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Published online by Cambridge University Press: 05 June 2012
Summary
A functional F[f] is a mapping of an entire function f onto a value. In electronic structure, functionals play a central role, not only in density functional theory, but also in the formulation of most of the theoretical methods as functionals of the underlying variables, in particular the wavefunctions. This appendix deals with the general formulation and derivation of variational equations from the functionals.
Basic definitions and variational equations
The difference between a function f(x) and a functional F[f] is that a function is defined to be a mapping of a variable x to a result (a number) f(x); whereas a functional is a mapping of an entire function f to a resulting number F[f]. The functional F[f], denoted by square brackets, depends upon the function f over its range of definition f(x) in terms of its argument x. Here we a describe some basic properties related to the functionals and their use in density functional theory; more complete description can be found in [93], App. A. A review of functional derivatives or the “calculus of variations” can be found in [861] and [862].
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