Functionals and Functional Derivatives of Wave Functions and Densities

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References

[1] Courant, R. and Hilbert, D. (1953) Methods of Mathematical Physics. Vol. 1, Interscience Publishers, New York.

[2] Giaquinta, M. and Hildebrandt, S. (1996) Calculus of Variations 1. The Lagrangian Formalism, Grundlehren der Mathematischen Wissenschaften 310. Springer-Verlag, Berlin.

[3] Parr, R.G. and Yang, C.Y. (1989) Density Functional Theory of Atoms and Molecules. Oxford University Press, Oxford.

[4] Gál, T. (2001) Differentiation of Density Functionals That Conserves the Normalization of the Density. Physical Review A, 63, Article ID: 022506.

http://dx.doi.org/10.1103/PhysRevA.63.022506

[5] Friedlander, G. and Joshi, M. (1998) Introduction to the Theory of Distributions. 2nd Edition, Cambridge University Press, UK.

[6] The recursion relation can be interpreted in terms of a susceptibility, whose inverse is defined by the relation, from which follows that the Dirac delta function is its own inverse.

[7] Trott, M. Functional Derivative. From MathWorld—A Wolfram Web Resource, Created by Eric W. Weisstein.

http://mathworld.wolfram.com/FunctionalDerivative.html

[8] Cioslowski, J. (1988) Density Functionals for the Energy of Electronic Systems: Explicit Variational Construction. Physical Review Letters, 60, 2141-2143.

http://dx.doi.org/10.1103/PhysRevLett.60.2141

[9] Cioslowski, J. (1988) Density Driven Self-Consistent Field Method. I. Derivation and Basic Properties. The Journal of Chemical Physics, 89, 4871-4874.

http://dx.doi.org/10.1063/1.455655

[10] 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. International Journal of Quantum Chemistry, 36, 255262.

[11] 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

[12] Zumbach, G. and Maschke, K. (1983) New Approach to the Calculation of Density Functionals. Physical Review A, 28, 544-554.

http://dx.doi.org/10.1103/PhysRevA.28.544

[13] Gonis, A., Dane, M., Nicholson, D.M. and Stocks, G.M. (2012) Computationally Simple, Analytic, Closed Form Solution of the Coulomb Self-Interaction Problem in Kohn-Sham Density Functional Theory. Solid State Communications, 152, 771-774.

http://dx.doi.org/10.1016/j.ssc.2012.01.048

[14] 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. arXiv:1302.4809 [cond-mat.mtrl-sci]

[15] Macke, W. (1955) Zur wellenmechanischen Behandlung von Vielkorperproblemen. Annalen der Physik, 452, 1-9.

[16] Harriman, J.E. (1981) Orthonormal Orbitals for the Representation of an Arbitrary Density. Physical Review A, 24, 680-682.

http://dx.doi.org/10.1103/PhysRevA.24.680

[17] Ludena, E.V. and López-Boada, R. (1996) Local-Scaling Transformation Version of Density Functional Theory: Generation of Density Functionals. Topics in Current Chemistry, 180, 169-224.

[18] Kohn, W. (1983) v-Representability and Density Functional Theory. Physical Review Letters, 51, 1596.

http://dx.doi.org/10.1103/PhysRevLett.51.1596