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

Coupled-trajectory surface hopping with sign consistency

The framework of exact factorization (XF) has inspired a series of trajectory-based nonadiabatic dynamics methods by introducing different approximations. Recently, the coupled-trajectory surface hopping (CTSH) method has been proposed to combine the key advantages of the coupled-trajectory mixed quantum-classical method based on XF and the fewest switches surface hopping. We here present a novel variant of CTSH, namely, sign-consistent CTSH (SC-CTSH), which considers proper trajectory clustering to reconstruct the nuclear density distribution and the consistency between wave function and active states to introduce decoherence. Using the exact quantum solutions as references, the high performance of SC-CTSH is benchmarked in the widely studied scattering models and compared with other related XF-based methods. Due to the incorporation of new trajectory clustering and sign consistency algorithms, SC-CTSH obtains more accurate quantum momentum and decoherence during the nonadiabatic dynamics, which makes the combination of XF and surface hopping more consistent and reliable. This study further highlights the significance of internal consistency between wave function and active states, which is important in the further development of mixed quantum-classical dynamics methods.

Nonadiabatic Field: A Conceptually Novel Approach for Nonadiabatic Quantum Molecular Dynamics

Reliable trajectory-based nonadiabatic quantum dynamics methods at the atomic/molecular level are critical for the practical understanding and rational design of many important processes in real large/complex systems, where the quantum dynamical behavior of electrons and that of nuclei are coupled. The paper reports latest progress of nonadiabatic field (NaF), a conceptually novel approach for nonadiabatic quantum dynamics with independent trajectories. Substantially different from the mainstreams of Ehrenfest-like dynamics and surface hopping methods, the nuclear force in NaF involves the nonadiabatic force arising from the nonadiabatic coupling between different electronic states, in addition to the adiabatic force contributed by a single adiabatic electronic state. NaF is capable of faithfully describing the interplay between electronic and nuclear motion in a broad regime, which covers where the relevant electronic states keep coupled in a wide range or all the time and where the bifurcation characteristic of nuclear motion is essential. NaF is derived from the exact generalized phase space formulation with coordinate-momentum variables, where constraint phase space (CPS) is employed for discrete electronic-state degrees of freedom (DOFs) and infinite Wigner phase space is used for continuous nuclear DOFs. We propose efficient integrators for the equations of motion of NaF in both adiabatic and diabatic representations. Since the formalism in the CPS formulation is not unique, NaF can in principle be implemented with various phase space representations of the time correlation function (TCF) for the time-dependent property. They are applied to a suite of representative gas-phase and condensed-phase benchmark models where numerically exact results are available for comparison. It is shown that NaF is relatively insensitive to the phase space representation of the electronic TCF and will be a potential tool for practical and reliable simulations of the quantum mechanical behavior of both electronic and nuclear dynamics of nonadiabatic transition processes in real systems.

Energy conservation in real-time nuclear-electronic orbital Ehrenfest dynamics

Real-time nuclear-electronic orbital Ehrenfest (RT-NEO-Ehrenfest) dynamics methods provide a first-principles approach for describing nonadiabatic molecular processes with nuclear quantum effects. For an efficient description of proton transfer within RT-NEO-Ehrenfest dynamics, the basis function center associated with the quantum proton can be allowed to move classically. This traveling proton basis (TPB) approach effectively captures proton quantum dynamics, although its energy conservation behavior is not yet fully satisfactory. Two recently proposed TPB approaches, in principle, conserve the extended energy, which includes both the system energy and the kinetic energy associated with the proton basis function center. Herein, a thermostatted TPB approach is proposed to improve the conservation of the system energy, excluding the kinetic energy associated with the proton basis function center. In this approach, the quantum proton dynamics are modulated by dynamically rescaling the proton momentum operator to maintain the system energy conservation. With the excited-state intramolecular proton transfer of o-hydroxybenzaldehyde as an example, this approach is shown to significantly improve the system energy conservation while preserving the accuracy of the quantum proton dynamics as achieved in the original TPB approach.

A basis-free phase space electronic Hamiltonian that recovers beyond Born-Oppenheimer electronic momentum and current density

We present a phase-space electronic Hamiltonian ĤPS (parameterized by both nuclear position X and momentum P) that boosts each electron into the moving frame of the nuclei that are closest in real space. The final form for the phase space Hamiltonian does not assume the existence of an atomic orbital basis, and relative to standard Born-Oppenheimer theory, the newly proposed one-electron operators can be expressed directly as functions of electronic and nuclear positions and momentum. We show that (i) quantum-classical dynamics along such a Hamiltonian maintains momentum conservation and that (ii) diagonalizing such a Hamiltonian can recover the electronic momentum and electronic current density reasonably well. In conjunction with other reports in the literature that such a phase-space approach can also recover vibrational circular dichroism spectra, we submit that the present phase-space approach offers a testable and powerful approach to post-BO electronic structure theory. Moreover, the approach is inexpensive and can be immediately applied to simulations of chiral induced spin selectivity experiments (where the transfer of angular momentum between nuclei and electrons is considered critical).

