A New Model Function

The philosophy behind the development undertaken here was to base the new model on the simple Gaussian function $f_1(u)$, as this was the most promising of the twelve functions studied previously, and then incorporate the Kimball cusp condition by altering only the short-range character of this function. The resulting model, called $G^{\hbox{{Cusp}}}[{\bf r},{\bf r'};\tilde{n}]$, consists of two terms,

$$G^{\hbox{{Cusp}}}[{\bf r},{\bf r'};\tilde{n}] = G^a[{\bf r},{\bf r'};\tilde{n}] \, + \, G^b_{\kappa,m}[{\bf r},{\bf r'};\tilde{n}] \, .$$ (5.7)

where the first term is the original Gaussian model,

$$G^a[{\bf r},{\bf r'};\tilde{n}]=\alpha(\tilde{n}) \, \hbox{e}^{\t... ...yle - \left(\frac{\vert{\bf r}-{\bf r'}\vert} {\beta(\tilde{n})}\right)^2} \, ,$$ (5.8)

and the second incorporates the cusp condition,

$$G^b_{\kappa,m}[{\bf r},{\bf r'};\tilde{n}] = (\alpha(\tilde{n}) +... ...left(\frac{\vert{\bf r}-{\bf r'}\vert} {\kappa \beta(\tilde{n})}\right)^m} \, .$$ (5.9)

Again, shorthand notation is used such that $\tilde{n}=\tilde{n}({\bf r})$. The role of the parameters $m$ and $\kappa$ is to constrain the influence of $G^b_{\kappa,m}[{\bf r},{\bf r'};\tilde{n}]$ to short-ranged interactions only - $m$ adjusts the general shape of the modification, while $\kappa$ directly alters its range, so that the behaviour of $G^a[{\bf r},{\bf r'};\tilde{n}]$ is left unchanged for large inter-electron separations. The range of influence of $G^b_{\kappa,m}[{\bf r},{\bf r'};\tilde{n}]$ is directly proportional to $\kappa$, so when $\kappa=0$, the range of the function is also zero, and the model reverts back to $G^a[{\bf r},{\bf r'};\tilde{n}]$. Different values for $m$ and $\kappa$ will be investigated in Sec. 5.5.4 - the general strategy for determining their values is to vary them in such a way as to give on-top pair-correlation values, $g_{\hbox {{XC}}}({\bf r},{\bf r})$, that stay within the range $[0,0.5]$.

Except in the obvious case where $\kappa=0$, the new model satisfies the Kimball cusp condition for all choices of $m$ and $\kappa$, i.e.

$$\frac{\partial g_{\hbox{{XC}}}^{\hbox{{Cusp}}}({\bf r},{\bf r'})}... ...}({\bf r},{\bf r'}) \bigg\vert _{{\bf r}={\bf r'}} = \alpha(\tilde{n}) + 1 \, ,$$ (5.10)

where $g_{\hbox{{XC}}}^{\hbox{{Cusp}}}({\bf r},{\bf r'})= G^{\hbox{{Cusp}}}[{\bf r},{\bf r'};\tilde{n}]+1$. The newly proposed model is probably the simplest way to incorporate the Kimball cusp condition within the existing WDA framework, and can be implemented within the original computer code with only a few minor adjustments. These points are discussed next.