Relation between Total Energy and Sum of One-electron Energies

Between equations (25) and (81), Fen+2Fee can be eliminated and applying once again the virial theorem in the form T= —E yields31 

Since for the neutral TF atom we have shown that jW=0, we obtain March and Plaskett s18 result that 

Equation (83) generalizes their result for positive ions the chemical potential then being given by equation (38). For some numerical estimates based on equations (38) and (83) the reader is referred to ref. 31. We shall return to relations (83) and (84) when we deal with molecules at equilibrium later. 

In this later connection, it will be of interest to examine the inhomogeneity correction to the relation (83). To understand the nature of this, we consider the general Euler equation (49). Multiplying this equation by the density p and integrating over the whole of space, yields 

For the model of a pure Coulomb field, the correction on the left-hand side of equation (86) can, in fact be estimated exactly, the result being given in Appendix 1 together with results for a harmonic well. We shall return to this discussion when we treat molecular energies below. 

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