By Holtje, Hans-Dieter
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Extra info for A Chemists Guide to Density Functional Theory
Since the Fermi hole is known to contain exactly one elementary charge (cf. equation (2-21)), the radius of this sphere is then given by 3 rS = 4π 1/ 3 H ρ( r1 ) −1 / 3 . (3-4) The radius rS is sometimes called the Wigner-Seitz radius and can be interpreted to a first approximation as the average distance between two electrons in the particular system. Regions of high density are characterized by small values of rS and vice versa. From standard electrostatics it is known that the potential of a uniformly charged sphere with radius rS is H proportional to 1/rS, or, equivalently, to ρ( r1 )1 / 3 .
We also give an alternative, more modern approach, namely the Levy constraint-search scheme. We go on and discuss the question whether a physically meaningful wave function can be uniquely associated with a certain density. The common denominator in our discussion is the primacy of the applicationoriented understanding over the puristic theoretical point of view. Readers who have also an affinity towards the latter and want to learn more about the many theoretical intricacies of the Hohenberg-Kohn theorems are recommended to consult the comprehensive and theoretically sound discussions contained in Parr and Yang, 1989, Kryachko and Ludeña, 1990, Dreizler and Gross, 1995 and Eschrig, 1996.
By this removal of one charge, the Fermi hole also takes care of the self-interaction problem. Further, due to the Pauli principle which ensures that two electrons of the same spin cannot be at the same position in space, the Fermi hole has to become equal to minus the density of r r electrons with this spin at the position of the reference electron for r2 → r1 , r r r r h X ( r2 → r1; r1 ) = −ρ( r1 ) . (2-22) What can we say about the shape of the Fermi hole? First, it can be shown that hX is negative everywhere, r r h X ( r1; r2 ) < 0 .
A Chemists Guide to Density Functional Theory by Holtje, Hans-Dieter
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