigned to the P−N2 bond in the

imino1 tautomer of PSO. In this molecule, due to the protonation of N1, the P−N1 bond is

long and weak and therefore a degree of bonding π interaction between the P and N2 atoms is

possible. The high positive charges on the S and P atoms along with the high negative charges

Chapter 8. Structure and B

onding in PSO

rn

274

Table 8.3 Calibration of Roby-Davidson population analysis results (shared electron populations and atomic charges) along with bond lengths

for P-N, P-O, S-N and S-O single, double and semipolar bonds.

P−N P−O

Molecule RPN /Å σ qP qN Molecule RPO /Å σ qP qO

Single Bond H2PNH2 1.73 1.22 0.24 −0.44 H2POH 1.68 1.04 0.33 −0.44

PO(NH2)3 1.69 1.01 1.30 −0.53 HOPO 1.64 1.00 0.64 −0.45

Semipolar Bond H3PNH 1.57 1.50 0.79 −0.66 H3PO 1.49 1.41 0.87 −0.60

PO(NH2)3 1.50 1.39 1.30 −0.66

Double Bond HPNH 1.59 1.83 0.30 −0.37 HOPO 1.48 1.75 0.64 −0.41

HPO 1.50 1.75 0.50 −0.37

S−N S−O

Molecule RSN /Å σ qS qN Molecule RSO /Å σ qS qO

Single Bond HSNH2 1.72 0.85 0.13 −0.41 HSOH 1.70 0.91 0.20 −0.41

H2SO4 1.63 0.76 1.87 −0.51

Semipolar Bond H2SNH 1.61 1.25 0.64 −0.70 H3SO 1.49 1.19 1.06 −0.72

H2SO4 1.45 1.21 1.87 −0.65

Double Bond SNH 1.58 1.64 0.13 −0.26 SO 1.52 1.47 0.31 −0.31

274


275

Table 8.4 Selected bond lengths (R in Å), Roby-Davidson shared electron numbers (σ in e)

and assigned bond types of amino and imino tautomers of PSO and of the PSO…Arg dimer

(Zw1)a

PSO (Amino1) PSO (Imino1) PSO…Arg Dimer (Zw1)

R σ Bond Type R σ Bond Type R σ Bond Type

S−O 1.45 1.3 Semipolar 1.63 b 0.8 Semipolar 1.48 1.1 Semipolar

S−O1 1.46 1.2 Semipolar 1.47 1.2 Semipolar 1.63 b 0.8 Semipolar

S−O2 1.61 b 0.8 Single 1.48 1.1 Single 1.46 1.2 Single

S−N2 1.70 0.9 Single 1.61 1.1 Part. Double 1.62 1.1 Single

P−N2 1.72 1.0 Single 1.60 1.3 Single 1.65 1.1 Single

P−O3 1.50 1.3 Semipolar 1.48 1.5 Semipolar 1.49 1.4 Semipolar

P−N 1.66 1.1 Single 1.67 1.1 Single 1.71 1.0 Single

P−N1 1.67 1.1 Single 1.92 0.6 Weak Single 1.74 0.9 Single

a Labelling of atoms as indicated in Figure 8.1.b Part of SOH group.

Table 8.5 Atomic charges (in e) on heavy atoms for PSO in amino and imino tautomers of

PSO and of the PSO…Arg dimer (Zw1) from Roby-Davidson analysis.a

PSO(Amino1)

PSO(Imino1)

PSO…Arg Dimer(Zw1)

S 1.79 1.77 1.84

O −0.72 −0.46b −0.82

O1 −0.59 −0.81 −0.59b

O2 −0.53b −0.64 −0.74

N2 −0.82 −1.08 −1.19

P 1.32 1.27 1.05

O3 −0.89 −0.78 −0.76

N −0.56 −0.62 −0.46

N1 −0.54 −0.28 −0.33

a Labelling of atoms as indicated in Figure 8.1.b Part of SOH group.


276

on the oxygens of PSO are consistent with semipolar S-O and P-O bonds. The high negative

charge on N2 is according to expectations in the case of imino tautomers, although it is quite

high in the amino1 form as well, due to the polar N-S, N-P, and especially N-H bonds.

Interestingly, there is an increased negative charge localisation on N2 in the case of the Zw1

dimer. This is probably due to the polarisation of PSO by the Arg+ residue.

The population analyses for the amino1 and imino2 tautomers were repeated with basis sets

containing two additional sets of d functions on the P and S atoms (with exponents chosen as

1/3 and 1/9 of those in the 6-31G(d) sets). This was done to ensure that the description of the

atomic 3d orbitals on these atoms is sufficiently accurate and flexible to resolve any incipient

π bonding. No significant changes in charges, shared electron numbers or relative energies

occurred. We conclude therefore that no appreciable π bonding is present in the various

tautomers of PSO and its complexes.

8.3.3.2 Hydrogen Bonding

A related issue is the hydrogen bonding potential of the various terminal oxygen and nitrogen

atoms in PSO. The traditional explanation of hydrogen bonding is that the protonic hydrogen

of the proton donor seeks out regions of high electron density, which are generally provided

by the lone pairs of the proton acceptor. As hydrogen bonds are usually (near) linear, it is

assumed that in general each lone pair is only capable of forming one hydrogen bond. It

would therefore be expected that a doubly bound oxygen atom, formally having two lone

pairs, would be able to form two hydrogen bonds. On the other hand, an oxygen atom

involved in a semipolar bond formally has three lone pairs and would therefore have the

potential to form three hydrogen bonds. The net effect of three lone pairs of electrons is an

effectively uniform (directionless) charge distribution on the oxygen atom. It is therefore

possible that this, coupled with the large negative charges, Oq , for oxygens involved in

semipolar bonding (as shown in Table 8.3) may allow the formation of more than three

hydrogen bonds.

The validity, or otherwise, of this view of hydrogen bonding is particularly important when

attempting to postulate and interpret which enzyme-inhibitor interactions in the crystal


277

structure of Langley et al.1 involve hydrogen bonds. Careful analysis of the crystal structure

data indicates that for each of the oxygen atoms attached to S (O, O1 and O2 as defined in

Figure 8.1) there are three potential proton donors with appropriate distances and relative

orientations for the formation of three hydrogen bonds. In addition there are four potential

proton donor residues which are appropriately placed to form hydrogen bonds to O3.

It is therefore important to determine whether it is realistic to assign such a large number of

hydrogen bonds to oxygen atoms in these environments. This section thus presents an

investigation of the stabilisation or destabilisation which is obtained for various model

compounds with a range of hydrogen bonding interactions. These model compounds include

H2CO (as the simplest and best understood example of a doubly bound oxygen atom), HPO

(in order to determine the effect of phosphorus on the hydrogen bonding), H3PO (as a simple

example of an oxygen in a semipolar bond) and PO(NH2)3 (as a model for the O3 oxygen in

PSOrn). Water molecules were used as proton donors where the orientation of these moieties

was fixed so there could be no additional stabilisation due to hydrogen bonds between

adjacent water molecules or to other parts of the model compound (see Figure 8.11). The

hydrogen bonds were also constrained to be linear. All other inter- and intramolecular

parameters were then optimised using MP2/cc-pVDZ, thus allowing the stabilisation energy,

stabE , to be calculated relative to the non-interacting monomers. It was, of course, particularly

important to estimate the possible contribution of basis set superposition error to this

stabilisation energy. This was done using the Boys-Bernardi method (described in Section

2.4.5) for each of the interacting molecules. The sum of the individual counterpoise

corrections was subtracted from stabE to give the counterpoise corrected estimate of the

hydrogen bonding energy, HBE . The successive hydrogen bond energies corresponding to the

introduction of 1, 2, 3 and 4 hydrogen bonding water molecules are presented in Table 8.6.

H2CO and HPO, as typical doubly bonded systems, were expected to have two lone pairs and

form two hydrogen bonds; this expectation is clearly confirmed by the results in Table 8.6. In

the case of H2CO a minimum energy structure was also found with the addition of a third

H2O, however two of the water molecules were within only 70° of each other (rather than

120°) indicating that they were binding to the same lone pair on the H2CO oxygen. Although

this structure was found to be stable by 1.7 kcal mol−1 before the application of counterpoise


278

Table 8.6 Successive stabilisation energies corresponding to the addition of water molecules

as proton donors to H2CO, HPO, H3PO and PO(NH2)3.

Hydrogen Bonds HBE /kcal mol−1

(H2O molecules) H2CO HPO H3PO PO(NH2)3

1 −3.0 −3.0 −4.7 −6.6

2 −2.3 −2.3 −4.1 −6.4

3 −0.9 − −2.7 −6.2

4 − − −0.0 −2.7

corrections, once superposition errors had been accounted for the tetramer was found to be

unstable by 0.9 kcal mol−1, suggesting that the structure is simply an artefact of the relatively

small cc-pVDZ basis set. In terms of its hydrogen bonding propensity, HPO is very similar to

H2CO, although for HPO no local minimum structure with three waters could be found. Thus

it appears that doubly bonded oxygen atoms will only form two hydrogen bonds and that

these align with the expected orientations of two equivalent localised lone pairs (See Figure

8.11).

H3PO formally has a semipolar P-O bond and, as expected, it can form up to three hydrogen

bonds. Local minima were also found on the PES with four and even five water molecules

apparently hydrogen bonded to the oxygen of H3PO. In both these structures the water

molecules appear evenly distributed around the oxygen atom suggesting that the lone pair

charge distribution is fairly directionless (that is, the lone pairs are not spatially discrete). The

stability of these structures could again be a BSSE artefact, however, as suggested by the

counterpoise corrected energies. For the structure with four water molecules the possibility of

attractive interaction between the water molecules was also investigated, however it was

found that the water-water interactions were, in fact, repulsive by 3.7 kcal mol−1.

Finally, binding to the oxygen atom in the hypothetical molecule PO(NH2)3 was investigated

as a model for the O3 atom (bound to P) in PSOrn. Even with the inclusion of the

counterpoise correction, this model compound is capable of forming four hydrogen bonds

with water molecules. The first three interactions are of comparable strength (and much

stronger than those in H2CO, HPO or H3PO) while the fourth seems significantly weaker

although still binding.


279

We therefore conclude that up to four hydrogen bonds can be attached to an oxygen atom

which is bound by a semipolar bond; it is therefore reasonable to interpret the crystal structure

of PSOrn as showing up to three hydrogen bonds to O, O1 and O2 and up to four hydrogen

bonds to O3.

Figure 8.11 Most stable hydrogen bonded structures for H2CO, HPO, H3PO and PO(NH2)3.

