Neural networks and kernel ridge regression for excited states dynamics of CH₂NH 2+ : From single-state to multi-state representations and multi-property machine learning models

Abstract

Excited-state dynamics simulations are a powerful tool to investigate photo-induced reactions of molecules and materials and provide complementary information to experiments. Since the applicability of these simulation techniques is limited by the costs of the underlying electronic structure calculations, we develop and assess different machine learning models for this task. The machine learning models are trained on ab initio calculations for excited electronic states, using the methylenimmonium cation (CH 2 NH + 2 ) as a model system. For the prediction of excited-state properties, multiple outputs are desirable, which is straightforward with neural networks but less explored with kernel ridge regression. We overcome this challenge for kernel ridge regression in the case of energy predictions by encoding the electronic states explicitly in the inputs, in addition to the molecular representation. We adopt this strategy also for our neural networks for comparison. Such a state encoding enables not only kernel ridge regression with multiple outputs but leads also to more accurate machine learning models for state-specific properties. An important goal for excited-state machine learning models is their use in dynamics simulations, which needs not only state-specific information but also couplings, i.e., properties involving pairs of states. Accordingly, we investigate the performance of different models for such coupling elements. Furthermore, we explore how combining all properties in a single neural network affects the accuracy. As an ultimate test for our machine learning models, we carry out excited-state dynamics simulations based on the predicted energies, forces and couplings and, thus, show the scopes and possibilities of machine learning for the treatment of electronically excited states.

Combining SchNet and SHARC: The SchNarc Machine Learning Approach for Excited-State Dynamics

“…As they are part of our everyday lives, their understanding can help to unravel fundamental processes of nature and to advance several research fields, such as photovoltaics [6,7], photocatalysis [8] or photosensitive drug design [9].Since the full quantum mechanical treatment of molecules remains challenging, exact quantum dynamics simulations are limited to systems containing only a couple of atoms, even if fitted potential energy surfaces (PESs) are used [10][11][12][13][14][15][16][16][17][18][19][20][21][22][23][24][25][26]. In order to treat larger systems in full dimensions, i.e., systems with up to 100s of atoms, and on long time scales, i.e., in the range of several 100 picoseconds, excited-state machine learning (ML) molecular dynamics (MD), where the ML model is trained on quantum chemistry data, has evolved as a promising tool in the last couple of years [27][28][29][30][31][32][33].Such nonadiabatic MLMD simulations are in many senses analog to excited-state ab initio molecular dynamics simulations. The only difference is that the costly electronic structure calculations are mostly replaced by a ML model, providing quantum properties like the PESs and the corresponding forces.…”

“…An arising difficulty compared to ground state energies and properties is that not only one, but several PESs as well as the couplings between them have to be taken into account. Additionally, the learning of couplings proves challenging due to the fact that properties resulting from electronic wave functions of two different states, Ψ i and Ψ j , have their sign dependent on the phase of the wave functions [32,33,82,83]. Since the wave function phase is not uniquely defined in quantum chemistry calculations, random phase jumps occur, leading to sign jumps of the coupling values along a reaction path.…”

“…In comparison, a SchNarc simulation with a ML model that applies the phase-less loss function is successful in reproducing the populations for both training sets. In those simulations, the NACs are multiplied with the corresponding energy gaps, i.e., C NAC ij = C NAC ij · ∆E ij , to get rid of singularities [21,33]. These smooth couplings C NAC ij are not directly learned, but rather constructed as the derivative of a virtual property, analogously to forces that are predicted as derivatives of an energy-ML model.…”

“…The virtual property is the multi-dimensional anti-derivative of the rightmost expression in equation (1), Ψ i | ∂H el R | Ψ j (a derivation is given in the SI). Compared to previous ML models for NACs [29,30,32,33], where NACs are learned and predicted as direct outputs or even single values, this approach provides rotational and translational covariance, which has recently been achieved in a similar way for the electronic friction tensor [85].However, even without the need of pre-processing the training set, the costly computations of NAC vectors for the training set generation remain. Approximations 4 Figure 1: Populations obtained from 90 QC (MR-CISD(6,4)/aug-cc-pVDZ) trajectories are shown by continuous lines and are compared to (A) populations resulting from 1000 initially excited trajectories obtained from SchNet (dotted lines) that is trained on not phase corrected data and takes the L 2 norm as loss function, (B) a similar SchNet model, but trained on phase corrected data, (C) a SchNet model trained on not phase corrected data, but using the new phase-less loss function, and (D) a SchNet model trained on phase corrected data using the new phase-less loss function .…”

“…Dynamics are shown of 1000 initially excited configurations considering only singlet states (upper line) and additionally triplet states (lower line). 8 Figure 4: Quantum populations of the methylenimmonium cation obtained from (A) 90 trajectories using MR-CISD/aug-cc-pVDZ (QC) and (B) 1000 trajectories using SchNarc (ML) as well as populations up to 3,000 fs of the thioformaldehyde molecule obtained from (C) 100 trajectories using CASSCF(6,5)/def2-SVP (QC) and (D) 9590 trajectories using SchNarc (ML).duce with accurate NACs [33]. In contrast, the CSH 2 molecule serves as a testsystem for slow populations transfer and shows intersystem crossing strongly dependent on the accuracy of the underlying potentials [96].…”