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 $\lambda \frac{d}{d\lambda} f([n_{\lambda m_0}],\mathbf{r},\mathbf{r'}) = \sum_{i=1}^3 \frac{d}{d x_i} [ x_i f([n_{\lambda m_0}],\mathbf{r},\mathbf{r'}) ] + \sum_{j=1}^3 \frac{d}{d x_j'} [ x_j' f([n_{\lambda m_0}],\mathbf{r},\mathbf{r'}) ]$. 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 $t_s(n,\nabla n)= n(\mathbf{r})^{3} g[ \frac{\partial_{x_1} n(\mathbf{r})}{n(\mathbf{r})^2}, \frac{\partial_{x_2} n(\mathbf{r})}{n(\mathbf{r})^2}, \frac{\partial_{x_3} n(\mathbf{r})}{n(\mathbf{r})^2}]$ .