UNCERTAINTIES OF MOLECULARSTRUCTURAL PARAMETERS
IAEA-ITAMP joint workshop on UncertaintyAssessment for Atomic and Molecular Data
Cambridge, MA, USA, July 7-9, 2014
Attila G. CsászárLaboratory of Molecular Structure and Dynamics &
MTA-ELTE Research Group on Complex Chemical SystemsEötvös University, Budapest, Hungary
OUTLINEIntroduction and motivation
The quantum chemical approach to re
Empirical approaches to re
The spectroscopic approach to re
Different structural typesUncertainties in experiment and theoryvia examples
Summary
Sources of uncertainties in structural parameters
(a) Many different experimental and first‐principles (QC) techniques
(b) Many different structure definitions
(c) Intrinsic uncertainties of the methods
quantum chemistry spectroscopy diffraction
empirical NMR
rotation X‐rays
electrons(GED)
neutrons
semi‐experimental (SE)
DFTab initio (BO)
J. Demaison
GED Spectroscopy ab initio (exl. nucl. motion)
ra B0 fijk re
rm semi‐exp ab initioexp.r,T
comparison
Principal interrelationships
Principal structural types
• equilibrium structures: reBO, read, reSE, reexp
• average structuresposition averages: rz = r,0; r,T (av. nuclear position in ground vibr. state)distance (and angle) averages: rg,T; ra,T; <r2>T1/2; <r ‐2>T‐1/2; <r3>T1/3; <r ‐3>T‐1/3
• mass‐dependent structures: rm; rc; rmρ; rm(1); rm(2)
• empirical structures: r0; rs; rs variants (ps‐Kr)
Accurate (equilibrium) structuresof gas-phase molecules
• Traditional debate (experiment):MW vs. GED
• Newer debate (experiment vs. theory): MW and GED vs. ab initio
• End of debate:experiment AND theory
Variability of structural parameters
• relatively large for angles(accurate determination of angles is usually slightly easier than that of distances)
• relatively small for bonded distances• 1.060 Å < re(C‐H) < 1.120 Å, variability is only about 0.06 Å• 1.153 Å < re(C=O) < 1.206 Å, variability is only about 0.05 Å(1.153 Å for OCSe, 1.160 Å for CO2, 1.205 Å for H2CO, 1.206 Å for CH3CHO)
• 1.10 Å < re(CO single, double, triple) < 1.43 Å, variability is about 0.33 Å(1.1056 Å for HCO+ and 1.4338 Å for HCOOCH3)
Structures of semirigid molecules:the experimental/empirical path
H. D. Rudolph, J. Demaison, A. G. Császár, J. Phys. Chem. A 2013, 117, 12969-12982.
Structures of semirigid molecules:experimental or theoretical paths
H. D. Rudolph, J. Demaison, A. G. Császár, J. Phys. Chem. A 2013, 117, 12969-12982.
Structures of semirigid molecules:joint experimental-theoretical path
H. D. Rudolph, J. Demaison, A. G. Császár, J. Phys. Chem. A 2013, 117, 12969-12982.
Theory: quantum chemistry• Quantum particles: nuclei and electrons• Full treatment is still impractical for all but the simplest many‐body systems
• Often it is sufficient to solve the time‐independentSchrödinger equation (TISE)
• Introduction of the Born‐Oppenheimer (BO) approximation results in electronic structure and nuclear motion theories
• No practical analytic solutions: variational and perturbative treatments
Number of independent internalcoordinate force constants
terms linear XY2 bent XY2 XY3Z X6Y6
Linear re 1 2 3 2
Quadratic fij 3 4 12 34
Cubic fijk 3 6 38 237
Quartic fijkl 6 9 102 1890
Quintic fijklm 6 12 249 12031
Sextic fijklmn 10 16 562 69 448
A. G. Császár, J. Phys. Chem. A 1992, 96, 7898-7904.
Motivations of the Focal-PointAnalysis (FPA) Approach
Get the right result for the right reason for polyatomicand polyelectronic systems.
Attach error bars to theoretical predictions (excellenthandle on uncertainties).
Consider small physical effects tacitly neglected in most quantum chemical studies, such as core correlation, relativistic effects, and corrections to the Born-Oppenheimer approximation.
Approach spectroscopic accuracy (1 cm-1) as opposed to chemical (1 kcal mol-1) or calibration (1 kJ mol-1) accuracy in predictions of relative energies and spectra.
