Scaling Invariance of Density Functionals

Abstract

Based on the homogeneity ($F[n_{\lambda m}]={\lambda }^{p(m)}F[n]$) and invariance ($F[n_{\lambda m_0}]=F[n]$) properties of a functional of the electron density under uniform scaling of the coordinates in the density $n_{\lambda m}(\mathbf{r})={\lambda }^{m}n(\lambda \mathbf{r}),(\lambda \in {\mathbb{R}}^{+},m\in \mathbb{R})$, it is proven that homogeneity implies invariace and therefore all homogeneous scaling functionals have the representation $F[n]=\frac{m-m_0}{p(m)}{\int }_V\frac{\delta F[n]}{\delta n(\mathbf{r})}n(\mathbf{r})d^{3}r$. Also, the homogeneity ($p(m)$) and invariant ($m_0$) degrees of density functionals related to the Kohn-Sham theory are calculated. Besides, it is shown that the functional density and the electron density itself satisfy the general equation representing the local scaling invariance of a functional $\lambda \frac{d}{d\lambda }f([n_{\lambda m_0}],\mathbf{r},{\mathbf{r}}^{\prime })={\sum }_{i=1}^3\frac{d}{d{x}_i}[x_{i}f([n_{\lambda m_0}],\mathbf{r},{\mathbf{r}}^{\prime })]+{\sum }_{j=1}^3\frac{d}{d{x}_j^{\prime }}[x_j^{\prime }f([n_{\lambda m_0}],\mathbf{r},{\mathbf{r}}^{\prime })]$. The equation simplifies for cases where the functional density depends only on the density and/or its gradient, and general forms of the solutions are provided, in particular for the non-interacting kinetic energy density is shown to take the form $t_s(n,\nabla n)=n(\mathbf{r}{)}^{3}g[\frac{{\partial }_{x_1}n(\mathbf{r})}{n(\mathbf{r}{)}^2},\frac{{\partial }_{x_2}n(\mathbf{r})}{n(\mathbf{r}{)}^2},\frac{{\partial }_{x_3}n(\mathbf{r})}{n(\mathbf{r}{)}^2}]$ .