Exact factorization of the time-dependent electron-nuclear wave function

Theoretical Advances in Polariton Chemistry and Molecular Cavity Quantum Electrodynamics

When molecules are coupled to an optical cavity, new light-matter hybrid states, so-called polaritons, are formed due to quantum light-matter interactions. With the experimental demonstrations of modifying chemical reactivities by forming polaritons under strong light-matter interactions, theorists have been encouraged to develop new methods to simulate these systems and discover new strategies to tune and control reactions. This review summarizes some of these exciting theoretical advances in polariton chemistry, in methods ranging from the fundamental framework to computational techniques and applications spanning from photochemistry to vibrational strong coupling. Even though the theory of quantum light-matter interactions goes back to the midtwentieth century, the gaps in the knowledge of molecular quantum electrodynamics (QED) have only recently been filled. We review recent advances made in resolving gauge ambiguities, the correct form of different QED Hamiltonians under different gauges, and their connections to various quantum optics models. Then, we review recently developed ab initio QED approaches which can accurately describe polariton states in a realistic molecule-cavity hybrid system. We then discuss applications using these method advancements. We review advancements in polariton photochemistry where the cavity is made resonant to electronic transitions to control molecular nonadiabatic excited state dynamics and enable new photochemical reactivities. When the cavity resonance is tuned to the molecular vibrations instead, ground-state chemical reaction modifications have been demonstrated experimentally, though its mechanistic principle remains unclear. We present some recent theoretical progress in resolving this mystery. Finally, we review the recent advances in understanding the collective coupling regime between light and matter, where many molecules can collectively couple to a single cavity mode or many cavity modes. We also lay out the current challenges in theory to explain the observed experimental results. We hope that this review will serve as a useful document for anyone who wants to become familiar with the context of polariton chemistry and molecular cavity QED and thus significantly benefit the entire community.

Coherent Mixing of Singlet and Triplet States in Acrolein and Ketene: A Computational Strategy for Simulating the Electron-Nuclear Dynamics of Intersystem Crossing

We present a theoretical study of intersystem crossing (ISC) in acrolein and ketene with the Ehrenfest method that can describe a superposition of singlet and triplet states. Our simulations illustrate a new mechanistic effect of ISC, namely, that a superposition of singlets and triplets yields nonadiabatic dynamics characteristic of that superposition rather than the constituent state potential energy surfaces. This effect is particularly significant in ketene, where mixing of singlet and triplet states along the approach to a singlet/singlet conical intersection occurs, with the spin-orbit coupling (SOC) remaining small throughout. In both cases, the effects require many recrossings of the singlet/triplet state crossing seam, consistent with the textbook treatment of ISC.

Ring Polymer Molecular Dynamics with Electronic Transitions

Full quantum dynamics of molecules and materials is of fundamental importance, which requires a faithful description of simultaneous quantum motions of the electron and nuclei. A new scheme is developed for nonadiabatic simulations of coupled electron-nuclear quantum dynamics with electronic transitions based on the Ehrenfest theorem and ring polymer molecular dynamics. Built upon the isomorphic ring polymer Hamiltonian, time-dependent multistate electronic Schrödinger equations are solved self-consistently with approximate equation of motions for nuclei. Each bead bears a distinct electronic configuration and thus moves on a specific effective potential. This independent-bead approach provides an accurate description of the real-time electronic population and quantum nuclear trajectory, maintaining a good agreement with the exact quantum solution. Implementation of first-principles calculations enables us to simulate photoinduced proton transfer in H_{2}O-H_{2}O^{+} where we find a good agreement with experiment.

Surface hopping modeling of charge and energy transfer in active environments

An active environment is any atomic or molecular system changing a chromophore's nonadiabatic dynamics compared to the isolated molecule. The action of the environment on the chromophore occurs by changing the potential energy landscape and triggering new energy and charge flows unavailable in the vacuum. Surface hopping is a mixed quantum-classical approach whose extreme flexibility has made it the primary platform for implementing novel methodologies to investigate the nonadiabatic dynamics of a chromophore in active environments. This Perspective paper surveys the latest developments in the field, focusing on charge and energy transfer processes.

Electronic Born-Oppenheimer approximation in nuclear-electronic orbital dynamics

Within the nuclear-electronic orbital (NEO) framework, the real-time NEO time-dependent density functional theory (RT-NEO-TDDFT) approach enables the simulation of coupled electronic-nuclear dynamics. In this approach, the electrons and quantum nuclei are propagated in time on the same footing. A relatively small time step is required to propagate the much faster electronic dynamics, thereby prohibiting the simulation of long-time nuclear quantum dynamics. Herein, the electronic Born-Oppenheimer (BO) approximation within the NEO framework is presented. In this approach, the electronic density is quenched to the ground state at each time step, and the real-time nuclear quantum dynamics is propagated on an instantaneous electronic ground state defined by both the classical nuclear geometry and the nonequilibrium quantum nuclear density. Because the electronic dynamics is no longer propagated, this approximation enables the use of an order-of-magnitude larger time step, thus greatly reducing the computational cost. Moreover, invoking the electronic BO approximation also fixes the unphysical asymmetric Rabi splitting observed in previous semiclassical RT-NEO-TDDFT simulations of vibrational polaritons even for small Rabi splitting, instead yielding a stable, symmetric Rabi splitting. For the intramolecular proton transfer in malonaldehyde, both RT-NEO-Ehrenfest dynamics and its BO counterpart can describe proton delocalization during the real-time nuclear quantum dynamics. Thus, the BO RT-NEO approach provides the foundation for a wide range of chemical and biological applications.