H2CO.(H2O)2

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280

8.4 Conclusion

With the aid of quantum chemistry, viz. density functional theory, the binding of PSOrn to the

enzyme OTCase was investigated and modelled through a study of PSO, a simplified model

for PSOrn, and its interaction with one and two arginine molecules. The PSO…(Arg)2 trimer

was found to be bound by ~ 171 kJ mol−1. Such high stability, due to the presence of four

hydrogen bonds as well as a large ionic interaction between the dinegative PSO2− and

protonated arginines, is consistent with the experimental observation that PSOrn is a powerful

enzyme inhibitor. The calculations confirm the proposals of Langley et al.1, inasmuch as

bound PSOrn is a dinegative imino tautomer. While in the case of free (neutral) PSO the most

stable tautomers were calculated to be amino types, when bound to one or two (protonated)

arginines PSO (as PSO− or PSO2−) is predicted to prefer an imino form. However, as in other

phosphorous and sulfur containing molecules, according to the population analyses that were

carried out the P-N, P-O, S-N and S-O bonds in PSO are generally best described as single or

semipolar bonds.


281

8.5 References

1. D. B. Langley, M. D. Templeton, B. A. Fields, R. E. Mitchell and C. A. Collyer, J.

Biol. Chem., 2000, 275, 20012.















6. A. E. Reed and P. v. R. Schleyer, J. Am. Chem. Soc., 1990, 112, 1434.

7. J. A. Dobado, H. Martínez-García, J. M. Molina and M. R. Sundberg, J. Am. Chem.

Soc., 1999, 121, 3156.

8. K. Jug and E. Fasold, Int. J. Quantum Chem., 1992, 41, 687.

9. W. Kutzelnigg, Angew. Chem. Int. Ed. Engl., 1984, 23, 272.

10. A. P. L. Rendell, G. B. Bacskay and N. S. Hush, J. Am. Chem. Soc., 1988, 110, 8343.


12. K. R. Roby, Mol. Phys., 1974, 27, 81.


14. C. Ehrhardt and R. Ahlrichs, Theor. Chim. Acta, 1985, 68, 231.

9 Conclusion

Chapter 9

Conclusion

Chapter 9. Conclusion

283

The major focus of the research work presented in this thesis is the calculation of accurate

thermochemical data, including atomisation energies and heats of formation. Gaussian-3 (G3)

and related methodologies have been found to be particularly useful for this purpose, yielding

chemically accurate heats of formation in most cases, with relatively modest computational

cost. Thus the G3 heats of formation of approximately 120 C1 and C2 fluorocarbons and

oxidised fluorocarbons (along with selected C3 fluorocarbons) were calculated. For the most

part, these showed good agreement with the best available theoretical and experimental

literature data, particularly when G3 was used in conjunction with isodesmic reaction

schemes. For molecules for which the G3 results were found to be in poor agreement with

experiment or for which the experimental values had large uncertainties (e.g., HCCF, FCCF,

CCH, CCF and HCOO), more extensive CCSD(T)/CBS calculations have confirmed the

validity of the G3 results, indicating that the literature values may need to be revised. Two

less computationally expensive approximations to G3 were also proposed: G3MP4(SDQ) and

G3[MP2(Full)]. These methods were found to reliably reproduce the G3 results, with mean

absolute deviations from G3 of ~ 0.4 and ~ 0.5 kcal mol−1 respectively for heats of formation

from atomisation energies. These deviations could be further reduced by the application of

isodesmic reaction schemes.

The G3 method was also successfully applied to the calculation of the heats of formation of

molecules containing third row atoms, in particular CFClBr2, and by extension to the

thermochemistry of its dissociation reactions. As G3 results for bromine containing species

had been previously only reported for very small molecules such as HBr, Br2 and CH3Br, it

was gratifying to find excellent agreement between the experimentally determined 0298f H∆ of

184 ± 5 kJ mol−1 and the G3 value of 188 ± 5 kJ mol−1. This work therefore provided further

evidence for the wide range of applicability of the G3 method. In addition, these results have

also provided valuable aid in the determination of the photolysis mechanism for CFClBr2 at

265 nm. Our calculations have predicted the dissociation energy for the two C-Br bonds to be

equal to the (experimental) energy required to produce the CFCl carbene, thus supporting the

hypothesis that the photolysis proceeds via a triple fragmentation pathway, releasing two Br

atoms, rather than by the concerted elimination of a Br2 molecule.

G3 type methods (including G2, G3, G3X and G3X2) as well as more extensive

CCSD(T)/CBS type calculations were also employed in the calculation of the


284

thermochemistry of 18 small phosphorus containing compounds (P2, P4, PH, PH2, PH3, P2H2,

P2H4, PO, PO2, PO3, P2O, P2O2, HPO, HPOH, H2POH, H3PO, HOPO and HOPO2). The

CCSD(T)/CBS results are consistent with the available experimental values and, as the

estimated uncertainties are quite small, they constitute the most accurate set of heats of

formation available for these molecules. They are therefore regarded as an excellent

benchmark for the testing of the more approximate G3n methods. The G3 and G3X methods

were found to consistently underestimate the benchmark atomisation energies, on average by

3.6 and 1.8 kcal mol−1 and by up to 6.5 and 5.6 kcal mol−1 respectively for this set of

molecules (excluding P4); G2 is comparable in performance to G3. The G3X2 method was

therefore proposed in an effort to improve the G3X results; G3X2 accounts for a higher

degree of electron correlation in comparison with G3 and G3X and, with the inclusion of

counterpoise corrections for BSSE on the phosphorus atoms in the core-valence correlation

calculations, it does show a modest improvement (with a mean absolute deviation from the

benchmark results of 1.5 kcal mol−1). Further investigation has also revealed that the

approximations underlying the Gaussian-n methods become unreliable for molecules which

involve unusual P-P bonding, such as double bonds (in P2 and P2H2), double and semipolar

bonds (in PPO) or large structural strain (in P4). We therefore recommend great care with the

application of these methods to molecules with bonds between second row elements unless

these are simple, unstrained single bonds.

Accurate thermochemical data, in particular those obtained by CCSD(T)/CBS calculations,

along with similar calculations for transition state structures (with appropriate approximations

for variational transition states) allowed us to reliably predict the kinetics of Twarowski’s

proposed catalytic schemes for H + OH recombination. The application of transition state and

RRKM theory resulted in rate coefficients which are consistent with the results of

experimental and modelling studies, although they are in most cases significantly lower than

Twarowski’s estimated values. At 2000K the rates of both catalytic schemes were found to be

comparable and significantly higher than the uncatalysed recombination; both cycles are

therefore expected to be catalytic at this temperature.

The potential energy surfaces for all possible reactions stemming from the NNH + O

recombination were investigated in detail along the appropriate reaction coordinates at the

B3LYP/6-31G(2df,p) level of theory. This study revealed the presence of several reaction


285

channels which had not been considered previously (including direct abstraction of H by O

and three product channels via the intermediate ONHN) and yielded improved descriptions of

channels which had hitherto been incompletely characterised (dissociation of cis- and trans-

ONNH into NO + NH). The heats of formation were determined for each of the species

corresponding to stationary points on the PES using both G3X theory and a CCSD(T)/CBS

type scheme. A reasonable level of consistency, corresponding to agreement of ~ 2 kcal mol−1

or better between the two sets of results, was observed. This thermochemical and geometric

data was further utilised to generate rate coefficients for the various reaction channels on the

potential energy surface via (variational) transition state and RRKM theories, thus allowing

the overall rate coefficients and the flux through each channel to be determined. We were

therefore able to conclude that the NNH + O channel is considerably less important in

combustion systems than had been previously believed. This was supported by modelling

studies of two combustion systems in the presence of N2 which yielded good agreement with

experiment when our revised thermochemical and kinetic data was employed.

In a study aimed at modelling the binding of the inhibitor PSOrn to OTCase, the relative

stabilities of various amino and imino tautomers of PSO, a model compound for PSOrn, were

investigated both in the gas phase and when bound to (model) residues from the active site of

OTCase. Gas phase calculations revealed that the amino tautomer is the most stable form of

free (neutral) PSO; in the presence of arginine residues, however, the imino structure becomes

lower in energy, the most stable structure being a doubly deprotonated PSO bound to two

Arg+ residues. Thus when bound to the active site of the enzyme, PSOrn would be expected to

adopt a dinegative imino form. Population analysis and hydrogen bonding studies have

revealed that the intramolecular bonds involving second row atoms are either single or

semipolar in nature and that the terminal oxygen atoms are capable of accommodating up to

four hydrogen bonds.

9

Appendices

Appendix 1

A1-1

Appendix 1: Fluorocarbons Supplementary

Information

A1 A1

Appendix 1.1 Atomic data: G3 energies and heats of formation at 0 K of atoms, and thermal

corrections to enthalpies of elements in their standard states, as used in this work.

E0 (G3) / Eh ∆ f H00/ kcal mol−1 H H298

000−c h / kcal mol−1

H −0.5010 51.63 1.01

C −37.8277 169.98 0.25

O −75.0310 58.99 1.04

F −99.6842 18.47 1.05

Appendix 1

A1-2

Appendix 1.2 C1 Hydrofluorocarbons: Rotational constants, vibrational frequencies (scaled by 0.8929), obtained at MP2(Full)/6-31G(d) and

HF/6-31G(d) levels of theory respectively and the resulting zero point energies and thermal correctionsa to the heats of formation.

