Hartree-Fock Variational with two spin aligned electrons

            The wave function is

        Eqn 1

and is here taken to be real.  Note that derivatives with respect to c_1,m and c_2,n will provide a naive introduction to functional derivatives.  The  Hamiltonian is

               Eqn 2

and its expectation value is given by

    Eqn 3

By definition the Hartree-Fock wave function is made up of the orthogonal ₁ and ₂ which minimize the expectation value of the Hamiltonian.  Utilizing this orthogonality

 

Eqn 4

which simplifies a bit to

 Eqn 5

Then taking the derivatives yields

   Eqn 6

Relabelling 1 and 2 in the numerator of the last term makes it equal to the second to the last term.  Then collecting the various expectation values and calling them E_1s.  The equation becomes

        Eqn 7

where the direct potential is

        Eqn 8

and the exchange potential is

   Eqn 9

            In the H.F. approximation including the exchange potential where

the equations to solve  become

  Eqn 10

            The exchange integral is a weighted overlap integral.  Suppose that the two electron are both likely to be in the region around r=β.  The integral for r₁ in the vicinity of β is roughly constant while that a distance away goes as 1/|r₁-β|.  An approximate form for this would thus be

  Eqn 11

where α is a measure of the average size of ₂(r₂) ₁(r₂) throughout the overlap region, β is the location of the overlap region and γ is a measure of the size of the overlap region.

            This has its principle effect for r₁ on the order of β.  For these values the ₂(r) enters the first of equation 10 squared and can thus be incorporated into the |A| while the ₁(β) can be made ₁(r) so that the equation for ₁(r) becomes approximately 

Eqn 12

so that the effect of the exchange potential is that of an attraction to the overlap region.

Examining the Hamiltonian

            The Hamiltonian is

          Eqn 13

where Δv(r₁, r₂) ≠ Δv(r₂, r₁).

            Note that the above is not only true for the H.F. potentials that it is also true for any and all others owing to the addition and subtraction of the same terms.  The one body wave functions are such that

            Eqn 14

            Consider the expectation value of h₁ + h₂  when operating on the antisymmetric product wave function

         Eqn 15

Next consider the expectation value of H-(E₁+E₂) utilizing the orthogonality of the orbitals

Eqn 16

with some variable renaming inside the integrals this becomes

                  Eqn 17

For the H.F. values of the v’s we see that the second term is

      Eqn 18

which is equal to the direct part of the 1/|r₁-r₂| potential as is the expectation of the first term.  The third term for the H.F. value is

   Eqn 19

which is the exchange part of the 1/|r₁-r₂| potential as is also the fourth term.  In the H.F. case, the expectation value of is thus

 Eqn 20

owing to the double counting of this term in both E₁ and E₂. 

            In the general case with v_d1 and v_ex1 left free the result is

          Eqn 21

Appendix A H.F. Orthogonality

            Equation 10, whose solution gives the H.F. orbitals, involves a different equation for each state.  This means that the proof that all non-degenerate solutions are orthogonal does not apply to it.  The fact that there is both a positive direct term and a negative exchange term allows us to rewrite these a bit.  Start with the direct potential, add a term to it and also add the same term to the exchange potential multiplied by ₁(r₁) 

   Eqn 22

and the exchange potential is

 Eqn 23

then rewrite the integral as

          Eqn 24

The new direct potiential and the non-local exchange potential are explicitly the same for both orbitals  and equation 10 becomes

        Eqn 25

and the usual proofs of orthogonality are explicitly valid.  Thus the exchange potential advances the H.F. method beyond the Hartree in that it automatically produces orthonormal orbitals.