It is occasionally argued [3] that the variational principle can be expressed in terms of the variational procedure, allowing, in principle, a fractionally normalized charge density. Here, the quantity, , is a Lagrange multiplier that guarantees the presence of a given normalization associated with wave functions not normalized to unity in terms of which the variational theorem takes the form. These considerations are both unnecessary as well as frought with formal danger. First, the wave functions considered with respect of the variational theorem can always be normalized to unity through division by the square root of its squared modulus. Second, a variational procedure based on a fractionally normalized density can easily misdirect developments along the treatment of ensembles of open systems in terms of formalism appropriate exclusively for pure states, as is discussed explicitly in the body of the paper. As a quick reminder, the wave functions used in the variational theorem must be of the form emerging as solutions to the Schr¨odinger equation, and those are normalized to unity.

has been cited by the following article:

• TITLE: 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

  AUTHORS: A. Gonis

  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

  JOURNAL NAME: World Journal of Condensed Matter Physics, Vol.4 No.3, August 29, 2014

  ABSTRACT: 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.