1. Calculating Molecular Binding Energies from Chemical Bonds to van der Waals Interactions Thom H. Dunning, Jr. Joint Institute for Computational Sciences University of Tennessee – Oak Ridge National Laboratory Oak Ridge, Tennessee

2. Joint Institute for Computational Sciences

3. Outline of Seminar Statement of the Problem Theoretical Methods Basis Sets and Error Analysis Correlation Consistent Basis Sets Errors in Molecular Calculations Errors in Molecular Binding Energies (for Four Classes) Intrinsic Errors of Methods Basis Set Convergence Errors Statistical Analysis of Errors in Binding Energies Conclusions

4. Statement of Problem Wide Range of Binding Energies 0.010.11.010.0100.01000.0 } He 2 D e (kcal/mol) Ne 2 Ar 2 Ar-HCl N 2 -HF H-COCO H-CO - H-C 2 H 2 H-C 2 H n H-CH n (HF) 2 Ar-HF N2N2

5. Statement of Problem Importance of Electron Correlation HFExpt’l HF100.3141.6 N 2 122.3228.4 F 2 -27.039.0 (HF) 2 3.74.6 N 2 -HF1.272.22 He 2 –0.0218 D e (kcal/mol) Chemical Bonds Hydrogen Bonds van der Waals “Bonds” Electrostatic “Bonds”

6. Theoretical Methods Configuration Interaction  e =  0 +   C i a  i a +   C ij ab  ij ab + … H e C = E e C RLong history in electronic structure theory RVery flexible, e.g., can describe both ground and excited states RNot size extensive/consistent Perturbation Theory H e = H 0 + H 1  e =  0 +  1 + 2  2 + … E e = E 0 + E 1 + 2 E 2 + … RMost widely used technique for including electron correlation RAssumes that electron correlation is perturbation to the HF hamiltonian RRecent studies have revealed serious convergence problems

7. Theoretical Methods Coupled Cluster Theory  e = e T  0 T = t 1 + t 2 + t 3 + … t 1 =   t i a a a + a i t 2 =   t ij ab a b + a a + a j a i t 3 =  RRecent addition to electronic structure theory RIncludes dominant higher-order terms as products of lower order terms RRapid convergence if wavefunction is dominated by well localized electron pairs RConvergence problems if HF wave- function provides poor zero-order description of molecule

8. Basis Sets and Error Analysis Correlation Consistent Basis Sets cc-Sets based on detailed study of electron correlation in atoms Correlation functions added in shells Hartree-Fock orbitals cc-pVDZ:+ (1s1p1d) cc-pVTZ:+ (2s2p2d1f) cc-pVQZ:+ (3s3p3d2f1g) … Augmented with diffuse functions for anions, long range interactions, etc. Molecular properties often exhibit systematic dependence Possible to extrapolate properties to complete basis set limit D e (CO) n (cc-pVnZ) 23456 240.0 245.0 250.0 255.0 260.0 CBS Limit HF Orbitals +1s1p1d +1s1p1d1f +1s1p1d1f1g cc-pVDZ cc-pVTZ cc-pVQZ

9. Basis Sets and Error Analysis Definition of Errors (Method “M”) Basis Set Convergence Error  Q bs M (n) = Q(M,n) – Q (M,  ) Intrinsic Error  Q M = Q(M,  ) – Q(expt’l) Calculational Error  Q calc’d M (n)= Q(M,n) – Q (expt’l) =  Q bs M (n) +  Q M

10. Basis Sets and Error Analysis Convergence Types n Type II Note:  Q calc’d M  0 n Type III QM()QM() n  Q bs M (n ) Type I QMQM Q(expt’l)  Q calc’d M

11. Molecular Binding Energies: Chemical Bonds T. H. Dunning, Jr., J. Phys. Chem. A 104, 9062- 9080 (2000) K. L. Bak, P. Jørgensen, J. Olsen, T. Helgaker, and W. Klopper, J. Chem. Phys. 112, 9229- 9242 (2000)

12. Intrinsic Errors in D e Chemical Bonds CHHFCON 2 D e (kcal/mol) a 83.9141.6258.6227.4 MP2-2.94.213.012.5 MP3-1.4-3.1-8.5-11.7 MP4-0.61.15.74.7 MP5-1.0 CCSD-1.0-2.2-8.1-9.8 CCSD(T)-0.2-0.10.0-0.3 CCSDT-0.1-0.2-0.6-1.1 a Corrected for core-valence and relativistic effects.

13. Basis Set Convergence Errors in D e Chemical Bonds  D e bs (n) (kcal/mol) n n

14. Region of “false positives” 0 Statistical Analysis of Binding Energies  Q (Error in Q)  (Q)  std = error variation  = average error

15. Intrinsic Errors in D e MP2, CCSD, CCSD(T) Methods (6Z Set)  D e (kcal/mol) (De)(De) MP2CCSD+(T)  6.0-8.3-1.0  std 7.54.50.5

16. Basis Set Convergence of D e CCSD(T) Method for Chemical Bonds  D e (kcal/mol) (De)(De)  = -1.0 kcal/mol  std = 0.5 kcal/mol

17. Extrapolation of Binding Energies Analysis of Electron Correlation in He Principal Expansion E = E HF(1s) + E 2(2s2p) corr + E 3(3s3p3d) corr + … Errors in He Atom Klopper et al. [J. Phys. B. 32, R103 (1999)] showed error in truncating series after n th term in principal expansion is Extrapolation Formula

