Properties of the Exchange-Correlation Hole

The pair-density $P({\bf r},{\bf r'})$ is a probability, consequently the definition given in (2.7) with the value $P({\bf r},{\bf r'})=0$, implies that,

$$n_{{\hbox{{XC}}}}({\bf r},{\bf r'}) \ge - n({\bf r'})$$ (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,

$$P({\bf r},{\bf r'}) = P({\bf r'},{\bf r}) \, ,$$ (2.12)

and this leads to,

$$n_{\hbox{{XC}}}({\bf r},{\bf r'}) = n_{\hbox{{XC}}}({\bf r'},{\bf r})\,\frac{n({\bf r'})}{n({\bf r})} \, .$$ (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,

$$n_{\hbox{{XC}}}({\bf r},{\bf r'}) = n_{\hbox{{X}}}({\bf r},{\bf r'})+n_{\hbox{{C}}}({\bf r},{\bf r'})$$ (2.14)

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

$$n_{\hbox{{X}}}({\bf r},{\bf r'}) = n_{{\hbox{{XC}}},\lambda=0}({\bf r},{\bf r'})$$ (2.15)

and subsequently the correlation (Coulomb) hole is,

$$n_{\hbox{{C}}}({\bf r},{\bf r'}) = n_{{\hbox{{XC}}},\lambda}({\bf r},{\bf r'}) - n_{\hbox{{X}}}({\bf r},{\bf r'}) \, .$$ (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),

$$E_{\hbox{{X}}} = \frac{1}{2} \int n({\bf r}) \, d{\bf r} \int \frac{n_{\hbox{{X}}}({\bf r},{\bf r'})}{\vert{\bf r}-{\bf r'}\vert} \, d{\bf r'}$$ (2.17)

where the exchange hole, given in terms of spin orbitals, $\psi_j({\bf r} s)$, is:

$$n_{\hbox{{X}}}({\bf r},{\bf r'}) = - \frac{1}{n({\bf r})} \sum_{s... ...[ \sum_{j}^{N} \vert \psi_j^*({\bf r} s) \psi_j({\bf r'} s) \vert\right]^2 \, .$$ (2.18)

This leads to the following sum rule condition,

$$\int n_{\hbox{{X}}}({\bf r},{\bf r'}) \, d{\bf r'} = -1 \, ,$$ (2.19)

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

$$\int n_{\hbox{{C}}}({\bf r},{\bf r'}) \, d{\bf r'} = 0 \, .$$ (2.20)

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

$$n_{\hbox{{X}}}({\bf r},{\bf r'}) \le 0 \, .$$ (2.21)

The on-top hole is the value of the hole density when the inter-electron distance is zero, i.e. when ${\bf r} = {\bf r'}$. 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,

$$n_{\hbox{{X}}}({\bf r},{\bf r}) = - n({\bf r}) \, .$$ (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 $n_{\hbox{{C}}}({\bf r},{\bf r'})$ can have positive or negative values, and so a correlation hole length scale is less well defined.