Geometric formulation of correlation in a many-electron system

Abstract

We present a systematic analysis of the introduction of correlation in the many-electron time-dependent problem with an accurate formulation based on geometric algebra notation. This provides a systematic definition of the configuration space, of the external potentials, of the one-electron operators for a many-electron system, and of the electron–electron interaction terms. We arrive both at a formal equation for the total energy and at the equation for the time-evolution of the wavefunction. From this, using the new geometric notation and the indistinguishability and equivalence of the electrons and the fact that we are interested either in the ground state or in states near the ground state, we formulate a variational problem from which a set of tractable equations, which self-consistently define the many-electron wavefunction and density, is obtained. The main emphasis is on the electron–electron correlation.

Keywords:

• many electron
• time dependent
• exchange
• correlation
• electronic structure
• ab-initio
• density functional theory

Acknowledgements

This paper is dedicated as a contribution to celebrate the Festschrift of Prof. Peter Weinberger. My academic exchange with him started with his visit to Mexico in 1973. The 35 years of collaboration have been fruitful, and also resulted in a very pleasant friendship and family relationship. A special opportunity for me came with his invitation to lecture and work in Vienna 1999–2000. This visit started a now nine-year old interaction with his group. The invitation in 1999 was open: ‘come and work in what ever subject you prefer’. The subject I chose was to use the mathematical theory of geometry of quadratic spaces to study physics from START. Of vital importance was the creative atmosphere that he and his colleges have kept as part of the tradition of Vienna. Vienna is both past and present; and more important creating the future. The atmosphere there is a mixture of Schubert, Rudolf von Alt, Sacher torte, Pauli and Schrödinger which provides an excellent setting for academic work. All the best, with my gratitude, to him.

The technical assistance of Mrs. Irma Vigil de Aragón is gratefully acknowledged. JK is a member of the Sistema Nacional de Investigadores, CoNaCyT Mexico.