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



Multi Configuration Dirac-Fock: Introduction

In any independent particle (IP) model, the motion of individual particles (e.g. electrons) is not correlated with the motion of other particles, appart from the Pauli exclusion principle. Interaction between particles correlate their motions. Thus, one can say that any good IP model (approximation), describing a well considered system of particles, includes only the part of the interaction which does not correlate the motions of the particles. The remaining part of the interaction is called the residual interaction or the correlation interaction. Conclusion; the single configuration Dirac-Fock model does NOT include electron correlation by definition. It can be be proved that all determinental wave functions formed from a complete set of single particle states make a complete set of N-particle states. Since each determinental wave function corresponds to a configuration, this means that all configurations make a complete set of N-particle states. The actual atomic wave functions could be developed in the configuration. One can formulate the variational priciple by using a linear combination of different configurations instead of a single configuration. In this way, one has a multi-configuration Hartree-Fock (MCHF) or multi-configuration Dirac-Fock (MCDF) approximation. This approach includes correlation in the atomic states. Single particle states are specified with quantum numbers tex2html_wrap_inline1005 in the non-relativistic case or tex2html_wrap_inline1007 in the relativistic case.

Some definitions;

Atomic shell.
All electrons with the same principle quantum number n. We will not use this term in this fashion.
Equivalent electrons.
All electrons in an atom (ion) with the same tex2html_wrap_inline1009 in the nonrelativistic case, or with the same tex2html_wrap_inline1011 in the relativistic case, i.e., electrons whose angular momentum projection quantum numbers only are different. These electrons have the same radial wave function but the angular momentum parts are different.
Atomic subshell.
Formed by all equivalent electrons. We refer to a subshell as a shell.
Occupation number q.
The number of equivalent electrons, i.e. the number of electrons in a shell tex2html_wrap_inline1013 in the non-relativistic case and tex2html_wrap_inline1015 in the relativistic case.
Closed shell.
Completely occupied (filled) shell. tex2html_wrap_inline1017 in the non-relativistic case or tex2html_wrap_inline1019 in the relativistic case.
Open shell.
Here the occupation numbers are tex2html_wrap_inline1021, in the non-relativistic case and tex2html_wrap_inline1023 in the relativistic case.
Configuration. Set of quantum numbers tex2html_wrap_inline1025 in the non-relativistic case and tex2html_wrap_inline1027 in the relativistic case, where tex2html_wrap_inline1029. These numbers characterize the atomic state in the case of the independent particle plus central field approximation. It is usually written in the form tex2html_wrap_inline1031. For many cases, the configuration with the total angular momentum quantum numbers JM specifies an atomic state completely for the independent particle plus central field approximation.



Author: Dr. Warren Perger and Ken Flurchick