Exploiting Machine Learning to Efficiently Predict Multidimensional Optical Spectra in Complex Environments

First-Principles Modeling of the Absorption Spectrum of Crystal Violet in Solution: The Importance of Environmentally Driven Symmetry Breaking

Theoretical spectroscopy plays a crucial role in understanding the properties of the materials and molecules. One of the most promising methods for computing optical spectra of chromophores embedded in complex environments from the first principles is the cumulant approach, where both (generally anharmonic) vibrational degrees of freedom and environmental interactions are explicitly accounted for. In this work, we verify the capabilities of the cumulant approach in describing the effect of complex environmental interactions on linear absorption spectra by studying Crystal Violet (CV) in different solvents. The experimental absorption spectrum of CV strongly depends on the nature of the solvent, indicating strong coupling to the condensed-phase environment. We demonstrate that these changes in absorption line shape are driven by an increased splitting between absorption bands of two low-lying excited states that is caused by a breaking of the D3 symmetry of the molecule and that in polar solvents, this symmetry breaking is mainly driven by electrostatic interactions with the condensed-phase environment rather than distortion of the structure of the molecule, in contrast with conclusions reached in a number of previous studies. Our results reveal the importance of explicitly including a counterion in the calculations in nonpolar solvents due to electrostatic interactions between CV and the ion. In polar solvents, these interactions are strongly reduced due to solvent screening effects, thus minimizing the symmetry breaking. Computed spectra in methanol are found to be in reasonable agreement with the experiment, demonstrating the strengths of the outlined approach in modeling strong environmental interactions.

Protein Effects on the Excitation Energies and Exciton Dynamics of the CP24 Antenna Complex

In this study, the site energy fluctuations, energy transfer dynamics, and some spectroscopic properties of the minor light-harvesting complex CP24 in a membrane environment were determined. For this purpose, a 3 μs-long classical molecular dynamics simulation was performed for the CP24 complex. Furthermore, using the density functional tight binding/molecular mechanics molecular dynamics (DFTB/MM MD) approach, we performed excited state calculations for the chlorophyll a and chlorophyll b molecules in the complex starting from five different positions of the MD trajectory. During the extended simulations, we observed variations in the site energies of the different sets as a result of the fluctuating protein environment. In particular, a water coordination to Chl-b 608 occurred only after about 1 μs in the simulations, demonstrating dynamic changes in the environment of this pigment. From the classical and the DFTB/MM MD simulations, spectral densities and the (time-dependent) Hamiltonian of the complex were determined. Based on these results, three independent strongly coupled chlorophyll clusters were revealed within the complex. In addition, absorption and fluorescence spectra were determined together with the exciton relaxation dynamics, which reasonably well agrees with experimental time scales.

Predicting Serotonin Detection with DNA-Carbon Nanotube Sensors across Multiple Spectral Wavelengths

Owing to the value of DNA-wrapped single-walled carbon nanotube (SWNT)-based sensors for chemically specific imaging in biology, we explore machine learning (ML) predictions DNA-SWNT serotonin sensor responsivity as a function of DNA sequence based on the whole SWNT fluorescence spectra. Our analysis reveals the crucial role of DNA sequence in the binding modes of DNA-SWNTs to serotonin, with a smaller influence of SWNT chirality. Regression ML models trained on existing data sets predict the change in the fluorescence emission in response to serotonin, ΔF/F, at over a hundred wavelengths for new DNA-SWNT conjugates, successfully identifying some high- and low-response DNA sequences. Despite successful predictions, we also show that the finite size of the training data set leads to limitations on prediction accuracy. Nevertheless, incorporating entire spectra into ML models enhances prediction robustness and facilitates the discovery of novel DNA-SWNT sensors. Our approaches show promise for identifying new chemical systems with specific sensing response characteristics, marking a valuable advancement in DNA-based system discovery.