Assessing the performance of coupled-trajectory schemes on full-dimensional two-state linear vibronic coupling models

We investigate the performance of coupled-trajectory methods for nonadiabatic molecular dynamics in simulating the photodynamics of 4-(dimethylamino)benzonitrile (DMABN) and fulvene, with electronic structure provided by linear vibrational coupling models. We focus on the coupled-trajectory mixed quantum-classical (CTMQC) algorithm and on the (combined) coupled-trajectory Tully surface hopping [(C)CTTSH] in comparison to independent-trajectory approaches, such as multi-trajectory Ehrenfest and Tully surface hopping. Our analysis includes not only electronic populations but also additional electronic and nuclear properties in position and momentum space. For both DMABN and fulvene, the recently developed CCTTSH algorithm successfully resolves the internal inconsistencies of coupled-trajectory Tully surface hopping. Instead, we find that DMABN highlights a significant weakness of CTMQC, which arises when the trajectories remain for a long time in the vicinity of a region of strong nonadiabaticity.

Ab initio spin-mapping non-adiabatic dynamics simulations of photochemistry

We perform on-the-fly non-adiabatic molecular dynamics simulations using the recently developed spin-mapping formalism. Two quantum dynamics approaches based on this mapping formalism, (i) the fully linearized Spin-LSC and (ii) the partially linearized Spin-PLDM, are explored using the quasi-diabatic propagation scheme. We have performed dynamics simulations in four ab initio molecular models for which benchmark ab initio multiple spawning (AIMS) data have been published. We find that the spin-LSC and the previously reported symmetric quasi-classical (SQC) approaches provide nearly equivalent population dynamics. While we expected the more involved spin-PLDM method to provide superior accuracy compared to the other mapping-based approaches, SQC and spin-LSC, we found that it performed with equivalent accuracy compared to the AIMS benchmark results. We further explore the underpinnings of the spin-PLDM correlation function by decomposing its N2 density matrix-focused initial conditions, where N is the number of states in the quantum subsystem. Finally, we found an approximate form of the spin-PLDM correlation function, which simplifies the simulation and reduces the computational costs from N2 to N.

Fermi's Golden Rule Rate Expression for Transitions Due to Nonadiabatic Derivative Couplings in the Adiabatic Basis

Starting from a general molecular Hamiltonian expressed in the basis of adiabatic electronic and nuclear position states, where a compact and complete expression for the nonadiabatic derivative coupling (NDC) Hamiltonian term is obtained, we provide a general analysis of the Fermi's golden rule (FGR) rate expression for nonadiabatic transitions between adiabatic states. We then consider a quasi-adiabatic approximation that uses crude adiabatic states and NDC couplings, both evaluated at the minimum potential energy configuration of the initial adiabatic state, for the definition of the zeroth and first-order terms of the Hamiltonian. Although the application of this approximation is rather limited, it allows deriving a general FGR rate expression without further approximation while accounting for non-Condon contribution to the FGR rate arising from momentum operators of NDC terms and its coupling with vibronic displacements. For a generic and widely used model where all nuclear degrees of freedom and environmental effects are represented as linearly coupled harmonic oscillators, we derive a closed-form FGR rate expression that requires only Fourier transform. The resulting rate expression includes quadratic contributions of NDC terms and their couplings to Franck-Condon modes, which require evaluation of two additional bath spectral densities in addition to the conventional one that appears in a typical FGR rate theory based on the Condon approximation. Model calculations for the case where nuclear vibrations consist of both a sharp high-frequency mode and an Ohmic bath spectral density illustrate new features and implications of the rate expression. We then apply our theoretical expression to the nonradiative decay from the first excited singlet state of azulene, which illustrates the utility and implications of our theoretical results.

