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

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 = ( k B T ) -1 is then explored. In particular, the effective potential U ( r β) which is essentially - k B T ln Z ( r β) 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 ).