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## The Exchange-Correlation Hole

The non-relativistic many-body electronic Hamiltonian for a system of interacting electrons is given by,

 (2.1)

The electron-electron interaction is a two-body operator and so the corresponding expectation value can be written as,
 (2.2)

where is the normalised antisymmetric groundstate wavefunction of the system. The pair-density gives the probability of simultaneously finding an electron at the point within volume element , and another electron at in volume element , among the other electrons in the system. Rigorously it is defined as,

The electron density is given by,
 (2.3)

since , this leads to the following condition,
 (2.4)

In a classical description the motions of electrons are not correlated, so the probability of finding the pair of electrons at the points and is simply given by a product of the density at the respective points, i.e.
 (2.5)

substituting into (2.2) yields the classical Coulomb repulsion, or Hartree energy. However this classical description violates (2.5). In reality, electrons obey Fermi statistics and so are kept apart quantum-mechanically by the Pauli-exclusion principle, and also from other non-classical Coulomb interactions. The effect of these exchange and correlation interactions is to reduce the classical value of the electron density at due to the instantaneous position of the second electron located at . Therefore each electron creates a depletion, or hole, of electron density around itself as a direct consequence of exchange-correlation effects. Taking account of the hole, the pair-density can be written as,
 (2.6)

where the quantum effects are accounted for by the exchange-correlation hole density, , surrounding each electron located at position .

From relation (2.5), the exchange-correlation hole satisfies an important normalisation condition known as a sum rule,

 (2.7)

This implies that the exchange-correlation hole itself has a deficit of exactly one electron, therefore an electron and its hole constitute an entity with no net charge.

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