A. G. Császár, W. D. Allen, H. F. Schaefer, J. Chem. Phys. 1998, 108, 9751-9764.
The Focal-Point Analysis (FPA) approachuse of a family of basis sets which systematically
approaches completeness (e.g., (aug-)cc-p(C)VnZ)applications of low levels of theory with prodigious basis sets (typically direct RHF and MP2 computations with up to thousands of basis functions)higher-order (valence) correlation (HOC) treatments [these days FCI, CCSDTQ, CCSDT, and CCSD(T)] with the largest possible basis setslayout of a two-dimensional extrapolation grid based on an assumed additivity of correlation incrementseschewal of empirical corrections/extrapolationsaddition of “small” correction terms (CC, Rel, DBOC)
A. G. Császár, W. D. Allen, H. F. Schaefer, J. Chem. Phys. 1998, 108, 9751-9764.
Electron correlation treatment
Hamilto
nian
One
-par
ticle
bas
is s
et
Exac
tan
swer
DZHF
TZHF
QZHF
5ZHF
Z(CBS)
HF
DZCCSD
DZCCSD(T)
DZFull-CI
DZCCSDT
(MP2, CISD) (MP4, CISDTQ)
CBSFull-CI
Rubik’s cube of BO electronic structure theory
Diatomic paradigms
• W. D. Allen and A. G. Császár, J. Chem. Phys. 98, 2983(1993).
• A. L. L. East, W. D. Allen, and A. G. Császár, inStructures and conformations of non‐rigid molecules, Kluwer: Dordrecht, 1993.
• A. G. Császár and W. D. Allen, J. Chem. Phys. 104,2746 (1996).
Percent error curves for the RHF/DZP electronic energy derivatives
___________________________________________________________________ W. D. Allen and A. G. Császár, J. Chem. Phys. 98, 2983 (1993).
Theory: incremental corrections to the equilibrium geometry of H2
16O
Level re/Å e/degreesaug‐cc‐pV6Z ICMRCI_FC 0.958 705 104.413CBS ICMRCI_FC 0.000 085 +0.010Core correlation 0.000 986 +0.134Relativistic (Breit) +0.000 164 0.074QED (1‐e Lamb) 0.000 002 +0.003DBOC +0.000 040 +0.015Final 0.957 837 104.500
A. G. Császár, G. Czakó, T. Furtenbacher, J. Tennyson, V. Szalay, S.V. Shirin, N. F. Zobov, O. L. Polyansky, J. Chem. Phys. 2005, 122, 214305.
“Empirical” structuresGuiding principle: I0 = Ie + ɛ, where I0 are rotational constantscorresponding to the ground vibrational state of an isotopologue
choice of ɛ symbol comment
0 r0 LSQ fitting
constant rs Costain, 1958; no LSQ
g g1/2 rm
(1) 1979; for each principal axis g
g g1/2+dg
∏ 1/(2N-2) rm(2) Watson, 1999
Traditional (semi)experimental approach(inverse, top down)
Determine (preferentially experimentally)rovibrational energy levels(ground state) effective rotational constantsvibration-rotation interaction constantscentrifugal and electronic corrections equilibrium geometry
Recommended first-principles approach(direct, bottom up)
Determine (preferentially fully ab initio) adiabatic potential energy hypersurface equilibrium geometryChecking the accuracy of re:rotational energy levelsvibrationally averaged spectroscopic constants
νννν EckartBBB
)2/1( vv ee BB
Be B0 idi
2i1
3N6
What we need
i B0 B(vi 1)
ith fundamental statei =1, …3N ‐ 6
Rotational constants ≤ 3 (per isotopologue)
Spectra = m isotopic species x 3N ‐ 5 vibrational states
Semiexperimental structure: principles
Spectroscopy (MW or IR) B0 (or I0)
Ab initio (or DFT) cubic force field i (or )
some important advantages:• small ( < 1%): high accuracy is ″not needed”• easy for isotopologues• fast
Semiexperimental structure: practice
Possible to study large molecules
• Proline (and other amino acids)• 17 atoms• 45 parameters• no. of cubic force constants: 62 835• W. D. Allen, E. Czinki, A. G. Császár, Chem. Eur. J. 2004, 10, 4512
• tropinone• 23 atoms• 34 parameters• J. Demaison,… J. Phys. Chem. A 2012
Semiexperimental structure: problems• accuracy of the ground‐state rotational constants• isotopic species: difficult or time consuming to produce• presence of small coordinates• strong Coriolis and/or anharmonic resonances• large rotation of the principal axes• accuracy of the force field
• form of the potential• H atoms: large amplitude motion or …
• ill‐conditioning (solution: mixed regression)• …
)1(0 ii vBB
Rotational constants (in cm-1) of D2
16O: “experiment” vs. theoryLABEL EXPT. #1 VAR DIFF.
