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We explore long lived electronic coherences in molecules using shaped ultrafast laser pulses to launch and probe entangled nuclear-electronic wave packets. We find that under certain conditions, the electronic phase remains well defined despite vibrational motion along many degrees of freedom. The experiments are interpreted with the help of electronic structure calculations which corroborate our interpretation of the measurements

We study alchemical atomic energy partitioning as a method to estimate atomisation energies from atomic contributions which are defined in physically rigorous and general ways through use of the uniform electron gas as a joint reference. We analyze quantitatively the relation between atomic energies and their local environment using a dataset of 1325 organic molecules. The atomic energies are transferable across various molecules, enabling the prediction of atomisation energies with a mean absolute error of 20 kcal/mol - comparable to simple statistical estimates but potentially more robust given their grounding in the physics-based decomposition scheme. A comparative analysis with other decomposition methods highlights its sensitivity to electrostatic variations, underlining its potential as representation of the environment as well as in studying processes like diffusion in solids characterized by significant electrostatic shifts.

We present a rare event sampling scheme applicable to coupled electronic excited states. In particular, we extend the forward flux sampling (FFS) method for rare event sampling to a nonadiabatic version (NAFFS) that uses the trajectory surface hopping (TSH) method for nonadiabatic dynamics. NAFFS is applied to two dynamically relevant excited-state models that feature an avoided crossing and a conical intersection with tunable parameters. We investigate how nonadiabatic couplings, temperature, and reaction barriers aspect transition rate constants in regimes that cannot be otherwise obtained with plain, traditional TSH. The comparison with reference brute-force TSH simulations for limiting cases of rareness shows that NAFFS can be several orders of magnitude cheaper than conventional TSH, and thus represents a conceptually novel tool to extend excited-state dynamics to time scales that are able to capture rare nonadiabatic events., Comment: 13 pages, 7 figures, submitted to Chemical Science

Finding accurate solutions to the Schr\"odinger equation is the key unsolved challenge of computational chemistry. Given its importance for the development of new chemical compounds, decades of research have been dedicated to this problem, but due to the large dimensionality even the best available methods do not yet reach the desired accuracy. Recently the combination of deep learning with Monte Carlo methods has emerged as a promising way to obtain highly accurate energies and moderate scaling of computational cost. In this paper we significantly contribute towards this goal by introducing a novel deep-learning architecture that achieves 40-70% lower energy error at 6x lower computational cost compared to previous approaches. Using our method we establish a new benchmark by calculating the most accurate variational ground state energies ever published for a number of different atoms and molecules. We systematically break down and measure our improvements, focusing in particular on the effect of increasing physical prior knowledge. We surprisingly find that increasing the prior knowledge given to the architecture can actually decrease accuracy., Comment: 10 pages + apppendix, 7 figures; V2: minor corrections to citations and reference energies for F, Ne, H2O; V3: final version for NEURIPS

Nonadiabatic ab initio molecular dynamics (MD) including spin-orbit coupling (SOC) and laser fields is investigated as a general tool for studies of excited-state processes. Up to now, SOCs are not included in standard ab initio MD packages. Therefore, transitions to triplet states cannot be treated in a straightforward way. Nevertheless, triplet states play an important role in a large variety of systems and can now be treated within the given framework. The laser interaction is treated on a non-perturbative level that allows nonlinear effects like strong Stark shifts to be considered. As MD allows for the handling of many atoms, the interplay between triplet and singlet states of large molecular systems will be accessible. In order to test the method, IBr is taken as a model system, where SOC plays a crucial role for the shape of the potential curves and thus the dynamics. Moreover, the influence of the nonresonant dynamic Stark effect is considered. The latter is capable of controlling reaction barriers by electric fields in timereversible conditions, and thus a control laser using this effect acts like a photonic catalyst. In the IBr molecule, the branching ratio at an avoided crossing, which arises from SOC, can be influenced.

