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Scaling Invariance of Density Functionals

Based on the homogeneity ($F[n_{λm}]=λ^{p(m)}F[n]$) and invariance ($F[n_{λm_0}]=F[n]$) properties of a functional of the electron density under uniform scaling of the coordinates in the density $n_{λm}(\mathbf{r})=λ^{m} n(λ\mathbf{r}),\,(λ\in\mathbb{R}^+,\, m\in\mathbb{R})$, it is proven that homogeneity implies invariace and therefore all homogeneous scaling functionals have the representation $F[n]=\frac{m-m_0}{p(m)} \int_V\,\frac{δF[n]}{δn(\mathbf{r})}\,n(\mathbf{r})\,d^3r$. 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 $λ\frac{d}{dλ} f([n_{λm_0}],\mathbf{r},\mathbf{r'}) = \sum_{i=1}^3 \frac{d}{d x_i} [ x_i f([n_{λm_0}],\mathbf{r},\mathbf{r'}) ] + \sum_{j=1}^3 \frac{d}{d x_j'} [ x_j' f([n_{λm_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 $t_s(n,\nabla n)= n(\mathbf{r})^{3} g[ \frac{\partial_{x_1} n(\mathbf{r})}{n(\mathbf{r})^2}, \frac{\partial_{x_2} n(\mathbf{r})}{n(\mathbf{r})^2}, \frac{\partial_{x_3} n(\mathbf{r})}{n(\mathbf{r})^2}]$ .