These authors contributed equally to this work. Abstract The accurate quantum chemical calculation of excited states is a challenging task, often requiring computationally demanding methods. When entire ground and excited potential energy surfaces (PESs) are desired, e.g., to predict the interaction of light excitation and structural changes, one is often forced to use cheaper computational methods at the cost of reduced accuracy. Here we introduce a novel method for the geometrically transferable optimization of neural network wave functions that leverages weight sharing and dynamical ordering of electronic states. Our method enables the efficient prediction of ground and excited-state PESs and their intersections at the highest accuracy, demonstrating up to two orders of magnitude cost reduction compared to single-point calculations. We validate our approach on three challenging excited-state PESs, including ethylene, the carbon dimer, and the methylenimmonium cation, indicating that transferable deep-learning QMC can pave the way towards highly accurate simulation of excited-state dynamics. Light-driven phenomena are also key to technological advancements, ranging from material design and chemical processing [4, 5] to biomedical technologies such as molecular motors and photo-controlled drug delivery [6, 7]. Despite the critical importance of these processes, their theoretical study is hindered by the need for accurate ab-initio descriptions of electronic excited states. Most quantum chemistry methods have been developed for the calculation of electronic ground states and their extensions to excited states are either limited or highly expensive and often require expert knowledge [8, 9].

We present a new nonlinear variational framework for simultaneously computing ground and excited states of quantum systems. Our approach is based on approximating wavefunctions in the linear span of basis functions that are augmented and optimized \emph{via} composition with normalizing flows. The accuracy and efficiency of our approach are demonstrated in the calculations of a large number of vibrational states of the triatomic H$_2$S molecule as well as ground and several excited electronic states of prototypical one-electron systems including the hydrogen atom, the molecular hydrogen ion, and a carbon atom in a single-active-electron approximation. The results demonstrate significant improvements in the accuracy of energy predictions and accelerated basis-set convergence even when using normalizing flows with a small number of parameters. The present approach can be also seen as the optimization of a set of intrinsic coordinates that best capture the underlying physics within the given basis set.

If we want to calculate the physical properties of matter (such as the electronic state), we need to describe the state of the electron. The equations of motion that we are familiar with cannot describe the states of small objects such as electrons, so we need to use something called quantum mechanics. In quantum mechanics, the state of an electron is described by a complex function, the "wave function". "wave function" is, roughly speaking, electrons' orbitals. The equation below, called the Schrödinger equation, is a basic equation in quantum mechanics that shows the relationship between the wave function and the energy (Hamiltonian, H-hat on the right side), where ψ is the wave function.

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.

Abstract: Photo-inducedprocesses 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. Introduction Machine learning (ML) is revolutionizing the most diverse domains, like image recognition [1], playing board games [2], or society integration of refugees [3]. Also in chemistry, anincreasing range of applications is being tackled with ML, for example, the design and discovery of new molecules and materials [4, 5, 6]. In the present study, we show how ML enables efficient photodynamics simulations. Photodynamics is the study of photo-induced processes that occur after a molecule is exposed to light. Photosynthesis or DNA photodamage leading to skin cancer are only two examples of phenomena that involve molecules interacting with light [7, 8, 9, 10, 11]. The simulation of such processes has been key to learn structure-dynamicsfunction relationshipsthat can be used to guide the design of photonic materials, such as photosensitive drugs [12], photocatalysts [4] and photovoltaics [13, 14].