The expectation energy is a functional of the wave function
,
written as,
where,
The
are the Pauli Spin matrices.
This can be re-written as,
where,
This can further be separated as follows, a one particle operator part,
and a two particle operator part,
with
given by
1.18 or 1.19.
Now apply the results of angular decomposition to the expectation energy result found above.
The radial one particle integrals are,
where,
This can be expressed as,
The two particle integrals can be expressed as;
where Y is Hartree's Y function given by,
and the angular coefficients for the exchange integral
is