Complementary variational principle for density-functional theories

A method for determining complementary energy functionals within a density-functional framework is derived. In particular, if $H\left[P\right]\mathrm{}$ is defined as a functional of the density equal to an energy functional $E_U\mathrm{}\left[P\right]\mathrm{}$ less a Coulomb energy term $\left(\frac{\nu }{2}\right)\int {1R^3}^{}\int {1R^3}^{}\times {\mathrm{}|x-y|\mathrm{}}^{-1\mathrm{}}P\left(x\right)P\left(y\right)\mathrm{}{d}^{3}y{d}^{3}x$ with $\nu$ a positive constant, then the functional $E_L\mathrm{}\left[P\right]\mathrm{}\equiv H\left[P\right]-\int \stackrel{}{1R^3}\frac{\delta H\left[P\right]\mathrm{}}{\delta P\left(x\right)\mathrm{}}P\left(x\right)\mathrm{}{d}^{3}x-\frac{1}{8\pi \nu }\int \stackrel{}{1R^3}{\left|\overrightarrow{\nabla }\frac{\delta H\left[P\right]\mathrm{}}{\delta P\left(x\right)\mathrm{}}\right|}^2{d}^{3}x+N\left[P\right]\mathrm{}limit\; of\frac{\delta H\left[P\right]\mathrm{}}{\delta P\left(x\right)\mathrm{}}\text{as}\mathrm{}\left|x\right|\mathrm{}\to \infty$ with $N\left[P\right]=\int {1R^3}^{}P\left(x\right)\mathrm{}{d}^{3}x$ is complementary to $E_U\mathrm{}\left[P\right]\mathrm{}$ whenever the second variation of $H\left[P\right]\mathrm{}$ is positive. Explicit forms of $E_L\mathrm{}\left[P\right]\mathrm{}$ for the Thomas-Fermi and the Thomas-Fermi—Dirac—von Weizsäcker theories are given and used to obtain both lower and upper bounds to the atomic energies of these theories. A discussion on the corresponding Hohenberg-Kohn complementary functional is also presented.