Hybrid quantum mechanics/molecular mechanics (QM/MM) simulations have advanced the field of computational chemistry tremendously. However, they require the partitioning of a system into two different regions that are treated at different levels of theory, which can cause artefacts at the interface. Furthermore, they are still limited by high computational costs of quantum chemical calculations. In this work, we develop BuRNN, an alternative approach to existing QM/MM schemes, which introduces a buffer region that experiences full electronic polarization by the inner QM region to minimize artefacts. The interactions between the QM and the buffer region are described by deep neural networks (NNs), which leads to high computational efficiency of this hybrid NN/MM scheme while retaining quantum chemical accuracy. We demonstrate the BuRNN approach by performing NN/MM simulations of the hexa-aqua iron complex., Comment: 37 pages, 9 figures

Although the amino acid tyrosine is among the main building blocks of life, its photochemistry is not fully understood. Traditional theoretical simulations are neither accurate enough, nor computationally efficient to provide the missing puzzle pieces to the experimentally observed signatures obtained via time-resolved pump-probe spectroscopy or mass spectroscopy. In this work, we go beyond the realms of possibility with conventional quantum chemical methods and develop as well as apply a new technique to shed light on the photochemistry of tyrosine. By doing so, we discover roaming atoms in tyrosine, which is the first time such a reaction is discovered in biology. Our findings suggest that roaming atoms are radicals that could play a fundamental role in the photochemistry of peptides and proteins, offering a new perspective. Our novel method is based on deep learning, leverages the physics underlying the data, and combines different levels of theory. This combination of methods to obtain an accurate picture of excited molecules could shape how we study photochemical systems in the future and how we can overcome the current limitations that we face when applying quantum chemical methods., Comment: 22 pages, 18 figures, 6 tables

Accurate numerical solutions for the Schr\"odinger equation are of utmost importance in quantum chemistry. However, the computational cost of current high-accuracy methods scales poorly with the number of interacting particles. Combining Monte Carlo methods with unsupervised training of neural networks has recently been proposed as a promising approach to overcome the curse of dimensionality in this setting and to obtain accurate wavefunctions for individual molecules at a moderately scaling computational cost. These methods currently do not exploit the regularity exhibited by wavefunctions with respect to their molecular geometries. Inspired by recent successful applications of deep transfer learning in machine translation and computer vision tasks, we attempt to leverage this regularity by introducing a weight-sharing constraint when optimizing neural network-based models for different molecular geometries. That is, we restrict the optimization process such that up to 95 percent of weights in a neural network model are in fact equal across varying molecular geometries. We find that this technique can accelerate optimization when considering sets of nuclear geometries of the same molecule by an order of magnitude and that it opens a promising route towards pre-trained neural network wavefunctions that yield high accuracy even across different molecules.

An extended Designed regression analysis of experimental data on density and refractive indices of several classes of ionic liquids yielded statistically averaged atomic volumes and polarizabilities of the constituting atoms. These values can be used to predict the molecular volume and polarizability of an unknown ionic liquid as well as its mass density and refractive index. Our approach does not need information on the molecular structure of the ionic liquid, but it turned out that the discrimination of the hybridization state of the carbons improved the overall result. Our results are not only compared to experimental data but also to quantum-chemical calculations. Furthermore, fractional charges of ionic liquid ions and their relation to polarizability are discussed.

Hydrogen dissociation is an unwanted competing pathway if a torsional motion around the C=C double bond in a chiral fluoroethylene derivative, namely (4-methylcyclohexylidene) fluoromethane (4MCF), is to be achieved. We show that the excited state H-dissociation can be drastically diminished on timescales long enough to initiate a torsion around the C=C double bond using the non-resonant dynamic Stark effect. Potential energy curves, dipoles and polarizabilities for the regarded one-dimensional reaction coordinate are calculated within the CASSCF method. The influence of the excitation and the laser control field is then simulated using wavepacket dynamics.

The photoinduced excited-state dynamics of the keto and enol forms of cytosine is investigated using ab initio surface hopping in order to understand the outcome of molecular beam femtosecond pump-probe photoionization spectroscopy experiments. Both singlet and triplet states are included in the dynamics. The results show that triplet states play a significant role in the relaxation of the keto tautomer, while they are less important in the enol tautomer. In both forms, the T$_1$ state minimum is found too low in energy to be detected in standard photoionization spectroscopy experiments and therefore experimental decay times should arise from a simultaneous relaxation to the ground state and additional intersystem crossing followed by internal conversion to the T$_1$ state. In agreement with available experimental lifetimes, we observe three decay constants of 7 fs, 270 fs and 1900 fs - the first two coming from the keto tautomer and the longer one from the enol tautomer. Deactivation of the enol form is due to internal conversion to the ground state via two S$_1$/S$_0$ conical intersections of ethylenic type.

