Inhomogeneity and Exchange Corrections to TF Theory

Before discussing the consequences of equation (48) for the total energy of positive ions, it is clearly of importance to understand how the density description has to be generalized beyond the TF approximation to account for the terms 0(Z2) and 0(Z5/3) in equation (48). 

This takes us back to the total TF energy in equation (20). When we consider again the basis of this, we note first that the kinetic energy density tr has been approximated by equation (18), which is a local free electron relation. Formally, it is straightforward to take the variation of T— J tr dr with respect to p, and then one can write the Euler equation (24) in the generalized form 

Clearly, equation (49) reduces to equation (24) if tr is replaced by the approximation (18) formally it now takes full account of the (rapid) variation of electron density in the atom, in contrast to the semiclassical TF Euler equation. Unfortunately, tr is only presently known in two special cases (i) to low order in gradient expansion corrections to equation (18) as in equation (76) below and (ii) in a perturbative development about the uniform electron assembly.13 Form (i) will be referred to again below. However, as Scott14 was first to argue for the neutral atom, the origin of the Z2 term in equation (48) resides in the inhomogeneity correction to the TF theory, which is formally contained in equation (49). Fortunately, an approximation based on the Coulomb field treatment of Section 4 suffices to gain a useful estimate of the order of the Z2 term in the neutral atom. 

Using this result, one can add on to the total energy a term for the exchange energy A = J ex dr and then minimizing with respect to the density p one finds 

Historically16 it is worthy of note that if one resorts in equation (51) to the TF approximation (18) for tr, then the Euler equation of the Thomas-Fermi-Dirac method results. We shall not go into the solutions of the Thomas-Fermi-Dirac equation in this review, though there has been recent interest in this area. Suffice it to say that in the full form of the Euler equation (51), we are working at the customary Hartree-Fock-Slater level. However, we shall content ourselves, until we come to Section 17 below, with understanding in a more intuitive, but inevitably less detailed, way how the corrections to the TF energy in equation (48) arise. 

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