Smoothed Density Matrix Exchange (SDMX)

Smoothed density matrix exchange (SDMX) features are nonlocal featurizations of the one-particle density matrix. The density matrix is smoothed around a real-space point and projected onto the \(\ell=0\) and \(\ell=1\) angular channels. Quadratic contractions of the projected quantities provide rotationally invariant proxies for the radial and angular structure of the exchange hole \(|n_1(\mathbf r,\mathbf r')|^2\). [1]

The construction was inspired by the Rung 3.5 functionals of Janesko et al. [2][3][4][5]. SDMX features are quadratic functionals of the density matrix. The packaged CIDER24X models use them to learn exchange; CIDER24Xe also uses orbital-occupation derivative data during training.

Scalar features

The scalar component starts from a smoothed, spherically averaged density matrix

\[\rho^0(R;\mathbf r) = \int \mathrm d^3\mathbf r'\, h(|\mathbf r'-\mathbf r|;R)n_1(\mathbf r',\mathbf r),\]

where \(R\) is a smoothing length. CIDER24X uses

\[h(u;R) = \left(\frac{2}{\pi}\right)^{3/2} \frac{4}{4-\sqrt{2}}\frac{e^{-2u^2/R^2}}{R^3} \left(1-e^{-2u^2/R^2}\right).\]

The kernel broadens with \(R\), so \(\rho^0(R;\mathbf r)\) is a smoothed approximation to the spherical average of the density matrix at distance \(R\). Integrating its square over the smoothing length gives

\[H_j^0(\mathbf r) = 4\pi\int \mathrm dR\,R^{2-j}|\rho^0(R;\mathbf r)|^2.\]

The radial derivative supplies a second scalar family,

\[H_j^{0\mathrm d}(\mathbf r) = 4\pi\int \mathrm dR\,R^{4-j} \left|\frac{\partial\rho^0(R;\mathbf r)}{\partial R}\right|^2.\]

Angular features

The \(\ell=1\) projection is represented by the vector

\[\boldsymbol{\rho}^1(R;\mathbf r) = \int \mathrm d^3\mathbf r'\, [\nabla h(|\mathbf r'-\mathbf r|;R)] n_1(\mathbf r',\mathbf r).\]

Its norm and radial derivative define two more scalar feature families,

\[\begin{split}H_j^1(\mathbf r) &= 4\pi\int \mathrm dR\,R^{4-j} |\boldsymbol{\rho}^1(R;\mathbf r)|^2, \\ H_j^{1\mathrm d}(\mathbf r) &= 4\pi\int \mathrm dR\,R^{6-j} \left|\frac{\partial\boldsymbol{\rho}^1(R;\mathbf r)} {\partial R}\right|^2.\end{split}\]

The radial-derivative terms reuse the principal contractions required by the corresponding \(H_j^0\) and \(H_j^1\) features. The vector channel adds the \(\ell=1\) angular information.

Uniform coordinate scaling

For the coordinate-scaled density matrix defined in Uniform Scaling, all four families obey

\[H_j[n_1^\lambda](\mathbf r) = \lambda^{3+j}H_j[n_1](\lambda\mathbf r).\]

The implemented uniform-electron-gas normalizations cover \(j\in\{0,1,2\}\). Model-specific normalizers convert these raw powers into the scale-invariant coordinates used by the exchange regression.

Numerical representation

CiderPress evaluates the smoothed density matrix at a discrete set of lengths \(R_i\) using Gaussian convolutions. It then represents the \(R\) dependence in a Gaussian basis, allowing the integrals that define the \(H_j\) features to be contracted analytically. The settings and contraction plans are implemented by SDMXFullSettings and SDMXFullPlan; the molecular contraction algorithm is described in PySCF Numerical and Derivative Implementation.

CIDER24X feature layout

Both packaged CIDER24X models contain the same 13 raw electronic features: three semilocal meta-GGA ingredients followed by ten SDMX features. The SDMX block is ordered as

\[\left( H_1^0,H_2^0, H_1^{0\mathrm d},H_2^{0\mathrm d}, H_1^1,H_2^1,H_0^1, H_1^{1\mathrm d},H_2^{1\mathrm d},H_0^{1\mathrm d} \right).\]

Feature order, normalization, bounded transforms, and mapped neural-network weights are stored in each model file. The optimized molecular implementation is available through PySCF. The periodic PySCF interface supports the CIDER24X methodology with pseudopotentials and uniform grids; GPAW evaluates the NLDF model families. Backend compatibility is listed in Choosing a CIDER Functional.

[1]

Kyle Bystrom, Stefano Falletta, and Boris Kozinsky. Training machine-learned density functionals on band gaps. Journal of Chemical Theory and Computation, 20(17):7516–7532, 2024. PMID: 39178337. URL: https://doi.org/10.1021/acs.jctc.4c00999, doi:10.1021/acs.jctc.4c00999.

[2]

Benjamin G. Janesko. Rung 3.5 density functionals. J. Chem. Phys., 133(10):104103, September 2010. doi:10.1063/1.3475563.

[3]

Benjamin G. Janesko. Rung 3.5 density functionals: Another step on Jacob’s ladder. Int. J. Quantum Chem., 113(2):83–88, 2013. doi:10.1002/qua.24256.

[4]

Benjamin G. Janesko, Giovanni Scalmani, and Michael J. Frisch. Practical auxiliary basis implementation of Rung 3.5 functionals. J. Chem. Phys., 141(3):034103, 07 2014. URL: https://doi.org/10.1063/1.4887085, doi:10.1063/1.4887085.

[5]

Benjamin G. Janesko, Emil Proynov, Giovanni Scalmani, and Michael J. Frisch. Long-range-corrected Rung 3.5 density functional approximations. J. Chem. Phys., 148(10):104112, 03 2018. URL: https://doi.org/10.1063/1.5017981, doi:10.1063/1.5017981.