Many approaches, which have been developed to express the potential energy of large systems, exploit the locality of the atomic interactions. A prominent example are fragmentation methods, in which quantum chemical calculations are carried out for overlapping small fragments of a given molecule that are then combined in a second step to yield the system's total energy. Here we compare the accuracy of the systematic molecular fragmentation approach with the performance of high-dimensional neural network (HDNN) potentials introduced by Behler and Parrinello. HDNN potentials are similar in spirit to the fragmentation approach in that the total energy is constructed as a sum of environment-dependent atomic energies, which are derived indirectly from electronic structure calculations. As a benchmark set we use all-trans alkanes containing up to eleven carbon atoms at the coupled cluster level of theory. These molecules have been chosen because they allow to extrapolate reliable reference energies for very long chains, enabling an assessment of the energies obtained by both methods for alkanes including up to 10 000 carbon atoms. We find that both methods predict high-quality energies with the HDNN potentials yielding smaller errors with respect to the coupled cluster reference.

Ab initio molecular dynamics including non-adiabatic and spin-orbit couplings on equal footing is used to unravel the deactivation of cytosine after UV light absorption. Intersystem crossing (ISC) is found to compete directly with internal conversion in tens of femtoseconds, thus making cytosine the organic compound with the fastest triplet population calculated so far. It is found that close degeneracy between singlet and triplet states can more than compensate for very small spin-orbit couplings, leading to efficient ISC. The femtosecond nature of the intersystem crossing process highlights its importance in photochemistry and challenges the conventional view that large singlet-triplet couplings are required for an efficient population flow into triplet states. These findings are important to understand DNA photostability and the photochemistry and dynamics of organic molecules in general.

Machine learning (ML) has shown to advance the research field of quantum chemistry in almost any possible direction and has recently also entered the excited states to investigate the multifaceted photochemistry of molecules. In this paper, we pursue two goals: i) We show how ML can be used to model permanent dipole moments for excited states and transition dipole moments by adapting the charge model of [Chem. Sci., 2017, 8, 6924-6935], which was originally proposed for the permanent dipole moment vector of the electronic ground state. ii) We investigate the transferability of our excited-state ML models in chemical space, i.e., whether an ML model can predict properties of molecules that it has never been trained on and whether it can learn the different excited states of two molecules simultaneously. To this aim, we employ and extend our previously reported SchNarc approach for excited-state ML. We calculate UV absorption spectra from excited-state energies and transition dipole moments as well as electrostatic potentials from latent charges inferred by the ML model of the permanent dipole moment vectors. We train our ML models on CH$_2$NH$_2^+$ and C$_2$H$_4$, while predictions are carried out for these molecules and additionally for CHNH$_2$, CH$_2$NH, and C$_2$H$_5^+$. The results indicate that transferability is possible for the excited states.

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 how machine learning is employed not only 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, approaches for nonadiabatic dynamics simulations and describe tricks and problems when using them in machine learning for excited states of molecules.

Machine learning is employed at an increasing rate in the research field of quantum chemistry. While the majority of approaches target the investigation of chemical systems in their electronic ground state, the inclusion of light into the processes leads to electronically excited states and gives rise to several new challenges. Here, we survey recent advances for excited-state dynamics based on machine learning. In doing so, we highlight successes, pitfalls, challenges and future avenues for machine learning approaches for light-induced molecular processes.

This perspective on laser control provides a quick overview over different schemes for coherent control of chemical reactions and photophysical processes. It originally appeared in Bunsen Magazin 1, 13 - 23 (2012).

In recent years, deep learning has become a part of our everyday life and is revolutionizing quantum chemistry as well. In this work, we show how deep learning can be used to advance the research field of photochemistry by learning all important properties for photodynamics simulations. The properties are multiple energies, forces, nonadiabatic couplings and spin-orbit couplings. The nonadiabatic couplings are learned in a phase-free manner as derivatives of a virtually constructed property by the deep learning model, which guarantees rotational covariance. Additionally, an approximation for nonadiabatic couplings is introduced, based on the potentials, their gradients and Hessians. As deep-learning method, we employ SchNet extended for multiple electronic states. In combination with the molecular dynamics program SHARC, our approach termed SchNarc is tested on a model system and two realistic polyatomic molecules and paves the way towards efficient photodynamics simulations of complex systems.

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 {\emph 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.