Electronic Excited States from Physically Constrained Machine Learning

Data-driven techniques are increasingly used to replace electronic-structure calculations of matter. In this context, a relevant question is whether machine learning (ML) should be applied directly to predict the desired properties or combined explicitly with physically grounded operations. We present an example of an integrated modeling approach in which a symmetry-adapted ML model of an effective Hamiltonian is trained to reproduce electronic excitations from a quantum-mechanical calculation. The resulting model can make predictions for molecules that are much larger and more complex than those on which it is trained and allows for dramatic computational savings by indirectly targeting the outputs of well-converged calculations while using a parametrization corresponding to a minimal atom-centered basis. These results emphasize the merits of intertwining data-driven techniques with physical approximations, improving the transferability and interpretability of ML models without affecting their accuracy and computational efficiency and providing a blueprint for developing ML-augmented electronic-structure methods.

Artificial-Intelligence-Enhanced On-the-Fly Simulation of Nonlinear Time-Resolved Spectra

Time-resolved spectroscopy is an important tool for unraveling the minute details of structural changes in molecules of biological and technological significance. The nonlinear femtosecond signals detected for such systems must be interpreted, but it is a challenging task for which theoretical simulations are often indispensable. Accurate simulations of transient absorption or two-dimensional electronic spectra are, however, computationally very expensive, prohibiting the wider adoption of existing first-principles methods. Here, we report an artificial-intelligence-enhanced protocol to drastically reduce the computational cost of simulating nonlinear time-resolved electronic spectra, which makes such simulations affordable for polyatomic molecules of increasing size. The protocol is based on the doorway-window approach for the on-the-fly surface-hopping simulations. We show its applicability for the prototypical molecule of pyrazine for which it produces spectra with high precision with respect to ab initio reference while cutting the computational cost by at least 95% compared to pure first-principles simulations.

Taming the third order cumulant approximation to linear optical spectroscopy

The second order cumulant method offers a promising pathway to predicting optical properties in condensed phase systems. It allows for the computation of linear absorption spectra from excitation energy fluctuations sampled along molecular dynamics (MD) trajectories, fully accounting for vibronic effects, direct solute-solvent interactions, and environmental polarization effects. However, the second order cumulant approximation only guarantees accurate line shapes for energy gap fluctuations obeying Gaussian statistics. A third order correction has recently been derived but often yields unphysical spectra or divergent line shapes for moderately non-Gaussian fluctuations due to the neglect of higher order terms in the cumulant expansion. In this work, we develop a corrected cumulant approach, where the collective effect of neglected higher order contributions is approximately accounted for through a dampening factor applied to the third order cumulant term. We show that this dampening factor can be expressed as a function of the skewness and kurtosis of energy gap fluctuations and can be parameterized from a large set of randomly sampled model Hamiltonians for which exact spectral line shapes are known. This approach is shown to systematically remove unphysical contributions in the form of negative absorbances from cumulant spectra in both model Hamiltonians and condensed phase systems sampled from MD and dramatically improves over the second order cumulant method in describing systems exhibiting Duschinsky mode mixing effects. We successfully apply the approach to the coumarin-153 dye in toluene, obtaining excellent agreement with experiment.

Two-Step Machine Learning Approach for Charge-Transfer Coupling with Structurally Diverse Data

Electronic coupling is important in determining charge-transfer rates and dynamics. Coupling strength is sensitive to both intermolecular, e.g., orientation or distance, and intramolecular degrees of freedom. Hence, it is challenging to build an accurate machine learning model to predict electronic coupling of molecular pairs, especially for those derived from the amorphous phase, for which intermolecular configurations are much more diverse than those derived from crystals. In this work, we devise a new prediction algorithm that employs two consecutive KRR models. The first model predicts molecular orbitals (MOs) from structural variation for each fragment, and coupling is further predicted by using the overlap integral included in a second model. With our two-step procedure, we achieved mean absolute errors of 0.27 meV for an ethylene dimer and 1.99 meV for a naphthalene pair, much improved accuracy amounting to 14-fold and 3-fold error reductions, respectively. In addition, MOs from the first model can also be the starting point to obtain other quantum chemical properties from atomistic structures. This approach is also compatible with a MO predictor with sufficient accuracy.