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.

Factorized Electron-Nuclear Dynamics with an Effective Complex Potential

We present a quantum dynamics approach for molecular systems based on wave function factorization into components describing the light and heavy particles, such as electrons and nuclei. The dynamics of the nuclear subsystem can be viewed as motion of the trajectories defined in the nuclear subspace, evolving according to the average nuclear momentum of the full wave function. The probability density flow between the nuclear and electronic subsystems is facilitated by the imaginary potential, derived to ensure a physically meaningful normalization of the electronic wave function for each configuration of the nuclei, and conservation of the probability density associated with each trajectory in the Lagrangian frame of reference. The imaginary potential, defined in the nuclear subspace, depends on the momentum variance in the nuclear coordinates averaged over the electronic component of the wave function. An effective real potential, driving the dynamics of the nuclear subsystem, is defined to minimize motion of the electronic wave function in the nuclear degrees of freedom. Illustration and the analysis of the formalism are given for a two-dimensional model system of vibrationally nonadiabatic dynamics.

Incorporating Nuclear Quantum Effects in Molecular Dynamics with a Constrained Minimized Energy Surface

The accurate incorporation of nuclear quantum effects in large-scale molecular dynamics (MD) simulations remains a significant challenge. Recently, we combined constrained nuclear-electronic orbital (CNEO) theory with classical MD and obtained a new approach (CNEO-MD) that can accurately and efficiently incorporate nuclear quantum effects into classical simulations. In this Letter, we provide the theoretical foundation for CNEO-MD by developing an alternative formulation of the equations of motion for MD. In this new formulation, the expectation values of quantum nuclear positions evolve classically on an effective energy surface that is obtained from a constrained energy minimization procedure when solving for the quantum nuclear wave function, thus enabling the incorporation of nuclear quantum effects in classical MD simulations. For comparison with other existing approaches, we examined a series of model systems and found that this new MD approach is significantly more accurate than the conventional way of performing classical MD and generally outperforms centroid MD and ring-polymer MD in describing vibrations in these model systems.

Quantum algorithms for quantum dynamics

Among the many computational challenges faced across different disciplines, quantum-mechanical systems pose some of the hardest ones and offer a natural playground for the growing field of quantum technologies. In this Perspective, we discuss quantum algorithmic solutions for quantum dynamics, reporting on the latest developments and offering a viewpoint on their potential and current limitations. We present some of the most promising areas of application and identify possible research directions for the coming years.

Equation-of-Motion Methods for the Calculation of Femtosecond Time-Resolved 4-Wave-Mixing and N-Wave-Mixing Signals

Femtosecond nonlinear spectroscopy is the main tool for the time-resolved detection of photophysical and photochemical processes. Since most systems of chemical interest are rather complex, theoretical support is indispensable for the extraction of the intrinsic system dynamics from the detected spectroscopic responses. There exist two alternative theoretical formalisms for the calculation of spectroscopic signals, the nonlinear response-function (NRF) approach and the spectroscopic equation-of-motion (EOM) approach. In the NRF formalism, the system-field interaction is assumed to be sufficiently weak and is treated in lowest-order perturbation theory for each laser pulse interacting with the sample. The conceptual alternative to the NRF method is the extraction of the spectroscopic signals from the solutions of quantum mechanical, semiclassical, or quasiclassical EOMs which govern the time evolution of the material system interacting with the radiation field of the laser pulses. The NRF formalism and its applications to a broad range of material systems and spectroscopic signals have been comprehensively reviewed in the literature. This article provides a detailed review of the suite of EOM methods, including applications to 4-wave-mixing and N-wave-mixing signals detected with weak or strong fields. Under certain circumstances, the spectroscopic EOM methods may be more efficient than the NRF method for the computation of various nonlinear spectroscopic signals.

Nonadiabatic Couplings Can Speed Up Quantum Tunneling Transition Path Times

Quantum tunneling is known to play an important role in the dynamics of systems with nonadiabatic couplings. However, until recently, the time-domain properties of nonadiabatic scattering have been severely under-explored. Using numerically exact quantum methods, we study the impact that nonadiabatic couplings have on the time it takes to tunnel through a barrier. We find that the Wigner phase time is the appropriate measure to use when determining the tunneling flight time also when considering nonadiabatic systems. The central result of the present study is that in an avoided crossing system in one dimension, the nonadiabatic couplings speed up the tunneling event, relative to the adiabatic case in which all nonadiabatic coupling is ignored. This has implications for both the study of quantum tunneling times and for the field of nonadiabatic scattering and chemistry.

