computational quantum chemistry for the che classroom - 2007 asee summer school for chemical...

Computational Quantum Chemistry for the ChE Classroom - 2007 ASEE Summer School for Chemical Engineers

Computational Quantum Chemistryfor the ChE ClassroomComputational Quantum Chemistryfor the Classroom

Phillip R. Westmoreland

University of Massachusetts Amherst

Amherst, MA 01003, USA

[email protected]

[2006-08: National Science Foundation, ENG/CBET]

ASEE Summer School for Chemical EngineersPullman WA, July 28 - Aug 2, 2007

Thanks to Gaussian Inc. for providing our software.


Overall goal: Show how you can teach practical use, value, and key principles of computational

quantum chemistry to undergraduates.

• Show examples; • Demonstrate codes;• Sketch needed theory & models from quantum

mechanics and statistical mechanics; • Discuss code availability and teaching materials;• Offer recommendations.


Approximate schedule

Hour 1 - How to do it.Demonstrating value: What we can and cannot do.3-D construction of chemical species and transition states.Computation of structures and frequencies.Various capabilities, such as solvation.

Break…Hour 2 - How it is done.

The underpinnings: Computational quantum chemistry.The underpinnings: Stat mechanics and transition-state theory.The underpinnings: Energy transfer, solvation, condensed phase.Scientific and industrial examples.Examine classroom materials and code availability.Recommendations.


Prologue: What do we want for engineering from quantum chemistry? Properties.

• Thermochemistry:– Heats of reaction.

• Compute enthalpies, bond energies, interaction energies.

– Reaction equilibria.• Obtain ∆Grxn from enthalpies, entropies, specific heats.

• Kinetics:– Identify reactants and corresponding products.– Compute rate constants.

• Various others:– Molecular shape/size, dipoles, spectroscopic info, force fields…


Can we obtain them?

• Yes, to a very useful extent.• The basis is ideal-gas thermochemistry for molecules

and radicals at zero kelvins, normally computable to impressive accuracy.

• Include higher temperatures easily by Rigid-Rotor Harmonic Oscillator approximation.– More accuracy requires treatment of anharmonicities.

• Also, solvated molecules and adsorbed species.


Accuracy is vital, but is precision?

• A given task may require accurate, precisely known numbers.– May be necessary for accurate design, costing, safety analysis.– Cost and time for calculation may be secondary.

• Often, accurate trends and estimates are at least as valuable.– Can be correlated with data to get high-accuracy predictions.– Can identify relationships between structure and properties.– A quick number or trend may be of enormous value in early stages

of product and process development, for operations, or for troubleshooting.

• Great data are best, but also use theory-based predictions.


What kind of properties come directly from computational quantum chemistry?

• Gives energies, structures optimized with respect to energy, harmonic frequencies, and other properties based on zero-kelvin electronic structures.

• Interpret with theory to get derived properties and properties at higher temperatures.

• The theoretical basis for most of this translation is:

Quantum-mechanical energies

Statistical mechanicsStatistical mechanics

Electronic structure theory Molecular simulations

Wavefunction methodsDensity-functional theory

Semi-empirical MO theory

"Ab initio":

Basis sets

Molecular dynamics,molecular mechanics

Monte Carlo simulations

(Computational quantum chemistry)

Quantum MD,Car-Parrinello

Potential energyfunctions

Molecular and material structure

Statisticalmechanics

Thermochemistry, kinetics, transport, materials properties, VLE, solutions


With quantum-chemistry tools, we can move from overall formulas... to sketches...

(C33N3H43)FeCl2,

a liganded di(methyl imide xylenyl) aniline ...

N

N

N

Fe

H3C

H3C

Cl

Cl

i-Pr

i-Pr

i-Pr

i-Pr


… To structures that represent 3-D functionality quantitatively.


Molecular size and shape can be optimized.

These calculations and graphics at HF/3-21G* weregenerated with MacSpartan Plus (Wavefunction Inc.).

Molecular shape corresponds to electron density:

HOMO(highest-energyoccupied molecularorbital)

LUMO(lowest-energyunoccupied molecularorbital)


The simplest properties are interaction energies: Here, the van der Waals well for an Ar dimer.

-150.00

-100.00

-50.00

0.00

50.00

100.00

0 2 4 6 8 10

Ar-Ar, angstroms

ε/κ,Κ

HF

2MP

3MP

4MP D

4MP DQ

4MP SDQ

CCSD

( )CCSD T

- 12-6L J-0.2 / = -0.8 /kcal mol kJ mol

:Basis set- - ;aug cc pVDZ

:Method

Here, calculate energy for each geometry (rAr-Ar).

Computed with Gaussian 03,Gaussian Inc.


Chemical bond energies:Simplest covalent bonds are much stronger.

-60

-40

-20

0

20

40

60

80

100

0 1 2 3

Br-Br, angstroms

UB3LYP/6-311++G(3df,3dp) withbasis-set-superposition error correction

-100

-50

0

50

100

150

0 1 2 3

.O-H, angstroms

HF

MP2

MP4SDQ

CCSD(T)

Basis set:aug-cc-pVDZ;Method:


Geometry is found by optimizing computed energy with respect to coordinates.