Beyond the Condon limit: Condensed phase optical spectra from atomistic simulations

While dark transitions made bright by molecular motions determine the optoelectronic properties of many materials, simulating such non-Condon effects in condensed phase spectroscopy remains a fundamental challenge. We derive a Gaussian theory to predict and analyze condensed phase optical spectra beyond the Condon limit. Our theory introduces novel quantities that encode how nuclear motions modulate the energy gap and transition dipole of electronic transitions in the form of spectral densities. By formulating the theory through a statistical framework of thermal averages and fluctuations, we circumvent the limitations of widely used microscopically harmonic theories, allowing us to tackle systems with generally anharmonic atomistic interactions and non-Condon fluctuations of arbitrary strength. We show how to calculate these spectral densities using first-principles simulations, capturing realistic molecular interactions and incorporating finite-temperature, disorder, and dynamical effects. Our theory accurately predicts the spectra of systems known to exhibit strong non-Condon effects (phenolate in various solvents) and reveals distinct mechanisms for electronic peak splitting: timescale separation of modes that tune non-Condon effects and spectral interference from correlated energy gap and transition dipole fluctuations. We further introduce analysis tools to identify how intramolecular vibrations, solute-solvent interactions, and environmental polarization effects impact dark transitions. Moreover, we prove an upper bound on the strength of cross correlated energy gap and transition dipole fluctuations, thereby elucidating a simple condition that a system must follow for our theory to accurately predict its spectrum.

Multifidelity Machine Learning for Molecular Excitation Energies

The accurate but fast calculation of molecular excited states is still a very challenging topic. For many applications, detailed knowledge of the energy funnel in larger molecular aggregates is of key importance, requiring highly accurate excitation energies. To this end, machine learning techniques can be a very useful tool, though the cost of generating highly accurate training data sets still remains a severe challenge. To overcome this hurdle, this work proposes the use of multifidelity machine learning where very little training data from high accuracies is combined with cheaper and less accurate data to achieve the accuracy of the costlier level. In the present study, the approach is employed to predict vertical excitation energies to the first excited state for three molecules of increasing size, namely, benzene, naphthalene, and anthracene. The energies are trained and tested for conformations stemming from classical molecular dynamics and density functional based tight-binding simulations. It can be shown that the multifidelity machine learning model can achieve the same accuracy as a machine learning model built only on high-cost training data while expending a much lower computational effort to generate the data. The numerical gain observed in these benchmark test calculations was over a factor of 30 but certainly can be much higher for high-accuracy data.

How to train a neural network potential

The introduction of modern Machine Learning Potentials (MLPs) has led to a paradigm change in the development of potential energy surfaces for atomistic simulations. By providing efficient access to energies and forces, they allow us to perform large-scale simulations of extended systems, which are not directly accessible by demanding first-principles methods. In these simulations, MLPs can reach the accuracy of electronic structure calculations, provided that they have been properly trained and validated using a suitable set of reference data. Due to their highly flexible functional form, the construction of MLPs has to be done with great care. In this Tutorial, we describe the necessary key steps for training reliable MLPs, from data generation via training to final validation. The procedure, which is illustrated for the example of a high-dimensional neural network potential, is general and applicable to many types of MLPs.

Elucidating the Role of Hydrogen Bonding in the Optical Spectroscopy of the Solvated Green Fluorescent Protein Chromophore: Using Machine Learning to Establish the Importance of High-Level Electronic Structure

Hydrogen bonding interactions with chromophores in chemical and biological environments play a key role in determining their electronic absorption and relaxation processes, which are manifested in their linear and multidimensional optical spectra. For chromophores in the condensed phase, the large number of atoms needed to simulate the environment has traditionally prohibited the use of high-level excited-state electronic structure methods. By leveraging transfer learning, we show how to construct machine-learned models to accurately predict the high-level excitation energies of a chromophore in solution from only 400 high-level calculations. We show that when the electronic excitations of the green fluorescent protein chromophore in water are treated using EOM-CCSD embedded in a DFT description of the solvent the optical spectrum is correctly captured and that this improvement arises from correctly treating the coupling of the electronic transition to electric fields, which leads to a larger response upon hydrogen bonding between the chromophore and water.