Towards the engineering of a photon-only two-stroke rotary molecular motor

The rational engineering of photoresponsive materials, e.g., light-driven molecular motors, is a challenging task. Here, we use structure-related design rules to prepare a prototype molecular rotary motor capable of completing an entire revolution using, exclusively, the sequential absorption of two photons; i.e., a photon-only two-stroke motor. The mechanism of rotation is then characterised using a combination of non-adiabatic dynamics simulations and transient absorption spectroscopy measurements. The results show that the rotor moiety rotates axially relative to the stator and produces, within a few picoseconds at ambient T, an intermediate with the same helicity as the starting structure. We discuss how such properties, that include a 0.25 quantum efficiency, can help overcome the operational limitations of the classical overcrowded alkene designs.

Nonadiabatic Excited State Dynamics of Organic Chromophores: Take-Home Messages

Nonadiabatic excited state dynamics are important in a variety of processes. Theoretical and experimental developments have allowed for a great progress in this area, while combining the two is often necessary and the best approach to obtain insight into the photophysical behavior of molecules. In this Feature Article we use examples of our recent work combining time-resolved photoelectron spectroscopy with theoretical nonadiabatic dynamics to highlight important lessons we learned. We compare the nonadiabatic excited state dynamics of three different organic molecules with the aim of elucidating connections between structure and dynamics. Calculations and measurements are compared for uracil, 1,3-cyclooctadiene, and 1,3-cyclohexadiene. The comparison highlights the role of rigidity in influencing the dynamics and the difficulty of capturing the dynamics accurately with calculations.

Quantum coherence in molecular photoionization

The study of onset and decay, as well as control of ultrafast quantum coherence in many-electron systems is in the focus of interest of attosecond physics. Interpretation of attosecond experiments detecting the ultrafast quantum coherence requires application of advanced theoretical and computational tools combining many-electron theory, description of the electronic continuum, including in the strong laser field scenario, as well as nuclear dynamics theory. This perspective reviews the recent theoretical advances in understanding the attosecond dynamics of quantum coherence in photoionized molecular systems and outlines possible future directions of theoretical and experimental study of coherence and entanglement in the attosecond regime.

Exact Factorization Adventures: A Promising Approach for Non-Bound States

Modeling the dynamics of non-bound states in molecules requires an accurate description of how electronic motion affects nuclear motion and vice-versa. The exact factorization (XF) approach offers a unique perspective, in that it provides potentials that act on the nuclear subsystem or electronic subsystem, which contain the effects of the coupling to the other subsystem in an exact way. We briefly review the various applications of the XF idea in different realms, and how features of these potentials aid in the interpretation of two different laser-driven dissociation mechanisms. We present a detailed study of the different ways the coupling terms in recently-developed XF-based mixed quantum-classical approximations are evaluated, where either truly coupled trajectories, or auxiliary trajectories that mimic the coupling are used, and discuss their effect in both a surface-hopping framework as well as the rigorously-derived coupled-trajectory mixed quantum-classical approach.

On the meaning of Berry force for unrestricted systems treated with mean-field electronic structure

We show that the Berry force as computed by an approximate, mean-field electronic structure can be meaningful if properly interpreted. In particular, for a model Hamiltonian representing a molecular system with an even number of electrons interacting via a two-body (Hubbard) interaction and a spin-orbit coupling, we show that a meaningful nonzero Berry force emerges whenever there is spin unrestriction-even though the Hamiltonian is real-valued and formally the on-diagonal single-surface Berry force must be zero. Moreover, if properly applied, this mean-field Berry force yields roughly the correct asymptotic motion for scattering through an avoided crossing. That being said, within the context of a ground-state calculation, several nuances do arise as far interpreting the Berry force correctly, and as a practical matter, the Berry force diverges near the Coulson-Fischer point (which can lead to numerical instabilities). We do not address magnetic fields here.

Excited state non-adiabatic dynamics of large photoswitchable molecules using a chemically transferable machine learning potential

Light-induced chemical processes are ubiquitous in nature and have widespread technological applications. For example, photoisomerization can allow a drug with a photo-switchable scaffold such as azobenzene to be activated with light. In principle, photoswitches with desired photophysical properties like high isomerization quantum yields can be identified through virtual screening with reactive simulations. In practice, these simulations are rarely used for screening, since they require hundreds of trajectories and expensive quantum chemical methods to account for non-adiabatic excited state effects. Here we introduce a diabatic artificial neural network (DANN), based on diabatic states, to accelerate such simulations for azobenzene derivatives. The network is six orders of magnitude faster than the quantum chemistry method used for training. DANN is transferable to azobenzene molecules outside the training set, predicting quantum yields for unseen species that are correlated with experiment. We use the model to virtually screen 3100 hypothetical molecules, and identify novel species with high predicted quantum yields. The model predictions are confirmed using high-accuracy non-adiabatic dynamics. Our results pave the way for fast and accurate virtual screening of photoactive compounds.

