Direct Variational Calculation of Two-Electron Reduced Density Matrices via Semidefinite Machine Learning

TL;DR

Overall, semidefinite machine learning interweaves data-driven boundary information with semidefinite positivity constraints to yield more accurate energies and 2-RDMs without explicit higher-order positivity conditions.

Abstract

We introduce a data-driven framework for approximating the convex set of $N$-representable two-electron reduced density matrices (2-RDMs). Traditional approaches characterize this set through linear matrix inequalities that define its supporting hyperplanes. Here, we instead learn a vertex-based approximation to its boundary from molecular data and use this information to improve the set defined by low-order positivity constraints, without explicitly constructing higher-order conditions. The resulting semidefinite machine learning approach -- combining an input convex neural network with semidefinite programming -- drives a direct variational calculation of the 2-RDM with enhanced accuracy at computational cost comparable to two-positivity calculations. Applications to the potential energy curves of ${\rm C}_2^{2-}$, ${\rm N}_2$, and ${\rm O}_2^{2+}$ demonstrate these systematic improvements as well as close agreement with complete active space configuration interaction results. Overall, semidefinite machine learning interweaves data-driven boundary information with semidefinite positivity constraints to yield more accurate energies and 2-RDMs without explicit higher-order positivity conditions.

Figures (3)

• Figure 1: Representations of a convex set: (A) V-representation and (B) H-representation. While the exact $N$-representable set is not a polytope and therefore does not admit a finite vertex representation, the convex hull of a finite set of data-driven boundary (exposed) points provides a V-representation that approximates the true set.
• Figure 2: Predicting ${\rm C}_2^{2-}$ dissociation curve. (a) Variational 2-RDM energies (v2RDM, green circles), complete active space configuration interaction energies (CASCI, black line), and energies predicted by the semidefinite ML algorithm (Table \ref{['table:algo']}) (SD-ML, blue triangles) are shown as its bond is stretched. (b) Additionally, the errors in v2RDM energies and SD-ML energies compared to CASCI energies are shown. As can be seen, the SD-ML methodology predicts a significantly more accurate potential energy surface for ${\rm C}_2^{2-}$ than v2RDM. Both v2RDM and CASCI energies are computed using the cc-pVDZ basis set and in a $[N_e= 10, N_o= 8]$ active space
• Figure 3: ${\rm C}_2^{2-}$ 2-RDM errors. The Frobenius norms of the difference matrices formed from each v2RDM and SD-ML 2-RDM relative to the CASCI 2-RDM are shown as a function of bond distance. The norms remain similar throughout the bond stretch, with a small improvement at the tail of the potential energy curve.