Machine Learning for Electronically Excited States of Molecules

Abstract

Electronically excited states of molecules are at the heart of photochemistry, photophysics, as well as photobiology and also play a role in material science. Their theoretical description requires highly accurate quantum chemical calculations, which are computationally expensive. In this review, we focus on not only how machine learning is employed to speed up such excited-state simulations but also how this branch of artificial intelligence can be used to advance this exciting research field in all its aspects. Discussed applications of machine learning for excited states include excited-state dynamics simulations, static calculations of absorption spectra, as well as many others. In order to put these studies into context, we discuss the promises and pitfalls of the involved machine learning techniques. Since the latter are mostly based on quantum chemistry calculations, we also provide a short introduction into excited-state electronic structure methods and approaches for nonadiabatic dynamics simulations and describe tricks and problems when using them in machine learning for excited states of molecules.

Figure 1

Targets of ML for the excited states of molecules (dashed: not yet achieved). All areas of excited-state quantum chemistry (QC) calculations can be enhanced with ML, ranging from input to primary outputs that are used in the computation of secondary outputs, which in turn are employed to calculate tertiary outputs. Analysis can be carried out at all stages. The classification is inspired by the one in ref (70).

Figure 2

Excited-state processes that can take place after excitation of a molecule by light. Absorption of light can make the molecule enter a higher electronic singlet state. Internal conversion to another state of same spin-multiplicity and/or intersystem crossing to a triplet state can take place. Radiative emission, i.e., fluorescence and phosphorescence, are possible reactions from an excited singlet and triplet state, respectively.

Figure 3

Different arrangements of electrons in molecular orbitals giving rise to the configuration interaction (CI) method. Inclusion of excited configurations in addition to the ground-state, reference determinant, ϕ₀, allows one to go beyond the Hartree–Fock method. Electrons are excited into higher electronic orbitals, and Slater determinants are indicated using the letters S, D, T, and Q, which refer to single, double, triplet, and quadruple excitations.

Figure 4

Electrons and orbitals of an arbitrary system to exemplify the active space needed for many multireference methods. (a) The highest, not considered, molecular orbitals are inactive and always empty. (c) The lowest, not considered, molecular orbitals are always doubly occupied. (b) The active space is shown with two active electrons in two active orbitals.

Figure 5

Potential energy curves of the three lowest singlet (S₀–S₂) and the four lowest triplet state (T₁–T₄) of the amino acid tyrosine along the O–H bond length of the hydroxy group located at the phenyl ring (Ph–OH) computed with CASSCF(12,11)/ano-rcc-pVDZ and CASPT2(12,11)/ano-rcc-pVDZ.

Figure 6

(a) Example of three potential energy curves ordered by their character along with respective potential couplings between different states shown by dashed lines. (b) Two singlets (E_i and E_j) and one triplet state (E_k) including coupling values (with vectorial properties, , shown by their norm) in the adiabatic basis, in which the triplet state crosses singlet states. (c) The diagonal, or spin-adiabatic, basis, in which all states are ordered by their energy and are spin-mixed. Kinetic couplings are shown by their norm. Note that the ground state is not shown. Potential energy curves are represented using solid lines and couplings using dashed lines.

Figure 7

Excited-state dynamics can be treated with (a) quantum approaches, where wave functions are used for the nuclei, or (b) classical approaches, based on trajectories.

Figure 8

Schematic representation of a multilayer feed-forward NN with inputs, X, nodes, n, and outputs, Y. In the usual implementation for the fitting of PESs, the NN maps a molecular geometry to the ground state, which could be similarly done for any other single state. In the case a manifold of excited states is described, one molecular input can also be mapped to a vector of different excited states, and additionally, other properties can be included. The forces are treated as derivatives of the NN potentials with respect to Cartesian coordinates.

Figure 9

Adaptive sampling scheme illustrated using two ML models (blue and red blocks). The active learning procedure starts from an initial, preliminary training set (yellow), which is used to train ML models. A sampling step, e.g., a time step of an MD simulation, is executed: The ML models take the molecular geometry of the sampling step as an input and predict the energies of the considered excited states, their derivatives, and additional required photochemical properties. In case the predictions of the ML models are deemed to be different, quantum chemical reference calculations are carried out, ML models are retrained, and the serial steps are carried out again. This procedure is executed until the desired quality of the ML PESs is attained in order to sufficiently describe the chemical problem under investigation.

Figure 10

NAC value between singlet state S_i and S_j in the MCH basis. A consistent sign along the reaction path of couplings is shown by blue dashed lines. The direct output of a quantum chemical calculation is shown by a magenta line.

Figure 11

Pie diagrams summarizing the reference methods used for the training set generation, the chosen ML models, and the type of descriptors for the description of the excited states with ML. Analysis is based on 45 studies of ML for the excited states.