Quantum Dynamics with Electronic Friction

A theory of electronic friction is developed using the exact factorization of the electronic-nuclear wave function. No assumption is made regarding the electronic bath, which can be made of independent or interacting electrons, and the nuclei are treated quantally. The ensuing equation of motion for the nuclear wave function is a nonlinear Schrödinger equation including a friction term. The resulting friction kernel agrees with a previously derived mixed quantum-classical result by Dou et al., [Phys. Rev. Lett. 119, 046001 (2017)]PRLTAO0031-900710.1103/PhysRevLett.119.046001, except for a pseudomagnetic contribution in the latter that is here removed. More specifically, it is shown that the electron dynamics generally washes out the gauge fields appearing in the adiabatic dynamics. However, these are fully re-established in the typical situation where the electrons respond rapidly on the slow time scale of the nuclear dynamics (Markov limit). Hence, we predict Berry's phase effects to be observable also in the presence of electronic friction. Application to a model vibrational relaxation problem proves that the proposed approach represents a viable way to account for electronic friction in a fully quantum setting for the nuclear dynamics.

Geometric energy transfer in two-component systems

Factoring a wave function into marginal and conditional factors partitions the subsystem kinetic energy into two terms. The first depends solely on the marginal wave function, through its gauge-covariant derivative, while the second depends on the quantum metric of the conditional wave function over the manifold of marginal variables. We derive an identity for the rate of change of the second term. This article is part of the theme issue 'Chemistry without the Born-Oppenheimer approximation'.

Theoretical perspectives on non-Born-Oppenheimer effects in chemistry

The Born-Oppenheimer approximation, which assumes that the electrons respond instantaneously to the motion of the nuclei, breaks down for a wide range of chemical and biological processes. The rate constants of such nonadiabatic processes can be calculated using analytical theories, and the real-time nonequilibrium dynamics can be described using numerical atomistic simulations. The selection of an approach depends on the desired balance between accuracy and efficiency. The computational expense of generating potential energy surfaces on-the-fly often favours the use of approximate, robust and efficient methods such as trajectory surface hopping for large, complex systems. The development of formally exact non-Born-Oppenheimer methods and the exploration of well-defined approximations to such methods are critical for providing benchmarks and preparing for the next generation of faster computers. Thus, the parallel development of rigorous but computationally expensive methods and more approximate but computationally efficient methods is optimal. This Perspective briefly summarizes the available theoretical and computational non-Born-Oppenheimer methods and presents examples illustrating how analytical theories and nonadiabatic dynamics simulations can elucidate the fundamental principles of chemical and biological processes. These examples also highlight how theoretical calculations are able to guide the interpretation of experimental data and provide experimentally testable predictions for nonadiabatic processes. This article is part of the theme issue 'Chemistry without the Born-Oppenheimer approximation'.

Quantum flux densities for electronic-nuclear motion: exact versus Born-Oppenheimer dynamics

We study the coupled electronic-nuclear dynamics in a model system to compare numerically exact calculations of electronic and nuclear flux densities with those obtained from the Born-Oppenheimer (BO) approximation. Within the adiabatic expansion of the total wave function, we identify the terms which contribute to the flux densities. It is found that only off-diagonal elements that involve the interaction between different electronic states contribute to the electronic flux whereas in the nuclear case the major contribution belongs to the BO electronic state. New flux densities are introduced where in both, the electronic and the nuclear case, the main contribution is contained in the component corresponding to the BO state. As a consequence, they can be determined within the BO approximation, and an excellent agreement with the exact results is found. This article is part of the theme issue 'Chemistry without the Born-Oppenheimer approximation'.

Nuclear-Electronic Orbital Approach to Quantization of Protons in Periodic Electronic Structure Calculations

The nuclear-electronic orbital (NEO) method is a well-established approach for treating nuclei quantum mechanically in molecular systems beyond the usual Born-Oppenheimer approximation. In this work, we present a strategy to implement the NEO method for periodic electronic structure calculations, particularly focused on multicomponent density functional theory (DFT). The NEO-DFT method is implemented in an all-electron electronic structure code, FHI-aims, using a combination of analytical and numerical integration techniques as well as a resolution of the identity scheme to enhance computational efficiency. After validating this implementation, proof-of-concept applications are presented to illustrate the effects of quantized protons on the physical properties of extended systems such as two-dimensional materials and liquid-semiconductor interfaces. Specifically, periodic NEO-DFT calculations are performed for a trans-polyacetylene chain, a hydrogen boride sheet, and a titanium oxide-water interface. The zero-point energy effects of the protons, as well as electron-proton correlation, are shown to noticeably impact the density of states and band structures for these systems. These developments provide a foundation for the application of multicomponent DFT to a wide range of other extended condensed matter systems.

Describing the photo-isomerization of a retinal chromophore model with coupled and quantum trajectories

The exact factorization of the electron-nuclear wavefunction is applied to the study of the photo- isomerization of a retinal chromophore model. We describe such an ultrafast nonadiabatic process by analyzing the time-dependent potentials of the theory and by mimicking nuclear dynamics with quantum and coupled trajectories. The time-dependent vector and scalar potentials are the signature of the exact factorization, as they guide nuclear dynamics by encoding the complete electronic dynamics and including excited-state effects. Analysis of the potentials is, thus, essential - when possible - to predict the time-dependent behavior of the system of interest. In this work, we employ the exact time-dependent potentials, available for the numerically-exactly solvable model used here, to propagate quantum nuclear trajectories representing the isomerization reaction of the retinal chromophore. The quantum trajectories are the best possible trajectory-based description of the reaction when using the exact-factorization formalism, and thus allow us to assess the performance of the coupled-trajectory, fully approximate, schemes derived from the exact-factorization equations.