Species Rotational Constants (cm−1) Vibrational frequencies (cm−1)ZPE

(kcal mol−1)

∆∆ f H2980

(kcal mol−1)

CH4 5.2850 5.2850 5.2848 1328 1328 1329 1520 1520 2856 26.78 2.39

2947 2949 2952

CH3F 5.2414 0.8490 0.8490 1061 1172 1172 1475 1476 1476 23.78 2.42

2886 2957 2957

CH2F2 1.6481 0.3479 0.3045 510 1106 1123 1164 1259 1464 20.16 2.56

1530 2941 3005

CHF3 0.3408 0.3407 0.1868 491 492 680 1126 1185 1187 15.76 2.78

1414 1414 3036

CF4 0.1883 0.1883 0.1883 422 422 610 610 610 896 10.74 3.08

1315 1315 1315

CH3 9.5909 9.5909 4.7955 275 1375 1375 2933 3090 3090 17.35 2.66

CH2F 8.7668 1.0151 0.9215 770 1133 1143 1443 2962 3088 15.07 2.46

CHF2 2.2359 0.3603 0.3152 521 1041 1150 1181 1343 3005 11.78 2.55

CF3 0.3573 0.3573 0.1850 491 491 677 1086 1286 1286 −7.60 2.77

CH2 20.0859 11.2493 7.2108 1397 2794 2850 10.06 2.37

CHF 15.669 1.2090 1.1224 1189 1405 2727 −7.61 2.39

CF2 2.8441 0.4132 0.3608 651 1155 1240 −4.35 2.48

CH 14.4520 2733 −3.91 2.07

CF 1.3802 1259 −1.80 2.08

Appendix 1

A1-3


(kcal mol−1)

∆∆ f H2980

(kcal mol−1)

CH2O 9.5774 1.2666 1.1187 1193 1235 1500 1811 2822 2886 16.36 2.39

CHFO 3.0430 0.3836 0.3406 659 1051 1115 1375 1878 2996 12.97 2.49

CF2O 0.3879 0.3823 0.1925 563 610 779 976 1305 1953 −8.85 2.67

CHO 23.3086 1.4662 1.3794 1118 1913 2601 −8.05 2.39

CFO 6.2938 0.3751 0.3540 632 1081 1912 −5.18 2.48

CH3OH 4.2466 0.8251 0.7954 311 1040 1061 1151 1346 1462 31.01 2.69

1475 1485 2844 2885 2951 3677

CH2FOH 1.5160 0.3413 0.3011 351 529 1020 1070 1132 1245 27.49 2.80

1363 1447 1522 2903 2990 3658

CHF2OH 0.3347 0.3327 0.1864 303 498 532 640 1025 1119 23.03 3.05

1192 1310 1384 1438 3040 3629

CF3OH 0.1890 0.1857 0.1852 234 428 441 583 607 617 17.94 3.41

888 1123 1234 1323 1414 3654

CH3OF 1.4291 0.3576 0.3028 244 437 922 1090 1154 1206 26.10 2.93

1433 1445 1484 2891 2971 2976

CH2FOF 0.6195 0.1865 0.1597 159 396 587 936 1087 1135 22.34 3.15

1157 1290 1429 1485 2949 3017

CHF2OF 0.2531 0.1570 0.1290 146 275 506 525 789 945 17.85 3.46

1124 1176 1195 1386 1406 3016

CF3OF 0.1833 0.1034 0.1014 133 262 424 430 572 597 12.84 3.81

676 887 1083 1283 1298 1337

CH3O 5.2530 0.9136 0.9097 727 990 1083 1414 1423 1488 22.56 2.49

2841 2900 2917

Appendix 1

A1-4


(kcal mol−1)

∆∆ f H2980

(kcal mol−1)

CH2FO 1.8277 0.3572 0.3172 528 912 1047 1109 1198 1413 19.26 2.58

1440 2885 2938

CHF2O 0.3559 0.3434 0.1903 443 500 645 1014 1147 1154 15.17 2.82

1356 1391 2959

CF3O 0.1995 0.1933 0.1847 225 411 572 583 607 884 10.21 3.26

1275 1278 1310

CH2OH 6.3381 0.9898 0.8696 368 763 1032 1149 1324 1452 22.54 2.68

2936 3059 3682

CHFOH 2.1107 0.3613 0.3132 198 515 1021 1045 1186 1253 18.88 2.92

1379 2928 3678

CF2OH 0.3591 0.3474 0.1828 257 483 491 674 1040 1113 14.83 3.09

1287 1363 3665

CH2OF 1.7795 0.3801 0.3162 204 463 762 935 1117 1169 17.34 2.99

1409 2973 3098

CHFOF 1.6064 0.1461 0.1349 80 341 499 1017 1051 1159 13.83 3.30

1194 1327 3008

CH3OOH 1.3976 0.3511 0.3029 182 240 445 926 1107 1155 33.53 3.31

1201 1397 1440 1453 1484 2873

2938 2961 3650

CF3OOH 0.1824 0.1041 0.1030 139 256 288 429 437 572 20.43 4.08

604 674 879 1067 1263 1294

1319 1438 3631

Appendix 1

A1-5


(kcal mol−1)

∆∆ f H2980

(kcal mol−1)

CH3OO 1.7316 0.3821 0.3332 162 478 950 1143 1143 1200 26.16 2.99

1439 1457 1470 2896 2972 2987

CF3OO 0.1848 0.1091 0.1076 122 279 420 442 571 593 12.86 3.80

687 879 1124 1251 1289 1341

HCOOH 2.5687 0.3968 0.3437 618 639 1065 1138 1286 1386 20.76 2.60

1817 2965 3609

FCOOH 0.3934 0.3757 0.1922 545 557 607 791 965 1215 16.59 2.80

1398 1881 3642

HCOO - A1 5.3878 0.3786 0.3537 509 821 1060 1218 1693 2010 10.45 2.58

FCOO - B2 0.4666 0.3636 0.2043 540 568 811 993 1561 2457 −9.91 2.68

CH2OHOH 1.3806 0.3410 0.3013 367 391 552 984 1042 1101

1178 1345 1365 1437 1515 2897 34.91 2.99

2941 3653 3654

CF2OHOH 0.1884 0.1839 0.1823 162 338 432 449 583 595 25.21 3.73

606 880 1101 1153 1158 1408

1451 3659 3659

OCH2OH 1.6489 0.3549 0.3147 301 532 877 1006 1110 1132 26.54 2.86

1329 1410 1444 2832 2930 3659

OCF2OH 0.1974 0.1950 0.1803 228 319 406 567 583 597 17.49 3.53

877 1092 1216 1287 1414 3647

a ∆∆ ∆ ∆f f fH H H2980

2980

00= −

Appendix 1

A1-6

Appendix 1.3 C2 Hydrofluorocarbons: Rotational constants and vibrational frequencies (scaled by 0.8929) obtained at MP2(Full)/6-31G(d)

and HF/6-31G(d) levels of theory respectively and the resulting zero point energies and thermal correctionsa to the heats of formation.


(kcal mol−1)

∆∆ f H2980

(kcal mol−1)

C2H6 2.6863 0.6695 0.6695 292 794 794 947 1194 1194 44.69 2.81

1382 1410 1468 1468 1473 1473

2857 2862 2901 2901 2923 2923

CH3CH2F 1.1984 0.3144 0.2749 244 392 784 867 1044 1107 40.98 3.04

1169 1270 1381 1418 1452 1469

1503 2869 2896 2924 2930 2948

CH2FCH2F 1.0612 0.1293 0.1205 129 273 444 787 1045 1065 37.16 3.41

1068 1157 1211 1268 1338 1444

1500 1506 2914 2916 2951 2975

CH3CHF2 0.3115 0.3017 0.1721 230 364 449 548 846 957 36.71 3.33

1120 1145 1158 1376 1391 1432

1455 1458 2883 2947 2952 2966

CHF2CH2F 0.3012 0.1222 0.0939 114 235 413 465 559 896 32.81 3.71

1081 1111 1124 1154 1238 1332

1402 1457 1486 2923 2971 2986

CH3CF3 0.1807 0.1724 0.1724 224 350 350 522 522 577 31.87 3.66

811 977 977 1263 1263 1277

1429 1453 1453 2898 2974 2974

Appendix 1

A1-7


(kcal mol−1)

∆∆ f H2980

(kcal mol−1)

CHF2CHF2 0.1679 0.1063 0.0689 85 196 348 406 476 523 28.35 4.09

607 1094 1123 1139 1148 1165

1306 1362 1390 1480 2985 2994

CH2FCF3 0.1767 0.0934 0.0923 105 209 341 398 516 530 27.94 4.07

646 828 980 1101 1200 1217

1303 1314 1448 1485 2935 2990

CHF2CF3 0.1210 0.0814 0.0671 72 201 235 351 406 505 23.45 4.48

559 569 706 859 1130 1169

1217 1256 1324 1385 1466 2995

CF3CF3 0.0931 0.0618 0.0618 62 205 205 337 370 370 18.46 4.88

504 504 602 602 691 793

1110 1274 1274 1280 1280 1453

CH3CH2 3.4574 0.7584 0.7040 148 407 778 967 994 1169 35.49 3.10

1386 1436 1455 1460 2822 2883

2914 2960 3047

CH2FCH2 1.3690 0.3355 0.2866 157 383 440 831 956 1061 31.83 3.27

1101 1228 1388 1426 1486 2857

2909 2974 3071

CH3CHF 1.5509 0.3172 0.2791 184 389 640 879 1023 1088 32.42 3.13

1152 1338 1405 1442 1458 2839

2903 2935 3000

Appendix 1

A1-8


(kcal mol−1)

∆∆ f H2980

(kcal mol−1)

CH2FCHF 1.2642 0.1306 0.1217 101 276 449 648 1005 1065 28.59 3.46

1075 1137 1208 1290 1426 1491

2875 2933 3019

CHF2CH2 0.3276 0.3174 0.1759 127 367 376 470 615 909 27.62 3.57

957 1142 1152 1369 1391 1427

2928 2993 3096

CH3CF2 0.3328 0.3074 0.1693 186 354 443 520 837 973 28.53 3.38

1081 1248 1250 1408 1447 1450

2863 2936 2966

CH2FCF2 0.3162 0.1234 0.0930 96 226 412 443 558 883 24.67 3.74

1047 1095 1208 1232 1292 1422

1481 2896 2967

CHF2CHF 0.3131 0.1241 0.0937 87 238 415 459 560 713 24.31 3.77

993 1127 1136 1174 1297 1389

1432 2945 3039

CF3CH2 0.1834 0.1817 0.1774 120 314 361 460 519 569 22.84 3.89

594 832 933 1183 1267 1284

1429 3001 3110

CHF2CF2 0.1724 0.1068 0.0684 70 194 345 392 476 524 20.27 4.11

617 1002 1127 1165 1240 1261

1379 1433 2954

Appendix 1

A1-9


(kcal mol−1)

∆∆ f H2980

(kcal mol−1)

C2F4H 0.1815 0.0939 0.0923 79 204 339 403 506 537 19.46 4.13

653 713 844 1161 1193 1219

1289 1424 3049

CF3CF2 0.1233 0.0809 0.0661 58 197 216 350 408 501 15.37 4.51

571 582 697 830 1134 1238

1271 1286 1418

CH2CH2 4.9089 0.9996 0.8305 801 978 982 1032 1208 1337 30.69 2.51

1438 1658 2964 2985 3030 3053

CH2CHF 2.1652 0.3509 0.3020 468 715 916 922 977 1148 26.70 2.71

1301 1393 1691 3002 3059 3085

CHFCHF−Z 1.8979 0.1329 0.1242 309 338 538 846 932 1127 22.44 3.10

1143 1275 1275 1745 3079 3088

CHFCHF−E 0.6950 0.1960 0.1529 225 492 748 804 906 1005 22.66 3.04

1112 1255 1381 1760 3071 3095

CH2CF2 0.3601 0.3459 0.1764 423 528 630 713 865 918 22.35 2.95

951 1332 1386 1748 3027 3115

CHFCF2 0.3492 0.1280 0.0937 223 307 469 586 605 816 18.06 3.42

922 1145 1265 1369 1824 3105

CF2CF2 0.1792 0.1078 0.0673 199 200 382 431 530 540 13.44 3.90

563 777 1164 1341 1355 1915

CH3CH 4.0717 0.8832 0.8312 356 646 915 1021 1231 1335 28.23 2.75

1395 1473 2775 2804 2864 2930

Appendix 1

A1-10


(kcal mol−1)

