We wish to find an extremum in the expectation energy functional subject to
the constraint that the single particle wave functions are orthonormalized.
This constraint is included using the method of Lagrange undetermined
multipliers
.
We will now evaluate each of the terms in the second expression, first the one particle terms;
The one particle terms become,
The coulomb term becomes,
Note
because of symmetry of a and b and;
And the exchange term becomes,
Now, the last term involving the Lagrange multiplier,
Finally, the Dirac-Fock radial SCF equations are,
where,
.
This gives the radial equations for the orbitals
where,
The above operators can be expressed as,
or in atomic units,
And,