This approximation starts from the one-electron equations of Eq(). is chosen to try to model the interaction terms in this equation. The ions contribute a potential
All the other electrons in the system also contribute to the potential. The potential due to the electrons is approximated by the electrostatic interaction with all the others, which can be written in terms of the electron density, , as
where the self-interaction potential due to electron i has been removed.
To actually calculate the Hartree potential it is necessary to know the electronic charge distribution of the system. If the electrons are assumed to be independent of each other, then it is straightforward to construct from the single electron eigenstates
where the summation over i includes all occupied states. Using this charge density the total one-electron potential is
The potential is different for each orbital, and therefore the orbitals are not orthogonal. Note that depends on all the other orbitals, , and so the solution of Eq.() must be found self-consistently.
The choice of in Eq.() all seems a bit like guesswork, but it can also be derived using the variational principle. We start with Eq.(). The electrons are assumed to be non-interacting, and so the N-electron wavefunction is just the product of the one-electron wavefunctions,
This can be used with Eq.() to find the expectation value of
Introducing a Lagrange multiplier, , for the condition that the one-electron wavefunctions are normalised, and minimising the above equation with respect to the wavefunctions, so that
leads to a set of single particle equations,
which are the same as substituting Eq.() in Eq.(). These equations are known as the Hartree equations.