For $\mathbb N^*:=\mathbb N \setminus \{0\}$, we consider the collection $\mathfrak M(N)$ of all the $N$ rows, for which, for $n=1,\cdots,N$, the $n-th$ row consists of an increasing sequence $(a_j^n)_j$ of real numbers. For $\mathfrak A \in \mathfrak M(N)$, we define its spectrum $\sigma(\mathfrak A)$ by $\sigma(\mathfrak A)=\{\lambda\in \mathbb R \;|\; \lambda=\sum_{n=1}^Na_{j_n}^n\}\,,$ where $(j_1,j_2,\dots,j_N)\in (\mathbb N^*)^N$. This spectrum is discrete and consists of an infinite sequence that can be ordered as a strictly increasing sequence $\lambda_k(\mathfrak A)$. For $\lambda \in \sigma (\mathfrak A)$ we denote by $m(\lambda,\mathfrak A) $ the number of representations of such a $\lambda$, hence the multiplicity of $\lambda$.\\ In this paper we investigate for given $N\in \mathbb N^*$ and $k\in \mathbb N^*$ the highest possible multiplicity (denoted by $\mathfrak m_k(N)$) of $\lambda_k(\mathfrak A)$ for $\mathfrak A \in \mathfrak M(N)$. We give the exact result for $N=2$ and for $N=3$ prove a lower bound which appears, according to numerical experiments, as a "good" conjecture. For the general case, we give examples demonstrating that the problem is quite difficult. \\ This problem is equivalent to the analogue eigenvalue multiplicity questions for Schr\"odinger operators describing a system of N non-interacting one-dimensional particles.

The gas phase relaxation dynamics of photoexcited 5-azacytosine has been investigated by means of SHARC (surface-hopping including arbitrary couplings) molecular dynamics, based on accurate multireference electronic structure computations. Both singlet and triplet states were included in the simulations in order to investigate the different internal conversion and intersystem crossing pathways of this molecule. It was found that after excitation, 5-azacytosine undergoes ultrafast relaxation to the electronic ground state with a time constant of about 1~picosecond. Two important conical intersections have been identified as the funnel responsible for this deactivation mechanism. The very low intersystem crossing yield of 5-azacytosine has been explained by the size of the relevant spin-orbit coupling matrix elements, which are significantly smaller than in related molecules like cytosine or 6-azauracil. This difference is due to the fact that in 5-azacytosine the lowest singlet state is of $n_\mathrm{N}\pi^*$ nature, whereas in cytosine and 6-azauracil it is of $n_\mathrm{O}\pi^*$ character., Comment: 9 pages, 7 figures, 1 table

The photodynamic properties of molecules determine their ability to survive in harsh radiation environments. As such, the photostability of heterocyclic aromatic compounds to electromagnetic radiation is expected to have been one of the selection pressures influencing the prebiotic chemistry on early Earth. In the present study, the gas-phase photodynamics of uracil, 5-methyluracil (thymine) and 2-thiouracil -- three heterocyclic compounds thought to be present during this era -- are assessed in the context of their recently proposed intersystem crossing pathways that compete with internal conversion to the ground state. Specifically, time-resolved photoelectron spectroscopy measurements evidence femtosecond to picosecond timescales for relaxation of the singlet 1$\pi\pi$* and 1n$\pi$* states as well as for intersystem crossing to the triplet manifold. Trapping in the excited triplet state and intersystem crossing back to the ground state are investigated as potential factors contributing to the susceptibility of these molecules to ultraviolet photodamage., Comment: 21 pages, 5 figures, 1 table

Molecular dynamics simulations are an important tool for describing the evolution of a chemical system with time. However, these simulations are inherently held back either by the prohibitive cost of accurate electronic structure theory computations or the limited accuracy of classical empirical force fields. Machine learning techniques can help to overcome these limitations by providing access to potential energies, forces and other molecular properties modeled directly after an electronic structure reference at only a fraction of the original computational cost. The present text discusses several practical aspects of conducting machine learning driven molecular dynamics simulations. First, we study the efficient selection of reference data points on the basis of an active learning inspired adaptive sampling scheme. This is followed by the analysis of a machine-learning based model for simulating molecular dipole moments in the framework of predicting infrared spectra via molecular dynamics simulations. Finally, we show that machine learning models can offer valuable aid in understanding chemical systems beyond a simple prediction of quantities.

