The mathematics of functional differentiation under conservation constraint

TL;DR

This paper clarifies the mathematics of K-conserving functional differentiation, deriving the most general form of constrained derivatives, and discusses their applications in density-functional theory and shape-conserving derivatives.

Contribution

It derives the general form of K-conserving functional derivatives and connects them with shape derivatives and practical applications in many-electron systems.

Findings

Derived the most general form of K-conserving derivatives.

Connected K-conserving derivatives with shape derivatives.

Provided an intuitive interpretation of K-conserving derivatives.

Abstract

The mathematics of K-conserving functional differentiation, with K being the integral of some invertible function of the functional variable, is clarified. The most general form for constrained functional derivatives is derived from the requirement that two functionals that are equal over a restricted domain have equal derivatives over that domain. It is shown that the K-conserving derivative formula is the one that yields no effect of K-conservation on the differentiation of K-independent functionals, which gives the basis for its generalization for multiple constraints. Connections with the derivative with respect to the shape of the functional variable and with the shape-conserving derivative, together with their use in the density-functional theory of many-electron systems, are discussed. Yielding an intuitive interpretation of K-conserving functional derivatives, it is also shown that K-conserving derivatives emerge as directional derivatives along K-conserving paths, which is achieved via a generalization of the Gateaux derivative for that kind of paths. These results constitute the background for the practical application of K-conserving differentiation.