Multilevel Methods Toward Accurate Estimation of Thermochemical Properties

1. 1782 II. József Robert Monckton Multilevel MethodsToward Accurate Estimation of Thermochemical Properties MilánSzőri 2014 The world’sfirstice-calorimeter 2014 „Virtualcalorimeter” http://en.wikipedia.org/wiki/Calorimetry http://gizmodo.com/298029/worlds-biggest-supercomputer-is-a-virus Viktor Orbán

2. Importance of thermochemical calculations • Predictive power • Check the experiment and its evaluation are done properly „The recommended values of this paper affect a large number of other thermochemical quantities which directly or indirectly rely on or refer to D0(H−OH), D0(OH) and ΔHf(OH).” J. Phys. Chem. A, 2002, 106, 2727–2747. • Check thesevaluesconsistency • Providemissing thermochemical properties • Forunknown species • Unable to measure • Estimation of rate constants • Input in atmospheric and combustion chemistry

3. ImportantProperties • ClassicalThermochemicalProperties G=H-TS H=U+pV=U+RT IE, IP, BDE, EA etc. • Needed • Geometry (rotationalcontribution and repulsionenergy) • Frequencies (vibrationalcontribution) • Energy Rewardaslittleaspossible!!!

4. Intermezzo The energy is a function of position and of electrons and nuclei and the time Very high dimension!!!! Decoupling (Separation) of coordinates is highly needed!

5. Separation in practise Exact nonrelativistic Hamiltonian in field-free space: Nuclei Electron Nuclei Nuclei + Electron Electron Small Constant R is only parameter if the coordinates of nuclei are fixed (PHYSICALLY!!!) Model: Classicaltreatmentfornuclei QM forelectron Clamped-nuclei Schrödinger equation:

6. Clamped-nuclei Schrödinger equation Electronic Schrödinger equation: Nuclear Schrödinger equation: Electronic energy the nuclei move in a potential set up by the electrons At fixed geometry (PHYSICALLY!!!): Total energy neglected Nuclear-nuclear repulsion Etot≈ Eel+ENN Total energy ≈ (Electronic energy)+ (Nuclear-Nuclear repulsion)

7. Model • Molecular SE • No generalsolution (highdimension), soinstead: • Clamped-nuclei SE: Classicalnuclei + QM electrons • It is possibletosolveitwithsomeapproximations(seeMolecularMethods) • Itgivessolutionfortheelectronicproblem • Correctionforthenuclei SE (+QM electrons) • Frequencycalculation (Vibrationalanalysis) Approximation of themolecular SE

8. One consequence of BO • Definition of a molecule: • An electrically neutral entity consisting of more than one atom which must correspond to a depression on the potential energy surface that is deep enough to confine at least one vibrational state. (IUPAC)

9. Molecular methods • In practiseaccuracy, robustness and system size are coupled • Neat ab initio or ideal DFT • Parametrized methods: • Scaling correlation energies (SAC-x, PCI-80, SCS-MPx) • Extrapolated methods (CBS) • Composite methods/Multilevel methods/Modelchemistry • Applied DFT functionals (B3LYP) • Semiempirical methods (PM6) • Force fields (CHARMM-AA)

10. Molecular methods • In practiseaccuracy, robustness and system size are coupled • Neat ab initio or ideal DFT • Parametrized methods: • Scaling correlation energies (SAC-x, PCI-80, SCS-MPx) • Extrapolated methods (CBS) • Composite methods/Multilevel methods/Modelchemistry • Applied DFT functionals (B3LYP) • Semiempirical methods (PM6) • Force fields (CHARMM-AA)

11. Definition • Quantum chemistry multilevelmethods /composite methods/modelchemistries: are computational chemistry methods that aim for high accuracy by combining the results of several(individual) calculations. ci and cjkcan be: dependent independent (!) fromexperimentaldata

12. Multilevel Methods G2, G3, G2MP2, G3MP2, G3B3, G3MP2B3, G3-RAD, … Purely additive protocols: Extrapolative/additive protocols: CBS-4, CBS-q, CBS-Q, CBS-APNO, W1, W1U, W1BD, W2, W3, W4, … SAC, MCQCISD, MCG3, G3S, G3S(MP2), G3X, … Scaled/additive protocols: BAC-MP4, PDDG/MNDO, PDDG/PM3 Bond-correcting protocols:

13. MultilevelMethods • Single-Point – trivial method • X1 –Xiamen • Additivity/Extrapolation/Scaled • MCCM - Multicoefficient (correlation) models • Gn – Gaussian • ccCA-x – CorrelationConsistentCompositeApproach • CBS-n – CompleteBasisSet • Wn – Weizmann • HEAT – HighaccuracyExtrapolatedAb initio Thermochemistry Today >1 NO! empiricalparameter(s)

