Physical Interpretation of Electron Correlation in the Local-Density Approximation
In this paper we explain how electron correlations due to the Pauli exclusion principle and Coulomb’s law are represented within the local-density approximation (LDA) of Hohenberg-Kohn-Sham density-functional theory (DFT). The description[2,3] of how electrons are correlated in the LDA is arrived at via the work formalism[4,5] for the determination of electronic structure due to Harbola and Sahni, and surpasses our current understanding of these correlations which is based on Kohn-Sham theory. In the description of the LDA via Kohn-Sham theory each point of an inhomogeneous electronic density system is approximated as being homogeneous, but with a density corresponding to the local value at that point. In other words, the correlations between electrons as described by this picture are those of the uniform electron gas. Equivalently, the wavefunction for the non-uniform electron gas system at each electron position corresponds to the wavefunction of a uniform electron gas with a density equal to the local value at that position. This description of the LDA leads naturally to the criticism which questions the efficacy of employing homogeneous electron gas wavefunctions in regions of space where the potential is rapidly varying and in the classically forbidden region where the wavefunction is exponential. On the other hand, the results of application of the LDA are remarkably accurate, considerably more so than can be expected on the basis of the above description. Are the LDA results then simply fortuitious? The answer to this question is negative, since as will be explained, the LDA in fact goes beyond uniform electron gas theory and explicitly incorporates the non-uniformity of the electronic density in its representation of electron correlations. The LDA wavefunction thus incorporates to a significant degree the physics apropos to regions where the potential is varying, and this thereby is the reason for the accuracy of the results achieved. Furthermore, the above mentioned criticism, based on an erroneous understanding of electron correlations in the approximation, is consequently invalid.
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