Scaling Invariance of Density Functionals

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}$),d^3r$. 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}}^{\mathbf{\prime }}$) = \sum_{i=1}^3 \frac{d}{d x_i} [ x_i f([n_{\lambda m_0}],$\mathbf{r}$,${\mathbf{r}}^{\mathbf{\prime }}$) ] + \sum_{j=1}^3 \frac{d}{d x_j’} [ x_j’ f([n_{\lambda m_0}],$\mathbf{r}$,${\mathbf{r}}^{\mathbf{\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}$)^2}]$ .