Energy Efrom

HΨ=EΨ

Position r

x

x

x

xx

Transitionstate

Ground state - minimum w.r.t. all coordinates

Minimum w.r.t. all but reaction coordinate


The entire minimum energy path may not be a simple motion, but the transition state is still separable.

Potential energy surface for O-O bond fission in CH2CHOO·B3LYP/6-31G(d);Kinetics analysis based on ourO-O reaction-coordinate-driving calculation at B3LYP/6-311+G(d,p)


Let’s examine a user interface, studying molecule construction, options, and results.

• Gaussview from Gaussian Inc. (www.gaussian.com).– Links to Gaussian ab initio codes.– Other useful codes include Spartan, Jaguar (Schroedinger

Inc.), Hyperchem (Hypercube); NWchem (Battelle Pacific NW Labs), GAMESS (US and UK), Columbus (Vienna).

• We will:– Construct various species, noting geometries.– Examine optimized results and information for molecules.– Study calculation options and set-up.– Examine transition states.


[Study using GaussView interface.]

[CH3CO2, a gas-phase Criegee intermediate]


Find transition-state geometries: Loose or tight.

• Terminology based on unimolecular TST expression:

• eκT/h = 1013.2s-1 (1.7.1013) at 298 K, so∆S > 0 (TS “looser” than reactant) A > 1013.2 s-1.

∆S < 0 (TS “tighter” than reactant) A < 1013.2 s-1.

€

kCTST = Aexp −Eact

RT

⎛

⎝ ⎜

⎞

⎠ ⎟=

eκT

hexp

Δ ′ S TS

R

⎛

⎝ ⎜

⎞

⎠ ⎟

⎡

⎣ ⎢

⎤

⎦ ⎥⋅exp −

Δ ′ H TS + RT

RT

⎛

⎝ ⎜

⎞

⎠ ⎟

Reaction A (s-1) Reaction A (s-1)

C2H6 2CH3 1017.0

(at 300 K)

CH3CH2CH2. C2H4+CH3

1013.0

Ethylbenzene Benzyl+CH3

1014.6

(at 950 K)

1011.8

(at 430 K)


Loose transition states: No classical TS;Can only map potential energy surface.

• Electronic energy is computed and increases monotonically, so no classical TS can be found by these codes.

• Rather, total energy = Eelec+Erotn

• Note: Using analytical expressions, differentiate E to find maximum is at about r/ro=2.5 to 3.

• Caution: Must use extended-range basis sets and correct for basis-set superposition error; can use variational TST.

Energy Efrom

HΨ=EΨ

Position r

x

x

x

xx

Transitionstate




Tight transition states: Use electronic energy.

• Compute electronic E and locate maximum by optimization.– Little change in external moment of inertia.

• Length of breaking single bond is about rTS = ro + (0.3 to 0.5 Å).

– From a double bond going to a single bond, rTS = (r + r )/2.

– Likewise, TS angles and dihedrals are intermediate between corresponding reactant and product angles.

– [An interesting difference for addition to pi bond (like H2C=CH2) is that H’s on the beta carbon flare upward in the TS.]

Energy Efrom

HΨ=EΨ

Position r

x

x

x

xx

Transitionstate




[Study using GaussView interface.]

[Diels-Alder addition of ethylene to 1,3-butadiene … or decomposition of cyclohexene]


Movie of CH3CH2F = C2H4+HF TS.


Computational Quantum Chemistryfor the Classroom

Part II. Principles and Methods.


“Ab initio” is widely but loosely used to mean “from first principles.”

• Actually, there is considerable use of assumed forms of functionalities and fitted parameters.

• John Pople noted that this interpretation of the Latin is by adoption rather than intent. In its first use:

• The two groups of Parr, Craig, and Ross [J Chem Phys 18, 1561 (1951)] had carried out some of the first calculations separately across the Atlantic - and thus described each set of calculations as being ab initio!


Three key features of theory are required for ab initio calculations.

• Understand how initial specification of nuclear positions is used to calculate energy:– Solving the Schrödinger equation

• Understand “basis sets” and how to choose them:– Functions that represent the atomic orbitals

– e.g., 3-21G, 6-311++G(3df,2pd), cc-pVTZ

• Understand levels of theory and how to choose them:– Wavefunction methods: Hartree-Fock, MP4, CI, CAS

– Density functional methods: LYP, B3LYP, etc.

– Compound methods: CBS, G3

– Semiempirical methods: AM1, PM3

• Then, calculate properties.


Energy E[from

EΨ=HΨ]

Position r

x

x

x

With energies,we can optimize

molecular structure

Initially, restrict our discussion to an isolated molecule.

• Equivalent to an ideal gas, but may be a cluster of atoms, strongly bonded or weakly interacting.

• Easiest to think of a small, covalently bonded molecule like H2 or CH4 in vacuo.

• Most simply, the goal of electronic structure calculations is energy.

• However, usually we want energy of an optimized structure and the energy’s variation with structure.


Begin with the Hamiltonian function, an effective, classical way to calculate energy.

