The extremum method in quantum chemistry: Part I. Extremum of tr(B TPC) and the trace algebra

Abstract

This is an elementary study on the extremum method in quantum chemistry. The definitions of the differentiation of a real matrix function F( t) with respect to the variable t and a scalar function F( X) =φ, which is the trace of a product of several matrices, with respect to a variable matrix X have been introduced. The mathematics is not novel but the suggested procedure employed to the extremum problems appearing in quantum chemistry may be different from the ordinary treatments. Some elementary applications will be reported in the second part. First of all, we shall show a general rule for finding the resultant expression for the derivative ∂φ/∂ X in this paper. In quantum chemistry, we usually have to find the extremum value of gY j ¦ Q¦Ψ j = C j T QC j subject to the restrictive condition that C j T M x C j = 1 for an arbitrary state Ψ j first, and extend the result to a set of states. A matrix eigenvalue equation can then be obtained, but here, a whole set of {Ψ j } will be dealt with at one time. The matrix eigenvalue equation Q( CW) = M x ( CW) Λ x can be obtained from the given rule very directly and very simply. Only one basis set is used in this situation. The other main problems discussed in this paper are the necessary and sufficient conditions for finding an extremum of tr L = tr ( B T PC) subject to the restrictive conditions of orthonormalities. There are two basis sets considered in this situation. An important conclusion we have found is that the matrix B T PC must be symmetric.