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



The Variational Procedure

We wish to find an extremum in the expectation energy functional subject to the constraint that the single particle wave functions are orthonormalized. This constraint is included using the method of Lagrange undetermined multipliers tex2html_wrap_inline1001.

eqnarray353

We will now evaluate each of the terms in the second expression, first the one particle terms;

eqnarray360

The one particle terms become,

equation377

The coulomb term becomes,

equation383

Note tex2html_wrap_inline1003 because of symmetry of a and b and;

eqnarray397

And the exchange term becomes,

eqnarray421

Now, the last term involving the Lagrange multiplier,

eqnarray432

Finally, the Dirac-Fock radial SCF equations are,

eqnarray448

where,

equation458.

This gives the radial equations for the orbitals

equation466

where,

equation474

The above operators can be expressed as,

equation479

or in atomic units,

equation289

And,

eqnarray509



Author: Dr. Warren Perger and Ken Flurchick