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

Variational Procedure To Determine The Multi-configurational Dirac-Fock Orbital Equations

To build a multi-configuration atomic state function (ASF) denoted by from configuration state functions (CSF), we proceed as follows,

where the are the configuration mixing coefficients. Using this, the expectation energy is written as,

and the Hamiltonian matrix elements are,

To prove this, note the expectation energy can be written as,

where is the identity matrix.

To evaluate the Hamiltonian matrix elements, the diagonal terms are,

Note, all f,g,F and G functions are symmetric with respect to interchange. Also . The off-diagonal terms are,

Now, the one particle radial integrals are,

The two particle radial integrals are,

these are also known as Slater integrals. Also,

Y is Hartree's Y function given by,

Now, perform a variation with respect to the mixing coefficients. In the multi-configuration scheme, the normalization condition becomes,

if Then the variation wrt the mixing coefficients can be written as,

This leads to the matrix eigenvalue problem for the mixing coefficents,

Variation with respect to orbital a gives four different terms; , , and .

First, some definitions,

generalized occupation number ¯

,

configuration averaged coefficients

and, ,

direct potential

,

single configuration exchange term

, ,

Kinetic terms

,

potential terms

.

Using these definitions, the can be expressed as,

Then,

The , similar to can be expressed as,

Then,

Next, the ,

Where,

Note,

Finally, the Lagrange multiplier term,

Finally, we obtain a multi-configuration Dirac-Fock orbital equation,

Therefore, the multiconfiguration orbital equations are,

where,

where, , and .