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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.
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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
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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]
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