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 Zumbach, G. and Maschke, K. (1983) New Approach to the Calculation of Density Functionals. Physical Review A, 28, 544-554.
 The strict mathematical definition of a functional is that of a function of coordinates mapped onto a number, usually in terms of an integral. The definition used here includes the mapping of a function onto a function, or a set of functions.
 Lieb, E.H. (1982) Density Functionals for Coulomb Systems. In: Shimony, A. and Feshbach, H., Eds., Physics as Natural Philosophy, Essays in Honor of Laszlo Tisza, MIT Press Cambridge, Massachusetts, 111-149.
See also, Lieb, E.H. (1985) Density Functional Methods in Physics. In: Dreizler, R.M., Ed., NATO ASI Series B 123, Plenum, New York, 31, and Lieb, E.H. (1983) in Int. J. Quantum Chem. 24, 243.
 Levy, M. (1979) Universal Variational Functionals of Electron Densities, First-Order Density Matrices, and Natural Spin-Orbitals and Solution of the v-Representability Problem. Proceedings of the National Academy of Sciences of the United States of America, 76, 6062-6065.
 It is occasionally argued  that the variational principle can be expressed in terms of the variational procedure, allowing, in principle, a fractionally normalized charge density. Here, the quantity, , is a Lagrange multiplier that guarantees the presence of a given normalization associated with wave functions not normalized to unity in terms of which the variational theorem takes the form. These considerations are both unnecessary as well as frought with formal danger. First, the wave functions considered with respect of the variational theorem can always be normalized to unity through division by the square root of its squared modulus. Second, a variational procedure based on a fractionally normalized density can easily misdirect developments along the treatment of ensembles of open systems in terms of formalism appropriate exclusively for pure states, as is discussed explicitly in the body of the paper. As a quick reminder, the wave functions used in the variational theorem must be of the form emerging as solutions to the Schr¨odinger equation, and those are normalized to unity.
 Cioslowski, J. (1988) Density Functionals for the Energy of Electronic Systems: Explicit Variational Construction. Physical Review Letters, 60, 2141-2143.
 Cioslowski, J. (1988) Density Driven Self-Consistent Field Method. I. Derivation and Basic Properties. The Journal of Chemical Physics, 89, 4871-4874.
 Cioslowski, J. (1989) Density Driven Self-Consistent Field Method. II. Construction of All One-Particle Wave Functions that Are Orthonormal and Sum up to a Given Density. Quantum Chemistry Symposium. International Journal of Quantum Chemistry, 36, 255-262.
 It is useful to use a function defined through a process to demonstrate the difficulties that can arise with respect to functionals defined by means of a procedure, rather than as functions of the independent variable. Consider the function, where is a real number, and is determined through the process of flipping a coin for any given, setting if the coin comes up heads, and if it comes up tales. Unlike a functional, however, this function of process has no parametric derivative (thus no derivative).
 Some exceptions are known. The ground-state wave function of a single particle system is given in terms of the density, and the corresponding kinetic energy functional takes the von Weizsacker form. Also the kinetic energy and exchange energy of an infinite number of plane waves forming a density are given explicitly by means of the expression. These cases are the exceptions that prove the rule that functionals of the density defy representation as explicit functions of the density.
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 This situation is not unlike that of the function defined over the set of natural numbers that exhibits a minimum but no derivative at the point of the minimum, e.g., defined over the integers and zero, has a minimum at 0, but no derivative there.
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 Indeed, one is hard put to find a case in quantum chemistry or condensed matter physics where functional differentiation is carried out in terms other that based on the Dirac delta function used as the arbitrary test function.
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