Independent Trajectory Mixed Quantum-Classical Approaches Based on the Exact Factorization

Mixed quantum-classical dynamics based on the exact factorization exploits the "derived" electron-nuclear correlation (ENC) term aiming for the description of quantum coherences. The ENC contains interactions between the phase of electronic states and nuclear quantum momenta which depend on the spatial shape of the nuclear density.The original surface hopping based on the exact factorization (SHXF) [\textit{J. Phys. Chem. Lett.} \textbf{2018}, \textit{9}, 1097] exploits frozen Gaussian functions to construct the nuclear density in the ENC term while the phase of electronic states is approximated as a fictitious nuclear momentum change.However, in reality, the width of nuclear wave packets varies in time depending on the shape of potential energy surfaces.In this work, we present a modified SHXF approach and a newly-developed Ehrenfest dynamics based on the exact factorization (EhXF) with time-dependent Gaussian functions and phases by enforcing total energy conservation.We perform numerical tests for various one-dimensional two-state model Hamiltonians.Overall, the time-dependent width of Gaussian functions and the energy conserving phase show a reliable decoherence compared to the original frozen Gaussian-based SHXF and the exact quantum mechanical calculation.Especially, the energy conserving phase is crucial for EhXF to reproduce the correct quantum dynamics.

Energy, Momentum, and Angular Momentum Transfer between Electrons and Nuclei

The recently developed exact factorization approach condenses all electronic effects on the nuclear subsystem into scalar and vector potentials that appear in an effective time dependent Schrödinger equation. Starting from this equation, we derive subsystem Ehrenfest identities characterizing the energy, momentum, and angular momentum transfer between electrons and nuclei. An effective electromagnetic force operator induced by the electromagnetic field corresponding to the effective scalar and vector potentials appears in all three identities. The effective magnetic field has two components that can be identified with the Berry curvature calculated with (a) different Cartesian coordinates of the same nucleus and (b) arbitrary Cartesian coordinates of two different nuclei. (a) has a classical interpretation as the induced magnetic field felt by the nucleus, while (b) has no classical analog. Subsystem Ehrenfest identities are ideally suited for quantifying energy transfer in electron-phonon systems. With two explicit examples we demonstrate the usefulness of the new identities.

The 2021 room-temperature superconductivity roadmap

Designing materials with advanced functionalities is the main focus of contemporary solid-state physics and chemistry. Research efforts worldwide are funneled into a few high-end goals, one of the oldest, and most fascinating of which is the search for an ambient temperature superconductor (A-SC). The reason is clear: superconductivity at ambient conditions implies being able to handle, measure and access a single, coherent, macroscopic quantum mechanical state without the limitations associated with cryogenics and pressurization. This would not only open exciting avenues for fundamental research, but also pave the road for a wide range of technological applications, affecting strategic areas such as energy conservation and climate change. In this roadmap we have collected contributions from many of the main actors working on superconductivity, and asked them to share their personal viewpoint on the field. The hope is that this article will serve not only as an instantaneous picture of the status of research, but also as a true roadmap defining the main long-term theoretical and experimental challenges that lie ahead. Interestingly, although the current research in superconductor design is dominated by conventional (phonon-mediated) superconductors, there seems to be a widespread consensus that achieving A-SC may require different pairing mechanisms.In memoriam, to Neil Ashcroft, who inspired us all.

A Photochemical Reaction in Different Theoretical Representations

The Born–Oppenheimer picture has forged our representation and interpretation of photochemical processes, from photoexcitation down to the passage through a conical intersection, a funnel connecting different electronic states. In this work, we analyze a full in silico photochemical experiment, from the explicit electronic excitation by a laser pulse to the formation of photoproducts following a nonradiative decay through a conical intersection, by contrasting the picture offered by Born–Oppenheimer and that proposed by the exact factorization. The exact factorization offers an alternative understanding of photochemistry that does not rely on concepts such as electronic states, nonadiabatic couplings, and conical intersections. On the basis of nonadiabatic quantum dynamics performed for a two-state 2D model system, this work allows us to compare Born–Oppenheimer and exact factorization for (i) an explicit photoexcitation with and without the Condon approximation, (ii) the passage of a nuclear wavepacket through a conical intersection, (iii) the formation of excited stationary states in the Franck–Condon region, and (iv) the use of classical and quantum trajectories in the exact factorization picture to capture nonadiabatic processes triggered by a laser pulse.

Exact-Factorization-Based Surface Hopping for Multistate Dynamics

A surface-hopping algorithm recently derived from the exact factorization approach, SHXF [Ha et al. J. Phys. Chem. Lett. 2018, 9, 1097], introduces an additional term in the electronic equation of surface hopping that couples electronic states through the quantum momentum. This term not only provides a first-principles description of decoherence, but here we show it is crucial to accurately capture nonadiabatic dynamics when more than two states are occupied at any given time. Using a vibronic coupling model of the uracil cation, we show that the lack of this term in traditional surface-hopping methods, including those with decoherence corrections, leads to failure to predict the dynamics through a three-state intersection, while SHXF performs similarly to the multiconfiguration time-dependent Hartree quantum dynamics benchmark.