• Express energy of a single classical particle or an N-particle collection as a Hamiltonian function of the 3N momenta pj and 3N coordinates qj (j=1,N) such that:

∂H∂qj

=−˙ p j∂H∂pj

=˙ q j

where:

H = Kinetic Energy (T) + Potential Energy (V) = Total Energy


For quantum mechanics, a Hamiltonian operator is used instead.

• Obtain a Hamiltonian function for a wave using the Hamiltonian operator:

H =−

h2

8πm∇2 +U(x,y,z)

to obtain:

where is the “wavefunction,” an eigenfunction of the equation.

EΨ =H Ψ(

v q ,t){ }=

ih2π

∂Ψ(v q ,t)

∂t

• Born recognized that 2 is the probability density function.


For quantum molecular dynamics, retain t; Otherwise, t-independent.

• Separation of variables gives (q) and thus the usual form of the Schroedinger or Schrödinger equation:

EΨ =H Ψ{ }

• If the electron motions can be separated from the nuclear motions (the Born-Oppenheimer approximation), then the electronic structure can be solved for any set of nuclear positions.


e-

proton

Easiest to consider H atom first as a prototype.

• Three energies:– Kinetic energy of the nucleus.– Kinetic energy of the electron.– Proton-electron attraction.

• With more atoms, also:– Internuclear repulsion– Electron-electron repulsion.

• Electrons are in specific quantum states (“orbitals”).• They can be in excited states (higher-energy orbitals).


Restate the nonrelativistic electronic Hamiltonian in atomic units.

• With distances in bohr (1 bohr = 0.529 Å) and with energies in hartrees (1 hartree = 627.5 kcal/mol),

EΨ =HΨ

H =−12

∂2

∂xi2 +

∂2

∂yi2 +

∂2

∂zi2

⎛

⎝ ⎜ ⎞

⎠ ⎟

i

Electrons

∑

−Zs

ris

⎛

⎝ ⎜ ⎞

⎠ ⎟

s

Nuclei

∑i

Electrons

∑ −1ris

⎛

⎝ ⎜ ⎞

⎠ ⎟

j

Electrons

∑i<j

Electrons

∑ −ZsZt

Ris

⎛

⎝ ⎜ ⎞

⎠ ⎟

s<t

Nuclei

∑s

Nuclei

∑(After Hehre et al., 1986)

where

• [Breaks down when electrons approach the speed of light,the case for innermost electrons around heavy atoms]


Set up Ψ, the system wavefunction.

• Need functionality (form) and parameters.• (1) Use one-electron orbital functions (“basis

functions”) to ...• (2) Compose the many-electron molecular orbitals

by linear combination, then ...• (3) Compose the system from ’s.• Wavefunction must be “antisymmetric”

– Exchanging identical electrons in Ψ should give -Ψ– Characteristic of a “fermion”; vs. “bosons” (symmetric).


H-atom eigenfunctions correspond to hydrogenic atomic orbitals.

m=0

2pz2py2px

1s

z

y

x

2s

3s 3px 3py 3pz 3d orbitals (5)

n=1

n=3

n=2

l=0 l=1 l=2

m=0 m=-1 m=+1 m=-2,-1,0,1,2


Construct each MO i by LCAO.

• Lennard-Jones (1929) proposed treating molecular orbitals as linear combinations of atomic orbitals (LCAO):

• Linear combination of p orbital on one atom with p orbital on another gives bond:

ψ i = Cμiφii=1

∞

∑

• Linear combination of s orbital on one atom with s or p orbital on another gives bond:


Molecular includes each electron.

• First, include spin (=-1/2,+1/2) of each e-. – Define a one-electron spin orbital, (x,y,z,) composed of a

molecular orbital (x,y,z) multiplied by a spin wavefunction or .

• Next, compose Ψ as a determinant of ’s.

Ψ =1n!

χ1(1) χ2(1) L χn(1)

χ1(2) χ2(2) L χn(2)

M M M

χ1(n) χ1(n) L χ1(n)

– Interchange row => Change sign; Functionally antisymmetric.

<- Electron 1 in all ’s;<- Electron 2 in all ’s;

<- Electron n in all ’s


However, basis functions i need not be purely hydrogenic - indeed, they cannot be.

• Form of basis functions must yield accurate descriptions of orbitals.

• Hydrogenic orbitals are reasonable starting points, but real orbitals:– Don’t have fixed sizes, – Are distorted by polarization, and – Involve both valence electrons (the outermost, “bonding”

shell) and non-valence electrons.

• Hydrogenic s-orbital has a cusp at zero, which turns out to cause problems.


Simulate the real functionality (1).

• Start with a function that describes hydrogenic orbitals well.

φ1s(

r r ;ζ1) =

ζ13

πexp(−ζ1

r r )

gs

r r ;α( )=

2απ

⎛ ⎝

⎞ ⎠

3/ 4

exp(−αr r 2)

r

True STO

3 gaussianfunctions

– Slater functions; e.g.,– Gaussian functions; e.g.,

• No s cusp at r=0

• However, all analytical integrals

– Linear combinations of gaussians; e.g., STO-3G

• 3 Gaussian “primitives” to simulate a STO

• (“Minimal basis set”)


– Alternatively, use size adjustment only for outermost electrons (“split-valence” set) to speed calculations.

