Some Unsolved Problems in Density Matrix Theory and Density Functional Theory

Abstract

Density functional theory (DFT) has its roots in density matrix theory (DMT). We therefore startwith a brief survey of the 1- and 2-electron density matrices (1-DM and 2-DM) and the densities derive from them, along with their main properties.

The problem of how to pass from the DMs to an energy density functional is considered in the light of simple examples: these show the richness of structure hidden within the 1-DM. In DFT, the diagonal element of the spinless 1-DM (i.e. the electron density, P) is deemed sufficient to determine everything else, including the 2-DM and the related electron interaction energy. In DMT, the 1-DM and 2-DM are indeed sufficient. But in either case the densities must be subject to the condition that they derive from an acceptable N-electron wave function — they must be N-representable. Otherwise, variational calculations will ‘collapse’, giving meaningless results.

In the absence of a general procedure for imposing N-representability, one may use an ansatz for the densities — forms which are known to arise (by integration) from an N-electron function of some suitable approximate form. An IPM approximation, with a 1-determinant wave function, is easy to handle but is too limited.

Procedures for obtaining the DMs are studied. The most general are essentially group theoretical and involve the separation of space and spin variables: one starts from the the Schrödinger equation HΦ = EΦ, asking which solutions are physically admissible (most of them will not be) and what conditions they bring with them.

The availability of an N-representable 2-DM appears to be the minimum requirement for deriving an acceptable electron density and avoiding all reference to an N-electron wave function.