Functionals and Functional Derivatives of Wave Functions and Densities
- 1 Physical and Life Sciences, Lawrence Livermore National Laboratory, Livermore, USA
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.
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- Consider all two-particle densities of the form, where denotes the ground state of the one-dimensional harmonic oscillator and on of its excited states (an infinite number). The orbital, can be developed in a power series of an equidensity basis (see following subsection) in terms of any of the densities, and differentiated within the space of that density through parametric differentiation of the elements of the basis. Each such differentiation gives the rate of change of a single orbital with respect to a particular density. But the rate of change at one density is not connected to that at another density. Although the coefficients of the expansion depend on the density, there is no functional derivative of the expansion coefficients. Indeed, they cannot be said to depend functionally on the density. In the expansion of in terms of plane waves, for example, the coefficients have no association with any particular density, (remember this one orbital contributes to an infinite number of densities). Furthermore, consider the (infinite) number of densities for the ground states of systems of N particles, with. The reader may wish to consider which of these densities is the independent variable whose functional is the orbital, Which of these densities would be reflected in the coefficients of a plane-wave expansion of At the same time, the rate of change of this orbital with respect to a given density depends only on that density and can be obtained through expansion of the orbital in the equidensity basis for that density