Uniform Scaling

Uniform scaling is a critical concept for understanding density functional design. Consider a given electron density distribution \(n(\mathbf{r})\). We can consider a transformed density of the form

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

This transformation is called uniform coordinate scaling,[1] because it essentially involves redefining \(\mathbf{r}\leftarrow\lambda\mathbf{r}\), which squishes (\(\lambda>0\)) or expands (\(\lambda<0\)) the density while maintaining its relative shape. The \(\lambda^3\) prefactor ensures that the particle number is conserved, i.e.

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

One can also consider uniform scaling of the density matrix:

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

NOTE: For orbital-dependent functionals, scaling the density and scaling the density matrix are not precisely equivalent, because there is a distribution of possible density matrices that can yield a given density. If the orbital-dependent potential causes the orbitals to rearrange themselves when uniform scaling occurs, the lowest-energy density matrix for a density \(n_\lambda\) will not necessarily be \(n_1^\lambda\), i.e. the scaled density matrix obtained from the lowest-energy density matrix with density \(n\). This is a very subtle point, however, and does not usually make a big impact, so in most cases we will always refer to scaling the density \(n_\lambda\) for simplicity, even when orbital-dependent quantities are involved. For more details, see Görling and Ernzerhof[2].

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 \(E_\text{c}[n]\) does not have such simple behavior under uniform scaling, but it does obey the limits[4]

\[\begin{split}\lim_{\lambda\rightarrow 0} E_\text{c}[n_\lambda] &= \lambda C_0[n] \\ \lim_{\lambda\rightarrow \infty} E_\text{c}[n_\lambda] &> -\infty\end{split}\]

Therefore, it can be helpful to design features with simple, well-understood behavior under uniform scaling. In particular, if the feature vector \(\mathbf{x}\) for the ML model is scale-invariant, i.e. if

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

then an exchange functional of the form

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

obeys the uniform scaling rule for exchange (\(E_\text{x}[n_\lambda] = \lambda E_\text{x}[n]\)). Similar, scale-invariant features can also be useful for correlation functionals because their behavior under uniform scaling will be the same as the behavior of the multiplicative baseline functional used for training. If the baseline model has reasonable behavior under uniform scaling (such as PBE/SCAN), this could help make more physically realistic models. (However, it could also needlessly restrict the model’s flexibility, so there are trade-offs involved).

[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.