The general Hartree-Fock equations

Suppose the configuration defining the configuration state $\gamma LS$has assigned w_a electrons to the n_al_a subshell and that msubshells are occupied. Then

$${\cal E}(\gamma LS) \sum_{a=1}^m w_a\left[ I(a,a) +\left(\frac{w_a-1}{2}\right)\sum_{k=0}^{2l_a} f_k(l_a)F^{k}(a,a)\right]+$$ =

$$\quad \sum_{a=2}^m \left\{\sum_{b=1}^{a-1} w_a w_b \left[F^{0}(a......+ \sum_{k=\vert l_a-l_b \vert}^{l_a+l_b} g_k(l_a,l_b)G^{k}(a,b)\right]\right\}.$$

This energy expression depends only on the radial integrals. The part of the expression that depends on P(a;r), for example, is the contribution to the energy of the entire (n_al_a)^w_a subshell, the negative of the removal energy of the subshell. Let us denote this quantity by $\bar{\cal E}((n_al_a)^{w_a})$. Then the stationary condition for a Hartree-Fock solution applies to this expression but since the variations must be constrained in order to satisfy orthonormality assumptions, Lagrange multipliers need to be introduced. The stationary condition then applies to the functional

$${\cal F}(P(a;r)) = -\bar{\cal E}((n_al_a)^{w_a}) +\sum_{b=1}^{m}\delta_{l_al_b}\lambda_{ab}\langle a \vert b \rangle.$$ (32)

Analyzing this energy more carefully, it can be shown that

$$-\bar{\cal E}((n_al_a)^{w_a}) w_a\left( I(a,a) +\left(\frac{w_a-1}{2}\right)\sum_{k=0}^{2l_a} f_k(l_a)F^{k}(a,a)\right)+$$ =

$$\quad \sum_{\begin{array}{c}b =1 \\b \ne a\end{array}}w_a w_......(a,b) + \sum_{k=\vert l_a-l_b \vert}^{(l_a+l_b)} g_k(l_a,l_b)G^{k}(a,b)\right].$$

Applying the variational conditions to each of the integrals, dividing by -w_a, we get the equation

$$\left(\frac{d^2}{dr^2} +\frac{2}{r}[Z-Y(a;r)] -\frac{l_a(l_a+1)}{r^2} -\varepsilon_{aa}\right)\P{a}{} =$$

$$\frac{2}{r}X(a;r) + \sum_{\begin{array}{c}b =1 \\b \ne a\end{array}} \delta_{l_al_b} \varepsilon_{ab}\P{b}{}$$ (33)

where

$$(w_a-1)\sum_{k=0}^{2l_a} f_k(l_a)Y^{k}({a}{a};r) +\sum^m_{\begin{array}{c}b =1 \\b \ne a\end{array}}w_bY^{0}({b}{b};r)$$ Y(a;r) =

$$\sum^m_{\begin{array}{c} b =1 \\ b \ne a \end{array}}w_b\sum_{k=\vert l_a-l_b \vert}^{(l_a+l_b)}g_k(l_a,l_b)Y^{k}({a}{b};r)\P{b}{} .$$ X(a;r) = (34)

Notice that a factor of two arises from the variational properties of the integrals, a factor that becomes four for the F^k(a,a)integrals that represent the ``self-interaction'' within a subshell.

2001-01-09