– For example, in the 6-31G set:• Inner orbitals of fixed size based on 6 primitives each.• Valence orbitals with 3 primitives for contracted limit, 1 primitive for

diffuse limit.

– Additional very diffuse limits may be added (e.g., 6-31+G or 6-311++G).

+λ =Contracted

limitDiffuse

limitAdjusted

size

Double-zetabasis setor more

(cc-pV5Z)


• Allow size variation.



• Allow shape distortion (polarization).– Usually achieved by mixing orbital types:

– For example, consider the 6-31G(d,p) or 6-31G** set:• Add d polarization to p valence orbitals, p character to s

• Can build complicated sets; e.g., 6-311++G(3df,2pd)

=+λd

+λp =



• Dunning’s correlation-consistent basis sets are among the best presentaly available.– cc-pVDZ (double-zeta), cc-pVTZ (large but a good

compromise), cc-pVQZ (at the edge), and cc-pV5Z.

• Another basis-set improvement is development of Effective Core Potentials.– For transition metals, innermost electrons are at relativistic

velocities.– Capture their energetics with effective core potentials.– For example, LANL2DZ (Los Alamos National Lab #2

Double Zeta).– Caution: Closed-shell only!


The third aspect is solution method.

• Hartree-Fock theory is the base level of wavefunction-based ab initio calculation.

• First crucial aspect of the theory: The variational principle.– If Ψ is the true wavefunction, then for any model

antisymmetric wavefunction , E()>E(Ψ). Therefore the problem becomes a minimization of energy with respect to the adjustable parameters, the Cµi’s and ’s.


The Hartree-Fock result omits electron-electron interaction (“electron correlation”).

• The variational principle led to the Roothaan-Hall equations (1951) for closed-shell wavefunctions:

Fμν −εiSμν( )cνi =0ν=1

N

∑ μ =1,2,L ,N

• εi is diagonal matrix of one-electron energies of the i.

• F, the Fock matrix, includes the Hamiltonian for a single electron interacting with nuclei and a self-consistent field of other electrons; S is an atomic-orbital overlap matrix.

• All electrons paired (RHF); there are analogous UHF equations.

or FC=SCε


One improvement is to use “Configuration Interaction”.

• Hartree-Fock theory is limited by its neglect of electron-electron correlation.– Electrons interact with a SCF, not individual e’s.

• “Full CI” includes the Hartree-Fock ground-state determinant and all possible variations.

Ψ =a0Ψ0 + asΨss=1

∞

∑– The wavefunction becomes a where s includes all combinations of substituting electrons into H-F virtual orbitals.

– The a’s are optimized; not so practical.


Partial CI calculations are feasible.

• CIS (CI with Singles substitutions), CISD, CISD(T) (CI with Singles, Doubles, and approximate Triples)– CI calculations where the occupied i elements in the SCF

determinant are substituted into virtual orbitals one and two at a time and excited-state energies are calculated.

• CASSCF (Complete Active Space SCF) is better: Only a few excited-state orbitals are considered, but they are re-optimized rather than the SCF orbitals.

• An important variant: Coupled Cluster methods, especially CCSD(T).


Perturbation Theory is an alternative.

• Møller and Plesset (1934) developed an electronic Hamiltonian based on an exactly solvable form H0 and a perturbation operator:

H =H 0 +λυ

• A consequence is that the wavefunction and the energy are perturbations of the Hartree-Fock results, including electron-electron correlation effects that H-F omits.

• Most significant: MP2 and MP4.


The important advances for practical calcs are in electronic density-functional theory.

• From Hohenberg and Kohn (1964)– Energy is a functional of electron density: E[]– Ground-state only, but the exact minimizes E[]

• Then Kohn and Sham (1965)– Variational equations for a “local” functional:

E[ρ]=Vnucl+T[ρ]+J elec[ρ]+Exc[ρ]

where Exc contains electron correlation:

Exc[ρ]=Ex[ρ],exchangefunctional(same−spininteractions)

+Ec[ρ],correlationfunctional(mixed−spininteractions)


• Need “nonlocal” effects of gradient,• Current approach: Hybrid functionals

– Combine Hartree-Fock Ex and DFT contributions Ex +Ec

– Numerous proposed functionals and combinations– Axel Becke’s BLYP, B3LYP, BH&HLYP; Truhlar’s M06– Give excellent structures and frequencies, poorer energies.

Nonlocal effects, introduced in early 1990’s, have made DFT powerful.

• Kohn and Sham had the local exchange functional:

Exc[ρ]= ε[ρ(r r )∫ ]d3r r

e.g., Exc,LDA[ρ]=−32

34π

⎛ ⎝

⎞ ⎠

1/3

ρ(r r )∫

4/3d3r r

r ∇ ρ(

r r )


For high-accuracy, practical calculations, Pople chose extrapolation from this matrix:

Improved electron correlation → H- MP2 MP4 QCISDT u CI

Morecomplete

basissets↓

STO-3G

3-21G

6-31Gd

6-311Gd,

6-311+Gd,

Ininitε i εt

Etoution


“Compound methods” aim at extrapolation.