Hybrid Quantum Mechanical/Molecular Mechanical Methods For Studying Energy Transduction in Biomolecular Machines

Hybrid quantum mechanical/molecular mechanical (QM/MM) methods have become indispensable tools for the study of biomolecules. In this article, we briefly review the basic methodological details of QM/MM approaches and discuss their applications to various energy transduction problems in biomolecular machines, such as long-range proton transports, fast electron transfers, and mechanochemical coupling. We highlight the particular importance for these applications of balancing computational efficiency and accuracy. Using several recent examples, we illustrate the value and limitations of QM/MM methodologies for both ground and excited states, as well as strategies for calibrating them in specific applications. We conclude with brief comments on several areas that can benefit from further efforts to make QM/MM analyses more quantitative and applicable to increasingly complex biological problems.

Recent progress in atomistic modeling of light-harvesting complexes: a mini review

In this mini review, we focus on recent advances in the atomistic modeling of biological light-harvesting (LH) complexes. Because of their size and sophisticated electronic structures, multiscale methods are required to investigate the dynamical and spectroscopic properties of such complexes. The excitation energies, in this context also known as site energies, excitonic couplings, and spectral densities are key quantities which usually need to be extracted to be able to determine the exciton dynamics and spectroscopic properties. The recently developed multiscale approach based on the numerically efficient density functional tight-binding framework followed by excited state calculations has been shown to be superior to the scheme based on pure classical molecular dynamics simulations. The enhanced approach, which improves the description of the internal vibrational dynamics of the pigment molecules, yields spectral densities in good agreement with the experimental counterparts for various bacterial and plant LH systems. Here, we provide a brief overview of those results and described the theoretical foundation of the multiscale protocol.

Machine Learning Exciton Hamiltonians in Light-Harvesting Complexes

We propose a machine learning (ML)-based strategy for an inexpensive calculation of excitonic properties of light-harvesting complexes (LHCs). The strategy uses classical molecular dynamics simulations of LHCs in their natural environment in combination with ML prediction of the excitonic Hamiltonian of the embedded aggregate of pigments. The proposed ML model can reproduce the effects of geometrical fluctuations together with those due to electrostatic and polarization interactions between the pigments and the protein. The training is performed on the chlorophylls of the major LHC of plants, but we demonstrate that the model is able to extrapolate well beyond the initial training set. Moreover, the accuracy in predicting the effects of the environment is tested on the simulation of the small changes observed in the absorption spectra of the wild-type and a mutant of a minor LHC.

Great Influence of Organic Cation Motion on Charge Carrier Dynamics in Metal Halide Perovskite Unraveled by Unsupervised Machine Learning

Unsupervised machine learning combined with time-dependent density functional theory reveals the significant influence of organic cation on the charge carrier lifetime of FAPbI₃ (FA = HC(NH₂)₂⁺) by analyzing their mutual information (MI) between the geometric features and the nonadiabatic coupling (NAC) and bandgap. Analysis of MI values demonstrates that the NAC and bandgap are dominated by the orientation and shape of the inorganic octahedron because iodine and lead atoms are composed of the band edge states. Counterintuitively, the correlated motion promotes the contribution of the FA cation to the NAC; in particular, one type of FA rotation even supersedes the influence of the velocities of the lead and iodine atoms due to the enhanced hydrogen bond interaction. Our study demonstrates the importance of the correlated motion on the excited-state lifetimes of FAPbI₃, which provides a guidance for optimizing the optoelectronic properties of metal halide perovskites.

Mapping Simulated Two-Dimensional Spectra to Molecular Models Using Machine Learning

Two-dimensional (2D) spectroscopy encodes molecular properties and dynamics into expansive spectral data sets. Translating these data into meaningful chemical insights is challenging because of the many ways chemical properties can influence the spectra. To address the task of extracting chemical information from 2D spectroscopy, we study the capacity of simple feedforward neural networks (NNs) to map simulated 2D electronic spectra to underlying physical Hamiltonians. We examined hundreds of simulated 2D spectra corresponding to monomers and dimers with varied Franck-Condon active vibrations and monomer-monomer electronic couplings. We find the NNs are able to correctly characterize most Hamiltonian parameters in this study with an accuracy above 90%. Our results demonstrate that NNs can aid in interpreting 2D spectra, leading from spectroscopic features to underlying effective Hamiltonians.