Coupled- and Independent-Trajectory Approaches Based on the Exact Factorization Using the PyUNIxMD Package

We present mixed quantum-classical approaches based on the exact factorization framework. The electron-nuclear correlation term in the exact factorization enables us to deal with quantum coherences by accounting for electronic and nuclear nonadiabatic couplings effectively within classical nuclei approximation. We compare coupled- and independent-trajectory approximations with each other to understand algorithms in description of the bifurcation of nuclear wave packets and the correct spatial distribution of electronic wave functions along with nuclear trajectories. Finally, we show numerical results for comparisons of coupled- and independent-trajectory approaches for the photoisomerization of a protonated Schiff base from excited state molecular dynamics (ESMD) simulations with the recently developed Python-based ESMD code, namely, the PyUNIxMD program.

Beyond Born-Oppenheimer based diabatic surfaces of 1,3,5-C6H3F3+ to generate the photoelectron spectra using time-dependent discrete variable representation approach

In this article, Beyond Born-Oppenheimer (BBO) treatment is implemented to construct diabatic potential energy surfaces (PESs) of 1,3,5-C₆H₃F₃⁺ over a series [eighteen (18)] of two-dimensional (2D) nuclear planes constituted with eleven normal modes (Q₂, Q_9x, Q_9y, Q_13x, Q_13y, Q_18x, Q_18y, Q_10x, Q_10y, Q_12x and Q_12y) to include all possible nonadiabatic interactions among six coupled electronic states (X̃²E'', , B̃²E' and ). We had formulated explicit expressions of adiabatic to diabatic transformation (ADT) equations [S. Mukherjee, J. Dutta, B. Mukherjee, S. Sardar and S. Adhikari, J. Chem. Phys., 2019, 150, 064308] for the same system forming six state sub-Hilbert space and at present, these ADT equations are solved by incorporating MRCI level ab initio adiabatic PESs and CP-MCSCF calculated nonadiabatic coupling terms (NACTs) to derive diabatic PESs and couplings. Such single-valued, smooth, symmetric and continuous diabatic surface matrices are utilized to carry out multi-state multi-mode nuclear dynamics with the help of time-dependent discrete variable representation (TDDVR) methodology to compute the photoelectron (PE) spectra of 1,3,5-C₆H₃F₃. Our theoretically calculated spectra for X̃²E'', and states using BBO treatment and TDDVR dynamics show peak by peak correspondence with the experimental results as well as better than the findings of the multi-configuration time-dependent Hartree (MCTDH) method.

Quantum Entanglement from Classical Trajectories

A long-standing challenge in mixed quantum-classical trajectory simulations is the treatment of entanglement between the classical and quantal degrees of freedom. We present a novel approach that describes the emergence of entangled states entirely in terms of independent and deterministic Ehrenfest-like classical trajectories. For a two-level quantum system in a classical environment, this is derived by mapping the quantum system onto a path-integral representation of a spin 1/2. We demonstrate that the method correctly accounts for coherence and decoherence and thus reproduces the splitting of a wave packet in a nonadiabatic scattering problem. This discovery opens up a new class of simulations as an alternative to stochastic surface-hopping, coupled-trajectory, or semiclassical approaches.

Gaussian-Type Orbital Calculations for High Harmonic Generation in Vibrating Molecules: Benchmarks for H2. J Chem Theory Comput 2021;17:7353-7365. [PMID: 34747605 DOI: 10.1021/acs.jctc.1c00837]

The response of the hydrogen molecular ion, H₂⁺, to few-cycle laser pulses of different intensities is simulated. To treat the coupled electron-nuclear motion, we use adiabatic potentials computed with Gaussian-type basis sets together with a heuristic ionization model for the electron and a grid representation for the nuclei. Using this mixed-basis approach, the time-dependent Schrödinger equation is solved, either within the Born-Oppenheimer approximation or with nonadiabatic couplings included. The dipole response spectra are compared to all-grid-based solutions for the three-body problem, which we take as a reference to benchmark the Gaussian-type basis set approaches. Also, calculations employing the fixed-nuclei approximation are performed, to quantify effects due to nuclear motion. For low intensities and small ionization probabilities, we get excellent agreement of the dynamics using Gaussian-type basis sets with the all-grid solutions. Our investigations suggest that high harmonic generation (HHG) and high-frequency response, in general, can be reliably modeled using Gaussian-type basis sets for the electrons for not too high harmonics. Further, nuclear motion destroys electronic coherences in the response spectra even on the time scale of about 30 fs and affects HHG intensities, which reflect the electron dynamics occurring on the attosecond time scale. For the present system, non-Born-Oppenheimer effects are small. The Gaussian-based, nonadiabatically coupled, time-dependent multisurface approach to treat quantum electron-nuclear motion beyond the non-Born-Oppenheimer approximation can be easily extended to approximate wavefunction methods, such as time-dependent configuration interaction singles (TD-CIS), for systems where no benchmarks are available.

