Solution of the Electronic Schrödinger Equation Using Basis Sets to Solve the Electronic Schrödinger Equation with Electron Correlation
Errors in HF Predictions: Binding Energies D e (kcal/mol) HF Expt l Chemical Bonds Hydrogen Bonds Electrostatic Bonds van der Waals Bonds HF 100.3 141.6 N 2 122.3 228.4 F 2-27.0 39.0 (HF) 2 3.7 4.6 N 2 -HF 1.27 2.22 He 2 NB 0.0218
Mathematical Models for Electron Correlation Configuration Interaction e = 0 + C ia i a + H e C = E e C C ij ab ij ab + Perturbation Theory R Long history in electronic structure theory R Very flexible, e.g., can describe both ground and excited states R Number of configurations grows rapidly with excitation level R Truncated CI not size extensive H e = H 0 + H 1 e = 0 + 1 + 2 2 + E e = E 0 + E 1 + 2 E 2 + R Most widely used technique for including electron correlation RH 0 usually taken to be the HF Hamiltonian R Recent studies have revealed serious convergence problems
Models for Electron Correlation (cont d) Coupled Cluster Theory e = e T 0 T = t 1 + t 2 + t 3 + t 1 = t ia a a+ a i t 2 = t ab ij a b+ a a+ a j a i t 3 =... R Recent addition to electronic structure theory R Includes dominant higher-order terms as products of lower order terms R Rapid convergence if wavefunction is dominated by well localized electron pairs, e.g., CCSD is exact if electron pairs are completely separate R Convergence problems if HF wavefunction provides a very poor zeroorder description of molecule
Notation for Correlated Calculations Perturbation Theory Methods H e = H 0 + H 1 = 0 + 1 + 2 2 +... Coupled Cluster Methods = e T 0 T = T 1 + T 2 + T 3 +... Variational Methods = 0 + C a i a i + C ab ij ab ij +... {ai} {ab}{ij} T 2 T 1 +T 2 T 1 +T 2 +T 3... MP2 MP3 MP4... CCD CCSD CCSDT... SDCI SDTCI... MRCI
SDCI Calculations on the Oxygen Atom -100.0 E n,n-1 = E corr (n, l) - E corr (n-1, l) E n,n+1 (me h ) -10.0-1.0 (1h) (ng) (nf) (nsnp) (nd) -0.1 1 2 3 4
Contributions to Correlation Energy (SDCI) Basis Function Groupings Contributions of basis functions to the correlation energy for the first row atoms fall into distinct groups with E 1,0 (sp) E 1,0 (d) E 2,1 (sp) E 2,1 (d) E 1,0 (f) E 3,2 (sp) E 3,2 (d) E 2,1 (f) E 1,0 (g) These grouping form the foundation for the construction of correlation consistent basis sets: cc-pvdz: cc-pvtz: cc-pvqz: HF Orbitals + (1s1p1d) HF Orbitals + (2s2p2d1f) HF Orbitals + (3s3p3d2f1g) where to balance the errors cc-pvdz: cc-pvtz: cc-pvqz: HF Set = (9s4p) HF Set = (10s5p) HF Set = (12s6p) -100.0-10.0-1.0-0.1 cc-pvdz cc-pvtz cc-pvqz cc-pv5z 1 2 3 4 N bf
B B P B J H Atomic Calculations with cc-sets E corr (me h ) -50.0-100.0-150.0-200.0-250.0-300.0 B J H P F R B B B J J J J H C H H H P F R F R P F R P F R Ne -350.0 2 3 4 5 6 P B N O F Exponential Convergence E corr (n) = E corr ( ) + E corr (2)e - (n-2) Inverse Powers of l max (=n) E corr (n) = E corr ( ) + A/n 3
Errors in Molecular Calculations Basis Set Convergence Error Q bsm (n) = Q(M,n) Q (M, ) Intrinsic Error Q M = Q(M, ) Q(expt l) Calculational Error Q calc dm (n) = Q(M,n) Q (expt l) = Q bsm (n) + Q M
Illustration of Types of Errors in Calculations Type I Type II Type III Q(expt l) Q M ( ) Q calc d M Q M Q bsm (n) Note: Q calc d M 0 n n n
Confusion: Convergence of D e (N 2 ) with MPn 240.0 D e (kcal/mol) 230.0 220.0 210.0 200.0 228.4 kcal/mol MP2 MP3 MP4 Basis set: cc-pvtz
Resolution of the N 2 Problem
Binding Energies: Chemically Bound Molecules CH HF N 2 CO D e (expt l) a 83.9 141.6 228.4 259.3 D e (core-valence) -0.2-0.2-0.8-0.9 D e (valence-only) 83.7 141.4 227.6 258.4 CCSD -0.8-2.0-9.9-7.5 CCSD(T) 0.0 0.1-0.3 0.1 CCSDT 0.1 0.0-0.9-0.3 MP2-2.7 4.4 12.4 13.6 MP3-1.2-3.3-11.8-7.9 MP4-0.4 1.3 4.2 5.9 MP5-0.8 a Huber, K. P.; Herzberg, G. Molecular Spectra and Molecular Structure IV. Constants of Diatomic Molecules; Van Nostrand,; Princeton, 1979.