2D spectroscopies from condensed phase dynamics: Accessing third-order response properties from equilibrium multi-time correlation functions

The third-order response lies at the heart of simulating and interpreting nonlinear spectroscopies ranging from two dimensional infrared (2D-IR) to 2D electronic (2D-ES), and 2D sum frequency generation (2D-SFG). The extra time and frequency dimensions in these spectroscopies provides access to rich information on the electronic and vibrational states present, the coupling between them, and the resulting rates at which they exchange energy that are obscured in linear spectroscopy, particularly for condensed phase systems that usually contain many overlapping features. While the exact quantum expression for the third-order response is well established it is incompatible with the methods that are practical for calculating the atomistic dynamics of large condensed phase systems. These methods, which include both classical mechanics and quantum dynamics methods that retain quantum statistical properties while obeying the symmetries of classical dynamics, such as LSC-IVR, Centroid Molecular Dynamics (CMD) and Ring Polymer Molecular Dynamics (RPMD) naturally provide short-time approximations to the multi-time symmetrized Kubo transformed correlation function. Here, we show how the third-order response can be formulated in terms of equilibrium symmetrized Kubo transformed correlation functions. We demonstrate the utility and accuracy of our approach by showing how it can be used to obtain the third-order response of a series of model systems using both classical dynamics and RPMD. In particular, we show that this approach captures features such as anharmonically induced vertical splittings and peak shifts while providing a physically transparent framework for understanding multidimensional spectroscopies.

UV-Visible Absorption Spectra of Solvated Molecules by Quantum Chemical Machine Learning

Predicting UV-visible absorption spectra is essential to understand photochemical processes and design energy materials. Quantum chemical methods can deliver accurate calculations of UV-visible absorption spectra, but they are computationally expensive, especially for large systems or when one computes line shapes from thermal averages. Here, we present an approach to predict UV-visible absorption spectra of solvated aromatic molecules by quantum chemistry (QC) and machine learning (ML). We show that a ML model, trained on the high-level QC calculation of the excitation energy of a set of aromatic molecules, can accurately predict the line shape of the lowest-energy UV-visible absorption band of several related molecules with less than 0.1 eV deviation with respect to reference experimental spectra. Applying linear decomposition analysis on the excitation energies, we unveil that our ML models probe vertical excitations of these aromatic molecules primarily by learning the atomic environment of their phenyl rings, which align with the physical origin of the π →π* electronic transition. Our study provides an effective workflow that combines ML with quantum chemical methods to accelerate the calculations of UV-visible absorption spectra for various molecular systems.

The atomistic modeling of light-harvesting complexes from the physical models to the computational protocol

The function of light-harvesting complexes is determined by a complex network of dynamic interactions among all the different components: the aggregate of pigments, the protein, and the surrounding environment. Complete and reliable predictions on these types of composite systems can be only achieved with an atomistic description. In the last few decades, there have been important advances in the atomistic modeling of light-harvesting complexes. These advances have involved both the completeness of the physical models and the accuracy and effectiveness of the computational protocols. In this Perspective, we present an overview of the main theoretical and computational breakthroughs attained so far in the field, with particular focus on the important role played by the protein and its dynamics. We then discuss the open problems in their accurate modeling that still need to be addressed. To illustrate an effective computational workflow for the modeling of light harvesting complexes, we take as an example the plant antenna complex CP29 and its H111N mutant.

Multi-fidelity prediction of molecular optical peaks with deep learning

Optical properties are central to molecular design for many applications, including solar cells and biomedical imaging. A variety of ab initio and statistical methods have been developed for their prediction, each with a trade-off between accuracy, generality, and cost. Existing theoretical methods such as time-dependent density functional theory (TD-DFT) are generalizable across chemical space because of their robust physics-based foundations but still exhibit random and systematic errors with respect to experiment despite their high computational cost. Statistical methods can achieve high accuracy at a lower cost, but data sparsity and unoptimized molecule and solvent representations often limit their ability to generalize. Here, we utilize directed message passing neural networks (D-MPNNs) to represent both dye molecules and solvents for predictions of molecular absorption peaks in solution. Additionally, we demonstrate a multi-fidelity approach based on an auxiliary model trained on over 28 000 TD-DFT calculations that further improves accuracy and generalizability, as shown through rigorous splitting strategies. Combining several openly-available experimental datasets, we benchmark these methods against a state-of-the-art regression tree algorithm and compare the D-MPNN solvent representation to several alternatives. Finally, we explore the interpretability of the learned representations using dimensionality reduction and evaluate the use of ensemble variance as an estimator of the epistemic uncertainty in our predictions of molecular peak absorption in solution. The prediction methods proposed herein can be integrated with active learning, generative modeling, and experimental workflows to enable the more rapid design of molecules with targeted optical properties.

