I would like to get the Slater-Koster integrals from formula:

$$ \sum{R_j}expi\textbf{k}\cdot(R_j-R_i)\int\psi^*_n(\textbf{r}-\textbf{R}_i)H\psi_m({r}-\textbf{R}_j)d\nu, $$ (1)

where the sum is over the N unit cells, $\textbf{R}_i$ rangs over the positions of the atoms on which orbitals $\psi_n$ are located

as mentioned in paper, Slater, John C., and George F. Koster. "Simplified LCAO method for the periodic potential problem." Physical Review 94, no. 6 (1954): 1498.

I am confused for the integral result for instance

$$E_{x,y}=lm(pp\sigma)-lm(pp\pi),$$ (2)

Let the direction cosines of the direction of the vector $\textbf{R}_j-\textbf{R}_i$, pointing from one atom to the other, be l, m, n.

How to deduce from fomula (1) to (2)?


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