Theory of Complex Spectra, Applying Slater-Condon Rules
<PERSON> and <PERSON> in Physical Review 1932, Vol. 42, pg. 167, separate the two $^2$D's of $d^3$. On page 173 (pg 7 of the PDF) of this article they determine the matrix of the electrostatic energy in eqn. 469
3.3 as: $$\begin{array}{ccc} - &a & b \ a & 3F_0+7F_2+63F_4 & 3(21)^{1/2}(F_2-5F_4) \ b & 3(21)^{1/2}(F_2-5F_4) & 3F_0+3F_2-57F_4 \ \end{array}$$
I'm trying to figure out how they actually determine this matrix. How do they do it?
Here is what I understand:
This matrix comes from a description of the two terms, $$^2D^a_{2,1/2} = 1/2 [-A+B-C+E]$$ $$^2D^b_{2,1/2} = (84)^{-1/2}[-5A-3B-C+4D-3E-2(6)^{1/2}F]$$
where $A,B,C,D,E,F$ are zero-order states of $d^3$ which are $$\begin{array}{cccc} A &\mathcal{A}u_1(2^+)u_2(2^-)u_3(-2^+) & B &\mathcal{A}u_1(2^+)u_2(1^+)u_3(-1^-) \ C &\mathcal{A}u_1(2^+)u_2(1^-)u_3(-1^+) & D &\mathcal{A}u_1(2^-)u_2(1^+)u_3(-1^+) \ E &\mathcal{A}u_1(2^+)u_2(0^+)u_3(0^-) & F &\mathcal{A}u_1(1^+)u_2(1^-)u_3(0^+) \ \end{array}$$
In this notation $u_i(a^j)$ indicates the one-electron central field eigenfunction for electron $i$ with the $j^{th}$ set of numbers $n^il^im_l^im_s^i$. $\mathcal{A}$ represents the antisymmetrizing operator $(N!)^{-1/2}\sum_P (-1)^p P$.
We can say that $(ab|1/r_{12}|cd)$ depends, in part, on $\sum_{k=0}^\infty c^k (l^a m_l^a, l^c m_l^c) c^k(l^dm_l^d,l^bm_l^b)$
There are tables of these values already determined, notably as $a^k$ and $b^k$, which can be used to evaluate the Coulomb and exchange integrals for $k=0,2,4$.
<PERSON> then extended <PERSON>'s method to determine the non-diagonal matrix elements in a paper (Physical Review, 1930, Vol. 36, pg. 1121) for separating multiplets of the same kind, as we have in our case.
In case you are confused by the notation, $J(a,b)=\sum a^k$ and $K(a,b)=\sum b^k$ and $a^k(l^am_l^a,l^bm_l^b)=c^k(l^am_l^a,l^am_l^a)c^k(l^bm_l^b,l^bm_l^b)$ and $b^k(l^am_l^a,l^b m_l^b) = [c^k(l^am_l^a,l^bm_l^b)]^2$. 490