Matrix Elements and Density Matrices for Many-Electron Spin Eigenstates Built from Orthonormal Orbitals

Abstract

This chapter discusses matrix elements and density matrices for many-electron spin eigenstates built from orthonormal orbitals. For an important class of many-electron problems, namely, those governed by spin-free or nearly spin-free Hamiltonians, it is practically and theoretically useful to formulate many-electron wavefunctions that, in addition to being antisymmetric, are also eigenfunctions of the total spin operators. The most widespread method of constructing antisymmetric wavefunctions is by an expansion in terms of Slater determinants of orthonormal orbitals. Thereby quantum mechanical problems are transformed into matrix problems and the matrix elements are integrals involving two Slater determinants and certain dynamical operators. Slater determinants have, however, one shortcoming: In general, they are not eigenfunctions of the total spin operator. A second route can be considered as originating with Dirac's vector model. A third route is to construct spin eigenfunctions with the help of projection operators that are not derived from group theory. In all of these methods, the construction of wavefunctions with the desired characteristics is the simpler task. It is in the evaluation of the expectation values and matrix elements of the many-electron operators that complexities and complications arise.