Conditional Wave Function Theory: A Unified Treatment of Molecular Structure and Nonadiabatic Dynamics

We demonstrate that a conditional wave function theory enables a unified and efficient treatment of the equilibrium structure and nonadiabatic dynamics of correlated electron-ion systems. The conditional decomposition of the many-body wave function formally recasts the full interacting wave function of a closed system as a set of lower-dimensional (conditional) coupled "slices". We formulate a variational wave function ansatz based on a set of conditional wave function slices and demonstrate its accuracy by determining the structural and time-dependent response properties of the hydrogen molecule. We then extend this approach to include time-dependent conditional wave functions and address paradigmatic nonequilibrium processes including strong-field molecular ionization, laser-driven proton transfer, and nuclear quantum effects induced by a conical intersection. This work paves the road for the application of conditional wave function theory in equilibrium and out-of-equilibrium ab initio molecular simulations of finite and extended systems.

Molecular Quantum Dynamics: A Quantum Computing Perspective

ConspectusSimulating molecular dynamics (MD) within a comprehensive quantum framework has been a long-standing challenge in computational chemistry. An exponential scaling of computational cost renders solving the time dependent Schrödinger equation (TDSE) of a molecular Hamiltonian, including both electronic and nuclear degrees of freedom (DOFs), as well as their couplings, infeasible for more than a few DOFs. In the Born-Oppenheimer (BO), or adiabatic, picture, electronic and nuclear parts of the wave function are decoupled and treated separately. Within this framework, the nuclear wave function evolves along potential energy surfaces (PESs) computed as solutions to the electronic Schrödinger equation parametrized in the nuclear DOFs. This approximation, together with increasingly elaborate numerical approaches to solve the nuclear time dependent Schrödinger equation (TDSE), enabled the treatment of up to a few dozens of degrees of freedom (DOFs). However, for particular applications, such as photochemistry, the BO approximation breaks down. In this regime of non-adiabatic dynamics, solving the full molecular problem including electron-nuclear couplings becomes essential, further increasing the complexity of the numerical solution. Although valuable methods such as multiconfigurational time-dependent Hartree (MCTDH) have been proposed for the solution of the coupled electron-nuclear dynamics, they remain hampered by an exponential scaling in the number of nuclear DOFs and by the difficulty of finding universal variational forms.In this Account, we present a perspective on novel quantum computational algorithms, aiming to alleviate the exponential scaling inherent to the simulation of many-body quantum dynamics. In particular, we focus on the derivation and application of quantum algorithms for adiabatic and non-adiabatic quantum dynamics, which include efficient approaches for the calculation of the BO potential energy surfaces (PESs). Thereafter, we study the time-evolution of a model system consisting of two coupled PESs in first and second quantization. In a first application, we discuss a recently introduced quantum algorithm for the evolution of a wavepacket in first quantization and exploit the potential quantum advantage of mapping its spatial grid representation to logarithmically many qubits. For the second demonstration, we move to the second quantization framework and review the scaling properties of two alternative time-evolution algorithms, namely, a variational quantum algorithm (VQA) (based on the McLachlan variational principle) and conventional Trotter-type evolution (based on a Lie-Trotter-Suzuki formula). Both methods clearly demonstrate the potential of quantum algorithms and their favorable scaling compared to the available classical approaches. However, a clear demonstration of quantum advantage in the context of molecular quantum dynamics may require the implementation of these algorithms in fault-tolerant quantum computers, while their application in near-term, noisy quantum devices is still unclear and deserves further investigation.

Investigation of nonadiabatic dynamics in the photolysis of methyl nitrate (CH3ONO2) by on-the-fly surface hopping simulation

The photolysis mechanism of methyl nitrate (CH₃ONO₂) was studied using the on-the-fly surface hopping dynamics at the XMS-CASPT2 level. Several critical geometries, including electronic state minima and conical intersections, were obtained, which play essential roles in the nonadiabatic dynamics of CH₃ONO₂. The ultrafast nonadiabatic decay dynamics to the ground state were simulated, which gives a proper explanation on the broad and structureless absorption spectra of CH₃ONO₂. The photodissociation channels, including CH₃O + NO₂, CH₃O + NO + O, and others, as well as their branching ratios, were identified. When the dynamics starts from the lowest two electronic states (S₁ and S₂), the CH₃O + NO₂ channel is the dominant photolysis pathway, although we observed the minor contributions of other channels. In contrast, when the trajectories start from the third excited state S₃, both CH₃O + NO₂ and CH₃O + NO + O channels become important. Here the CH₃O-NO₂ bond dissociation takes place first, and then for some trajectories, the N-O bond of the NO₂ part breaks successively. The quasi-degeneracy of electronic states may exist in the dissociation limits of both CH₃O + NO₂ and CH₃O + NO + O channels. The current work provides valuable information in the understanding of experimental findings of the wavelength-dependent photolysis mechanism of CH₃ONO₂.

