Form of spinless first- and second-order density matrices in atoms and molecules, derived from eigenfunctions of S2 and Sz

Many-electron theory of atoms and molecules starts out from a spin-independent Hamiltonian H. In principle, therefore, one can solve for simultaneous eigenfunctions Ψ of Hand the spin operators S2 and Sz. The fullest possible factorization into space and spin parts is here exploited to construct the spinless second-order density matrix Γ, and hence also the first-order density matrix. After invoking orthonormality of spin functions, and independently of the total number of electrons, the factorized form of Ψ is shown to lead to Γ as a sum of only two terms for S = 0, a maximum of three terms for S = 1/2 and four terms for S ≥ 1. These individual terms are characterized by their permutational symmetry. As an example, theground state of the Be atom is discussed.