An Approximate Differential Equation for Calculating the Electron Density in Closed Shell Atoms and in Molecules

Abstract

It is well known from density functional theory that the ground-state electron density ρ(r) in an atom or molecule can be calculated exactly in principle, from a one-body potential energy V(r) into which exchange and correlation interactions are incorporated. Therefore, it is of considerable interest to set up a differential equation which will enable ρ(r) to be calculated directly from V(r) for atoms and molecules with many electrons, without recourse to wave function determination.The problem will be tackled as follows:(i) A careful discussion will be given of the differential equation in one-dimension. Its relation to local density approximations will be demonstrated and it will be shown that the linear harmonic oscillator is a case when the local density equation becomes exact.(ii) A treatment of a closed shell atom with many-electrons will be presented. Here, central field separability is utilized, and two methods are developed. In the first, a separate differential equation is to be solved for each value of the orbital angular momentum l. The second, and presently less accurate, approach solves directly for the total density ρ(r).(iii) For three-dimensional potentials V(r), with arbitrarily low symmetry, as in the general molecular problem, the route to ρ(r) via a generalization of the partition function Z(r, β) with β = (kBT)-1 is then explored. In particular, the effective potential U(rβ) which is essentially - kBT ln Z(rβ) is the tool employed. Such a route is first shown to afford an exact formulation of (i) and (ii) above, which are essentially one-dimensional problems. For a general V(r), a generalization which is approximate is proposed, based on a linearized equation for the effective potential U. This approximation can then be refined by (a) use of the lowest bound state wave function or (b) incorporating non-linear gradient terms in V(r).