Computational spectroscopy of complex systems

Numerous linear and non-linear spectroscopic techniques have been developed to elucidate structural and functional information of complex systems ranging from natural systems, such as proteins and light-harvesting systems, to synthetic systems, such as solar cell materials and light-emitting diodes. The obtained experimental data can be challenging to interpret due to the complexity and potential overlapping spectral signatures. Therefore, computational spectroscopy plays a crucial role in the interpretation and understanding of spectral observables of complex systems. Computational modeling of various spectroscopic techniques has seen significant developments in the past decade, when it comes to the systems that can be addressed, the size and complexity of the sample types, the accuracy of the methods, and the spectroscopic techniques that can be addressed. In this Perspective, I will review the computational spectroscopy methods that have been developed and applied for infrared and visible spectroscopies in the condensed phase. I will discuss some of the questions that this has allowed answering. Finally, I will discuss current and future challenges and how these may be addressed.

Artificial Neural Networks as Propagators in Quantum Dynamics

The utilization of artificial neural networks (ANNs) provides strategies for accelerating molecular simulations. Herein, ANNs are implemented as propagators of the time-dependent Schrödinger equation to simulate the quantum dynamics of systems with time-dependent potentials. These ANN propagators are trained to map nonstationary wavepackets from a given time to a future time within the discrete variable representation. Each propagator is trained for a specified time step, and iterative application of the propagator enables the propagation of wavepackets over long time scales. Such ANN propagators are developed and applied to one- and two-dimensional proton transfer systems, which exhibit nuclear quantum effects such as hydrogen tunneling. These ANN propagators are trained for either a specific time-independent potential or general potentials that can be time-dependent. Hierarchical, multiple time step algorithms enable parallelization, and the extension to higher dimensions is straightforward. This strategy is applicable to quantum dynamical simulations of diverse chemical and biological processes.

Influence of non-adiabatic effects on linear absorption spectra in the condensed phase: Methylene blue

Modeling linear absorption spectra of solvated chromophores is highly challenging as contributions are present both from coupling of the electronic states to nuclear vibrations and from solute-solvent interactions. In systems where excited states intersect in the Condon region, significant non-adiabatic contributions to absorption line shapes can also be observed. Here, we introduce a robust approach to model linear absorption spectra accounting for both environmental and non-adiabatic effects from first principles. This model parameterizes a linear vibronic coupling (LVC) Hamiltonian directly from energy gap fluctuations calculated along molecular dynamics (MD) trajectories of the chromophore in solution, accounting for both anharmonicity in the potential and direct solute-solvent interactions. The resulting system dynamics described by the LVC Hamiltonian are solved exactly using the thermalized time-evolving density operator with orthogonal polynomials algorithm (T-TEDOPA). The approach is applied to the linear absorption spectrum of methylene blue in water. We show that the strong shoulder in the experimental spectrum is caused by vibrationally driven population transfer between the bright S₁ and the dark S₂ states. The treatment of the solvent environment is one of many factors that strongly influence the population transfer and line shape; accurate modeling can only be achieved through the use of explicit quantum mechanical solvation. The efficiency of T-TEDOPA, combined with LVC Hamiltonian parameterizations from MD, leads to an attractive method for describing a large variety of systems in complex environments from first principles.

Machine Learning for Electronically Excited States of Molecules

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.

Machine Learning for Electronically Excited States of Molecules

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.