A (2 2 0) 17.456 17.536 -0.080
A (0 2 2) 16.857 16.619 0.238
A (3 0 0) 14.354 14.667 -0.313
A (1 0 2) 14.551 14.261 0.290
B (0 2 2) 7.292 7.352 -0.060
B (3 0 0) 6.956 7.020 -0.064
B (1 0 2) 7.112 7.099 0.087
Rotational constants (in cm-1) of D2
16O: “experiment” vs. theoryLABEL EXPT. #1 VAR DIFF. EXPT. #2 DIFF.
A (2 2 0) 17.456 17.536 -0.080 17.545 0.008
A (0 2 2) 16.857 16.619 0.238 16.631 0.012
A (3 0 0) 14.354 14.667 -0.313 14.677 0.010
A (1 0 2) 14.551 14.261 0.290 14.255 0.006
B (0 2 2) 7.292 7.352 -0.060 7.359 0.007
B (3 0 0) 6.956 7.020 -0.064 7.018 0.002
B (1 0 2) 7.112 7.099 0.087 7.113 0.014
Large rotation of the principal axis system upon isotopic substitution
• important for oblate molecules (A B >> C)• non‐constant systematic error : A B• see J. Demaison, A. G. Császár, L. D. Margulés, H. D. Rudolph,
J. Phys. Chem. A 2011, 115, 14078.
Ia Ib angleparent 0.0001 -0.0001 018Osyn +0.0151 -0.0160 41°
example: residuals of HNO3 (mean value: 0.0001)
How about uncertainties?
The structure of HCO+
r(HC) r(CO) method reference1.09725(4) 1.10474(2) expt. Woods 881.0919(5) 1.1058(2) ab initio Sebald 901.09215 1.10545 ab initio Martin 931.0919(9) 1.1053(3) semi-expt. Puzzarini 96
1.0924 1.10558 expt. Dore 03
Vibrationally averaged vs. equilibrium geometries
• effective
• average
• rms
• substitution rs
2/12 rrv
r
2/12r
223
e 1 r
1er
221
e 1 r
223
e 1 fr
ee / rrr
C-H bond length in cyclopropane
Method value ref. Method value ref.
r0 1.083(3) Butch 1973 r0 1.0774 Endo 1987
ra 1.089(3) Bastiansen 1964 rz 1.080(3)
rg 1.099(2) Yamada 1985 re 1.074(1)
rz 1.084(2) reSE 1.079(1) Gauss 2000
re 1.083(5) reBO 1.079 Demaison 2000
Discrepancies: usual between different methods
Molecule bond re method r
OCCl2 C=O 1.176(2) rs 1.1852(5)
OCSe C=O 1.1533(3) rs 1.1561(2)
GeH3Cl Ge-Cl 2.1447(2) r0 2.14947(5)
Discrepancies:not rare within the “same” method (re)
molecule parameter I II
HCO+ C-H 1.09725(4) 1.0919(9)
FCP C-F 1.272 1.284
H2C=C=C: C=C: 1.291(1) 1.287
PH3 P-H 1.41161(2) 1.4168(2)
Isotopic effects on the equilibrium geometry of water
Isotopomer re/Å e/degreesH2
16O 0.957 837 104.500H2
17O 0.000 001 +0.0001H2
18O 0.000 001 +0.0003D2
16O 0.000 020 0.010D2
18O 0.000 023 0.009HD16O 0.000 001/ 0.002
0.000 022
A. G. Császár, G. Czakó, T. Furtenbacher, J. Tennyson, V. Szalay, S.V. Shirin, N. F. Zobov, O. L. Polyansky, J. Chem. Phys. 2005, 122, 214305.
Isotopic effects on the equilibrium geometry of water
Isotopomer re/Å e/degreesH2
16O 0.957 837 104.500H2
17O 0.000 001 +0.0001H2
18O 0.000 001 +0.0003D2
16O 0.000 020 0.010D2
18O 0.000 023 0.009HD16O 0.000 001/ 0.002
0.000 022
Conclusion: isotopic effects are large for non‐equilibrium but almost negligible for equilibrium structural parameters.
