Functionals and Functional Derivatives of Wave Functions and Densities

A. Gonis

Physical and Life Sciences, Lawrence Livermore National Laboratory, Livermore, USA.

DOI: 10.4236/wjcmp.2014.43022                

Abstract

It is shown that the process of conventional functional differentiation does not apply to functionals whose domain (and possibly range) is subject to the condition of integral normalization, as is the case with respect to a domain defined by wave functions or densities, in which there exists no neighborhood about a given element in the domain defined by arbitrary variations that also lie in the domain. This is remedied through the generalization of the domain of a functional to include distributions in the form of , where  is the Dirac delta function and is a real number. This allows the determination of the rate of change of a functional with respect to changes of the independent variable determined at each point of the domain, with no reference needed to the values of the functional at different functions in its domain. One feature of the formalism is the determination of rates of change of general expectation values (that may not necessarily be functionals of the density) with respect to the wave functions or the densities determined by the wave functions forming the expectation value. It is also shown that ignoring the conditions of conventional functional differentiation can lead to false proofs, illustrated through a flaw in the proof that all densities defined on a lattice are -representable. In a companion paper, the mathematical integrity of a number of long-standing concepts in density functional theory are studied in terms of the formalism developed here.

Keywords

Functional, Functional Derivative, Functional Derivative of Wave Functions, Functional Derivatives with Respect to Wave Functions, Functional Derivatives with Respect to Density, Rate of Change of a Functional, Rate of Change of Expectation Values, Rate of Change of a Wave Function

Conflicts of Interest

The authors declare no conflicts of interest.

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