The Exchange-Correlation Hole

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

$$\hat{H} = \hat{T} \, + \, \hat{V}_{\hbox{{ext}}} \, + \, \hat{V}_... ... \; \sum_{i=1}^N \sum_{j > i}^{N} \frac{1}{\vert{\bf r}_i-{\bf r}_j \vert} \, .$$ (2.1)

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

$$\langle \Psi \vert \hat{V}_{\hbox{{ee}}} \vert \Psi \rangle = \fr... ...\bf r},{\bf r'})} {\vert\,{\bf r} - {\bf r'}\,\vert} \, d{\bf r'} d{\bf r} \, .$$ (2.2)

where $\Psi$ is the normalised antisymmetric groundstate wavefunction of the system. The pair-density $P({\bf r},{\bf r'})$ gives the probability of simultaneously finding an electron at the point ${\bf r}$ within volume element $d{\bf r}$, and another electron at ${\bf r'}$ in volume element $d{\bf r'}$, among the other $N-2$ electrons in the system. Rigorously it is defined as,

$$P({\bf r},{\bf r'}) = N(N-1) \, \int \cdots \int \, \vert\Psi({\b... ...r}_N s_N)\vert^2 \, d{\bf r}_3 s_3 \ldots d{\bf r}_{N} s_N \, . \hspace{-1.0cm}$$

The electron density is given by,

$$n({\bf r}) = \frac{1}{N-1} \, \int P({\bf r},{\bf r'}) \, d{\bf r'} \, ,$$ (2.3)

since $\int n({\bf r}) \, d{\bf r}=N$, this leads to the following condition,

$$\int \int P({\bf r},{\bf r'}) \, d{\bf r'} d{\bf r} = N(N-1) \, .$$ (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 ${\bf r}$ and ${\bf r'}$ is simply given by a product of the density at the respective points, i.e.

$$P^{\hbox{{class}}}({\bf r},{\bf r'}) = n({\bf r}) n({\bf r'}) \, ,$$ (2.5)

substituting $P^{\hbox{{class}}}({\bf r},{\bf r'})$ 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 ${\bf r}$ due to the instantaneous position of the second electron located at ${\bf r'}$. 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,

$$P({\bf r},{\bf r'}) = n({\bf r}) n({\bf r'}) + n({\bf r}) n_{\hbox{{XC}}}({\bf r},{\bf r'}) \, ,$$ (2.6)

where the quantum effects are accounted for by the exchange-correlation hole density, $n_{\hbox {{XC}}}({\bf r},{\bf r'})$, surrounding each electron located at position ${\bf r}$.

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

$$\int \, n_{\hbox{{XC}}}({\bf r},{\bf r'}) \, d{\bf r'} =-1 \, .$$ (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.