• The G1, G2, and G3 methods of Pople and co-workers calculate energies in cells of the matrix, then project more accurately.– G2 gave ave. error in ∆Hf of ±1.59 kcal/mol.

– G3 gives ave. error in ∆Hf of ±1.02 kcal/mol.

• CBS methods of Petersson and co-workers are compound methods that give impressive results.– Extrapolate to “Complete Basis Set” limit.– The best for normal use: CBS-Q (based on MP2 geometries

and frequencies) and CBS-QB3 (based on B3LYP).


Besides energy, calculations give electron density, HOMO, LUMO.

• Electron density (from electron probability density function = 2) is an effective representation of molecular shape.

• Each molecular orbital is calculated, including highest-energy occupied MO (HOMO) and lowest-energy unoccupied MO (LUMO)

• HOMO-LUMO gap is useful for Frontier MO theory and for band gap analysis.


Use these methods to calculate thermochemistry and kinetics.

• Use quantum-mechanical calculations in statistical-mechanics relations.


At zero K, define the dissociation energy D0 as the well depth less zero-point energy.

-125

0

0 1 2 3

Bond length, angstroms

E (relative

todissociatedpartners)

ZPE=0.5hv

∆E(0)D(0)

Alternate view is that

D0 =

E0(dissociated partners)

- [E0(molecule) + ZPE],

where ZPE is the zero-K energy of the stretching vibration.


Vibrational frequencies (at 0 K) are calculated using parabolic approximation to well bottom.

• How many? Need 3Natoms coordinates to define molecule.– If free translational motion in 3 dimensions, then three translational

degrees of freedom– Likewise for free rotation: 3 d.f. if nonlinear, 2 if linear

– Thus, 3Natoms-5 (linear) or 3Natoms-6 (nonlinear) vibrations

• For diatomic, ∂2E/∂r2 = force constant k [for r dimensionless].– F (= ma = m∂2r/∂t2) = -kr is a harmonic oscillator in Newtonian

mechanics (Hooke’s law)– Harmonic frequency is (k/m)1/2/2π s-1 or (k/m)1/2/2πc cm-1

(wavenumbers)

• For polyatomic, analyze Hessian matrix [∂2E/∂ri∂rj] instead.


Next, determine ideal-gas thermochemistry.

• Start with ∆fH0° and understand how energies are given.– We recognize that energies are not absolute, but rather must be

defined relative to some reference.– We use the elements in their equilibrium states at standard

pressure, typically 1 atm or 1 bar (0.1 MPa):

Δ f Hoo =Ho

o(CaHbOcL , idealgas,0K, 0.1MPa)

−aHoo(Cgraphite)−

b2

Hoo(H2(g))−

c2

Hoo(O2(g))L

– From ab initio calculations, energy is typically referenced to the constituent atoms, fully dissociated. Get ∆fH0° from:

−Total atomizationenergyof themolecule, Do∑[ ] [abinitioresults]

=Δf Hoo − a⋅Δf Ho

o(C)+b⋅Δ fHoo(H)+c⋅Δf Ho

o(O)[ ] tabulated data[ ]


For S298o and Cp

o, apply statistical mechanics.

• Quantum mechanics gives information to compute the partition functions q(V,T)=∑exp(-εi / T) for:– Translational degrees of freedom– External rotational degrees of freedom (linear or nonlinear

rotors)– Rovibrational degrees of freedom (stretches, bends, other

harmonic oscillators, and internal rotors)

• Electronic degrees of freedom require only εelectronic and degeneracy.– Both calculable from quantum mechanics.


Use expressions of entropy, energy, and heat capacity in terms of partition function(s).

S=Nκ 1+lnq(V,T)

N⎛ ⎝

⎞ ⎠

+T∂ lnq(V,T)

∂T⎛ ⎝

⎞ ⎠

V

⎡

⎣ ⎢ ⎤

⎦ ⎥

S(molar)=R⋅ ln qtransqrotqvibrqelece( )+T∂ lnqtransqrotqvibrqelec

∂T⎛ ⎝

⎞ ⎠ V

⎡

⎣ ⎢ ⎤

⎦ ⎥

=Strans+Srot +Svibr +Selec

H =E +PV =E + RT( )idealgas

E =RT2 ∂ lnq(V,T)∂T

⎛ ⎝

⎞ ⎠ V

Cp( )idealgas=

∂E∂T

⎛ ⎝

⎞ ⎠ N,V

+R =Cv,trans+Cv,rot +Cv,vibr +Cv,elec+R


For ideal gas, begin with the translational degrees of freedom.

• Quantum mechanics for pure translation in 3-D gives:

εtranslation=h2n2

8ml2wheren=1,2,K [particleinabox]

qtrans(3D) =2πmκT

h2⎛ ⎝

⎞ ⎠

3/2

⋅V

Cv,trans(3D)o =3

R2

Strans(3D)o =R ln

2πmTh2

⎛ ⎝

⎞ ⎠

3/2

⋅RTP

⎡

⎣ ⎢ ⎤

⎦ ⎥ +

52

⎧ ⎨ ⎩

⎫ ⎬ ⎭

• Note the standard-state pressure in the last equation.


