Efficient and Accurate Simulations of Vibrational and Electronic Spectra with Symmetry-Preserving Neural Network Models for Tensorial Properties

Abstract

Machine learning has revolutionized the highdimensional representations for molecular properties such as potential energy. However, there are scarce machine learning models targeting tensorial properties, which are rotationally covariant. Here, we propose tensorial neural network (NN) models to learn both tensorial response and transition properties in which atomic coordinate vectors are multiplied with scalar NN outputs or their derivatives to preserve the rotationally covariant symmetry. This strategy keeps structural descriptors symmetry invariant so that the resulting tensorial NN models are as efficient as their scalar counterparts. We validate the performance and universality of this approach by learning response properties of water oligomers and liquid water and transition dipole moment of a model structural unit of proteins. Machine-learned tensorial models have enabled efficient simulations of vibrational spectra of liquid water and ultraviolet spectra of realistic proteins, promising feasible and accurate spectroscopic simulations for biomolecules and materials.

Perspective on integrating machine learning into computational chemistry and materials science

“…The concept of ML-based Hamiltonian and densityfunctional surrogate models directly leads to the construction of approximate electronic structure models based on ML. Recently reported approaches include an ML-based Hückel model, 141 parametrized Frenkel 102,[142][143][144][145] and Tight-binding (TB) Hamiltonians 146 as well as semi-empirical methods with ML-tuned parameters. 147,148 .…”

“…This approach has further been applied to model absorption spectra. 77,144 Raimbault et al 278 introduced a kernel approach for predicting the Raman spectra of organic crystals based on molecular polarizabilities. Using a NN based approach, Sommers et al 279 have demonstrated that ML can also be used to simulate Raman spectra of extended systems such as liquid water, which would be computationally unfeasible when done with DFT.…”

“…In addition to vibrational spectra, ML models are also capable of modeling response properties, allowing the simulation of electronic excitations using, e.g., MQCD approaches (see Section VI). For example, Zhang et al 144 use NN models to obtain transition dipole moments, which in turn could be used to predict UV and visible light spectra. ML approaches have further been used to predict nuclear magnetic resonance (NMR) spectra from molecular simulations.…”

Physically inspired deep learning of molecular excitations and photoemission spectra

“…12,13 Machine learning (ML) methods have the ability to overcome the gap between experiment and theory for spectroscopic characterization by reducing the computational effort of spectroscopic simulations without sacricing prediction accuracy. 14,15 ML methods in the context of spectroscopy have previously focused on predicting single energy levels, [15][16][17][18][19] oscillator strengths, 20,21 dipole moments, [22][23][24] highest occupied molecular orbital (HOMO) and lowest unoccupied molecular orbital (LUMO) energies [25][26][27][28] or band gaps. [29][30][31] They have also been applied successfully to identify and characterize structures from X-ray absorption spectra.…”

“…In contrast to previous efforts, [42][43][44][45] this matrix representation is not based on local atomic orbital features and the elements of this matrix have no direct physical correspondence beyond the fact that the matrix eigenvalues correspond to the learned molecular resonances. As we are only training on rotationally invariant quantities, the model achieves this without the need to explicitly encode vectorial [46][47][48][49] or tensorial equivariance properties 23,25 beyond the rotationally invariant representation of the input molecular coordinates. 28 The simple algebraic modication of describing vectorial targets by diagonalization of a matrix output leads to increased learning rates, reduced prediction errors, and increased transferability in predicting electron addition and removal energies across molecular composition space.…”