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The Pursuit of Fallacy in Density Functional Theory: The Quest for Exchange and Correlation, the Rigorous Treatment of Exchange in the Kohn-Sham Formalism and the Continuing Search for Correlation

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As pointed out in the paper
preceding this one, in the case of functionals whose independent variable must
obey conditions of integral normalization, conventional functional
differentiation, defined in terms of an arbitrary test function, is generally
inapplicable and functional derivatives with respect to the density must be
evaluated through the alternative and widely used limiting procedure based on
the Dirac delta function. This leads to the determination of the rate of change
of the dependent variable with respect to its independent variable at each
isolated pair, , that may not be part of a
functional (a set of ordered pairs). This extends the concept of functional
derivative to expectation values of operators with respect to wave functions
leading to a density even if the wave functions (and expectation values) do not
form functionals. This new formulation of functional differentiation forms the
basis for the study of the mathematical integrity of a number of concepts in
density functional theory (DFT) such as the existence of a universal functional
of the density, of orbital-free density functional theory, the derivative
discontinuity of the exchange and correlation functional and the extension of
DFT to open systems characterized by densities with fractional normalization.
It is shown that no universal functional exists but, rather, a universal
process based only on the density and independent of the possible existence of
a potential, leads to unique functionals of the density determined through the
minimization procedure of the constrained search. The mathematical integrity of
two methodologies proposed for the treatment of the Coulomb interaction, the
self-interaction free method and the optimized effective potential method is
examined and the methodologies are compared in terms of numerical calculations.
As emerges from this analysis, the optimized effective potential method is
found to be numerically approximate but formally invalid, contrary to the
rigorously exact results of the self-interaction-free method.

KEYWORDS

Density Functional Theory, Variational Properties of Density Functional Theory, Self-Interaction Error, Optimized Effective Potential, Functional Derivative, Parametric Derivative, Functional Rate of Change, Derivative Discontinuity, Orbital-Free Density Functional Theory, Exchange and Correlation Functional

Conflicts of Interest

The authors declare no conflicts of interest.

Cite this paper

Gonis, A. (2014) The Pursuit of Fallacy in Density Functional Theory: The Quest for Exchange and Correlation, the Rigorous Treatment of Exchange in the Kohn-Sham Formalism and the Continuing Search for Correlation.

*World Journal of Condensed Matter Physics*,**4**, 200-225. doi: 10.4236/wjcmp.2014.43023.

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