Smooth Density Matrix Exchange (SDMX)
SDMX features[1] are nonlocal featurizations of the density matrix. The basic idea of SDMX is to “smooth” out the density matrix around a reference point \(\mathbf{r}\) and also project it onto low-order spherical harmonics (i.e. principle quantum numbers of \(\ell=0\) and \(\ell=1\)). As a result, one can obtain very simple (but nonlocal) proxies for the shape of the exchange hole \(|n_1(\mathbf{r},\mathbf{r}')|^2\), which can be used for fitting both the exchange and correlation energies. The formulas and details for the different SDMX versions are described below. These features take significant inspiration from the “Rung 3.5” functionals developed by Janesko et al..[2][3][4][5] We also plan to implement Rung 3.5 functionals and add them as an alternative to SDMX features, but this development has not been done yet. The key difference between SDMX and the original Rung 3.5 approach is that SMDX features are strictly quadratic functionals of the density matrix, while Rung 3.5 functionals are not.
One key consideration in the design of the SDMX features is their behavior under uniform scaling of the density. In keeping with the design objective of enabling the incorporation of exact constraints into ML functionals, all SDMX features have power law behavior under uniform scaling, as discussed below.
The construction of the most basic SDMX feature starts with a smoothed, spherically averaged density matrix
where \(R\) is a length-scale parameter, and \(h\) is a smooth convolution kernel. Then, one can construct the set of features \(H_j^0(\mathbf{r})\) as
where \(j\) is real. Currently we use \(j\in\{0,1,2\}\), but other values of \(j\) (including non-integers) are possible. However, \(j<0\) and \(j>2\) might cause numerical stability and normalization issues. Currently, \(h(u; R)\) takes the form
Different functions could be chosen, but the important aspect of the smoothing functions is that convolving the density matrix with it yields a smoothed approximation to the spherically averaged density matrix at distance \(R\), and that the level of smoothing increases as \(R\) increases. These properties are necessary 1) to obey uniform scaling rules and 2) to ensure that \(\rho^0(R; \mathbf{r})\) can be expanded efficiently using a Gaussian basis. To evaluate \(H_j^0(\mathbf{r})\), the smoothed density matrix \(\rho^0(R; \mathbf{r})\) is evaluated at a discrete set of distances \(R_i\) and then interpolated over \(R\) using a Gaussian basis; \(H_j^0\) can then be evaluated analytically within the Gaussian basis. Note that under uniform scaling, this feature scales as \(\lambda^{3+j}\), i.e.
The complexity of the SDMX features can be increased by also calculating expectation values of the gradients of the smoothed density matrix with respect to the length-scale \(R\), as follows:
In practice, these features can be obtained at almost no computational overhead compared to the original \(H_j^0\) features because the bottleneck computational operations are the same as those needed to compute \(H_j^0\). The uniform scaling behavior of \(H_j^{0\text{d}}\) is also \(\lambda^{3+j}\).
The angular complexity of the SDMX features can be increased by computing (in addition to \(\rho^0\)) the following higher-order quantity
which involves the gradient of \(h(|\mathbf{r}'-\mathbf{r}|; R)\) with respect to \(\mathbf{r}\). Then, one can compute new features
These are more costly to compute than \(H_j^0\) or \(H_j^\text{0d}\), but they provide significantly more information to the model because they contain \(\ell=1\) angular information. The uniform scaling behavior of \(H_j^1\) is also \(\lambda^{3+j}\).
Finally, one can combine the radial and Cartesian derivatives to construct one more type of feature
which also scales as \(\lambda^{3+j}\) under uniform scaling and comes at very little additional computational cost if \(\boldsymbol{\rho}^1(R; \mathbf{r})\) was already computed to evaluate \(H_j^\text{1}(\mathbf{r})\).