Post–Hartree–Fock Explained

In computational chemistry, post–Hartree–Fock^{[1] [2]} (post-HF) methods are the set of methods developed to improve on the Hartree–Fock (HF), or self-consistent field (SCF) method. They add electron correlation which is a more accurate way of including the repulsions between electrons than in the Hartree–Fock method where repulsions are only averaged.

Details

In general, the SCF procedure makes several assumptions about the nature of the multi-body Schrödinger equation and its set of solutions:

• For molecules, the Born–Oppenheimer approximation is inherently assumed. The true wavefunction should also be a function of the coordinates of each of the nuclei.
• Typically, relativistic effects are completely neglected. The momentum operator is assumed to be completely nonrelativistic.
• The basis set is composed of a finite number of orthogonal functions. The true wavefunction is a linear combination of functions from a complete (infinite) basis set.
• The energy eigenfunctions are assumed to be products of one-electron wavefunctions. The effects of electron correlation, beyond that of exchange energy resulting from the anti-symmetrization of the wavefunction, are completely neglected.

For the great majority of systems under study, in particular for excited states and processes such as molecular dissociation reactions, the fourth item is by far the most important. As a result, the term post–Hartree–Fock method is typically used for methods of approximating the electron correlation of a system.

Usually, post–Hartree–Fock methods^[3] give more accurate results than Hartree–Fock calculations, although the added accuracy comes with the price of added computational cost.

Post–Hartree–Fock methods

• Configuration interaction (CI)^{[4] [5]}
• Coupled cluster (CC)^{[6] [7] [8]}
• Multi-configuration time-dependent Hartree (MCTDH,^[9])
• Møller–Plesset perturbation theory (MP2,^[10] MP3, MP4,^[11] etc.)
• Quadratic configuration interaction (QCI)^[12]
• Quantum chemistry composite methods (G2,^[13] G3,^[14] CBS, T1.^[15] etc.)

Related methods

Methods that use more than one determinant are not strictly post–Hartree–Fock methods, as they use a single determinant as reference, but they often use similar perturbation, or configuration interaction methods to improve the description of electron correlation. These methods include:

• Multi-configurational self-consistent field (MCSCF)
• Multireference single and double configuration interaction (MRCISD)
• N-electron valence state perturbation theory (NEVPT).

Further reading

• Book: Jensen, F. . Introduction to Computational Chemistry . John Wiley & Sons . New York . 1999 . 0471980854.

Notes and References

1. Book: Cramer, Christopher J. . Essentials of Computational Chemistry . John Wiley & Sons . 2002 . 0-470-09182-7.
2. Book: Jensen, Frank . Introduction to Computational Chemistry 2nd edition . John Wiley & Sons . 1999 . 0-470-01187-4.
3. Book: DaCosta, Herbert. Rate Constant Calculation for Thermal Reactions : Methods and Applications. 2011. John Wiley & Sons. 9781118166123. 769342424.
4. Analytical second derivatives for excited electronic states using the single excitation configuration interaction method: theory and application to benzo[a]pyrene and chalcone . Molecular Physics . 96 . 10 . 1533–1541 . Taylor & Francis . May 10, 1999 . 10.1080/00268979909483096 . David Maurice . Martin Head-Gordon . amp . 1999MolPh..96.1533M.
5. A doubles correction to electronic excited states from configuration interaction in the space of single substitutions . Chemical Physics Letters . 219 . 1–2 . 21–29 . Elsevier . 1994 . 10.1016/0009-2614(94)00070-0 . Martin Head-Gordon . Rudolph J. Rico . Manabu Oumi . Timothy J. Lee . amp . 1994CPL...219...21H.
6. A full coupled‐cluster singles and doubles model: The inclusion of disconnected triples . The Journal of Chemical Physics . 76 . 4 . 1910–1919 . The American Institute of Physics . 1982 . 10.1063/1.443164 . George D. Purvis . Rodney J. Bartlett . amp . 1982JChPh..76.1910P.
7. A fifth-order perturbation comparison of electron correlation theories . Chemical Physics Letters . 157 . 6 . 479–483 . Elsevier Science . March 24, 1989 . 10.1016/S0009-2614(89)87395-6 . Krishnan Raghavachari . Gary W. Trucks . John A. Pople . Martin Head-Gordon . amp . 1989CPL...157..479R.
8. Two-body coupled cluster expansions . The Journal of Chemical Physics . 115 . 11 . 5033–5041 . The American Institute of Physics . June 19, 2001 . 10.1063/1.1390516 . Troy Van Voorhis . Martin Head-Gordon . amp . 2001JChPh.115.5033V.
9. The multi-configurational time-dependent Hartree approach . Chem. Phys. Lett. . 165 . 73 . 73–78 . 1990 . 10.1016/0009-2614(90)87014-I . H. D. Meyer . U. Manthe . L. S. Cederbaum . amp. 1990CPL...165...73M .
10. Note on an Approximation Treatment form Many-Electron Systems . Physical Review . 46 . 7 . 618–622 . The American Physical Society . October 1934 . 10.1103/PhysRev.46.618 . Chr. Møller . M. S. Plesset . amp . 1934PhRv...46..618M.
11. Approximate fourth-order perturbation theory of the electron correlation energy . International Journal of Quantum Chemistry . 14 . 1 . 91–100 . Wiley InterScience . February 22, 1978 . 10.1002/qua.560140109 . amp . Krishnan Raghavachari . John A. Pople.
12. Quadratic configuration interaction. A general technique for determining electron correlation energies . amp . John A. Pople . Martin Head‐Gordon . Krishnan Raghavachari . The Journal of Chemical Physics . 87 . 10 . 5968–35975 . American Institute of Physics . 1987 . 10.1063/1.453520 . 1987JChPh..87.5968P.
13. Gaussian‐2 theory for molecular energies of first‐ and second‐row compounds . The Journal of Chemical Physics . 94 . 11 . 7221–7231 . The American Institute of Physics . February 15, 1991 . 10.1063/1.460205 . Larry A. Curtiss . Krishnan Raghavachari . Gary W. Trucks . John A. Pople . amp . 1991JChPh..94.7221C. free .
14. Gaussian-3 (G3) theory for molecules containing first and second-row atoms . The Journal of Chemical Physics . 109 . 18 . 7764–7776 . The American Institute of Physics . July 22, 1998 . 10.1063/1.477422 . Larry A. Curtiss . Krishnan Raghavachari . Paul C. Redfern . Vitaly Rassolov . John A. Pople . amp . 1998JChPh.109.7764C.
15. Efficient Calculation of Heats of Formation . The Journal of Physical Chemistry A . 113 . 10 . 2165–2175 . ACS Publications . January 2009 . 10.1021/jp810144q . William S. Ohlinger . Philip E. Klunzinger . Bernard J. Deppmeier . Warren J. Hehre . amp . 19222177. 2009JPCA..113.2165O .