• degrees of freedom large• no isotopic substitution (e.g., F)• no systematic error
– , electron correlation, resonances, …• independent errors (autocorrelation)• small condition number • problem of weighting
Estimation of the uncertainty from the standard deviations of the fit
Solution strategies
• increase the accuracy of the data• experimental: accuracy of B0, , …• ab initio: accuracy of B0 ‐ Be
• decrease the condition number • increase the variety of data (semi‐exp. !)
• clever isotopic substitution• "mixed" regression (or predicate observations)
Mixed regression
y = X
• weighted least squares
• predicate
S wi yiexp ˆ y i
calc i 2
min
S wi yiexp ˆ y i
calc i 2
w j jpred j
calc 2 min
where wk 1 k2
Equilibrium and effective structures of the H2
16O molecule
r/Å /degrees
BornOppenheimer 0.957 82 104.485Adiabatic (H2
16O) 0.957 85 104.500(D2
16O) 0.957 83 104.490Spectroscopic 0.957 77 104.48Effective (rg) (H2
16O) 0.976 25 103.96(300 K) (D2
16O) 0.971 36 104.03_______________________________________________________ A. G. Császár, G. Czakó, T. Furtenbacher, J. Tennyson, V. Szalay, S. V. Shirin,N. F. Zobov, O. L. Polyansky, J. Chem. Phys. 122, 214305 (2005).G. Czakó, E. Mátyus, A. G. Császár, J. Phys. Chem. A 113, 11665 (2009).
________________________________
_________________________________________________________________________
rg structures of water (T = 293 K)H2
16O D216O
rg(OH)/Å rg(HH)/Å rg(OD)/Å rg(DD)/Årg calc.
(GED)(Spectr.)
0.97566(0.9763)(0.9745)
1.53812(1.567)(1.537)
0.97136(0.9700)(0.9702)
1.53122(1.526)(1.531)
ra calc.(Adiabatic re
0.971380.95785
1.529561.51472
0.967830.95783
1.525161.51460)
rg re 0.01781(0.0182)cubic
0.01353(0.0131)cubic__________________________________________________________
G. Czakó, E. Mátyus, A. G. Császár, J. Phys. Chem. A 113, 11665 (2009).
Temperature dependence ofrg(OH) of water
________________________________________________________
G. Czakó, E. Mátyus, A. G. Császár, J. Phys. Chem. A 113, 11665 (2009).
H216O
rg(OH)/Å rg(HH)/Å0 K
300 K700 K
0.975650.976250.97718
1.538231.538121.53825
1000 K1500 K
0.978130.98048
1.539071.54070
GED parameters, root-mean-square amplitudelg (Å) and anharmonicity parameter Å
T/K lg(OX) lg(XX) (OX) (XX)
0 0.0690 0.1142 7.0 ‐1.7
500 0.0690 0.1145 6.5 ‐1.9
1500 0.0713 0.1280 5.3 ‐24.7
0 0.0586 0.0958 3.8 ‐0.2
500 0.0587 0.0972 3.6 ‐1.2
1500 0.0630 0.1172 3.0 ‐25.0
H2O
D2O
NH3 and ND3 structures (300 K)
_______________________________________________________ I. Szabó, C. Fábri, G. Czakó, E. Mátyus, A.G. Császár, J. Phys. Chem. A 116, 4356 (2012).
A system with tunneling motion: NH3
T/K rg(NH)rovib
rg(NH)vib
ra(NH)rovib
ra(NH)vib
rg(HNH)rovib
rg(HNH)vib
0 1.02962 1.02962 1.02454 1.02454 106.733 106.733
10 1.02966 1.02965 1.02458 1.02457 106.717 106.717
30 1.02969 1.02965 1.02461 1.02457 106.713 106.717
50 1.02972 1.02965 1.02464 1.02457 106.711 106.717
re(NH) = 1.01117 and e(HNH) = 106.362. All distances in angstrom, all angles in degrees.
VII. SummaryIt is possible to obtain very accurate molecular structures by experimental means(spectroscopy or GED) when the number of molecular parameters is not large (notlarger than the number of rotational constants for spectroscopy) but the standarddeviation of the fit is definitely not a good indicator of the accuracy obtained.
It is possible to obtain first‐principles predictions of structural parameters of anykind with reliable uncertainty estimates but this procedure is constrained at the moment to molecules containing no more than about 4 atoms and requireenormous computations is accuracy a specific goal.