Symmetry Properties of Reduced Density Matrices

Abstract

Publisher Summary This chapter describes the symmetry properties of reduced density matrices (RDM) and their eigen functions. The chapter proposes a unified rigorous approach to the study of the symmetry properties of RDM. In the RDM approach, there arises the problem of finding the integral of motion and discrete symmetries for the density matrix D p Ψ and their relation with the motion integrals for the wave function Ψ. This problem is more general and profound than the so-called N-representability problem, which consists of finding a complete set of characteristics of RDM arising from the contraction of either antisymmetric or symmetric wave functions. A more complex problem is to find the symmetry properties of the transition density matrices from the known symmetry properties of the wave functions that determine it. The chapter reviews that the known RDM symmetry allows the simplification of the calculation of matrix elements, to quasi-diagonalize a secular problem. It is important to note that the symmetry can be recovered in two ways: either by assembly averaging over all densities of a complete set of degenerated states or by averaging the density for a certain state over the whole symmetry group of a system. These procedures are absolutely equivalent and give the invariant part of RDM.