Lecture: ab initio Methods Derivation of the Hartree-Fock Equations

Determination of the differential equation for tex2html_wrap_inline677.

We wish to minimize tex2html_wrap_inline705 subject to tex2html_wrap_inline707. This is called the isoparametric problem. Introduce a single Lagrange multiplier tex2html_wrap_inline709, such that;

equation246

equation248

Now show that terms in which tex2html_wrap_inline711 is varied are just the complex conjugate of the terms in which tex2html_wrap_inline713 are varied, thus the variation in tex2html_wrap_inline713, alone insures the minimization is satisfied.

Proof: Let H = T + V, where tex2html_wrap_inline717

Note tex2html_wrap_inline719

Also tex2html_wrap_inline721 if tex2html_wrap_inline723 (V is real)

Now the kinetic energy part, tex2html_wrap_inline725. Using Green's Theorem,

equation250

The surface is out at tex2html_wrap_inline727, because the volume V is all space and tex2html_wrap_inline729 at tex2html_wrap_inline727. Thus the surface integral is zero. (evaluate this by partial integration.) Then,

equation255

Now perform the variation with respect to tex2html_wrap_inline733 only subject to tex2html_wrap_inline707.

eqnarray258

the constraint tex2html_wrap_inline737 does not have to be diagonal but one can perform a unitary transformation on tex2html_wrap_inline677 to make the constraint diagonal.

Then

eqnarray281

for an arbitrary variation in tex2html_wrap_inline741, the integrand [ ] must vanish. This implies that we now have a PDE for tex2html_wrap_inline677,

eqnarray292

or rewriting the last term,

eqnarray305

This is an eigenvalue equation of the form,

equation319

tex2html_wrap_inline745 is the same single particle Hamiltonian, this implies that tex2html_wrap_inline747. This is the energy required to remove the tex2html_wrap_inline631 particle from the system, which is known as Koopman`s Theorem.



Author: Ken Flurchick