Interelectronic angle densities of equivalent electrons in Hartree–Fock theory of atoms

The interelectronic angle density $A(θ12)$ is the probability density function that the angle $θij$ $(0⩽θij⩽π)$ subtended by the vectors $ri$ and $rj$ of any two electrons i and j becomes $θ12.$ For equivalent electrons in atoms, it is shown that the density $A(θ12)$ in the Hartree–Fock theory is given by a simple polynomial of $cos θ12.$ Detailed expressions are reported for all LS terms arising from $s2,$ $pN$ $(N=2–6),$ $dN$ $(N=2–10),$ and $fN$ $(N=2,12)$ electron configurations. With no modifications, the present results apply as well to the interelectronic angle density $Ā(θ̄12)$ in momentum space, where $θ̄12$ is the angle between two electron momenta.