14. Measuringthecalcs performance • Inthecase of a singlemolecule: • Deviation/Error (D): D=experimentalvalue – calculatedvalue • Absolutedeviation/absoluteerror (AD): AD=abs(D) Relativedeviation (RD): RD=AD/experimentalvalue • Forset of molecules (usually standard test sets): • MeanUnsignedError (MUE): MUE=mean(ADi) • Maximum absolutedeviation (MAD): MAD=max(ADi) • Root-Mean-SquaredError (RMSE): • Standard Deviation (SD): • Errordistribution (Histogram): Chemicalaccuracy:AD< 1 kcal/mol Spectroscopicaccuracy: AD< 1 kJ/mol • MeanDeviation (MD): • MD=mean(Di) • Largestdeviation (LD): • LD=max(Di) RMSE SD

15. Measuring the calcs performance • Setof molecules (standard test sets): • Often transition metals are not well-represented • (the largest experimental error > 40 kJ/mol) http://www.cmt.anl.gov/OldCHMwebsiteContent/compmat/g3-05.htm http://www.cmt.anl.gov/OldCHMwebsiteContent/compmat/g2geoma.htm http://www.cmt.anl.gov/OldCHMwebsiteContent/compmat/g3-99.htm http://www.begdb.com/

16. MultilevelMethods

17. Single-Point • Experience: geometry isnot thatsensitive to the level of theory as energy. • Example:ozone(1O3) a non-trivialcase • Notation: QCISD(T)/6-31G(d)//B3LYP/6-31G(d) Properties//Geometry It is notalwaysa simpleenergycalculation! (NMR) John A. Pople http://cccbdb.nist.gov/

18. Single-Point Length of a line betweentwopointsdoesnotalwaysgiveyouthesmallestdistanceincomputationalchemistry.

19. G3MP2B3 Larry A. Curtiss John A. Pople Johann Carl Friedrich Gauss

20. G3MP2B3modelchemistry Itselements: Geometry and frequencies: B3LYP/6-31G* ΔE(ZPE)=0.96*ZPE (2)Additional higher polarization:ΔE(G3Large)=E(MP2/G3Large))-E(MP2/6-31G(d)) (3) Correction forMP2truncation: ΔE(QCI)=E(QCISD(T)/6-31G*)-E(MP2/6-31G*) (4) Remaining deficiencies:ΔE(HLC)=-Anβ -B(nα -nβ) nα ≥nβ ΔE(SO) ΔE(SO)arethesameasusedin G3 E0 (G3MP2B3)=E(MP2/6-31G*)+ΔE(G3Large)+ΔE(QCI)+E(HLC)+E(ZPE) +E(SO) E0(G3MP2B3)=E(QCISD(T)/6-31G*)+ΔE(G3Large)+E(HLC)+E(ZPE) +E(SO) Fromtable A. G. Baboul, L. A. Curtiss, P. C. Redfern, and K. Raghavachari, J. Chem. Phys., 1999, 110, 7650-7657.

21. G3MP2B3model chemistry E E(MP2/6-31G(d)//B3LYP/6-31G(d)) ΔE(G3Large) ΔE(QCI) E(MP2/G3MP2Large//B3LYP/6-31G(d)) ‘Vector sum’ E(QCISD(T)/6-31G(d)//B3LYP/6-31G(d)) G3MP2B3 energy-(ΔE(HLC)+ΔE(ZPE)) QCISD(T) limit 6-31G(d) 6-311+G(3df,2p) An estimation of QCISD(T)/6-311++G(3df,2p) level of theory Curtiss, L. A.; Raghavachari, K.; Pople, J. A. J. Chem. Phys. 1999,110, 7650.

22. ΔE(HLC) correction • E.g. OH radical O 1s12s2 2p4valence: 6e- H 1s1valence: 1e- Ms=2 nα= 4 ≥ nβ= 3 ΔE(HLC)=-Anβ -B(nα -nβ) ΔE(HLC)=E(empiric)= -0.010041×3-0.004995×(4-3)=-0.035118 Hartree

23. ΔE(SO) correctionforatoms Unit conversion!!! It is notincludedinthe G3MP2 energyinthe output file !!! Thiscorrectionneedsto be added manually!!! J. Chem. Phys., 1998, 109, 7764-7776.

24. Practise T = 298.15 K E(QCISD(T)/6-31G(d)) Ecorr Temperature= 298.150000 Pressure= 1.000000 E(ZPE)= 0.007972 E(Thermal)= 0.010332 E(QCISD(T))= -75.537195 E(Empiric)= -0.035118 DE(MP2)= -0.093259 G3MP2(0 K)= -75.657600 G3MP2 Energy= -75.655240 G3MP2 Enthalpy= -75.654295 G3MP2 Free Energy= -75.674542 ΔE(HLC) ΔE(G3Large) E(G3MP2B3)=Etot(G3MP2B3) + Ecorr T = 298.15 K and P = 1 atm H(G3MP2B3)=Etot(G3MP2B3) + Hcorr G(G3MP2B3)=Etot(G3MP2B3) + Gcorr

25. MUE fromexperiment 1 kcal/mol (=4.184 kJ/mol) is thechemicalaccuracy

26. MultilevelMethods Performance MeasuredbyPublications

27. G2

28. G3

29. G3MP2B3

30. Thankforyourattention!