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The Journal of Physical Chemistry A

Machine-Learning Coupled Cluster Properties through a Density Tensor Representation

Benjamin G. Peyton, Connor G. Briggs, Ruhee D’Cunha, Johannes T. Margraf, T. Daniel Crawford

Published May 15, 2020• 33 citations
Publisher

Abstract

The introduction of machine-learning (ML) algorithms to quantum mechanics enables rapid evaluation of otherwise intractable expressions at the cost of prior training on appropriate benchmarks. Many computational bottlenecks in the evaluation of accurate electronic structure theory could potentially benefit from the application of such models, from reducing the complexity of the underlying wave function parameter space to circumventing the complications of solving the electronic Schrödinger equation entirely. Applications of ML to electronic structure have thus far been focused on learning molecular properties (mainly the energy) from geometric representations. While this line of study has been quite successful, highly accurate models typically require a “big data” approach with thousands of training data points. Herein, we propose a general, systematically improvable scheme for wave function-based ML of arbitrary molecular properties, inspired by the underlying equations that govern the canonical approach to computing the properties. To this end, we combine the established ML machinery of the t-amplitude tensor representation with a new reduced density matrix representation. The resulting model provides quantitative accuracy in both the electronic energy and dipoles of small molecules using only a few dozen training points per system.

DOI
10.1021/acs.jpca.0c02804

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The introduction of machine-learning (ML) algorithms to quantum mechanics enables rapid evaluation of otherwise intractable expressions at the cost of prior training on appropriate benchmarks. Many computational bottlenecks in the evaluation of accurate electronic structure theory could potentially benefit from the application of such models, from reducing the complexity of the underlying wave function parameter space to circumventing the complications of solving the electronic Schrödinger equation entirely. Applications of ML to electronic structure have thus far been focused on learning molecular properties (mainly the energy) from geometric representations. While this line of study has been quite successful, highly accurate models typically require a “big data” approach with thousands of training data points. Herein, we propose a general, systematically improvable scheme for wave function-based ML of arbitrary molecular properties, inspired by the underlying equations that govern the canonical approach to computing the properties. To this end, we combine the established ML machinery of the t-amplitude tensor representation with a new reduced density matrix representation. The resulting model provides quantitative accuracy in both the electronic energy and dipoles of small molecules using only a few dozen training points per system.
The introduction of machine-learning (ML) algorithms to quantum mechanics enables rapid evaluation of otherwise intractable expressions at the cost of prior training on appropriate benchmarks. Many computational bottlenecks in the evaluation of accurate electronic structure theory could potentially benefit from the application of such models, from reducing the complexity of the underlying wave function parameter space to circumventing the complications of solving the electronic Schrödinger equation entirely. Applications of ML to electronic structure have thus far been focused on learning molecular properties (mainly the energy) from geometric representations. While this line of study has been quite successful, highly accurate models typically require a “big data” approach with thousands of train- ing data points. Herein, we propose a general, systematically improvable scheme for wave function-based ML of arbitrary molecular properties, inspired by the underlying equations that govern the canonical approach to computing the properties. To this end, we combine the established ML machinery of the t-amplitude tensor representation with a new reduced density matrix representation. The resulting model provides quantitative accuracy in both the electronic energy and dipoles of small molecules using only a few dozen training points per system.
Calculating the electronic structure of molecules and solids has become an important pillar of modern research in diverse fields of research from biology and materials science to chemistry and physics. Unfortunately, increasingly accurate and thus reliable approximate solution schemes to the underlying Schrödinger equation scale steeply in computational cost, rendering most accurate approaches like "gold standard" coupled cluster theory, CC, quickly intractable for larger systems of interest. Here we show that this scaling can be significantly reduced by applying machine-learning to the CC correlation energy. We introduce a vector-based representation of CC wave functions and use potential energy surfaces of a small molecule test set to learn the correlation energy from this representation. Our results show that the CC correlation energy can be efficiently learned, even when the representation is constructed from approximate amplitudes provided by computationally less demanding Møller-Plesset (MP2) perturbation theory. Exploiting existing linear scaling MP2 implementations, this potentially opens the door to CC-quality molecular dynamics simulations.