Rigid-rotor model for external rotation introduces the moment of inertia I and rotational symmetry ext.

εrotationina plane=h2J J +1( )

8π 2IwhereJ =0,1,2,K

qrot = 2J +1( )exp−εJ /κT( )∑linear(2D) ⏐ → ⏐ ⏐ ⏐ ⏐ 1

σ ext

8π2κT ⋅ Ih2

⎛ ⎝ ⎜ ⎞

⎠ ; Cv,rot(2D)

o =2R2

; Srot(2D)o =R lnqrot(2D)+1( )

nonlinear(3D) ⏐ → ⏐ ⏐ ⏐ ⏐ ⏐ 1σext

π1/ 2 8π2κT( )3/2

⋅ IA IBIC( )3/2

h3

⎛

⎝ ⎜

⎞

⎠ ⎟

Cv,rot(3D)o =3

R2

; Srot(3D)o =R lnqrot(3D) +

32

⎛ ⎝

⎞ ⎠


Add harmonic oscillators with frequencies ni and electronic degeneracy of go.

• For each harmonic oscillator,

€

εvibration = n +1

2

⎛

⎝ ⎜

⎞

⎠ ⎟hν = nhν +

1

2hν where n = 0,1,2,K

qvib =1

1−e−x wherex=hν /κT;

Cv,vibo =R

x2ex

ex −1( )2 and Svib

o =R ln1

1−e−x⎛ ⎝

⎞ ⎠

+x

ex +1⎡ ⎣ ⎢

⎤ ⎦ ⎥

– It is convenient to redefine zero for vibrational energy as zero rather than 0.5hn; this shift requires the zero-point energy correction to energy. As a result,

• If only the ground electronic state contributes, then (Cvo)elec=0 and

(So)elec=R·ln go. Otherwise, need g1 & ε1.


Obtain reaction equilibrium constant from ∆Grxno.

• With ∆fH298o, S298

o, Cpo for each reactant and product,

€

DHrxn = α i Δ f H298,io + Cp,i

o dT298

T

∫ ⎛

⎝ ⎜

⎞

⎠ ⎟∑

€

DSrxn = α i S298,io +

Cp,io dT

T298

T

∫ ⎛

⎝ ⎜

⎞

⎠ ⎟∑

€

DGrxn = α i Δ f H298,io − 298 ⋅S298

o( ) + Cp,i

o dT298

T

∫ − TCp,i

o

TdT

298

T

∫ ⎛

⎝ ⎜

⎞

⎠ ⎟

⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥

∑

€

Krxn = exp(−ΔGrxn

RT) = exp −

ΔHrxn − TΔSrxn

RT

⎛

⎝ ⎜

⎞

⎠ ⎟= exp

ΔSrxn

R

⎛

⎝ ⎜

⎞

⎠ ⎟⋅exp −

ΔHrxn

RT

⎛

⎝ ⎜

⎞

⎠ ⎟

• Note “compensation effect”:


Find transition state with quantum chemistry.

• We have already discussed how to locate transition states along the “minimum energy path”: – A stationary point (∂E/∂ = 0) with respect to all displacements.– A minimum with respect to all displacements except the one

corresponding to the reaction coordinate.– More precisely, all but one eigenvalue of the Hessian matrix of

second derivatives are positive (real frequencies) or zero (for the overall translational and rotational degrees of freedom.

• The exception: Motion along the reaction coordinate– It corresponds to a frequency ‡ that is an imaginary number.– If eit is a sinusoidal oscillation, then ‡ is exponential change.


Apply transition-state theory for high-P limit.

• TS has a geometrical structure, electronic state, and vibrations, so assume we can calculate q‡, H‡, S‡, Cp

‡

• For classical transition-state theory, Eyring assumed:– At equilibrium, TS would obey equilibrium relations with reactant;– The reaction coordinate would be a separable degree of freedom;– Thus, it was treated as a 1-D translation or a vibration,

KTS =TS[ ]

ReactantA[ ]

⎛

⎝ ⎜ ⎞

⎠ ⎟

eq

=qTS

qA

exp−εTS −εA

κT⎛ ⎝

⎞ ⎠ ⇒ kCTST =

κTh

⋅′ q TS

qA

exp−ETS −EA

RT⎛ ⎝

⎞ ⎠

• Using the thermochemical form of Keq,

kCTST =κTh

⋅expΔ ′ S TS

R⎛ ⎝

⎞ ⎠ exp −

Δ ′ H TS

RT⎛ ⎝

⎞ ⎠

=eκTh

expΔ ′ S TS

R⎛ ⎝

⎞ ⎠

⋅ exp−Δ ′ H TS +RT

RT⎛ ⎝

⎞ ⎠


Computational quantum chemistry gives very useful numbers for Eact, also can give good A-factor.

• For gas kinetics, calculate H‡, S‡, Cp‡, ∆S‡(T), ∆H‡(T).

– Reaction coordinate contributes zero to S‡

– Standard-state correction is necessary for bimolecular reactions.

– Eact, like bond energy, may be adequate for comparisons.

