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

Area of Science:

• Quantum Chemistry
• Computational Chemistry
• Machine Learning

Background:

• Traditional methods for N-representable 2-RDMs rely on linear matrix inequalities.
• Approximating the N-representable set is crucial for accurate electronic structure calculations.

Purpose of the Study:

• To develop a data-driven framework for approximating the convex set of N-representable 2-RDMs.
• To improve the accuracy of 2-RDM calculations using machine learning and semidefinite programming.

Main Methods:

• A vertex-based approximation of the 2-RDM boundary is learned from molecular data.
• An input convex neural network is combined with semidefinite programming.
• A direct variational calculation of the 2-RDM is performed.

Main Results:

• The semidefinite machine learning approach achieves enhanced accuracy in 2-RDM calculations.
• Computational cost is comparable to traditional two-positivity calculations.
• Accurate potential energy curves were obtained for various molecules (C2^2-, N2, O2^2+, CO, NO+, CN-).

Conclusions:

• Semidefinite machine learning effectively combines data-driven insights with positivity constraints.
• This method yields more accurate energies and 2-RDMs without requiring explicit higher-order positivity conditions.