∆∆ f H2980

(kcal mol−1)

CH3CF 1.7801 0.3549 0.3129 98 485 751 923 1045 1192 24.87 3.12

1356 1412 1444 2831 2903 2955

CH2FCH 1.6163 0.3499 0.3038 259 449 798 905 1056 1105 24.57 2.97

1249 1351 1384 2861 2871 2901

CH2FCF 1.4920 0.1385 0.1299 147 333 496 712 993 1117 21.10 3.30

1162 1218 1380 1422 2867 2913

CHF2CH 0.3603 0.3458 0.1764 366 414 499 536 798 1006 20.50 3.15

1120 1170 1340 1374 2831 2884

CHF2CF 0.3285 0.1391 0.1034 58 260 420 540 556 860 16.96 3.69

1133 1164 1224 1361 1377 2912

CF3CH 0.2050 0.1898 0.1731 153 339 420 521 528 580 15.83 3.65

821 1047 1182 1268 1319 2900

CF3CF 0.1823 0.0990 0.0985 11 267 363 398 520 529 12.34 4.07

675 827 1223 1227 1261 1327

CH2CH 7.8272 1.1191 0.9791 740 790 856 1068 1257 1460 21.69 2.55

2927 3012 3062

CHFCH−Z 2.3878 0.3837 0.3306 437 629 825 853 1053 1253 18.11 2.75

1452 3062 3100

CHFCH−E 3.0421 0.3641 0.3252 481 644 756 810 1080 1247 18.03 2.74

1475 3017 3104

CH2CF 3.9358 0.3430 0.3155 427 614 797 905 1093 1367 18.29 2.75

1537 2975 3077

Appendix 1

A1-11


(kcal mol−1)

∆∆ f H2980

(kcal mol−1)

CHFCF−Z 0.8850 0.1829 0.1516 204 402 719 727 945 1120 14.46 3.09

1325 1577 3099

CHFCF−E 2.8566 0.1322 0.1264 296 298 518 685 1049 1163 14.14 3.15

1260 1569 3052

CF2CH 0.3880 0.3650 0.1881 416 519 528 624 819 904 14.02 3.00

1279 1586 3130

CF2CF 0.3742 0.1260 0.0943 204 279 414 459 588 864 10.00 3.51

1204 1317 1663

CH3C 5.3133 0.9550 0.9485 662 774 997 1334 1362 1433 21.59 2.54

2797 2859 2887

CH2FC 1.8557 0.3715 0.3287 441 511 914 1060 1159 1339 17.93 2.75

1388 2845 2885

CHF2C 0.3723 0.3384 0.1916 362 421 527 811 1115 1125 14.15 2.99

1307 1362 2868

CF3C 0.2087 0.1974 0.1795 280 299 515 524 570 816 9.60 3.36

1201 1218 1293

HCCH 1.1605 710 710 789 789 2006 3219 16.50 2.31

3320

HCCF 0.3167 442 442 684 684 1043 2262 12.65 2.58

3294

FCCF 0.1156 291 291 408 408 771 1321 8.57 3.06

2502

CH2C 9.5141 1.3022 1.1454 505 828 1235 1650 2970 3045 14.63 2.56

Appendix 1

A1-12


(kcal mol−1)

∆∆ f H2980

(kcal mol−1)

CHFC 3.4765 0.3821 0.3443 360 780 1044 1157 1691 3018 11.51 2.68

CF2C 0.4245 0.3592 0.1946 351 494 585 928 1297 1706 7.67 2.88

CCH 1.5328 496 496 1654 3234 8.41 2.36

CCF 0.3697 218 218 1003 2072 5.02 2.77

CH2CO 9.4884 0.3370 0.3254 439 555 646 993 1131 1399 19.17 2.79

2124 3015 3105

CHFCO 1.7613 0.1517 0.1396 242 472 561 668 1024 1169 15.44 3.07

1408 2152 3106

CF2CO 0.3491 0.1288 0.0941 207 273 378 442 673 794 11.07 3.52

1315 1451 2209

CHCO 27.3099 0.3613 0.3565 485 553 606 1172 2207 3013 11.49 2.75

CFCO 37.0140 0.1318 0.1314 204 433 590 940 1691 2286 8.78 3.03

CH3CHO 1.8856 0.3366 0.3016 136 488 764 861 1099 1129 33.58 3.08

1371 1398 1434 1443 1815 2812

2863 2912 2964

CH2FCHO 1.3230 0.1429 0.1322 85 315 515 721 1019 1078 29.72 3.38

1105 1222 1328 1392 1461 1821

2858 2911 2960

CHF2CHO 0.3075 0.1291 0.0998 76 316 369 416 592 978 25.57 3.70

1082 1128 1131 1312 1369 1395

1841 2875 3003

CF3CHO 0.1806 0.0993 0.0977 71 251 309 423 515 515 20.73 4.05

688 835 981 1225 1225 1330

1383 1858 2894

Appendix 1

A1-13


(kcal mol−1)

∆∆ f H2980

(kcal mol−1)

CH3CFO 0.3606 0.3236 0.1761 130 394 561 585 837 1002 29.72 3.32

1059 1222 1398 1438 1444 1894

2887 2945 2991

CH2FCFO 0.3440 0.1286 0.0953 112 243 443 528 635 870 25.84 3.62

1034 1121 1167 1238 1421 1468

1925 2913 2960

CHF2CFO 0.1786 0.1107 0.0745 53 224 311 427 542 586 21.50 4.02

747 924 1116 1183 1211 1381

1424 1932 2975

CF3CFO 0.1256 0.0834 0.0686 47 219 233 375 417 501 16.57 4.41

578 678 763 801 1122 1244

1293 1373 1944

CH3CO 2.7440 0.3313 0.3128 87 454 826 938 1037 1356 26.08 3.12

1432 1433 1911 2871 2946 2950

CH2FCO 1.6312 0.1424 0.1343 137 312 496 842 895 1100 22.29 3.30

1215 1346 1451 1925 2908 2964

CHF2CO 0.3213 0.1323 0.1002 57 367 407 411 590 947 18.11 3.65

1133 1150 1319 1366 1940 2977

CF3CO 0.1856 0.1000 0.1000 57 234 388 412 523 531 13.27 4.01

667 795 1203 1241 1268 1960

a ∆∆ ∆ ∆f f fH H H2980

2980

00= −

b Computed from HF/6-31G(d) geometry.

Appendix 1

A1-14

Appendix 1.4 C3 Hydrofluorocarbons: Rotational constants and vibrational frequencies (scaled by 0.8929) obtained at MP2(Full)/6-31G(d)

and HF/6-31G(d) levels of theory respectively and the resulting zero point energies and thermal correctionsa to the heats of formation.


(kcal mol−1)

∆∆ f H2980

(kcal mol−1)

C3H6 1.5495 0.3109 0.2720 189 407 572 880 919 954 47.88 3.21

1008 1060 1158 1289 1389 1424

1451 1464 1679 2852 2895 2923

2964 2976 3039

C3H7 1.0384 0.2983 0.2674 118 241 314 481 713 855 52.93 3.76

860 997 1049 1171 1277 1315

1388 1432 1458 1466 1475 2853

2858 2881 2912 2917 2952 3039

C3F6 0.0840 0.0419 0.0328 36 125 170 238 245 351 21.39 5.61

358 457 497 566 586 638

660 751 1025 1198 1228 1254

1342 1404 1832

C3F6H 0.0760 0.0460 0.0398 34 89 143 204 234 287 28.08 5.82

348 447 509 517 567 623

740 840 924 1123 1176 1239

1247 1296 1314 1379 1403 2963

a ∆∆ ∆ ∆f f fH H H2980

2980

00= −

Appendix 2

A2-1

Appendix 2: Phosphorus Compounds

Supplementary Information

Appendix 2.1 Geometric data for molecules in Table 4.1. Bond lengths in Å, angles in

degrees, rotational constants and vibrational frequencies in cm−1.

H2 - 1ΣΣΣΣg - D∞h

G3

Z-matrix

H(1)H(2) 1 0.7375

Rotational Constant

61.5135

Vibrational Frequency

4149

G3X

Z-matrix

H(1)H(2) 1 0.7427

Rotational Constant

60.6414


4401

H(2)H(1)

Appendix 2

A2-2

O2 - 3ΣΣΣΣg - D∞h

G3

Z-matrix

O(1)O(2) 1 1.2460

Rotational Constant

1.3578


1784

G3X

Z-matrix

O(1)O(2) 1 1.2064

Rotational Constant

1.4484


1640

O(2)O(1)

Appendix 2

A2-3

P2 - 1ΣΣΣΣg - D∞h

G3

Z-matrix

P(1)P(2) 1 1.9324

Rotational Constant0.2915


811

G3X

Z-matrix

P(1)P(2) 1 1.8952

Rotational Constant0.3031


793

P(2)P(1)

Appendix 2

A2-4

P4 - 1A1 - Td

G3

Z-matrix

P(1)P(2) 1 2.1948P(3) 1 2.1948 2 60.0P(4) 1 2.1948 2 60.0 3 -70.5

Rotational Constants

0.1130, 0.1130, 0.1130

Vibrational Frequencies

363, 363, 466, 466, 466, 615

G3X

Z-matrix

P(1)P(2) 1 2.2112P(3) 1 2.2112 2 60.0P(4) 1 2.2112 2 60.0 3 -70.5


0.1113, 0.1113, 0.1113


363, 363, 454, 454, 454, 599

P(3) P(4)

P(1)

P(2)

Appendix 2

A2-5

HO - 2ΠΠΠΠ - C∞v

G3

Z-matrix

O(1)H(2) 1 0.9790

Rotational Constant

18.5531


3569

G3X

Z-matrix

O(1)H(2) 1 0.9761

Rotational Constant

18.6606


3642

H(2)O(1)

