A new diabatization scheme for direct quantum dynamics: Procrustes diabatization

Representation and conservation of angular momentum in the Born-Oppenheimer theory of polyatomic molecules

This paper concerns the representation of angular momentum operators in the Born-Oppenheimer theory of polyatomic molecules and the various forms of the associated conservation laws. Topics addressed include the question of whether these conservation laws are exactly equivalent or only to some order of the Born-Oppenheimer parameter κ = (m/M)^1/4 and what the correlation is between angular momentum quantum numbers in the various representations. These questions are addressed in both problems involving a single potential energy surface and those with multiple, strongly coupled surfaces and in both the electrostatic model and those for which fine structure and electron spin are important. The analysis leads to an examination of the transformation laws under rotations of the electronic Hamiltonian; of the basis states, both adiabatic and diabatic, along with their phase conventions; of the potential energy matrix; and of the derivative couplings. These transformation laws are placed in the geometrical context of the structures in the nuclear configuration space that are induced by rotations, which include the rotational orbits or fibers, the surfaces upon which the orientation of the molecule changes but not its shape, and the section, an initial value surface that cuts transversally through the fibers. Finally, it is suggested that the usual Born-Oppenheimer approximation can be replaced by a dressing transformation, that is, a sequence of unitary transformations that block-diagonalize the Hamiltonian. When the dressing transformation is carried out, we find that the angular momentum operator does not change. This is a part of a system of exact equivalences among various representations of angular momentum operators in Born-Oppenheimer theory. Our analysis accommodates large-amplitude motions and is not dependent on small-amplitude expansions about an equilibrium position. Our analysis applies to noncollinear configurations of a polyatomic molecule; this covers all but a subset of measure zero (the collinear configurations) in the nuclear configuration space.

Diabatic States of Molecules

Quantitative simulations of electronically nonadiabatic molecular processes require both accurate dynamics algorithms and accurate electronic structure information. Direct semiclassical nonadiabatic dynamics is expensive due to the high cost of electronic structure calculations, and hence it is limited to small systems, limited ensemble averaging, ultrafast processes, and/or electronic structure methods that are only semiquantitatively accurate. The cost of dynamics calculations can be made manageable if analytic fits are made to the electronic structure data, and such fits are most conveniently carried out in a diabatic representation because the surfaces are smooth and the couplings between states are smooth scalar functions. Diabatic representations, unlike the adiabatic ones produced by most electronic structure methods, are not unique, and finding suitable diabatic representations often involves time-consuming nonsystematic diabatization steps. The biggest drawback of using diabatic bases is that it can require large amounts of effort to perform a globally consistent diabatization, and one of our goals has been to develop methods to do this efficiently and automatically. In this Feature Article, we introduce the mathematical framework of diabatic representations, and we discuss diabatization methods, including adiabatic-to-diabatic transformations and recent progress toward the goal of automatization.

Path Integrals for Nonadiabatic Dynamics: Multistate Ring Polymer Molecular Dynamics

This review focuses on a recent class of path-integral-based methods that simulate nonadiabatic dynamics in the condensed phase using only classical molecular dynamics trajectories in an extended phase space. Specifically, a semiclassical mapping protocol is used to derive an exact, continuous, Cartesian variable path-integral representation for the canonical partition function of a system in which multiple electronic states are coupled to nuclear degrees of freedom. Building on this exact statistical foundation, multistate ring polymer molecular dynamics methods are developed for the approximate calculation of real-time thermal correlation functions. The remarkable promise of these multistate ring polymer methods, their successful applications, and their limitations are discussed in detail.Expected final online publication date for the Annual Review of Physical Chemistry, Volume 73 is April 2022. Please see http://www.annualreviews.org/page/journal/pubdates for revised estimates.

Predicting Molecular Photochemistry Using Machine-Learning-Enhanced Quantum Dynamics Simulations