Strategies for the construction of machine-learning potentials for accurate and efficient atomic-scale simulations

Recent advances in machine-learning interatomic potentials have enabled the efficient modeling of complex atomistic systems with an accuracy that is comparable to that of conventional quantum-mechanics based methods. At the same time, the construction of new machine-learning potentials can seem a daunting task, as it involves data-science techniques that are not yet common in chemistry and materials science. Here, we provide a tutorial-style overview of strategies and best practices for the construction of artificial neural network (ANN) potentials. We illustrate the most important aspects of (a) data collection, (b) model selection, (c) training and validation, and (d) testing and refinement of ANN potentials on the basis of practical examples. Current research in the areas of active learning and delta learning are also discussed in the context of ANN potentials. This tutorial review aims at equipping computational chemists and materials scientists with the required background knowledge for ANN potential construction and application, with the intention to accelerate the adoption of the method, so that it can facilitate exciting research that would otherwise be challenging with conventional strategies.

Molecular excited states through a machine learning lens

Theoretical simulations of electronic excitations and associated processes in molecules are indispensable for fundamental research and technological innovations. However, such simulations are notoriously challenging to perform with quantum mechanical methods. Advances in machine learning open many new avenues for assisting molecular excited-state simulations. In this Review, we track such progress, assess the current state of the art and highlight the critical issues to solve in the future. We overview a broad range of machine learning applications in excited-state research, which include the prediction of molecular properties, improvements of quantum mechanical methods for the calculations of excited-state properties and the search for new materials. Machine learning approaches can help us understand hidden factors that influence photo-processes, leading to a better control of such processes and new rules for the design of materials for optoelectronic applications.

Vibronic and Environmental Effects in Simulations of Optical Spectroscopy

Including both environmental and vibronic effects is important for accurate simulation of optical spectra, but combining these effects remains computationally challenging. We outline two approaches that consider both the explicit atomistic environment and the vibronic transitions. Both phenomena are responsible for spectral shapes in linear spectroscopy and the electronic evolution measured in nonlinear spectroscopy. The first approach utilizes snapshots of chromophore-environment configurations for which chromophore normal modes are determined. We outline various approximations for this static approach that assumes harmonic potentials and ignores dynamic system-environment coupling. The second approach obtains excitation energies for a series of time-correlated snapshots. This dynamic approach relies on the accurate truncation of the cumulant expansion but treats the dynamics of the chromophore and the environment on equal footing. Both approaches show significant potential for making strides toward more accurate optical spectroscopy simulations of complex condensed phase systems.

Explicit environmental and vibronic effects in simulations of linear and nonlinear optical spectroscopy

Accurately simulating the linear and nonlinear electronic spectra of condensed phase systems and accounting for all physical phenomena contributing to spectral line shapes presents a significant challenge. Vibronic transitions can be captured through a harmonic model generated from the normal modes of a chromophore, but it is challenging to also include the effects of specific chromophore-environment interactions within such a model. We work to overcome this limitation by combining approaches to account for both explicit environment interactions and vibronic couplings for simulating both linear and nonlinear optical spectra. We present and show results for three approaches of varying computational cost for combining ensemble sampling of chromophore-environment configurations with Franck-Condon line shapes for simulating linear spectra. We present two analogous approaches for nonlinear spectra. Simulated absorption spectra and two-dimensional electronic spectra (2DES) are presented for the Nile red chromophore in different solvent environments. Employing an average Franck-Condon or 2DES line shape appears to be a promising method for simulating linear and nonlinear spectroscopy for a chromophore in the condensed phase.

Machine Learning for Absorption Cross Sections

We present a machine learning (ML) method to accelerate the nuclear ensemble approach (NEA) for computing absorption cross sections. ML-NEA is used to calculate cross sections on vast ensembles of nuclear geometries to reduce the error due to insufficient statistical sampling. The electronic properties-excitation energies and oscillator strengths-are calculated with a reference electronic structure method only for a relatively few points in the ensemble. The KREG model (kernel-ridge-regression-based ML combined with the RE descriptor) as implemented in MLatom is used to predict these properties for the remaining tens of thousands of points in the ensemble without incurring much of additional computational cost. We demonstrate for two examples, benzene and a 9-dicyanomethylene derivative of acridine, that ML-NEA can produce statistically converged cross sections even for very challenging cases and even with as few as several hundreds of training points.