Appendix 2

A2-6

H2O - 1A1 - C2v

G3

Z-matrix

O(1)H(2) 1 0.9686H(3) 1 0.9686 2 103.9


26.4131, 14.3801, 9.3109


1631, 3635, 3740

G3X

Z-matrix

O(1)H(2) 1 0.9620H(3) 1 0.9620 2 103.7


26.6798, 14.6088, 9.4399


1647, 3749, 3852

H(2)

O(1)

H(3)

Appendix 2

A2-7

HO2 - 2A″″″″ - Cs

G3

Z-matrix

O(1)H(2) 1 0.9835O(3) 1 1.3247 2 104.6


20.2165, 1.1318, 1.0718


1117, 1450, 3583

G3X

Z-matrix

O(1)H(2) 1 0.9774O(3) 1 1.3238 2 105.4


20.6569, 1.1316, 1.0728


1168, 1428, 3529

H(2)

O(1) O(3)

Appendix 2

A2-8

PH - 3ΣΣΣΣ - C∞v

G3

Z-matrix

P(1)H(2) 1 1.4256

Rotational Constant

8.4977


2283

G3X

Z-matrix

P(1)H(2) 1 1.4321

Rotational Constant

8.4209


2302

H(2)P(1)

Appendix 2

A2-9

PH2 - 2B1 - C2v

G3

Z-matrix

P(1)H(2) 1 1.4199H(3) 1 1.4199 2 92.5


9.2479, 7.9442, 4.2733


1123, 2302, 2303

G3X

Z-matrix

P(1)H(2) 1 1.4265H(3) 1 1.4265 2 91.6


9.0041, 7.9990, 4.2359


1113, 2328, 2338

H(2)

P(1)

H(3)

Appendix 2

A2-10

PH3 - 1A′′′′ - Cs

G3

Z-matrix

P(1)H(2) 1 1.4146H(3) 1 1.4146 2 94.6H(4) 1 1.4146 2 94.6 3 95.0


4.5307, 4.5304, 3.8680


1018, 1135, 1135, 2323, 2325,2330

G3X

Z-matrix

P(1)H(2) 1 1.4219H(3) 1 1.4219 2 93.3H(4) 1 1.4219 2 93.3 3 93.5


4.4376, 4.4374, 3.9147


1010, 1126, 1126, 2355, 2364,2367

H(4)H(2)

P(1)

H(3)

Appendix 2

A2-11

P2H2 - 1Ag - Cs

G3

Z-matrix

P(1)P(2) 1 2.0429H(3) 1 1.4221 2 94.7H(4) 2 1.4221 1 94.7 3 180.0


4.3429, 0.2501, 0.2364


626, 686, 775, 961, 2299, 2311

G3X

Z-matrix

P(1)P(2) 1 2.0351H(3) 1 1.4294 2 94.2H(4) 2 1.4294 1 94.2 3 180.0


4.2893, 0.2522, 0.2382


611, 678, 766, 962, 2298, 2313

H(4)

P(1)

P(2)

H(3)

Appendix 2

A2-12

P2H4 - 1Ag - Cs

G3

Z-matrix

P(1)P(2) 1 2.2311H(3) 1 1.4170 2 95.2H(4) 1 1.4170 2 95.2 3 -94.1H(5) 2 1.4170 1 95.2 3 -85.9H(6) 2 1.4170 1 95.2 3 180.0


2.1786, 0.1937, 0.1917


58, 436, 622, 649, 869, 908,1103, 1106, 2311, 2319, 2321,2324

G3X

Z-matrix

P(1)P(2) 1 2.2594H(3) 1 1.4242 2 93.8H(4) 1 1.4242 1 93.8 3 -92.4H(4) 2 1.4242 1 93.8 3 -87.6H(4) 2 1.4242 1 93.8 3 180.0


2.1468, 0.1894, 0.1881


103, 422, 613, 629, 854, 896,1084, 1093, 2333, 2338, 2340,2351

H(4)

H(5)

P(1)

P(2)

H(3)

H(6)

Appendix 2

A2-13

PO - 2ΠΠΠΠ - C∞v

G3

Z-matrix

P(1)O(2) 1 1.4715

Rotational Constant

0.7381


1254

G3X

Z-matrix

P(1)O(2) 1 1.4831

Rotational Constant

0.7266


1241

O(2)P(1)

Appendix 2

A2-14

PO2 - 2A1 - C2v

G3

Z-matrix

P(1)O(2) 1 1.4924O(3) 1 1.4924 2 136.4


3.4850, 0.2745, 0.2544


409, 1079, 1301

G3X

Z-matrix

P(1)O(2) 1 1.4743O(3) 1 1.4743 2 134.4


3.2851, 0.2852, 0.2625


378, 1061, 1314

O(2)

P(1)

O(3)

Appendix 2

A2-15

PO3 - 2A2′′′′ - D3h

G3

Z-matrix

P(1)O(2) 1 1.5025O(3) 1 1.5025 2 120.0O(4) 1 1.5025 2 120.0 3 180.0


0.3112, 0.3112, 0.1556


442, 442, 443, 1028, 1690, 1690

G3X

Z-matrix

P(1)O(2) 1 1.4792O(3) 1 1.4792 2 120.0O(4) 1 1.4792 2 120.0 3 180.0


0.3211, 0.3211, 0.1606


152, 152, 424, 1009, 1107, 1108

O(4)

P(1)

O(2)

O(3)

Appendix 2

A2-16

PO3 - 2B2 - C2v

Z-matrix

P(1)O(2) 1 1.6076O(3) 1 1.4726 2 111.4O(4) 1 1.4726 2 111.4 3 180.0


0.3394, 0.2804, 0.1535


343, 431, 434, 856, 1176, 1465

O(3)

P(1)

O(2)

O(4)

Appendix 2

A2-17

PPO - 1ΣΣΣΣ - C∞v

G3

Z-matrix

P(1)P(2) 1 1.9217O(3) 1 1.5029 2 180.0

Rotational Constant

0.1255


179, 179, 669, 1275

G3X

Z-matrix

P(1)P(2) 1 1.8926O(3) 1 1.4730 2 180.0

Rotational Constant

0.1298


214, 215, 657, 1294

P(2) P(1)O(3)

Appendix 2

A2-18

P2O - 1A1 - C2v

Z-matrix

P(1)O(2) 1 1.7755P(3) 2 1.7755 1 69.0


0.6193, 0.2691, 0.1876


241, 655, 825

O(2)

P(3)P(1)

Appendix 2

A2-19

P2O2 - 1Ag - D2h

G3

Z-matrix

P(1)O(2) 1 1.6916P(3) 2 1.6916 1 96.6O(4) 3 1.6916 2 83.4 1 0.0


0.4158, 0.1707, 0.1210


24, 446, 615, 720, 759, 912

G3X

Z-matrix

P(1)O(2) 1 1.6611P(3) 2 1.6611 1 96.5O(4) 3 1.6620 2 83.6 1 0.0


0.4298, 0.1773, 0.1255


445, 552, 619, 708, 731, 876

O(2)

P(3)P(1)

O(4)

Appendix 2

A2-20

P2O2 - C1

Z-matrix

P(1)O(2) 1 1.7500P(3) 2 1.7479 1 70.9O(4) 3 1.7491 2 83.9 1 50.2


0.2922, 0.2168, 0.1569


385, 396, 513, 652, 785, 877

O(4)

P(1) P(3)

O(2)

Appendix 2

A2-21

P2O2 - 3A″″″″ - Cs

Z-matrix

P(1)O(2) 1 1.6977P(3) 2 1.6559 1 129.6O(4) 3 1.4938 2 111.5 1 0.0


0.4799, 0.0959, 0.0799


129, 168, 463, 591, 875, 1275

P(1)

O(2)

P(3)

O(4)

Appendix 2

A2-22

P2O2 - 3A″″″″ - Cs

Z-matrix

P(1)O(2) 1 1.6836P(3) 2 1.6645 1 128.0O(4) 3 1.4895 2 108.6 1 180.0


1.2663, 0.0729, 0.0690


93, 137, 346, 646, 893, 1291

P(1)

O(2)

P(3)

O(4)

Appendix 2

A2-23

P2O3 - 1A - C2

G3

Z-matrix

O(1)P(2) 1 1.4920O(3) 2 1.6787 1 111.1P(4) 3 1.6787 2 135.3 1 28.3O(5) 4 1.4920 3 111.1 2 28.3


0.2676, 0.0693, 0.0584


76, 86, 98, 389, 497, 571,833, 1279, 1296

G3X

Z-matrix

O(1)P(2) 1 1.4665O(3) 2 1.6499 1 110.6P(4) 3 1.6499 2 143.8 1 34.5O(5) 4 1.4665 3 110.6 2 34.5


0.3024, 0.0636, 0.0567


52, 72, 94, 390, 459, 555,841, 1267, 1282

O(1)

P(4)

O(3)

P(2)

O(5)

Appendix 2

A2-24

HPO - 1A′′′′ - Cs

G3

Z-matrix

P(1)O(2) 1 1.5170H(3) 1 1.4530 2 105.6


9.0393, 0.6702, 0.6239


1007, 1228, 2153

G3X

Z-matrix

P(1)O(2) 1 1.4851H(3) 1 1.4727 2 104.9


8.7388, 0.6994, 0.6475


998, 1212, 2052

O(2)

P(1)

H(3)

Appendix 2

A2-25

POH - 3A″″″″ - Cs

Z-matrix

P(1)O(2) 1 1.6668H(3) 2 0.9741 1 112.6


22.6958, 0.5352, 0.5228


798, 928, 3651

O(2)P(1)

H(3)

Appendix 2

A2-26

HPOH - 2A″″″″ - Cs

G3

Z-matrix

P(1)O(2) 1 1.6759H(3) 1 1.4161 2 93.8H(4) 2 0.9730 1 108.9 3 180.0


6.2768, 0.5224, 0.4823


364, 795, 891, 1091, 2322, 3669

G3X

Z-matrix

P(1)O(2) 1 1.6585H(3) 1 1.4276 2 93.9H(4) 2 0.9637 1 109.7 3 180.0


6.2639, 0.5326, 0.4909


441, 812, 913, 1115, 2312, 3772

H(3)

P(1)

O(2)

H(4)

Appendix 2

A2-27

HPOH - 2A″″″″ - Cs

Z-matrix

P(1)O(2) 1 1.6701H(3) 1 1.4284 2 98.8H(4) 2 0.9714 1 113.7 3 0.0


6.2564, 0.5222, 0.4820


178, 797, 878, 1069, 2248, 3685

H(3)

P(1)O(2)

H(4)