18. Extrapolation of D e CCSD(T) Method for Chemical Bonds  D e (kcal/mol) (De)(De) DZ-TZ TZ-QZ QZ-5Z 5Z-6Z DTTQQ556  -3.5-0.10.0-0.1  std 2.00.50.30.2 1.6

19. Molecular Binding Energies: Hydrogen Bonds T. H. Dunning, Jr., J. Phys. Chem. A 104, 9062- 9080 (2000) A. Halkier, W. Klopper, T. Helgaker, P. Jørgensen, and P. R. Taylor, J. Chem. Phys. 111, 9157 (1999)

20. Intrinsic Errors in D e Hydrogen Bond in HF Dimer (HF) 2 D e (kcal/mol)4.56  0.05 a MP2-0.09 MP3-0.03 MP4-0.02 CCSD-0.16 CCSD(T)-0.02 a W. Klopper, M. Quack, and M. Suhm, J. Chem. Phys. 108, 10096 (1998).

21. Basis Set Convergence Errors in D e Hydrogen Bonds in (H 2 O) 2 and Others  D e bs (n) (kcal/mol) n n -1.5 -0.5 0.0 2345 cc-pVnZ aug-cc-pVnZ d-aug-cc-pVnZ (H 2 O) 2

22. Errors in D e Hydrogen Bonds D e (n) (kcal/mol) n BSSE BSCE D e (n=∞)

23. Extrapolation of D e Hydrogen Bond in HF Dimer D e (n) (kcal/mol) n Calculated Extrapolated

24. Molecular Binding Energies: Electrostatic Interactions T. H. Dunning, Jr., J. Phys. Chem. A 104, 9062- 9080 (2000)

25. Intrinsic Errors in D e Electrostatic Interactions N 2 -HFAr-HFAr-FHAr-HClAr-ClH D e (cm -1 )776 ± 30 a 211 ± 4 b 109 ± 10 b 176 ± 5 c 148 ± 10 c MP235-10-163133 MP3-36-31-31 MP4387-10107 CCSD-52-45-36 CCSD(T)170-1507 a R. J. Bemish, E. J. Bohac, M. Wu, and R. E. Miller, J. Chem. Phys. 101, 9457 (1994) and references therein. b J. M. Huston, J. Chem. Phys. 96, 6752 (1992) and references therein. c J. M. Huston, J. Chem. Phys. 89, 4550 (1988); J. M. Hutson, J. Chem. Phys. 96, 4237 (1992) and references therein.

26. Basis Set Convergence Errors in D e Electrostatic Interactions  D e bs (n) (cm -1 ) n n

27. Molecular Binding Energies: van der Waals Interactions T. H. Dunning, Jr., J. Phys. Chem. A 104, 9062- 9080 (2000)

28. Intrinsic Errors in D e van der Waals Interactions He 2 Ne 2 Ar 2 D e (cm -1 )7.59 a 29.4 b 99.6 c MP2-2.7-10.513.4 MP3-1.1-7.1-17.6 MP4-0.5-1.90.4 MP5-0.2 CCSD-1.1-6.8-27.6 CCSD(T)-0.2-1.0-2.6 CCSDT-0.0 a R. A. Aziz and M. J. Slaman, J. Chem. Phys. 94, 8047 (1991); R. A. Aziz, A. R. Janzen, and R. Moldover, Phys. Rev. Lett. 74, 1586 (1995). b R. A. Aziz, W. J. Meath, and A. R. Allnatt, Chem. Phys. 78, 295 (1983); R. A. Aziz and M. J. Slaman, Chem. Phys. 130, 187 (1989). c R. A. Aziz and M. J. Slaman, Mol. Phys. 58, 679 (1986); R. A. Aziz, J. Chem. Phys. 99, 4518 (1993).

29. Basis Set Convergence Errors in D e van der Waals Interactions  D e bs (n) (cm -1 ) n n

30. Conclusions Critical Assessment of Methods Coupled cluster method provides reliable means of computing molecular properties for molecules well described by single configuration Perturbation method is poorly convergent or even non- convergent; often does not achieve chemical accuracy for chemical bonds Critical Assessment of Basis Sets Correlation consistent basis sets systematically approach complete basis set limit, extrapolation possible Choice of cc-basis set family depends on molecular system  Chemically bound covalent molecules—standard sets

31. Conclusions (cont’d)  Chemically bound ionic, hydrogen-bonded, and electrostaticly bound molecules—singly augmented sets  van der Waals bound molecules—doubly augmented sets Convergence with basis set is slow  Difficult to describe coulomb hole using expansions in one- electron functions  Rate of convergence depends on molecular details  Single, double or triple bonds  Chemically bound, hydrogen-bonded, electrostatically bound or van der Waals bound  Extrapolation substantially improves convergence rate

32. Acknowledgements It is a pleasure to acknowledge contributions of … Kirk Peterson, David Woon, David Feller, Ricky Kendall, Tanja van Mourik, and Angela Wilson to this work It is also a pleasure to acknowledge work of … Poul Jørgensen, Trygve Helgaker, Wim Klopper, Jeppe Olsen and coworkers, whose work has also contributed greatly to calibrating the methods used for molecular calculations Finally, I would like to thank … Division of Chemical Sciences, Office of Science, U.S. Department of Energy for their support of this work.

33. End of Presentation

34. Definitions D e and D 0 E DeDe D0D0 A+B AB Zero Point Energy Separated Atoms Molecule

35. Higher Order Effects in He 2