Dynamics near a conical intersection-A diabolical compromise for the approximations of ab initio multiple spawning

Full multiple spawning (FMS) offers an exciting framework for the development of strategies to simulate the excited-state dynamics of molecular systems. FMS proposes to depict the dynamics of nuclear wavepackets by using a growing set of traveling multidimensional Gaussian functions called trajectory basis functions (TBFs). Perhaps the most recognized method emanating from FMS is the so-called ab initio multiple spawning (AIMS). In AIMS, the couplings between TBFs-in principle exact in FMS-are approximated to allow for the on-the-fly evaluation of required electronic-structure quantities. In addition, AIMS proposes to neglect the so-called second-order nonadiabatic couplings and the diagonal Born-Oppenheimer corrections. While AIMS has been applied successfully to simulate the nonadiabatic dynamics of numerous complex molecules, the direct influence of these missing or approximated terms on the nonadiabatic dynamics when approaching and crossing a conical intersection remains unknown to date. It is also unclear how AIMS could incorporate geometric-phase effects in the vicinity of a conical intersection. In this work, we assess the performance of AIMS in describing the nonadiabatic dynamics through a conical intersection for three two-dimensional, two-state systems that mimic the excited-state dynamics of bis(methylene)adamantyl, butatriene cation, and pyrazine. The population traces and nuclear density dynamics are compared with numerically exact quantum dynamics and trajectory surface hopping results. We find that AIMS offers a qualitatively correct description of the dynamics through a conical intersection for the three model systems. However, any attempt at improving the AIMS results by accounting for the originally neglected second-order nonadiabatic contributions appears to be stymied by the hermiticity requirement of the AIMS Hamiltonian and the independent first-generation approximation.

Multi-configuration electron-nuclear dynamics: An open-shell approach

The multi-configuration electron-nuclear dynamics for open shell systems with a spin-unrestricted formalism is described. The mean fields are evaluated using second-order reduced density matrices for electronic and nuclear degrees of freedom. Applications to light-element diatomics including equilibrium geometries, electronic energies, dipole moments, and absorption spectra are presented. The von Neumann entropies for different spin states of a LiH molecule are compared.

Nonadiabatic Dynamics with Coupled Trajectories

In this paper, we discuss coupled-trajectory schemes for molecular-dynamics simulations of excited-state processes. New coupled-trajectory strategies to capture decoherence effects, revival of coherence and nonadiabatic interferences in long-time dynamics are proposed, and compared to independent-trajectory schemes. The working framework is provided by the exact factorization of the electron-nuclear wave function, and it exploits ideas emanating from various surface-hopping schemes. The new coupled-trajectory algorithms are tested on a one-dimensional two-state system using different model parameters which allow one to induce different dynamics. The benchmark is provided by the numerically exact solution of the time-dependent Schrödinger equation.

Nucleus-electron correlation revising molecular bonding fingerprints from the exact wavefunction factorization

We present a novel theory and implementation for computing coupled electronic and quantal nuclear subsystems on a single potential energy surface, moving beyond the standard Born-Oppenheimer (BO) separation of nuclei and electrons. We formulate an exact self-consistent nucleus-electron embedding potential from the single product molecular wavefunction and demonstrate that the fundamental behavior of the correlated nucleus-electron can be computed for mean-field electrons that are responsive to a quantal anharmonic vibration of selected nuclei in a discrete variable representation. Geometric gauge choices are discussed and necessary for formulating energy invariant biorthogonal electronic equations. Our method is further applied to characterize vibrationally averaged molecular bonding properties of molecular energetics, bond lengths, and protonic and electron densities. Moreover, post-Hartree-Fock electron correlation can be conveniently computed on the basis of nucleus-electron coupled molecular orbitals, as demonstrated for correlated models of second-order Møllet-Plesset perturbation and full configuration interaction theories. Our approach not only accurately quantifies non-classical nucleus-electron couplings for revising molecular bonding properties but also provides an alternative time-independent approach for deploying non-BO molecular quantum chemistry.

Simulating electronic excitation and dynamics with real-time propagation approach to TDDFT within plane-wave pseudopotential formulation

We give a perspective on simulating electronic excitation and dynamics using the real-time propagation approach to time-dependent density functional theory (RT-TDDFT) in the plane-wave pseudopotential formulation. RT-TDDFT is implemented in various numerical formalisms in recent years, and its practical application often dictates the most appropriate implementation of the theory. We discuss recent developments and challenges, emphasizing numerical aspects of studying real systems. Several applications of RT-TDDFT simulation are discussed to highlight how the approach is used to study interesting electronic excitation and dynamics phenomena in recent years.

Fock-Space Embedding Theory: Application to Strongly Correlated Topological Phases

A many-body wave function can be factorized in Fock space into a marginal amplitude describing a set of strongly correlated orbitals and a conditional amplitude for the remaining weakly correlated part. The marginal amplitude is the solution of a Schrödinger equation with an effective Hamiltonian that can be viewed as embedding the marginal wave function in the environment of weakly correlated electrons. Here, the complementary equation for the conditional amplitude is replaced by a generalized Kohn-Sham equation, for which an orbital-dependent functional approximation is shown to reproduce the topological phase diagram of a multiband Hubbard model as a function of crystal field and Hubbard parameters. The roles of band filling and interband fluctuations are elucidated.

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.