Appendix 2

A2-28

H3PO - 1A1 - C3v

G3

Z-matrix

P(1)O(2) 1 1.4977H(3) 1 1.4112 2 117.5H(4) 1 1.4112 2 117.5 3 120.0H(5) 1 1.4112 2 117.5 3 -120.0


3.5612, 0.5645, 0.565


855, 855, 1112, 1112, 1147, 1247,2380, 2380, 2405

G3X

Z-matrix

P(1)O(2) 1 1.4787H(3) 1 1.4187 2 117.3H(4) 1 1.4187 2 117.3 3 120.0H(5) 1 1.4187 2 117.3 3 -120.0


3.5060, 0.5764, 0.5764


834, 834, 1104, 1104, 1143,1264, 2345, 2345, 2366

H(4) H(5)

P(1)

O(2)

H(3)

Appendix 2

A2-29

H2POH - 1A′′′′ - Cs

G3

Z-matrix

P(1)O(2) 1 1.6803H(3) 2 0.9705 1 108.0H(4) 1 1.4159 2 99.1 3 132.5H(5) 1 1.4159 2 99.1 3 -132.5


3.7301, 0.4754, 0.4722


262, 785, 895, 899, 1117, 1142,2321, 2331, 3688

G3X

Z-matrix

P(1)O(2) 1 1.6634H(3) 2 0.9610 1 108.6H(4) 1 1.4267 2 98.8 3 133.4H(5) 1 1.4267 2 98.8 3 -133.4


3.6995, 0.4845, 0.4801


257, 788, 912, 925, 1137, 1144,2313, 2313, 3804

H(4)

P(1)

O(2)

H(3)

H(5)

Appendix 2

A2-30

H2POH - 1A′′′′ - Cs

Z-matrix

P(1)O(2) 1 1.6701H(3) 2 0.9721 1 113.4H(4) 1 1.4234 2 101.8 3 47.7H(5) 1 1.4234 2 101.8 3 -47.7


3.6989, 0.4778, 0.4741


410, 793, 892, 893, 1096, 1144, 2274, 2285, 3663

H(5)

P(1) O(2)

H(3)H(4)

Appendix 2

A2-31

HOPO - 1A′′′′ - Cs

G3

Z-matrix

P(1)O(2) 1 1.6380O(3) 1 1.4960 2 110.5H(4) 2 0.9816 1 112.1 3 0.0


1.1834, 0.3082, 0.2445


394, 546, 848, 956, 1274, 3591

G3X

Z-matrix

P(1)O(2) 1 1.6163O(3) 1 1.4726 2 111.0H(4) 2 0.9706 1 113.6 3 0.0


1.2306, 0.3150, 0.2508


379, 556, 837, 944, 1263, 3681

O(3)

P(1)O(2)

H(4)

Appendix 2

A2-32

HOPO - 1A′′′′ - Cs

Z-matrix

P(1)O(2) 1 1.6432O(3) 1 1.4900 2 108.8H(4) 2 0.9771 1 110.6 3 180.0


1.2970, 0.2980, 0.2423


408, 450, 828, 952, 1296, 3637

O(3)

P(1)

O(2)

H(4)

Appendix 2

A2-33

HPO2 - 1A1 - C2v

Z-matrix

P(1)O(2) 1 1.4793O(3) 1 1.4793 2 134.9H(4) 1 1.4034 2 112.5 3 180.0


2.1344, 0.2823, 0.2493


475, 663, 1053, 1151, 1454, 2476

O(2)

P(1)

H(4)

O(3)

Appendix 2

A2-34

HOPO2 - 1A′′′′ - Cs

G3

Z-matrix

P(1)O(2) 1 1.6047O(3) 1 1.4728 2 112.1O(4) 1 1.4788 2 113.5 3 180.0H(5) 2 0.9779 1 110.5 3 180.0


0.3110, 0.2799, 0.1473


391, 433, 450, 515, 899, 1028,1180, 1445, 3623

G3X

Z-matrix

P(1)O(2) 1 1.5846O(3) 1 1.4536 2 112.0O(4) 1 1.4591 2 113.9 3 180.0H(5) 2 0.9677 1 111.2 3 180.0


0.3175, 0.2883, 0.1511


391, 424, 435, 502, 889, 1036,1179, 1452, 3736

O(3)

P(1)

O(4)

O(2)

H(5)

Appendix 2

A2-35

Appendix 2.2 Computed Heats of Reaction for Twarowski’sa data set.

Reaction No. Reaction0298r H∆ (kcal mol−1)

G3 G3Xb Literaturec Diff(G3X−Lit)

