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Theory of Electron Correlation

  • Chapter
Explicitly Correlated Wave Functions in Chemistry and Physics

Part of the book series: Progress in Theoretical Chemistry and Physics ((PTCP,volume 13))

Abstract

Various aspects of electron correlation are reviewed, from a statistical interpretation of the concept of electron correlation to modern methods to treat correlation effects. Popular inconsistencies in the concept of electron correlation (in part related to an inappropriate normalization) are clarified. Current claims, e.g. that ‘there is a Fermi correlation between electrons of the same spin, and no correlation between electrons of different spin’, or that the ‘Fermi hole integrates to -1’, must at least be modified. It is stressed that Fermi-correlation has more to do with the removal of self-pairing than with genuine exchange. The usefulness of correlation coefficients to describe electron correlation is pointed out. Some less-known facts are stressed, like the possibility of positive (attractive) correlation, and the extremely strong negative correlation in unnatural parity singlet states.

The importance of a formulation of the n-electron problem in Fock space is stressed, and a modern Fock space theory is presented. Excitation operators and k-particle density matrices play a central role. An important aspect of Fock-space theory is separability of operators, which is closely related to extensivity of properties. It is preferable to have a theory entirely in terms of additively separable quantities, such as e.g. the cluster amplitudes of coupled-cluster theory. While the k-particle density matrices are not additively separable (except for k=1), the cumulants of the k-particle density matrices are additively separable. These cumulants have additional attractive properties, which are likely to make them promising tools in the many-electron theory of the future. A generalization of normal ordering with respect to an arbitrary reference function is presented, that contains the traditional particle-hole formalism as a special case, namely for a single Slater determinant reference function. Related to this generalized normal ordering is a generalized Wick theorem.

Some special aspects of correlation are discussed, like democratic vs. autocratic correlation and the relation of correlation to the Born-Oppenheimer separation, the correlation in open-shell states, and the short-range correlation related to singularities in the Hamiltonian. The role of the correlation cusp, especially the slow convergence of a basis expansion, as well as their solutions are discussed.

This chapter ends with a formulation of guiding principles for a satisfactory theory of n-electron states. A state should not be parametrized with more information content than is necessary. This automatically implies linear scaling with the particle number in a molecule in terms of localized orbitals. The challenge of density functional methods is to a large extent related to its restricted information content. Finally density-matrix functional methods are discussed.

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  1. Lehrstuhl für Theoretische Chemie, Ruhr-Universität Bochum, D-44780, Bochum, Germany

    Werner Kutzelnigg

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  1. Institute of Bioorganic Chemistry PAN, Adam Mickiewicz University in Poznań, Poznań, Poland

    Jacek Rychlewski

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Kutzelnigg, W. (2003). Theory of Electron Correlation. In: Rychlewski, J. (eds) Explicitly Correlated Wave Functions in Chemistry and Physics. Progress in Theoretical Chemistry and Physics, vol 13. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-0313-0_1

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  • DOI: https://doi.org/10.1007/978-94-017-0313-0_1

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