The processes which occur after molecules absorb light underpin an enormous range of fundamental technologies and applications, including photocatalysis to enable new chemical transformations, sunscreens to protect against the harmful effects of UV overexposure, efficient photovoltaics for energy generation from sunlight, and fluorescent probes to image the intricate details of complex biomolecular structures. Reflecting this broad range of applications, an enormously versatile set of experiments are now regularly used to interrogate light-driven chemical dynamics, ranging from the typical ultrafast transient absorption spectroscopy used in many university laboratories to the inspiring central facilities around the world, such as the next-generation of X-ray free-electron lasers.Computer simulations of light-driven molecular and material dynamics are an essential route to analyzing the enormous amount of transient electronic and structural data produced by these experimental sources. However, to date, the direct simulation of molecular photochemistry remains a frontier challenge in computational chemical science, simultaneously demanding the accurate treatment of molecular electronic structure, nuclear dynamics, and the impact of nonadiabatic couplings.To address these important challenges and to enable new computational methods which can be integrated with state-of-the-art experimental capabilities, the past few years have seen a burst of activity in the development of "direct" quantum dynamics methods, merging the machine learning of potential energy surfaces (PESs) and nonadiabatic couplings with accurate quantum propagation schemes such as the multiconfiguration time-dependent Hartree (MCTDH) method. The result of this approach is a new generation of direct quantum dynamics tools in which PESs are generated in tandem with wave function propagation, enabling accurate "on-the-fly" simulations of molecular photochemistry. These simulations offer an alternative route toward gaining quantum dynamics insights, circumventing the challenge of generating ab initio electronic structure data for PES fitting by instead only demanding expensive energy evaluations as and when they are needed.In this Account, we describe the chronological evolution of our own contributions to this field, focusing on describing the algorithmic developments that enable direct MCTDH simulations for complex molecular systems moving on multiple coupled electronic states. Specifically, we highlight active learning strategies for generating PESs during grid-based quantum chemical dynamics simulations, and we discuss the development and impact of novel diabatization schemes to enable direct grid-based simulations of photochemical dynamics; these developments are highlighted in a series of benchmark molecular simulations of systems containing multiple nuclear degrees of freedom moving on multiple coupled electronic states. We hope that the ongoing developments reported here represent a major step forward in tools for modeling excited-state chemistry such as photodissociation, proton and electron transfer, and ultrafast energy dissipation in complex molecular systems.

Analyzing Grid-Based Direct Quantum Molecular Dynamics Using Non-Linear Dimensionality Reduction

Grid-based schemes for simulating quantum dynamics, such as the multi-configuration time-dependent Hartree (MCTDH) method, provide highly accurate predictions of the coupled nuclear and electronic dynamics in molecular systems. Such approaches provide a multi-dimensional, time-dependent view of the system wavefunction represented on a coordinate grid; in the case of non-adiabatic simulations, additional information about the state populations adds a further layer of complexity. As such, wavepacket motion on potential energy surfaces which couple many nuclear and electronic degrees-of-freedom can be extremely challenging to analyse in order to extract physical insight beyond the usual expectation-value picture. Here, we show that non-linear dimensionality reduction (NLDR) methods, notably diffusion maps, can be adapted to extract information from grid-based wavefunction dynamics simulations, providing insight into key nuclear motions which explain the observed dynamics. This approach is demonstrated for 2-D and 9-D models of proton transfer in salicylaldimine, as well as 8-D and full 12-D simulations of cis-trans isomerization in ethene; these simulations demonstrate how NLDR can provide alternative views of wavefunction dynamics, and also highlight future developments.

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.

Perspective on integrating machine learning into computational chemistry and materials science

Machine learning (ML) methods are being used in almost every conceivable area of electronic structure theory and molecular simulation. In particular, ML has become firmly established in the construction of high-dimensional interatomic potentials. Not a day goes by without another proof of principle being published on how ML methods can represent and predict quantum mechanical properties-be they observable, such as molecular polarizabilities, or not, such as atomic charges. As ML is becoming pervasive in electronic structure theory and molecular simulation, we provide an overview of how atomistic computational modeling is being transformed by the incorporation of ML approaches. From the perspective of the practitioner in the field, we assess how common workflows to predict structure, dynamics, and spectroscopy are affected by ML. Finally, we discuss how a tighter and lasting integration of ML methods with computational chemistry and materials science can be achieved and what it will mean for research practice, software development, and postgraduate training.

Machine Learning-Assisted Excited State Molecular Dynamics with the State-Interaction State-Averaged Spin-Restricted Ensemble-Referenced Kohn-Sham Approach

We present a machine learning-assisted excited state molecular dynamics (ML-ESMD) based on the ensemble density functional theory framework. Since we represent a diabatic Hamiltonian in terms of generalized valence bond ansatz within the state-interaction state-averaged spin-restricted ensemble-referenced Kohn-Sham (SI-SA-REKS) method, we can avoid singularities near conical intersections, which are crucial in excited state molecular dynamics simulations. We train the diabatic Hamiltonian elements and their analytical gradients with the SchNet architecture to construct machine learning models, while the phase freedom of off-diagonal elements of the Hamiltonian is cured by introducing the phase-less loss function. Our machine learning models show reasonable accuracy with mean absolute errors of ∼0.1 kcal/mol and ∼0.5 kcal/mol/Å for the diabatic Hamiltonian elements and their gradients, respectively, for penta-2,4-dieniminium cation. Moreover, by exploiting the diabatic representation, our models can predict correct conical intersection structures and their topologies. In addition, our ML-ESMD simulations give almost identical result with a direct dynamics at the same level of theory.