Electron Density Description of Atoms and Molecules

The problem of calculating accurate ground-state wave functions for atoms, and to a lesser degree for molecules, is tractable with modern computers for relatively small numbers of electrons N. 

However, for large numbers of electrons, a different approach is clearly required. This is afforded by the density description. Instead of a wave function with 3N spatial co-ordinates, one works with the ground-state electron density p(r), where this is explicitly the number of electrons per unit volume at position r. This is evidently a three-dimensional quantity, independent of the number of electrons, and is therefore a favourable tool for really large molecules. 

2 Density-Potential Relation of Thomas-Fermi Statistical Theory 

As already remarked, the idea underlying the Thomas-Fermi (TF) statistical theory is to treat the electrons around a point r in the electron cloud as though they were a completely degenerate electron gas. Then the lowest states in momentum space are all doubly occupied by electrons with opposed spins, out to the Fermi sphere radius corresponding to a maximum or Fermi momentum pt(r) at this position r. Therefore if we consider a volume dr of configuration space around r, the volume of occupied phase space is simply the product dr 47ipf(r)/3. However, we know that two electrons can occupy each cell of phase space of volume h3 and hence we may write for the number of electrons per unit volume at r, 

This is the first basic relation of the TF theory. It amounts to taking the result for a uniform Fermi gas and applying it locally at r. 

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Electron Density Description of Atoms and Molecules from which it follows that... 

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