TITLE:
Functionals and Functional Derivatives of Wave Functions and Densities
AUTHORS:
A. Gonis
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
JOURNAL NAME:
World Journal of Condensed Matter Physics,
Vol.4 No.3,
August
29,
2014
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.