Photo-induced processes are fundamental in nature, but accurate simulations are seriously limited by the cost of the underlying quantum chemical calculations, hampering their application for long time scales. Here we introduce a method based on machine learning to overcome this bottleneck and enable accurate photodynamics on nanosecond time scales, which are otherwise out of reach with contemporary approaches. Instead of expensive quantum chemistry during molecular dynamics simulations, we use deep neural networks to learn the relationship between a molecular geometry and its high-dimensional electronic properties. As an example, the time evolution of the methylenimmonium cation for one nanosecond is used to demonstrate that machine learning algorithms can outperform standard excited-state molecular dynamics approaches in their computational efficiency while delivering the same accuracy.

We introduce weighted atom-centered symmetry functions (wACSFs) as descriptors of a chemical system's geometry for use in the prediction of chemical properties such as enthalpies or potential energies via machine learning. The wACSFs are based on conventional atom-centered symmetry functions (ACSFs) but overcome the undesirable scaling of the latter with increasing number of different elements in a chemical system. The performance of these two descriptors is compared using them as inputs in high-dimensional neural network potentials (HDNNPs), employing the molecular structures and associated enthalpies of the 133855 molecules containing up to five different elements reported in the QM9 database as reference data. A substantially smaller number of wACSFs than ACSFs is needed to obtain a comparable spatial resolution of the molecular structures. At the same time, this smaller set of wACSFs leads to significantly better generalization performance in the machine learning potential than the large set of conventional ACSFs. Furthermore, we show that the intrinsic parameters of the descriptors can in principle be optimized with a genetic algorithm in a highly automated manner. For the wACSFs employed here, we find however that using a simple empirical parametrization scheme is sufficient in order to obtain HDNNPs with high accuracy.

Machine learning has emerged as an invaluable tool in many research areas. In the present work, we harness this power to predict highly accurate molecular infrared spectra with unprecedented computational efficiency. To account for vibrational anharmonic and dynamical effects -- typically neglected by conventional quantum chemistry approaches -- we base our machine learning strategy on ab initio molecular dynamics simulations. While these simulations are usually extremely time consuming even for small molecules, we overcome these limitations by leveraging the power of a variety of machine learning techniques, not only accelerating simulations by several orders of magnitude, but also greatly extending the size of systems that can be treated. To this end, we develop a molecular dipole moment model based on environment dependent neural network charges and combine it with the neural network potentials of Behler and Parrinello. Contrary to the prevalent big data philosophy, we are able to obtain very accurate machine learning models for the prediction of infrared spectra based on only a few hundreds of electronic structure reference points. This is made possible through the introduction of a fully automated sampling scheme and the use of molecular forces during neural network potential training. We demonstrate the power of our machine learning approach by applying it to model the infrared spectra of a methanol molecule, n-alkanes containing up to 200 atoms and the protonated alanine tripeptide, which at the same time represents the first application of machine learning techniques to simulate the dynamics of a peptide. In all these case studies we find excellent agreement between the infrared spectra predicted via machine learning models and the respective theoretical and experimental spectra., Comment: 12 pages, 9 figures

Mixed quantum-classical dynamics simulations show that intersystem crossing between singlet and triplet states in SO$_2$ and in nucleobases takes place on an ultrafast timescale (few 100~fs), directly competing with internal conversion., Comment: 4 pages, 2 figures

This Chapter is devoted to unravel the relaxation processes taking place after photoexcitation of isolated DNA/RNA nucleobases in gas phase from a time-dependent perspective. To this aim, several methods are at hand, ranging from full quantum dynamics to various flavours of semiclassical or ab initio molecular dynamics, each with its advantages and its limitations. As this contribution shows, the most common approach employed up-to-date to learn about the deactivation of nucleobases in gas phase is a combination of the Tully surface hopping algorithm with on-the-fly CASSCF calculations. Different methods or, even more dramatically, different electronic structure methods can provide different dynamics. A comprehensive review of the different mechanisms suggested for each nucleobase is provided and compared to available experimental time scales. The results are discussed in a general context involving the effects of the different applied electronic structure and dynamics methods. Mechanistic similarities and differences between the two groups of nucleobases---the purine derivatives (adenine and guanine) and the pyrimidine derivatives (thymine, uracil, and cytosine)---are elucidated. Finally, a perspective on the future of dynamics simulations in the context of nucleobase relaxation is given., Comment: 54 pages, 19 figures