• Most other factors can be handled.– If reaction coordinate involves H motion and low T, quantum-

mechanical tunneling may occur (use calculated barrier shape).– High-pressure limit is required (use RRKM, Master Equation).– Low-frequency modes like internal rotors give the most

uncertainty in ∆S‡, but we can calculate barriers.– In principle, the same for anharmonicity of vibrations.


B

A

Consider three types of environmental effects:First, rate-limiting collisional energy transfer, a central issue

for gas kinetics (low-pressure limit).

• Classical fall-off for simple decomposition or isomerization

• For association reactions: Classical falloff if simple stabilization; Inverse falloff if chemically activated decomposition

(C+D A)

P

ln kass

oci

ati

on

ka/s,

k [M]a/s,o

ln [M] or ln

ka/d (C+D B+B’)

ln [M] or ln P

ln k

deco

mp

osi

tion

k [M]o

k

A

B+B’

C+D


Each adduct / isomer is treated in a Master Equation.

• Solve for concentrations of state-resolved species A at constant density:

• Microcanonical kdestruction of A(Ei) from Quantum-RRK or RRKM theory.

• Solve numerically or stochastically (e.g., MultiWell code of Barker).

• At steady state, a closed-form solution is found if downward E-transfer u =Zcollisional[M].

– Westmoreland et al., J Phys Chem A 93, 8371 (1989).– E-transfer efficiency is correlated.

€

d A E i( )[ ]

dt= kA

j≠i

∞

∑ E i ← E j( ) ⋅ A E j( )[ ] − kA

j≠i

∞

∑ E j ← E i( ) ⋅ A E i( )[ ]

− kdestruction,r

r=1

n

∑ ⋅ A E i( )[ ] + k formation,r

r=1

n

∑ ⋅ Reactant(s)[ ]

€

kA E j ← E i( )


Recall the CH2CHO---O potential energy surface:

O

O O

·

CHO +CH O

OO

+ H

·O

HO

·O

+ O2

2

+ HO2

C H2 2

-91.5 kcal/mol

+ OO

· + O

O

·

OO·

O

O·

O

O·

O

O·

H transferC H2 3

10

11

12

13

log k,cc/mol s

0 1 2 3 4

1000/T(K)

CH2O+CHO

CH2CHO+O

C2H2+HO 2

• Proved critical to resolving the interactions among 73 stationary states (species and TS’s) for C2H3+O2.


Second, solvation effects.

• Solute-solute and solute-solvent interactions.• One approach: Cavity in dielectric solvent.• COSMO is another useful approach:

– Molecule-shaped cavity.– Solute screening by perfect conductor.– Continuum dielectric solvent.

• COSMO-RS (COSMOlogic):– Compute surface-charge distribution.– Compare to solvent’s charge dist’n.– Compute chemical potential. – More details from Andreas Klammt (here) and in literature.


Third, atomistic surface effects.

• Model surface as:– Cluster of atoms (bigger = better) or– Periodic domain of some depth.

• Hybrid methods have proven powerful:– Spatial extrapolation such as embedded-

atom models of catalysts (Haldor Topsøe) and Morokuma’s ONIOM method; connect or extrapolate domains of different-level calculations.

– QM/MM for biomolecule structure and ab-initio molecular dynamics for ordered condensed phases; calculate interactions as dynamics calculations proceed.

(Neurock)


Different approach: Ab Initio Molecular Dynamics.

• Car-Parrinello Molecular Dynamics code, UKCP or CPMD.• CASTEP (CAmbridge Serial Total Energy Package).• Vienna plane-wave pseudopotential code VASP.• DACAPO (Nørskov, CAMP at T.U. Denmark).• Codes in Japan by CAMM/CAMP industry collaboration (16

companies) and JRCAT (Joint Research Center for Atom Technology, Tsukuba).

• Also from physics community, MD with electronic structure by tight-binding MD (Kaxiras and co-workers).


Also, our Reactive Molecular Dynamics.

• Our method is MDReact [Stoliarov et al., Polymer 44, 883 (2003)].

• Requires reactive force field like our RMDff or (newly developed) BEBOP.

• Another: Goddard’s ReaxFF.


A few examples…


First, our synchrotron VUV-photoionization at Lawrence Berkeley National Laboratory.

• Our group’s molecular-beam mass spectrometry has discovered new species (enols) and yielded new rate constants, aided by computational chemistry (k’s, ionization energies).

10

11

12

13

14

0.4 0.6 0.8 1 1.2 1.41/T*1000

Log k

f

Yampolski (1974)

DATA

Just (1977)

ab-initio calculation

H+C2H4 -> H2+C2H3


VUV-photoionization MBMS only detected i-forms of C4H3 and C4H5 in diverse flames (computed IE’s).

Hansen, Klippenstein, Taatjes, Miller, Wang, Cool, Yang, Yang, Wei, Huang, Wang, Qi, Law, Westmoreland, Kasper, Kohse-Höinghaus, J. Phys. Chem. A 110(10), 3670 (2006).


With synchrotron VUV-photoionization, see C6H6’s.

