Post-Hartree-Fock methods

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In computational chemistry , the post-Hartree-Fock methods (post-HF methods) are the series of methods that have been developed to complement the Hartree-Fock method (HF) or the self-consistent field method (SCF ) to improve. These methods are based on the HF method and have the advantage that they are also pure ab initio methods (i.e. do not contain any empirical parameters) and can be systematically improved. In the post-HF methods, the electron correlation is added to describe the Coulomb repulsion between electrons. In the Hartree-Fock method, in contrast, the repulsion is only considered averaged ( mean field theory ).

Basics

In general, the SCF method makes several assumptions about the nature of the many-body Schrödinger equation and its solution theorem:

  • The Born-Oppenheimer approximation is assumed to be inherent for molecules . However, the true wave function should also be a function of the nuclear coordinates.
  • Typically, relativistic effects are completely neglected, the momentum operator is assumed to be completely non-relativistic.
  • The basic set consists of a finite number of atom-centered functions (typically Gaussian Type Orbitals ). The true wave function, however, is a linear combination of functions from a complete (i.e. infinite) basis set.
  • The eigen-energy functions are assumed to be the products of one-electron wave functions. The effects of the electron correlation that go beyond the exchange energy resulting from the anti-symmetrization of the wave function are completely neglected.

For the vast majority of the systems studied, especially for excited states and processes such as dissociation reactions, the fourth point is by far the most important. Therefore, the term "post-Hartree-Fock method" is typically used for methods that approximate the electron correlation of a system.

Typically, post-Hartree-Fock methods provide more accurate results than Hartree-Fock calculations, although the additional accuracy comes at the cost of additional costs (i.e., higher numerical complexity).

approach

The Post-Hartree-Fock methods are all based on the fact that the electronic wave function is described not just by one, but by a linear combination of several Slater determinants . In addition to the determinant, which is used in the Hartree-Fock method and which represents the ground state, there are also so-called excited determinants, since electrons occupy virtual orbitals here. For electrons and 2n spin orbitals there are in principle possible determinants. From this set the determinants for the approach of the wave function have to be selected. The individual Post-Hartree-Fock methods differ from one another essentially in how these determinants are selected and the expansion coefficients are determined.

There are variational methods that are based on the principle of variation and provide an upper limit of the exact energy (such as Configuration Interaction) and perturbative methods such as Møller – Plesset.

Methods

  • Configuration Interaction (CI)
  • Coupled Cluster (CC)
  • Multi-Configuration Time-Dependent Hartree (MCTDH)
  • Møller – Plesset perturbation theory of various orders (MP2, MP3, MP4 etc.)
  • Quadratic Configuration Interaction (QCI)
  • Compound methods (G2, G3, T1 etc.)

Related methods

Methods that use more than one determinant as a starting point are not post-Hartree-Fock methods in the strict sense, as the latter use a single determinant as a reference. However, they often use approaches based on perturbation theory or configuration interaction to improve the description of the electron correlation. Therefore there is a close relationship to the post-HF methods. These methods include:

  • Multi-configurational self-consistent field (MCSCF), e.g. B. CASSCF
  • Multi-reference configuration interaction (MRCI)
  • N-electron valence state perturbation theory (NEVPT)

Individual evidence

  1. ↑ Christopher J. Cramer: Essentials of Computational Chemistry . John Wiley & Sons, 2002, ISBN 0-470-09182-7 .
  2. ^ Frank Jensen: Introduction to Computational Chemistry 2nd edition . John Wiley & Sons, 1999, ISBN 0-470-01187-4 .
  3. ^ Chr. Møller, MS Plesset: Note on an Approximation Treatment for Many-Electron Systems . In: Physical Review . tape 46 , no. 7 , October 1, 1934, p. 618-622 , doi : 10.1103 / PhysRev.46.618 .
  4. ↑ Björn O. Roos, Peter R. Taylor, Per EM Sigbahn: A complete active space SCF method (CASSCF) using a density matrix formulated super-CI approach . In: Chemical Physics . tape 48 , no. 2 , May 15, 1980, ISSN  0301-0104 , p. 157-173 , doi : 10.1016 / 0301-0104 (80) 80045-0 .
  5. ↑ Björn O. Roos, Per EM Siegbahn: A direct CI method with a multiconfigurational reference state . In: International Journal of Quantum Chemistry . tape 17 , no. 3 , 1980, ISSN  1097-461X , pp. 485-500 , doi : 10.1002 / qua . 560170310 .
  6. ↑ C. Angeli, R. Cimiraglia, S. Evangelisti, T. Leininger, J.-P. Malrieu: Introduction of n-electron valence states for multireference perturbation theory . In: The Journal of Chemical Physics . tape 114 , no. 23 , June 4, 2001, ISSN  0021-9606 , p. 10252-10264 , doi : 10.1063 / 1.1361246 .