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The Hartree Approximation

This approximation starts from the one-electron equations of Eq(gif). tex2html_wrap_inline5845 is chosen to try to model the interaction terms in this equation. The ions contribute a potential

equation169

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, tex2html_wrap_inline5861 , as

equation179

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 tex2html_wrap_inline5861 from the single electron eigenstates

equation189

where the summation over i includes all occupied states. Using this charge density the total one-electron potential is

  equation193

The potential tex2html_wrap_inline5869 is different for each orbital, and therefore the orbitals are not orthogonal. Note that tex2html_wrap_inline5869 depends on all the other orbitals, tex2html_wrap_inline5873 , and so the solution of Eq.(gif) must be found self-consistently.

The choice of tex2html_wrap_inline5845 in Eq.(gif) all seems a bit like guesswork, but it can also be derived using the variational principle. We start with Eq.(gif). The electrons are assumed to be non-interacting, and so the N-electron wavefunction is just the product of the one-electron wavefunctions,

  equation211

This tex2html_wrap_inline5833 can be used with Eq.(gif) to find the expectation value of tex2html_wrap_inline5881

eqnarray219

Introducing a Lagrange multiplier, tex2html_wrap_inline5849 , for the condition that the one-electron wavefunctions are normalised, and minimising the above equation with respect to the wavefunctions, so that

equation238

leads to a set of single particle equations,

  equation244

which are the same as substituting Eq.(gif) in Eq.(gif). These equations are known as the Hartree equations.


next up previous contents
Next: The Hartree-Fock Approximation Up: One-Electron Methods Previous: One-Electron Methods

Andrew Williamson
Tue Nov 19 17:11:34 GMT 1996