Uniform Scaling

Uniform coordinate scaling[1] relates a density \(n(\mathbf r)\) to

\[n_\lambda(\mathbf{r}) = \lambda^3 n(\lambda \mathbf{r})\]

The transformation compresses (\(\lambda>1\)) or expands (\(0<\lambda<1\)) the density and preserves its shape and particle number:

\[\int \mathrm{d}^3\mathbf{r}\, n_\lambda(\mathbf{r}) = \int \mathrm{d}^3\mathbf{r}\, n(\mathbf{r}).\]

The corresponding coordinate-scaled one-particle density matrix is

\[n_1^\lambda(\mathbf{r}, \mathbf{r}') = \lambda^3 n_1(\lambda \mathbf{r}, \lambda \mathbf{r}')\]

Density-matrix scaling

For an orbital-dependent functional, scaling a density and scaling one particular density matrix are distinct operations because multiple density matrices can yield the same density. Orbital relaxation under the scaled potential can therefore make the minimizing density matrix at \(n_\lambda\) differ from \(n_1^\lambda\), the coordinate-scaled form of the original minimizing density matrix.

This distinction enters statements that depend explicitly on orbitals or the one-particle density matrix; see Görling and Ernzerhof[2]. The density notation \(n_\lambda\) is used below for the density-functional scaling relations.

The exact exchange functional \(E_\text{x}[n]\) has a simple, exact behavior under uniform scaling:[3]

\[E_\text{x}[n_\lambda] = \lambda E_\text{x}[n]\]

The correlation functional \(E_\text{c}[n]\) obeys the low- and high-density limits[4]

\[\begin{split}E_\text{c}[n_\lambda] &\rightarrow \lambda\,\mathcal{C}_0[n] \quad (\lambda\rightarrow 0) \\ \lim_{\lambda\rightarrow \infty} E_\text{c}[n_\lambda] &> -\infty\end{split}\]

where \(\mathcal{C}_0[n]\) is a density functional that does not depend on \(\lambda\).

A scale-invariant feature vector satisfies

\[\mathbf{x}[n_\lambda](\mathbf{r}) = \mathbf{x}[n](\lambda \mathbf{r})\]

An exchange functional of the form

\[E_\text{x}[n] = \int \mathrm{d}^3\mathbf{r}\, e_\text{x}^\text{LDA}(n(\mathbf{r}))\, F_\text{x}^\text{ML}(\mathbf{x}(\mathbf{r}))\]

then obeys the exchange scaling rule \(E_\text{x}[n_\lambda] = \lambda E_\text{x}[n]\). The LDA exchange energy density \(e_\text{x}^\text{LDA}\propto n^{4/3}\) supplies the required \(\lambda\) scaling. The learned enhancement factor \(F_\text{x}^\text{ML}\) is scale invariant because its inputs are.

With a correlation baseline, the multiplicative energy form inherits the baseline’s scaling behavior. Exact correlation has nonhomogeneous coordinate scaling, so a full-XC model may include explicit density dependence alongside scale-invariant descriptors. The CIDER26XC construction follows this route, as described in From Exchange Models to Full XC.

[1]

Richard M. Martin. Electronic Structure: Basic Theory and Practical Methods. Cambridge University Press, Cambridge, 2004.

[2]

Andreas Görling and Matthias Ernzerhof. Energy differences between Kohn-Sham and Hartree-Fock wave functions yielding the same electron density. Phys. Rev. A, 51(6):4501–4513, jun 1995. URL: https://link.aps.org/doi/10.1103/PhysRevA.51.4501, doi:10.1103/PhysRevA.51.4501.

[3]

Mel Levy and John P. Perdew. Hellmann-Feynman, virial, and scaling requisites for the exact universal density functionals. Shape of the correlation potential and diamagnetic susceptibility for atoms. Phys. Rev. A, 32(4):2010–2021, 1985. doi:10.1103/PhysRevA.32.2010.

[4]

Aaron D. Kaplan, Mel Levy, and John P. Perdew. The Predictive Power of Exact Constraints and Appropriate Norms in Density Functional Theory. Annu. Rev. Phys. Chem., 74(1):193–218, April 2023. doi:10.1146/annurev-physchem-062422-013259.