Hartree-Fock solutions of the ground state of lithium

Angular momentum theory assumes spin-orbitals are orthonormal. Often orthogonality follows from spin-angular quantum numbers being different, but in the $1s^22s\;\Term 2 S/$ ground  state of lithium, for example, this assumption requires orthogonality of the 1s and 2s radial functions. The stationary conditions now apply to the functional

$$\begin{array}{l l}{\cal F}(P(1s;r),P(2s;r)) ={\cal E}(1s^22s\;\T......le 2s \vert 2s \rangle +\lambda_{1s,2s}\langle 1s \vert 2s \rangle\end{array}$$ (21)

where

$${\cal E}(1s^22s\;\Term 2 S/) = 2I(1s,1s) + I(2s,2s) +F^{0}(1s,1s)+ 2 F^{0}(1s,2s) - G^{0}(1s,2s) .$$ (22)

Applying the variational condition we get equations of the form

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

$$\frac{2}{r}X(nl;r) +\quad \varepsilon_{nl,n'l}P(n'l;r)$$ (23)

where $\varepsilon_{1s,2s} = \lambda_{1s,2s}/2$ and $\varepsilon_{2s,1s} =\lambda_{1s,2s}$ or

$$\varepsilon_{1s,2s} = (1/2)\varepsilon_{2s,1s}.$$ (24)

Multiplying eq:Li by $P(n^\prime l^\prime ;r)$

2001-01-09