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. (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.
 Perdew, J.P., Ernzerhof, M. and Burke, K. (1996) Rationale for Mixing Exact Exchange with Density Functional Approximations. The Journal of Chemical Physics, 105, 9982.
 Kim, K. and Jordan, K.D. (1994) Comparison of Density Functional and MP2 Calculations on the Water Monomer and dimer. The Journal of Physical Chemistry, 98, 10089-10094.
 Lee, C., Yang, W.T. and Parr, R.G. (1988) Development of the Colle-Salvetti Correlation-Energy Formula into a Functional of the Electron Density. Physical Review B, 37, 785-789.
 Zhao, Y. and Truhlar, D.G. (2008) The M06 Suite of Density Functionals for Main Group Thermochemistry, Thermochemical Kinetics, Noncovalent Interactions, Excited States, and Transition Elements: Two New Functionals and Systematic Testing of Four M06-Class Functionals and 12 Other Functionals. Theoretical Chemistry Accounts, 120, 215241.
 Zhao, Y. and Truhlar, D.G. (2006) Density Functional for Spectroscopy: No Long-Range Self-Interaction Error, Good Performance for Rydberg and Chargetransfer States, and Better Performance on Average than B3LYP for Ground States. The Journal of Physical Chemistry A, 110, 13126-13130.
 Dane, M., Gonis, A., Nicholson, D.M. and Stocks, G.M. (2013) On the Solution of the Self-Interaction Problem in Kohn-Sham Density Functional Theory. Journal of Physics and Chemistry of Solids (Online Version), 75, 1160-1178.
 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.
 Perdew, J.P., Parr, R.G., Levy, M. and Balduz, J.L. (1982) Density-Functional Theory for Fractional Particle Number: Derivative Discontinuities of the Energy. Physical Review Letters, 49, 1691-1694.
 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.
 Perdew, J.P. and Zunger, A. (1981) Self-Interaction Correction to Density-Functional Approximations for Many-Electron Systems. Physical Review B, 23, 5048-5079.
 Kümmel, S. and Perdew, J.P. (2003) Simple Iterative Construction of the Optimized Effective Potential for Orbital Functionals, Including Exact Exchange. Physical Review Letters, 90, Article ID: 043004.
 Grabo, T., Kreibich, T., Kurth, S. and Gross, E. (2000) Strong Coulomb Correlations in Electronic Structure Calculations: Beyond the Local Density Approximation. Gordon and Breach Science Publishers, Amsterdam, 203-311.
 Yang, W.T. and Wu, Q. (2002) Direct Method for Optimized Effective Potentials in Density-Functional Theory. Physical Review Letters, 89, Article ID: 143002.
 Yang, W.T., Ayers, P.W. and Wu, Q. (2004) Potential Functionals: Dual to Density Functionals and Solution to the v-Representability Problem. Physical Review Letters, 92, 146404.
 Gidopoulos, N.I. and Lathiotakis, N.N. (2012) Constraining Density Functional Approximations to Yield Self-Interaction Free Potentials. The Journal of Chemical Physics, 136, Article ID: 224109. http://dx.doi.org/10.1063/1.4728156
 Heaton-Burgess, T., Bulat, F.A. and Yang, W.T. (2007) Optimized Effective Potentials in Finite Basis Sets. Physical Review Letters, 98, Article ID: 256401.
 Harbola, M.K. and Sahni, V. (1989) Quantum-Mechanical Interpretation of the Exchange-Correlation Potential of Kohn-Sham Density Functional Theory. Physical Review Letters, 62, 489-492.
 Wang, A. and Carter, E.A. (2000) Chapter 5 of “Theoretical Methods in Condensed Phase Chemistry”. In: Schwartz, S.D., Ed., A New Book Series of “Progress in Theoretical Chemistry and Physics”, Kluwer, Dordrecht, 117-184.
 Mirtschink, A., Seidl, M. and Gori-Giorgi, P. (2013) Derivative Discontinuity in the Strong-Interaction Limit of Density-Functional Theory. Physical Review Letters, 111, Article ID: 126402.