A Tour of the Foundations of
Computational Chemistry and Material Science
Lecture 1: ab initio Methods



Dirac-Fock Expectation Energy Evaluation

The expectation energy is a functional of the wave function tex2html_wrap_inline981, tex2html_wrap_inline993 written as,

equation172

where,

equation177

The tex2html_wrap_inline1004 are the Pauli Spin matrices. This can be re-written as,

equation189

where,

equation206

This can further be separated as follows, a one particle operator part,

equation220

and a two particle operator part,

eqnarray234

with tex2html_wrap_inline983 given by 1.18 or 1.19.

Now apply the results of angular decomposition to the expectation energy result found above.

equation263

The radial one particle integrals are,

equation266

where,

equation270

This can be expressed as,

equation289

The two particle integrals can be expressed as;

  1. Coulomb Integral

    equation300

  2. Exchange Integral

    equation310

    where Y is Hartree's Y function given by,

    equation318

The angular coefficients tex2html_wrap_inline997 for the coulomb integrals, non-relativistic case are,

eqnarray324

and the angular coefficients for the exchange integral tex2html_wrap_inline999 is

equation340



Author: Dr. Warren Perger and Ken Flurchick