A long-standing goal of science is to accurately solve the Schrödinger equation for large molecular systems. The poor scaling of current quantum chemistry algorithms on classical computers imposes an effective limit of about a few dozen atoms for which we can calculate molecular electronic structure. We present a machine learning (ML) method to break through this scaling limit and make quantum chemistry calculations of very large systems possible. We show that Euclidean Neural Networks can be trained to predict the electron density with high fidelity from limited data. Learning the electron density allows us to train a machine learning model on small systems and make accurate predictions on large ones. We show that this ML electron density model can break through the quantum scaling limit and calculate the electron density of systems of thousands of atoms with quantum accuracy.
We present a novel machine learning algorithm for the many-electron problem, predicting the convex combination of two-electron reduced density matrices (2-RDMs)─obtained from upper- and lower-bound energy calculations─that closely approximates the exact energy. In contrast to other recently developed approaches based on the wave function or one-electron density, our 2-RDM machine-learning approach predicts energies and properties without steep scaling or functional approximation. As conjectured by Preskill and co-workers, a small amount of data in a physics-based machine learning algorithm─in this case, information about the RDMs and their violation of selected higher-order N -representability conditions─yields highly accurate electronic energies that capture both dynamic and static correlation. We demonstrate the method by predicting the potential energy curves for BH and N 2 within a few millihartrees of results from exact diagonalization. This machine learning algorithm provides a general framework for improving electronic structure calculations, with the potential for wide-reaching applications to both moderately and strongly correlated molecular systems.
Density functional theory (DFT) is the most widely used method for calculating molecular properties; however, its accuracy is often insufficient for quantitative predictions. Coupled-cluster (CC) theory is the most successful method for achieving accuracy beyond DFT and for predicting properties that closely align with experiment. It is known as the ''gold standard'' of quantum chemistry. Unfortunately, the high computational cost of CC limits its widespread applicability. In this work, we present the Molecular Orbital Learning (MōLe) architecture, an equivariant machine learning model that directly predicts CC's core mathematical objects, the excitation amplitudes, from the mean-field Hartree-Fock molecular orbitals as inputs. We test various aspects of our model and demonstrate its remarkable data efficiency and out-of-distribution generalization to larger molecules and off-equilibrium geometries, despite being trained only on small equilibrium geometries. Finally, we also examine its ability to reduce the number of cycles required to converge CC calculations. MōLe can set the foundations for high-accuracy wavefunction-based ML architectures to accelerate molecular design and complement force-field approaches.
Machine learning has recently been introduced into the arsenal of tools that are available to computational chemists. In the past few years, we have seen an increase in the applicability of these tools on a plethora of applications, including the automated exploration of a large fraction of the chemical space, the reduction of repetitive computational tasks, the detection of outliers on large databases, and the acceleration of molecular simulations. An attractive application of machine learning in molecular electronic structure theory is the “recycling” of molecular wave functions for faster and more accurate completion of complex quantum chemical calculations. Along these lines, we have developed hybrid quantum chemical/machine learning workflows that utilize information from low-level wave functions for the accurate prediction of higher-level wave functions. The data-driven coupled-cluster (DDCC) family of methods is discussed in this article together with the importance of the inclusion of physical properties in such hybrid workflows. After a short introduction to the philosophy and the capabilities of DDCC, we present our recent progress in extending its applicability to larger and more complex molecular structures and data sets. A significant advantage offered by DDCC is its transferability, with respect to different molecular systems and different excitation levels. As we show here, predicted wave functions at the coupled-cluster singles and doubles level of theory can be used for the accurate prediction of the perturbative triples of the CCSD(T) scheme. We conclude with some personal considerations with respect to future directions related to the development of the next generation of such hybrid quantum chemical/machine learning models.