Binding Energies: Hydrogen-bonded Molecules (HF) 2 D e (expt l) a, kcal/mol 4.56 CCSD -0.16 CCSD(T) -0.02 MP2-0.09 MP3-0.03 MP4-0.02 a Cayton, D. C.; Jucks, K. W.; Miller, R. E. J. Chem. Phys. 1989, 90, 2631; Klopper, W.; Quack, M.; Suhm, M. A. J. Chem. Phys. 1998, 108, 10096.
Binding Energies: Weakly Bound Molecules N 2 -HF a Ar-HF b Ar-FH b Ar-HCl c Ar-ClH c D e (expt l), cm -1 776.±30 211.±4 109.±10 176.±5 148.±10 CCSD -52-45 -36 CCSD(T) 17 0-15 0-1 MP2 35-10 -16 31 33 MP3-36 -31-31 MP4 38 7-10 10 7 a Lovejoy, C. M.; Nesbitt, D. J. J. Chem. Phys. 1987, 86, 3151; Nesbitt, D. J.; Child, M. S. J. Chem. Phys. 1993, 98, 478; Nesbitt, D. J.; Lindeman, T. G.; Farrell, J. T., Jr.; Lovejoy, C. M. J. Chem. Phys. 1994, 100, 775; Bemish, R. J.; Bohac, E. J.; Wu, M.; Miller, R. E. J. Chem. Phys. 1994, 101, 9457; Farrell, J. T.; Sneh, O.; Nesbitt, D. J. J. Phys. Chem. 1994, 98, 6068; Tang, S. N.; Chang, H-C.; Klemperer, W. J. Phys. Chem. 1994, 98, 7313. b Hutson, J. M. J. Chem. Phys. 1992, 96, 6752 and references therein. c Hutson, J. M. J. Chem. Phys. 1988, 89, 4550; Hutson, J. M. J. Phys. Chem. 1992, 96, 4237; and references therein.
Binding Energies: Very Weakly Bound Molecules He 2 a Ne 2 b Ar 2 c D e (expt l), cm -1 7.59 29.4 99.6 D e (core-valence) +0.05-0.8 D e (valence-only) 29.4 98.8 CCSD -1.1-6.8-26.8 CCSD(T) -0.2-1.0-1.8 CCSDT 0.0 MP2-2.7-10.5 13.2 MP3-1.1-7.1-16.8 MP4-0.5-1.9 1.2 MP5-0.2 a Aziz, R. A.; Slaman, M. J. J. Chem. Phys. 1991, 94, 8047. Aziz, R. A.; Janzen, A. R.; Moldover, R. Phys. Rev. Lett. 1995, 74, 1586. b Aziz, R. A.; Meath, W. J.; Allnatt, A. R. Chem. Phys. 1983, 78, 295. Aziz, R. A.; Slaman, M. J. Chem. Phys. 1989, 130, 187. c Aziz, R. A.; Slaman, M. J. Mol. Phys. 1986, 58, 679. Aziz, R. A. J. Chem. Phys. 1993, 99, 4518
References 1. Gaussian basis sets for use in correlated molecular calculations. I. The atoms boron through neon and hydrogen, T. H. Dunning, Jr., J. Chem. Phys. 90, 1007-1023 (1989). 2. Electron affinities of the first-row atoms revised. Systematic basis sets and wave functions, R. A. Kendall, T. H. Dunning, Jr., and R. J. Harrison, J. Chem. Phys. 96, 6796-6806 (1992). 3. Gaussian basis sets for use in correlated molecular calculations. III. The second row atoms, Al-Ar, D. E. Woon and T. H. Dunning, Jr., J. Chem. Phys. 980,1358-1371 (1993). 4. Gaussian basis sets for use in correlated molecular calculations. IV. Calculation of static electrical response properties, D. E. Woon and T. H. Dunning, Jr., J. Chem. Phys. 100, 2975-2988 (1994). 5. Gaussian basis sets for use in correlated molecular calculations. V. Core-valence basis sets for boron through neon, D. E. Woon and T. H. Dunning, Jr., J. Chem. Phys. 103, 4572-4585 (1995).
References (cont d) 6. Gaussian basis sets for use in correlated molecular calculations. VI. Sextuple-zeta correlation-consistent sets for boron through neon, A. K. Wilson, T. van Mourik, and T. H. Dunning, Jr., J. Molec. Struct. (Theochem) 388, 339-349 (1996). 7. Gaussian basis sets for use in correlated molecular calculations. VII. The atoms aluminum through argon revisted, T. H. Dunning, Jr., K. A. Peterson, and A. K. Wilson, J. Chem. Phys. 114, 9244-9253 (2001). 8. Gaussian basis sets for use in correlated molecular calculations. VIII. Standard and augmented sextuple zeta correlation consistent basis sets for aluminum through argon, T. van Mourik and T. H. Dunning, Jr., Intern. J. Quant. Chem. 76, 205-221 (2000). 9. Gaussian basis sets for use in correlated molecular calculations. IX. Correlation consistent sets for the atoms gallium through krypton, T. H. Dunning, Jr., J. Chem. Phys. 110, 7667-7676 (1999). 10. Accurate correlation consistent basis sets for molecular core-valence effects: The second row atoms, Al-Ar, and the first row atoms B-Ne revisited, K. A. Peterson and T. H. Dunning, Jr., J. Chem. Phys. 117, 10548-10560 (2002).