Proof of the Hartree-Fock equations

As per the proof of the Hartree equations, the proof of the Hartree-Fock equations involves the use of an analogue of eq4 to find an expression similar to eq18. Applying the Lagrange method of undetermined multipliers, the resultant expression is then minimised to obtain the non-canonical form of the Hartree-Fock equations.

The analogue of eq4 is:

where  is given by eq66.

Substituting eq86 and eq89 in eq93


Since  (see this article for explanation),

For , we must have . To find the minimum value of E subject to the constraints of , we apply the Lagrange method of undetermined multipliers to form the new functional , where

Substituting eq95 in eq96,

Applying the functional variation method as per the proof of the Hartree equations and noting that the minimum energy corresponding to F is when a small change in the functional’s input yields no change in the functional’s output, i.e. when or

It can be easily shown, by expanding the Coulomb and exchange terms in eq98, that the two terms within each square bracket are the same. Therefore,

where we have changed the dummy variables and to and respectively.

Let’s carry out the following for the above equation:

1) Use the Hermitian property of the operator on the first term.
2) Switch the dummy indices i and j to j and i respectively for the 2nd exchange integral and the 1st undetermined multiplier term.
3) Rearrange.

The result is:



Show that , i.e. the Lagrange undetermined multipliers are elements of a Hermitian matrix.


Changing the dummy variables and of eq96 to and respectively and taking the complex conjugate throughout, noting that E and hence F is real (recall that the functional F is obtained by subtracting the function , where from E),

Using eq96 again and switching the dummy indices i and j to j and i respectively,

Comparing eq100 and eq101,

or equally


Substitute eq101a in eq99,


Using the same logic described in the steps taken from eq20 to eq21, we have

and its complex conjugate

Eq104 and eq105 are the Hartree-Fock equations. However, they are not in the canonical form (eigenvalue form). For the derivation of the canonical Hartree-Fock equations, see the next article.


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