The Functional Form

As with all types of functional, the manner in which the (coupling-constant averaged) exchange-correlation (XC) hole is modelled characterises the functional form. To recap, the exact hole $n_{\hbox {{XC}}}({\bf r},{\bf r'})$ has the general form given by relation (2.24),

$$n_{\hbox{{XC}}}({\bf r},{\bf r'}) = n({\bf r'}) [g_{\hbox{{XC}}}({\bf r},{\bf r'}) -1] \, .$$ (4.1)

In the LDA the, non-local dependence on the density, $n({\bf r'})$, is replaced by the local density $n({\bf r})$, and the pair-correlation function is that obtained from the homogeneous electron gas $g_{\hbox{{XC}}}^{\hbox{{hom}}}(r)$, which uses the modulus of the distance $r = \vert\,{\bf r}-{\bf r'}\,\vert$,

$$n_{\hbox{{XC}}}^{\hbox{{LDA}}}({\bf r},{\bf r'}) = n({\bf r}) [g_{\hbox{{XC}}}^{\hbox{{hom}}}(r) - 1] \, .$$ (4.2)

The local density dependence in (4.2) means that for finite inhomogeneous systems such as atoms and molecules, the LDA prescription will lead to an overestimate of the true XC hole value at points of high density, and an underestimate at low densities. However, the LDA in general yields sensible energies for many systems as result of the partial cancellation of these errors in the XC hole [87]. The LDA hole also obeys the fundamental sum rule given in (2.8) that constrains the XC energy to sensible bounds. In an attempt to improve upon the LDA description, Gunnarsson et al. proposed the average density approximation (ADA) [84], whereby $n({\bf r'})$ is replaced by a density average $\bar{n}({\bf r})$ over the extent of the hole, rather than just $n({\bf r})$. The ADA ansatz is:

$$n_{\hbox{{XC}}}^{\hbox{{ADA}}}({\bf r},{\bf r'}) = \bar{n}({\bf r}) [\tilde{g}_{\hbox{{XC}}}^{\hbox{{hom}}}(r,\bar{n}({\bf r})) - 1] \,$$ (4.3)

where $\tilde{g}_{\hbox{{XC}}}^{\hbox{{hom}}}(r,\bar{n}({\bf r}))$ is the homogeneous electron gas form, but the averaged density $\bar{n}({\bf r})$ is used instead of the actual density. The average density is determined by,

$$\bar{n}({\bf r}) = \int W[r,\bar{n}({\bf r})] \, n({\bf r'}) \, d{\bf r'} \, ,$$ (4.4)

where the weight function $W[r,\bar{n}({\bf r})]$ must satisfy the normalisation condition,

$$\int W[r,\bar{n}({\bf r})] \, d{\bf r'} = 1 \, .$$ (4.5)

The ADA however has certain deficiencies and the error cancellation is usually not as complete as in the LDA [87].

In the same paper that the ADA was proposed, Gunnarsson et al. devised the weighted density approximation, which was also arrived at independently by Alonso and Girifalco [85]. The WDA hole retains the same non-local density dependence as the exact result given in (2.24), and is usually modelled in terms of a simple analytic function, $G^{\hbox{{WDA}}}[{\bf r},{\bf r'};\tilde{n}({\bf r}))]$,

$$n_{\hbox{{XC}}}^{\hbox{{WDA}}}({\bf r},{\bf r'}) = n({\bf r'}) \, G^{\hbox{{WDA}}}[{\bf r},{\bf r'};\tilde{n}({\bf r})] \, .$$ (4.6)

The specific choice of $G^{\hbox{{WDA}}}[{\bf r},{\bf r'};\tilde{n}({\bf r})]$ is investigated in this work, suffice it to say that it must obey a minimum number of exact limiting conditions, which will be elaborated on in Sec. 4.2.3, in order to be physically sensible. The quantity $\tilde{n}({\bf r})$ is a non-local parameter called the weighted density, and is determined at each point in space by satisfying the sum rule relation (2.8) on $n_{\hbox{{XC}}}^{\hbox{{WDA}}}({\bf r},{\bf r'})$,

$$\int n({\bf r'}) \, G^{\hbox{{WDA}}}[{\bf r},{\bf r'};\tilde{n}({\bf r})] \, d{\bf r'} = -1 \, .$$ (4.7)

Substituting (4.6) into the exact formula for $E_{\hbox{{XC}}}[n({\bf r})]$, given by (2.9), leads to the definition of the WDA energy functional,

$$E_{\hbox{{XC}}}^{\hbox{{WDA}}}[n({\bf r})] = \frac{1}{2}\in... ...,{\bf r'})} {\vert\,{\bf r}- {\bf r'}\,\vert}\, d{\bf r'} \, .$$ (4.8)

Form (2.11), the corresponding WDA energy-density $\varepsilon_{\hbox{{XC}}}^{\hbox{{WDA}}}({\bf r})$ is,

$$\varepsilon_{\hbox{{XC}}}^{\hbox{{WDA}}}({\bf r}) = \frac{1... ...},{\bf r'})} {\vert\,{\bf r}-{\bf r'}\,\vert}\, d{\bf r'} \, .$$ (4.9)

The WDA is a conceptually simple XC functional that is based on the exact expression given by the adiabatic connection method. The WDA essentially models the exact, but unknown, XC hole by retaining the correct non-local dependence on the density and through the use of analytic expressions, $G^{\hbox{{WDA}}}[{\bf r},{\bf r'};\tilde{n}({\bf r})]$. The form of the WDA functional is therefore immediately understandable - this should be contrasted with the complicated GGA forms given in the previous chapters.