1 H + H → H2 −104.7 −104.6 −104.2 −0.4

2 O + O → O2 −118.0 −119.1 −119.1 0.0

3 H + O → OH −103.2 −103.2 −102.8 −0.4

4 H + OH → H2O −118.0 −118.0 −118.7 0.7

5 H + O2 → HO2 −49.9 −48.9 −48.8 −0.1

6 H + O2 → OH + O 14.8 15.9 16.3 −0.4

7 H2 + OH → H2O + H −13.3 −13.4 −14.5 1.1

8 OH + OH → O + H2O −14.7 −14.8 −15.9 1.1

9 O + H2 → OH + H 1.4 1.4 1.4 0.0

10 HO2 + H → OH + OH −38.6 −38.4 −37.7 −0.7

11 HO2 + H → H2 + O2 −54.8 −55.7 −55.4 −0.3

12 HO2 + O → O2 + OH −53.3 −54.3 −54.0 −0.3

13 HO2 + OH → H2O + O2 −68.1 −69.1 −69.9 0.8

14 H + PO → HPO −65.1 −65.4 −66.9 1.5

15 H + PO2 → HOPO −92.9 −92.2 −94.2 2.0

16 H + PO3 → HOPO2 −115.2 −113.2 −116.0 2.8

17 H + HPO → HPOH −54.1 −53.5

18 H + P → PH −71.7 −72.0

19 PH2 + H → PH3 −81.6 −81.8

20 PH + H → PH2 −75.5 −75.6

21 HPOH + H → H2POH −76.5 −77.3

22 O + PO → PO2 −119.5 −120.9 −122.0 1.1

23 PO2 + O → PO3 −93.7 −95.8 −96.8 1.0

24 HOPO + O → HOPO2 −116.0 −116.7 −118.6 1.9

25 P + O → PO −142.7 −143.9 −143.0 −0.9

26 P2 + O → P2O −89.5 −72.5

27 PH + O → HPO −136.1 −137.3

28 O + P2O → P2O2 −124.8 −126.2

29 OH + PO → HOPO −109.1 −110.0 −113.4 3.5

30 OH + PO2 → HOPO2 −105.7 −105.7 −109.9 4.2

31 OH + PH2 → H2POH −87.9 −89.2

32 PH + OH → HPOH −86.9 −87.5

33 PO + PO → P2O2 −44.6 −46.0

34 PO + PO2 → P2O3 −76.0 −76.4

35 PO + P → P2O −62.5 −63.6

36 P + P → P2 −115.8 −135.0

Appendix 2

A2-36



37 P2 + P2 → P4 −52.7 −52.3

38 H2O + PO3 → OH + HOPO2 2.8 4.9 2.7 2.1

39 H2 + PO3 → HOPO2 + H −10.5 −8.6 −11.8 3.2

40 O2 + PO → O + PO2 −1.5 −1.8 −2.9 1.1

41 P + O2 → O + PO −24.8 −24.8 −23.9 −0.9

42 O2 + PH → O + HPO −18.1 −18.2

43 H + HOPO → H2O + PO −8.8 −8.0 −5.3 −2.7

44 H + HOPO → H2 + PO2 −11.8 −12.4 −10.0 −2.4

45 H + HOPO2 → H2O + PO2 −12.3 −12.3 −8.8 −3.5

46 H + PO3 → O + HOPO 0.9 3.5 2.5 1.0

47 H + PO3 → OH + PO2 −9.5 −7.4 −6.1 −1.4

48 H + P2O3 → HOPO + PO −16.9 −15.8

49 H + HPO → H2 + PO −39.6 −39.2 −37.3 −1.9

50 H + PH3 → H2 + PH2 −23.0 −22.8

51 H + PH2 → H2 + PH −29.2 −29.0

52 H + PH → H2 + P −32.9 −32.6

53 H + P2O → OH + P2 −13.7 −30.7

54 H + P2O → PO + PH −9.2 −8.4

55 H + P2O → HPO + P −2.5 −1.8

56 H + P2O2 → PO + HPO −20.4 −19.5

57 H + H2POH → H2O + PH2 −30.0 −28.8

58 H + H2POH → H2 + HPOH −28.2 −27.3

59 H + HPOH → H2O + PH −31.0 −30.5

60 H + HPOH → H2 + HPO −50.6 −51.1

61 O + HOPO → OH + PO2 −10.4 −11.0 −8.6 −2.4

62 O + HOPO2 → O2 + HOPO −1.9 −2.4 −0.6 −1.8

63 O + PO3 → O2 + PO2 −24.3 −23.3 −22.3 −1.0

64 O + P2O3 → PO + PO3 −17.7 −19.4

65 O + P2O3 → PO2 + PO2 −43.6 −44.5

66 O + HPO → H + PO2 −54.5 −55.5 −55.1 −0.4

67 O + HPO → OH + PO −38.2 −37.8 −35.9 −1.9

68 O + P2 → P + PO −27.0 −8.9

69 O + PH3 → OH + PH2 −21.6 −21.4

70 O + PH2 → H + HPO −60.6 −61.6

71 O + PH2 → OH + PH −27.8 −27.6

72 O + PH → H + PO −71.0 −71.9

73 O + PH → OH + P −31.5 −31.2

Appendix 2

A2-37



74 O + P2O → O2 + P2 −28.5 −46.6

75 O + P2O → PO + PO −80.2 −80.3

76 O + P2O → PO2 + P −57.0 −57.3

77 O + P2O2 → O2 + P2O 6.8 7.1

78 O + P2O2 → PO + PO2 −74.9 −75.0

79 O + H2POH → OH + HPOH −26.8 −25.9

80 O + HPOH → H + HOPO −93.2 −94.3

81 O + HPOH → OH + HPO −49.1 −49.7

82 OH + PO → H + PO2 −16.3 −17.7 −19.2 1.5

83 OH + HOPO → H2O + PO2 −25.1 −25.8 −24.5 −1.3

84 OH + HOPO → H + HOPO2 −12.8 −13.5 −15.7 2.2

85 OH + PO3 → O + HOPO2 −12.0 −10.0 −13.2 3.2

86 OH + P2O3 → PO + HOPO2 −29.7 −29.3

87 OH + P2O3 → PO2 + HOPO −33.2 −33.6

88 OH + HPO → H2O + PO −52.9 −52.6 −51.8 −0.8

89 OH + HPO → H + HOPO −44.1 −44.6 −46.5 2.0

90 OH + P → H + PO −39.5 −40.7 −40.1 −0.5

91 OH + PH3 → H2O + PH2 −36.4 −36.2

92 OH + PH3 → H + H2POH −6.3 −7.3

93 OH + PH2 → H2O + PH −42.5 −42.4

94 OH + PH2 → H + HPOH −11.5 −11.9

95 OH + PH → H2O + P −46.3 −46.0

96 OH + PH → H + HPO −32.8 −34.1

97 OH + P2O → H + P2O2 −21.6 −23.0

98 OH + P2O → P + HOPO −46.6 −46.4

99 OH + P2O2 → PO + HOPO −64.5 −64.0

100 OH + H2POH → H2O + HPOH −41.5 −40.7

101 OH + HPOH → H2O + HPO −63.9 −64.6

102 HO2 + PO → O2 + HPO −15.1 −16.5 −18.1 1.6

103 HO2 + PO → O + HOPO −44.5 −45.2 −48.4 3.2

104 HO2 + PO → OH + PO2 −54.9 −56.2 −57.0 0.8

105 HO2 + PO2 → O2 + HOPO −42.9 −43.4 −45.4 2.1

106 HO2 + PO2 → O + HOPO2 −41.0 −41.0 −44.9 3.9

107 HO2 + PO2 → OH + PO3 −29.1 −31.0 −31.7 0.7

108 HO2 + HOPO → OH + HOPO2 −51.4 −51.9 −53.5 1.6

109 HO2 + PO3 → O2 + HOPO2 −65.3 −64.3 −67.2 2.9

110 HO2 + HPO → O2 + HPOH −4.2 −4.6

Appendix 2

A2-38



111 HO2 + P → O2 + PH −21.8 −23.1

112 HO2 + P → OH + PO −78.1 −79.1 −77.9 −1.2

113 HO2 + P2 → OH + P2O −24.9 −7.7

114 HO2 + PH2 → O2 + PH3 −31.7 −32.9

115 HO2 + PH2 → O + H2POH −23.3 −24.4

116 HO2 + PH → O2 + PH2 −25.6 −26.8

117 HO2 + PH → O + HPOH −22.3 −22.8

118 HO2 + PH → OH + HPO −71.4 −72.5

119 HO2 + P2O → OH + P2O2 −60.2 −61.4

120 HO2 + HPOH → O2 + H2POH −26.6 −28.4

121 PO + HOPO2 → HOPO + PO2 −3.5 −4.2 −3.5 −0.7

122 PO + PO3 → PO2 + PO2 −25.8 −25.2 −25.3 0.1

123 PO + P2O → PO2 + P2 −30.0 −48.4

124 PO + P2O2 → PO2 + P2O 5.3 5.3

125 PO + H2POH → HOPO + PH2 −21.2 −20.8

126 PO + HPOH → HOPO + PH −22.2 −22.4

127 PO + HPOH → HPO + HPO −11.0 −11.9

128 PO2 + HPO → H + P2O3 −10.9 −11.0

129 PO2 + HPO → PO + HOPO −27.8 −26.8 −27.3 0.5

130 PO2 + P → PO + PO −23.2 −23.0 −20.9 −2.0

131 PO2 + PH3 → HOPO + PH2 −11.2 −10.4

132 PO2 + PH2 → HOPO + PH −17.4 −16.6

133 PO2 + PH → PO + HPO −16.6 −16.3

134 PO2 + PH → HOPO + P −21.1 −20.2

136 PO2 + P2O → PO3 + P2 −4.2 −23.3

136 PO2 + P2O → P2O3 + P −13.4 −12.8

137 PO2 + P2O → PO + P2O3 −31.4 −30.5

138 PO2 + H2POH → HOPO + HPOH −16.4 −15.0

139 PO2 + H2POH → HOPO2 + PH2 −17.7 −16.6

140 PO2 + HPOH → HOPO + HPO −38.8 −38.8

141 PO2 + HPOH → HOPO2 + PH −18.7 −18.2

142 HOPO + PO3 → PO2 + HOPO2 −22.3 −20.9 −21.8 0.9

143 HOPO + P2O → HOPO2 + P2 −26.5 −44.2

145 HOPO + P2O2 → HOPO2 + P2O 8.8 9.5

145 HOPO2 + P → PO + HOPO −26.7 −27.2 −24.4 −2.8

146 HOPO2 + PH → HOPO + HPO −20.0 −20.6

147 PO3 + HPO → PO + HOPO2 −50.1 −47.8 −49.1 1.3

Appendix 2

A2-39



148 PO3 + P → PO + PO2 −49.0 −48.1 −46.2 −1.9

149 PO3 + PH3 → HOPO2 + PH2 −33.6 −31.3

150 PO3 + PH2 → HOPO2 + PH −39.7 −37.5

152 PO3 + PH → HPO + PO2 −42.4 −41.5

152 PO3 + PH → HOPO2 + P −43.5 −41.1

153 PO3 + P2O → PO2 + P2O2 −31.1 −30.4

154 PO3 + H2POH → HOPO2 + HPOH −38.7 −35.9

156 PO3 + HPOH → HOPO2 + HPO −61.1 −59.7

156 HPO + P → PO + PH −6.7 −6.6

157 HPO + PH2 → PO + PH3 −16.6 −16.4

158 HPO + PH → PO + PH2 −10.4 −10.2

159 HPO + HPOH → PO + H2POH −11.4 −11.9

161 P + PH → H + P2 −44.0 −63.0

161 P + P2O → PO + P2 −53.2 −71.4

162 P + P2O2 → PO + P2O −17.9 −17.7

164 P + HPOH → HPO + PH −17.6 −18.6

164 PH + PH3 → PH2 + PH2 6.1 6.2

165 PH3 + HPOH → PH2 + H2POH 5.1 4.6

166 PH2 + PH → P + PH3 −9.9 −9.8

168 PH2 + HPOH → HPO + PH3 −27.5 −28.4

168 PH + PH → P + PH2 −3.7 −3.6

169 PH + P2O → HPO + P2 −46.5 −64.8

170 PH + P2O2 → HPO + P2O −11.3 −11.1

171 PH + H2POH → PH2 + HPOH 1.0 1.6

172 PH + HPOH → HPO + PH2 −21.4 −22.2

173 PH + HPOH → P + H2POH −4.7 −5.3

174 P2O + P2O → P2 + P2O2 −35.3 −53.7

175 HPOH + HPOH → HPO + H2POH −22.4 −23.8

a A. Twarowski, Combustion and Flame, 1995, 102, 41.

b Using experimental 0298f H∆ for H, O and P and G3X(RAD) type

0298f H∆ for PO and PO2.

c Using experimental 0298f H∆ for H, O, P, OH, H2O and Bauschlicher’s CBS data for PO, PO2, PO3, HPO,

HOPO, HOPO2 (C. W. Bauschlicher, Jr., J. Phys. Chem. A, 1999, 103, 11126.).

Appendix 2

A2-40

Appendix 2.3 Transition State Geometries (Z-matrices with bond lengths in Å and angles in

degrees).

1a: H + PO2 →→→→ HOPO

1000, 1250 K 1500, 1750, 2000 K

P P

O 1 1.497 O 1 1.500

O 1 1.464 2 129.0 O 1 1.464 2 128.1

H 2 2.501 1 95.6 3 0.0 H 2 2.426 1 96.8 3 0.0

1b: HOPO + H →→→→ PO2 + H2

P

O 1 1.564

O 1 1.491 2 121.4

H 2 1.310 1 122.2 3 0.0

H 4 0.858 2 178.9 1 0.0

1c: H2 + OH →→→→ H2O + H

O

H 1 0.979

H 1 1.293 2 98.8

H 3 0.841 1 167.3 2 0.0

Appendix 2

A2-41

2a: PO2 +OH →→→→ HOPO2

1000 K 1250 K

P P

O 1 1.481 O 1 1.480

O 1 1.482 2 129.4 O 1 1.482 2 129.3

O 1 2.927 2 118.7 3 −180.0 O 1 2.839 2 117.8 3 −180.0H 4 0.961 1 112.0 2 180.0 H 4 0.962 1 110.7 2 180.0

1500 K 1750 K

P P

O 1 1.480 O 1 1.479

O 1 1.482 2 129.2 O 1 1.481 2 129.2

O 1 2.794 2 117.4 3 −180.0 O 1 2.747 2 116.9 3 −180.0H 4 0.962 1 110.3 2 180.0 H 4 0.963 1 110.0 2 180.0

2000 K

P

O 1 1.479

O 1 1.481 2 129.3

O 1 2.699 2 116.5 3 −180.0H 4 0.963 1 110.0 2 180.0

2b: HOPO2 + H →→→→ PO2 + H2O

P

O 1 1.483

O 1 1.487 2 135.7

O 1 1.761 2 112.3 3 156.1

H 4 0.989 1 107.2 2 21.3

H 4 1.285 1 135.3 2 −106.8

Appendix 2

A2-42

Appendix 2.4 Transition State Vibrational Frequencies and Rotational Constants (in cm−1).

Reaction Temperature(s) Rotational Constants Vibrational Frequencies

1a: H + PO2 → HOPO 1000, 1250 K 1.2566 0.2936 0.2380 340i 82 152 406 1022 1329

1500, 1750, 2000 K 1.2632 0.2946 0.2389 444i 81 169 407 997 1321

1b: HOPO + H → PO2 + H2 All 1.0019 0.2722 0.2140 2668i 224 270 469 774 787

899 1276 1477

1c: H2 + OH → H2O + H All 18.5846 2.9867 2.5732 2813i 628 675 1283 1443 3602

2a: PO2 +OH → HOPO2 1000 K 0.2919 0.1124 0.0812 200i 86 126 164 426 470

1087 1212 3982

1250 K 0.2918 0.1154 0.0827 211i 93 134 175 428 504

1089 1211 3978

1500 K 0.2916 0.1219 0.0860 236i 108 152 199 432 579

1095 1207 3967

1750 K 0.2915 0.1255 0.0877 250i 117 162 212 434 620

1097 1201 3961

2000 K 0.2914 0.1294 0.0896 265i 127 173 226 437 663

1100 1193 3953

2b: HOPO2 + H → PO2 + H2O All 0.2708 0.2481 0.1321 3665i 244 279 327 438 531

632 738 1115 1238 1405 3530

Appendix 3

A3-1

Appendix 3: NNH + O Supplementary

Information

Appendix 3.1 B3LYP/6-31G(2df,p) geometries (Z-matrices) for minima and first order

saddle points on the N2OH potential energy surface (in Å and degrees).

