Circumventing the many-body problem by learning the two-body reduced density matrix

Accomplishing the goal of machine learning wavefunctions of arbitrary quantum systems would open a plethora of crucial and useful applications. While there are many notable attempts at achieving this goal [1, 2], the current state of the art is limited in the degrees of freedom it can handle. We present an alternative that focuses on the two-body reduced density matrix (2-rdm) as a central quantity to be targeted by machine learning. Inspired by previous work on the electron density [3, 4], density matrix [5] and spectral density [6], we develop models for the electronic 2-rdm of small molecular systems such as water, ammonia and formaldehyde that reproduce results from high-level multireference wavefunction methods. The model 2-rdm delivers electronic energies, atomic forces, and any conceivable one and two-electron molecular properties. We discuss opportunities and roadblocks for developing the model 2-rdms with a particular emphasis on imposing N-representability conditions.

[1] J. Hermann, J. Spencer, Kenny Choo, Antonio Mezzacapo, W. M. C. Foulkes, David Pfau, Giuseppe Carleo, and Frank Noé. Ab initio quantum chemistry with neural-network wavefunctions. Nature Reviews Chemistry, 7(10):692–709, August 2023.

[2] F. Noé, A. Tkatchenko, Klaus-Robert Müller, and Cecilia Clementi. Machine Learning for Molecular Simulation. Annual Review of Physical Chemistry, 71(1):361–390, April 2020.

[3] F. Brockherde, L. Vogt, Li Li, Mark E. Tuckerman, Kieron Burke, and Klaus-Robert Müller. By-passing the Kohn-Sham equations with machine learning. Nature Communications, 8(1):872, October 2017.

[4] Y. Bai, L. Vogt-Maranto, Mark E. Tuckerman, and William J. Glover. Machine learning the Hohenberg-Kohn map for molecular excited states. Nature Communications, 13(1):7044, November 2022.

[5] X. Shao, L. Paetow, Mark E. Tuckerman, and Michele Pavanello. Machine learning electronic structure methods based on the one-electron reduced density matrix. Nature Communications, 14(1):6281, October 2023

[6] L. Fiedler, N. A. Modine, Steve Schmerler, Dayton J. Vogel, Gabriel A. Popoola, Aidan P. Thompson, Sivasankaran Rajamanickam, and Attila Cangi. Predicting electronic structures at any length scale with machine learning. npj Computational Materials, 9(1):115, June 2023.