Exchange and correlation treatment

Schwarz, K.

doi:10.1107/97809553602060000639

International
Tables for
Crystallography
Volume D
Physical properties of crystals
Edited by A. Authier

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International Tables for Crystallography (2006). Vol. D. ch. 2.2, p. 299

Section 2.2.9.1. Exchange and correlation treatment

K. Schwarz^a ^*

^a Institut für Materialchemie, Technische Universität Wien, Getreidemarkt 9/165-TC, A-1060 Vienna, Austria
Correspondence e-mail: kschwarz@theochem.tuwein.ac.at

2.2.9.1. Exchange and correlation treatment

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Hartree–Fock-based (HF-based) methods (for a general description see, for example, Pisani, 1996) are based on a wavefunction description (with one Slater determinant in the HF method). The single-particle HF equations (written for an atom in Rydberg atomic units) can be written in the following form, which is convenient for further discussions: $[\displaylines{\left[-\nabla^2+V_{Ne}({\bf r})+ \sum\limits_{j=1}^N \int\left| \psi_j ^{\rm HF}(r^\prime)\right| ^2{2\over |{\bf r-r}^\prime|}\,\,{\rm d}{\bf r}^\prime\right.\hfill\cr\left.\quad- \sum\limits_{j=1}^N \int\psi_j^{\rm HF}({\bf r}^\prime)^\ast{1\over|{\bf r-r}^\prime|}P_{rr^\prime}\psi_j^{\rm HF}({\bf r}^\prime)\,\,{\rm d}{\bf r}^\prime\right]\psi_i^{\rm HF}({\bf r})\hfill\cr\quad\quad = \epsilon_i^{\rm HF}\psi_i^{\rm HF}({\bf r}),\hfill(2.2.9.1)}]$ with terms for the kinetic energy, the nuclear electronic potential, the classical electrostatic Coulomb potential and the exchange, a function potential which involves the permutation operator $[P_{rr^{\prime}}]$ , which interchanges the arguments of the subsequent product of two functions. This exchange term can not be rewritten as a potential times the function $[\psi _{i}^{\rm HF}({\bf r})]$ but is truly non-local (i.e. depends on $[{\bf r}]$ and $[{\bf r}^{\prime}]$ ). The interaction of orbital j with itself (contained in the third term) is unphysical, but this self-interaction is exactly cancelled in the fourth term. This is no longer true in the approximate DFT method discussed below. The HF method treats exchange exactly but contains – by definition – no correlation effects. The latter can be added in an approximate form in post-HF procedures such as that proposed by Colle & Salvetti (1990).

Density functional theory (DFT) is an alternative approach in which both effects, exchange and correlation, are treated in a combined scheme but both approximately. Several forms of DFT functionals are available now that have reached high accuracy, so many structural problems can be solved adequately. Further details will be given in Section 2.2.10.

References

Colle, R. & Salvetti, O. (1990). Generalisation of the Colle–Salvetti correlation energy method to a many determinant wavefunction. J. Chem. Phys. 93, 534–544.Google Scholar

Pisani, C. (1996). Quantum-mechanical ab-initio calculation of properties of crystalline materials. Lecture notes in chemistry, 67, 1–327. Berlin, Heidelberg, New York: Springer-Verlag.Google Scholar

International Tables for Crystallography (2006). Vol. D. ch. 2.2, p. 299