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## Properties of the Exchange-Correlation Hole

The pair-density is a probability, consequently the definition given in (2.7) with the value , implies that,

 (2.11)

and so the magnitude of the hole density can never be greater than the density at the site of the electron. The pair-density is also symmetric under the interchange of electron coordinates,
 (2.12)

 (2.13)

This has consequences for the pair-correlation function described in the next subsection. The exchange-correlation hole can be conveniently separated into a summation of exchange and correlation contributions, also known as the Fermi and Coulomb holes,
 (2.14)

whereby the exchange (Fermi) hole is defined in terms of the -dependent hole as,
 (2.15)

and subsequently the correlation (Coulomb) hole is,
 (2.16)

The exchange hole can be defined exactly from the Hartree-Fock expression for the exchange energy, given by the last term in relation (1.26),
 (2.17)

where the exchange hole, given in terms of spin orbitals, , is:
 (2.18)

This leads to the following sum rule condition,
 (2.19)

and so the corresponding sum rule on the correlation hole must be,
 (2.20)

From relation (2.19), the exchange hole satisfies the inequality,
 (2.21)

The on-top hole is the value of the hole density when the inter-electron distance is zero, i.e. when . Considering exchange interactions only, relation (2.19) is zero for a pair of opposite-spin electrons, whereas for same spin electrons it leads to the exact on-top condition,
 (2.22)

The sum rule and negativity constraints on the exchange hole represent stringent requirements, which together, determine a well defined spatial range for the exchange hole. A deep exchange hole, i.e. one with a large on-top value, will have a short spatial extent, and vice-versa. The corresponding sum rule on the correlation hole is a much weaker constraint in comparison, since can have positive or negative values, and so a correlation hole length scale is less well defined.

Next: The Pair-Correlation Function Up: Fundamentals Previous: Exact Definition of Exchange   Contents
Dr S J Clark
2003-05-04