Analytic construction of Hartree-Fock density matrices

A method is presented for variational calculation of the energy and the spin densities derived from a single-determinant wavefunction. Sum and difference coordinates [unk]R = (1/2)([unk]r(1) + [unk]r(1)) and [unk]r = [unk]r(1) - [unk]r(1) are introduced, and the density matrix P([unk]r(1),[unk]r(1)) is expanded in partial waves in the new coordinate frame: [Formula: see text] The functions h(L)(epsilon,r) are bound or continuum hydrogenic functions with energy epsilon.It is shown that the spin densities depend on the s partial waves only, and a Euler equation for these partial waves is derived: [Formula: see text] in which U([unk]R) is the electrostatic potential, a(epsilon) = h(0)(epsilon,0), and mu is chosen to normalize the spin density to N electrons. Further, the electronic energy can be expressed in terms of the s partial waves and the constant mu in the above equations.The idempotent density matrix that ensues from a particular choice of functions {B(00)[unk](epsilon,[unk]R)} is generated by choosing partial density waves {B(LM)(epsilon,[unk]R), L > 0} so that tr[P(2)(1 - P)(2)] is minimized.