N2 - 1ΣΣΣΣg NH - 3ΣΣΣΣ

N N

N 1 1.099 H 1 1.045

NO - 2ΠΠΠΠ OH - 2ΠΠΠΠ

N O

O 1 1.151 H 1 0.976

NNH - 2A′′′′ NNO - 1ΣΣΣΣ

N N

N 1 1.175 N 1 1.128

H 1 1.059 2 117.3 O 1 1.186 2 180.0

HNO - 1A′′′′ NN-H - 2A′′′′

N N

O 1 1.201 N 1 1.117

H 1 1.069 2 108.7 H 1 1.542 2 119.0

Appendix 3

A3-2

trans-ONNH - 2A′′′′ cis-ONNH - 2A′′′′

N N

N 1 1.245 N 1 1.229

O 1 1.201 2 132.2 O 1 1.210 2 139.3

H 2 1.024 1 107.5 3 180.0 H 2 1.037 1 110.0 3 0.0

ONHN - 2A′′′′ ONN-H - 2A′′′′

N N

N 1 1.251 N 1 1.142

O 1 1.242 2 129.7 O 1 1.188 2 173.0

H 1 1.045 2 112.7 3 180.0 H 2 1.650 1 113.7 3 0.0

ON2-H - 2A′′′′ NNOHsq - 2A′′′′

N N

N 1 1.163 N 1 1.215

O 1 1.207 2 155.6 O 1 1.408 2 96.8

H 1 1.512 2 100.3 3 180.0 H 2 1.268 1 89.0 3 0.0

NNOHtr - 2A′′′′ ONNH c-t TS - 2A

N N

N 1 1.155 N 1 1.259

O 1 1.420 2 141.8 O 1 1.201 2 133.0

H 1 1.171 3 61.6 2 180.0 H 2 1.019 1 116.7 3 90.9

Appendix 3

A3-3

ONHN-ONNHt - 2A′′′′

N

N 1 1.251

O 1 1.212 2 142.3

H 1 1.216 2 65.7 3 180.0

Appendix 3

A3-4

Appendix 3.2 B3LYP/6-31G(2df,p) rotational constants and harmonic vibrational frequencies for minima and first order saddle points on the

N2OH potential energy surface (in cm−1).

Species Rotational Constant(s) Scaled Vibrational Frequencies

N2 1.9942 2395

NH 16.4281 3199

NO 1.7036 1956

HO 18.6606 3642

NNH 22.0461 1.5489 1.4472 1108 1854 2757

NNO 0.4207 615 615 1321 2328

HNO 18.4723 1.4287 1.3262 1549 1658 2768

trans-ONNH 6.3081 0.4135 0.3881 653 770 1258 1348 1711 3383

cis-ONNH 5.5599 0.4162 0.3872 573 755 1211 1331 1724 3190

ONHN 3.6496 0.4427 0.3948 550 939 1249 1404 1536 3077

NN-H 11.6966 1.5727 1.3863 1100i 641 2191

ONN-H 7.3044 0.3850 0.3657 946i 381 617 663 1284 2213

ON2-H 4.9909 0.4196 0.3871 1479i 668 727 815 1274 1949

NNOHsq 2.0273 0.5747 0.4477 1868i 689 905 1000 1605 2030

NNOHtr 5.9012 0.3731 0.3509 1635i 498 514 714 1803 2211

ONNH c-t TS 5.3473 0.4073 0.3910 1133i 629 832 1244 1637 3464

ONHN-ONNHt 6.0159 0.4075 0.3816 2225i 445 665 1257 1689 2255

Appendix 3

A3-5

Appendix 3.3 B3LYP/6-31G(2df,p) geometries (Z-matrices) for variational transition states

on the N2OH potential energy surface (in Å and degrees).

trans-ONNH →→→→ NNH + O - 2A′′′′

1000 K 1500 K

N N

N 1 1.170 N 1 1.169

O 1 2.799 2 107.2 O 1 2.574 2 107.0

H 2 1.055 1 118.0 3 180.0 H 2 1.053 1 117.8 3 180.0

2000 K 2500 K

N N

N 1 1.169 N 1 1.170

O 1 2.374 2 106.5 O 1 2.199 2 106.3

H 2 1.051 1 117.3 3 180.0 H 2 1.047 1 116.2 3 180.0

cis-ONNH →→→→ NNH + O - 2A′′′′

1000 K 1500 K

N N

N 1 1.170 N 1 1.169

O 1 3.001 2 112.7 O 1 2.901 2 111.4

H 2 1.054 1 118.9 3 0.0 H 2 1.053 1 119.1 3 0.0

2000 K 2500 K

N N

N 1 1.168 N 1 1.167

O 1 2.751 2 110.1 O 1 2.551 2 108.3

H 2 1.051 1 119.5 3 0.0 H 2 1.049 1 119.9 3 0.0

Appendix 3

A3-6

ONHN →→→→ NNH + O - 2A′′′′

1000 K 1500 K

N N

N 1 1.168 N 1 1.165

O 1 3.024 2 134.9 O 1 2.799 2 134.6

H 1 1.055 2 119.5 3 180.0 H 1 1.052 2 120.7 3 180.0

2000 K 2200 K

N N

N 1 1.157 N 1 1.147

O 1 2.393 2 134.3 O 1 2.093 2 131.4

H 1 1.045 2 124.9 3 180.0 H 1 1.035 2 131.2 3 180.0

trans-ONNH →→→→ NO + NH - 2A

1000 K 1500 K

N N

N 1 2.349 N 1 2.224

O 1 1.142 2 110.0 O 1 1.141 2 110.1

H 2 1.042 1 89.6 3 154.8 H 2 1.042 1 90.9 3 145.0

2000 K 2400 K

N N

N 1 2.124 N 1 2.099

O 1 1.141 2 110.1 O 1 1.147 2 110.2

H 2 1.041 1 92.2 3 140.9 H 2 1.041 1 92.3 3 139.2

Appendix 3

A3-7

cis-ONNH →→→→ NO + NH - 2A

1000 K 1500 K

N N

N 1 2.324 N 1 2.199

O 1 1.142 2 109.9 O 1 1.141 2 110.1

H 2 1.042 1 90.3 3 154.0 H 2 1.041 1 91.3 3 144.9

2000 K 2500 K

N N

N 1 2.099 N 1 2.074

O 1 1.141 2 110.1 O 1 1.141 2 110.1

H 2 1.041 1 92.4 3 139.9 H 2 1.041 1 92.7 3 141.5

ONHN →→→→ HNO + N - 4A

1000 K 1500 K

N N

N 1 2.169 N 1 2.119

O 1 1.207 2 117.4 O 1 1.208 2 117.0

H 1 1.051 2 99.5 3 121.5 H 1 1.050 2 99.8 3 122.0

2000 K 2500 K

N N

N 1 2.069 N 1 2.044

O 1 1.200 2 116.6 O 1 1.211 2 116.4

H 1 1.048 2 100.2 3 122.5 H 1 1.047 2 100.4 3 122.8

Appendix 3

A3-8

NNH + O →→→→ N2 + OH - 2A′′′′

1000 K 1500 K

N N

N 1 1.165 N 1 1.162

O 1 3.991 2 119.0 O 1 3.841 2 119.5

H 1 1.075 2 119.0 3 0.0 H 1 1.081 2 119.5 3 0.0

2000 K 2500 K

N N

N 1 1.158 N 1 1.156

O 1 3.691 2 120.4 O 1 3.641 2 120.6

H 1 1.091 2 120.4 3 0.0 H 1 1.095 2 120.6 3 0.0

Appendix 3

A3-9

Appendix 3.4 B3LYP/6-31G(2df,p) rotational constants and harmonic vibrational frequencies for variational transition states on the N2OH

potential energy surface (in cm−1).

Reaction Temp. /K Rotational Constants Scaled Vibrational Frequencies

NNH + O → trans-ONNH 1000 2.2571 0.1664 0.1550 118i 124 253 1064 1874 2826

1500 2.3111 0.1918 0.1771 130i 164 334 1051 1882 2866

2000 2.3510 0.2200 0.2012 146i 210 408 1041 1885 2920

2500 2.4076 0.2490 0.2256 216i 260 499 1038 1878 2988

NNH + O → cis-ONNH 1000 1.9769 0.1492 0.1387 118i 100 242 1068 1878 2838

1500 1.9520 0.1597 0.1476 129i 106 290 1064 1884 2855

2000 1.9367 0.1768 0.1620 139i 122 319 1059 1892 2881

2500 1.9186 0.2042 0.1846 155i 142 392 1052 1902 2923

NNH + O → ONHN 1000 3.0576 0.1308 0.1254 102i 103 171 1057 1875 2822

1500 3.0968 0.1488 0.1420 115i 129 227 1038 1892 2858

2000 3.2448 0.1919 0.1812 119i 199 312 998 1941 2968

2200 3.1956 0.2403 0.2235 177i 303 448 991 1996 3120

trans-ONNH → NO + NH 1000 2.3006 0.2307 0.2105 155i 97 256 575 1944 3223

1500 2.3483 0.2510 0.2287 227i 116 300 651 1935 3226

2000 2.3895 0.2693 0.2446 306i 131 343 722 1923 3231

2400 2.4007 0.2740 0.2489 325i 129 355 738 1918 3232

cis-ONNH → NO + NH 1000 2.3069 0.2347 0.2140 166i 103 264 590 1943 3223

1500 2.3584 0.2555 0.2325 245i 116 310 668 1933 3228

2000 2.3990 0.2742 0.2489 326i 134 355 738 1919 3232

2500 2.4147 0.2791 0.2530 346i 134 368 763 1914 3235

Appendix 3

A3-10

Reaction Temp. /K Rotational Constants Scaled Vibrational Frequencies

ONHN → HNO + N 1000 2.1767 0.2518 0.2296 248i 252 614 1392 1586 2957

1500 2.1734 0.2615 0.2375 290i 273 658 1382 1573 2977

2000 2.1686 0.2717 0.2457 336i 295 698 1373 1555 2999

2500 2.1656 0.2770 0.2500 358i 306 717 1368 1545 3011

NNH + O → N2 + OH 1000 2.6851 0.0879 0.0851 118i 103 307 1065 1844 2532

1500 2.7579 0.0940 0.0909 133i 125 361 1059 1849 2454

2000 2.8617 0.1006 0.0972 155i 147 424 1053 1847 2338

2500 2.8909 0.1030 0.0994 166i 153 453 1053 1840 2287

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