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

8.0 8.6 9.2 9.8 10.4

Photon Energy (eV)

Arbitrary Signal

Benzene

Fulvene

1,5-Hexadiyne

In fuel-lean allene-doped C2H4 flame at 5.0 mm

Law, Carrière, Westmoreland, Proc. Combust. Inst. 30(1), 1353 (2005).

20%20%

45%45%

35%35%

In stoichiometric cyclohexane flame:


Molecular modeling has had notable industrial successes; See www.wtec.org/loyola/molmodel

• “Blue-ribbon” panel was set up by US Government agencies.

• Detailed site reports on ~100 organizations, mostly companies.– Based on literature, contacts, and

site visits to companies in Europe, Japan, and US.

– Info on over 500 more.


Quantum chemistry is increasingly valuable.

• Molecular modeling for homogeneous catalysis has been a big success (EniChem, Mitsubishi, Phillips, etc.).– Has followed adaptive development, altering ligands.

• ExxonMobil calculates thermochemistry for safety analyses (heats of reaction and combustion).

• Toyota reports use for predicting kinetics of soot formation.

• BASF uses semi-empirical and ab initio calculations for excited states to give pigment and dye behaviors.


Rohm and Haas developed group-contribution parameters with quantum-chemical calculations.

• Approach: Application properties for many products can be rationalized and predicted by relative hydrophobicities of the constituent polymers.

• Needed to add groups for group-contribution methods.• Used computational quantum chemistry with continuum

solvation model.• Developed a protocol for deriving relative

hydrophobicity parameters of each group.• Predictions agreed well with experiment.• [Dow also reports use for new Benson groups.]


Even failures can be instructive.

• A new modeler at Eastman Chemical was assigned to get mechanism for “acid-catalyzed acyl transfer” in alcoholysis of carboxylic anhydride.

• Searched and searched, but couldn’t find TS.• In discussions with the process engineers, he found

that in process, base was added.• He then quickly found a base-catalyzed transition

state -- and proved the process was not kinetically limited but transport-limited.


A classroom example: Senior Design homework.

• [Preface: The needed material was introduced in four 1 hr 15 min classes, including background and in-class calculations.]

• We’re developing a possible process using dimethyloxiranes as precursors for a special antifreeze based on 2,3-dihydroxybutane. No heats of combustion or heats of formation have been reported in the literature for the dimethyloxiranes, but we need the information for designing safety systems.

• Specifically, we need accurate estimates of heats of combustion for two dimethyloxiranes, cis-2,3-epoxybutane and trans-2,3-epoxybutane. Using computational quantum chemistry, we can generate these numbers at an acceptable level of accuracy using only a couple of hours of PC time and a little bit of spreadsheet-based analysis.



• Calculate the LHV (lower heating value) or enthalpy of combustion based on the reaction:

C4H8O(g) + 5.5O2(g) = 4CO2(g) + 4H2O(g)

so LHV = ∆Hreaction = 4HCO2 +4 HH2O – HC4H8O – 5.5HO2. Note that you don’t have to get the heat of formation for each species, but just the overall heat of reaction.

• That will require optimizing the structure and calculating the frequencies for each species. We will use the Gaussian code. Instructions for determining heat of reaction from computational quantum chemistry are available online (http://www.gaussian.com/g_whitepap/thermo.htm). You will need to open the text output file in a word processor, but the instructions (especially the sections “Thermochemistry output from Gaussian” and “Worked-Out Examples”) are very helpful.



• To assess uncertainty, we need calculations at several different levels. At most, using the slowest methods, calculations for the C4H8O molecule should take about 24.5 min for optimization and 1 hr 14 min for frequencies on the computer-lab computers.

[Each student was then assigned a relatively fast, low-level method and a small basis set to use. – All the students completed the assignment, some with more help to get

past data entry. – All except two reported great satisfaction and a sense of the utility of

these methods.– Those two had experienced such instruction previously and more

extensively in PChem, finding this segment redundant.]


Recommendations for normal use:

• Molecular structures and frequencies: B3LYP/6-31G(d,p)

• Energetics: CBS-QB3 [uses B3LYP optimization]– High-end calcs - but expensive: CCSD(T)/cc-pVQZ etc.

– May need multireference codes - Caution!

• Collisional energy transfer: MultiWell (Barker, U Mich)

• Solvent effects: COSMO-RS methods (CosmoLogic)

• Surface kinetics: VASP (Vienna)


Coming soon to a code near you:

• Better parallel and massively parallel codes.– Columbus is one attempt.

• Greatly improved electronic DFT calculations.– Zhao and Truhlar’s M06 (Minnesota), Scuseria (Rice).– However, different forms for different needs.

• Better handling of anharmonicities.• Explicit generation of high-pressure-limit rate

constants.• Advanced ab-initio molecular dynamics.


Long-term?


In closing,

• Computational thermochemistry can be set up easily and has very useful accuracy.– Binding energies, heats of reaction, chemical equilibria.

• Computational kinetics is approaching the same level of effectiveness.

• Computation size is inevitably an issue.• Frontiers: Excited states, condensed phase, surfaces.

• These slides are available at: www.ecs.umass.edu/che/westm/qc_che.pdf

• You can contact me at [email protected]

computational quantum chemistry for the che classroom - 2007 asee summer school for chemical...

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