Completing density functional theory by machine learning hidden messages from molecules

Abstract

Kohn–Sham density functional theory (DFT) is the basis of modern computational approaches to electronic structures. Their accuracy heavily relies on the exchange-correlation energy functional, which encapsulates electron–electron interaction beyond the classical model. As its universal form remains undiscovered, approximated functionals constructed with heuristic approaches are used for practical studies. However, there are problems in their accuracy and transferability, while any systematic approach to improve them is yet obscure. In this study, we demonstrate that the functional can be systematically constructed using accurate density distributions and energies in reference molecules via machine learning. Surprisingly, a trial functional machine learned from only a few molecules is already applicable to hundreds of molecules comprising various first- and second-row elements with the same accuracy as the standard functionals. This is achieved by relating density and energy using a flexible feed-forward neural network, which allows us to take a functional derivative via the back-propagation algorithm. In addition, simply by introducing a nonlocal density descriptor, the nonlocal effect is included to improve accuracy, which has hitherto been impractical. Our approach thus will help enrich the DFT framework by utilizing the rapidly advancing machine-learning technique.

Quantum chemical accuracy from density functional approximations via machine learning

“…The construction of such an ML approach requires a molecular descriptor flexible enough to accomplish both types of tasks, and for this, it seems natural to employ the electron density. It is worth noting that as molecular descriptors have evolved from objects such as SMILES strings 44,45 , molecular graphs 46,47 , and molecular graphs with feature vectors 24,25,48 , there has been a progression toward descriptors that attempt to capture key features of the electron density in a simple manner 15,[48][49][50][51] . Admittedly, employing the full electron density carries with it a considerable computational cost; nevertheless, it is useful to develop such frameworks, considering that more optimal algorithms could follow.…”

“…The second map from density to energy predicts the result of plugging that solution back into the Hohenberg-Kohn functional to obtain the ground-state energy. While other machine-learning methods for the prediction of electron densities or density functionals have appeared recently 50,51,[55][56][57][58][59][60][61][62] , the ML-HK map facilitates the use of both machine-learned densities, from which electronic properties could be computed, and density functionals for obtaining total energies and gradients for geometry optimization and MD simulation.…”

Machine learning accurate exchange and correlation functionals of the electronic density

“…Both approaches show great promise to significantly speed up ab initio calculations as they completely circumvent solving the cubic-scaling self-consistent field (SCF) equations. Other works, including the one presented here, have attempted to parametrize an XC functional with ML, and we discuss related methods [15][16][17] in detail in Supplementary Note 1.…”

Retrospective on a decade of machine learning for chemical discovery

“…The problem of XC is harder. Nagai et al 8 showed that accurate densities of just three small molecules are sufficient to create machine-learned approximations that are comparable to those created by people. In ab initio quantum chemistry, Welborn et al 9 have shown how to use features from Hartree-Fock calculations to accurately predict CCSD energies, while an intriguing alternative is to map to spin problems and use a restricted Boltzmann machine 10 .…”