Exact factorization method for bound vibrational states: An analytical tool for accurate approximations

The Exact Factorization (XF) method represents an interesting formulation of the Schrödinger equation where subsystem wavefunctions are exactly coupled. Here, I show that the XF method can be employed as an analytical tool to study the quantum vibrational problem of bound systems. In particular, after elaborating suitable XF-based wavefunction Ansätze, the ground-state energy approximated expression for bilinearly and quartically coupled harmonic oscillators is estimated. The XF-based analytical solution is compared with adiabatic and perturbative ones, and it is usually found to be an order of magnitude more accurate than these for estimating the anharmonic and coupling correction part of the ground-state energy. This procedure will possibly increase the numerical stability and accuracy of perturbative or Hartree-product based methods when applied to bound state calculations.

Vibronic Interactions in the Photoelectron Spectra of CAl3Ge-: A Theoretical Study

This study probes the vibronic interactions in the photoelectron spectra of CAl3Ge-, exploring its six excited electronic states through an approach that combines the ab initio electronic structure calculations and the quantum nuclear dynamics. Central to this investigation is utilizing a model diabatic Hamiltonian, which allows for the exact evaluation of Hamiltonian parameters and fitting potential energy cuts (PECs). Notably, the analysis of these PECs uncovers pronounced nonadiabatic effects within the photoelectron spectra, emphasized by the presence of multiple conical intersections. The investigation of nuclear dynamics, with and without the same spatial symmetry coupling, is achieved by employing time-dependent (TD) and time-independent (TI) quantum mechanical methods. These nuclear dynamical studies effectively simulated all the photoelectron spectral bands. Eventually, the theoretical findings conformed well with available experimental observations, highlighting the nonadiabatic effects among the closely situated spectral bands.

A semiclassical non-adiabatic phase-space approach to molecular translations and rotations: Surface hopping with electronic inertial effects

We demonstrate that working with a correct phase-space electronic Hamiltonian captures electronic inertial effects. In particular, we show that phase space surface hopping dynamics do not suffer (at least to very high order) from non-physical non-adiabatic transitions between electronic eigenstates during the course of pure nuclear translational and rotational motion. This work opens up many new avenues for quantitatively investigating complex phenomena, including angular momentum transfer between chiral phonons and electrons as well as chiral-induced spin selectivity effects.

Electronic Coherences in Molecules: The Projected Nuclear Quantum Momentum as a Hidden Agent

Electronic coherences are key to understanding and controlling photoinduced molecular transformations. We identify a crucial quantum-mechanical feature of electron-nuclear correlation, the projected nuclear quantum momenta, essential to capture the correct coherence behavior. For simulations, we show that, unlike traditional trajectory-based schemes, exact-factorization-based methods approximate these correlation terms and correctly capture electronic coherences in a range of situations, including their spatial dependence, an important aspect that influences subsequent electron dynamics and that is becoming accessible in more experiments.

Nuclear Quantum Effects in Quantum Mechanical/Molecular Mechanical Free Energy Simulations of Ribonucleotide Reductase

The enzyme ribonucleotide reductase plays a critical role in DNA synthesis and repair. Its mechanism requires long-range radical transfer through a series of proton-coupled electron transfer (PCET) steps. Nuclear quantum effects such as zero-point energy, proton delocalization, and hydrogen tunneling are known to be important in PCET. We present a strategy for efficiently incorporating nuclear quantum effects into multidimensional free energy surfaces and real-time dynamical simulations for condensed-phase systems such as enzymes. This strategy is based on the nuclear-electronic orbital (NEO) method, which treats specified protons quantum mechanically on the same level as the electrons. NEO density functional theory (NEO-DFT) is combined with the quantum mechanical/molecular mechanical finite temperature string method with umbrella sampling via a simple reweighting procedure. Application of this strategy to PCET between two tyrosines, Y731 and Y730, in ribonucleotide reductase illustrates that nuclear quantum effects could either raise or lower the free energy barrier, leading to a range of possible kinetic isotope effects. Real-time time-dependent DFT (RT-NEO-TDDFT) simulations highlight nuclear-electronic quantum dynamics. These approaches enable the incorporation of nuclear quantum effects into a wide range of chemically and biologically important processes.

An electronic phase-space Hamiltonian approach for electronic current density and vibrational circular dichroism

The Born-Oppenheimer framework stipulates that chemistry and physics occur on potential energy surfaces VBO(X) parameterized by a nuclear coordinate X, which are built by diagonalizing a BO Hamiltonian ĤBO(X). However, such a framework cannot recover many measurable chemical and physical features, including vibrational circular dichroism spectra. In this article, we show that a phase-space electronic Hamiltonian ĤPS(X,P), parameterized by both nuclear position X and momentum P, with a similar computational cost as solving ĤBO(X), can recover not just experimental vibrational circular dichroism signals but also a meaningful electronic current density that explains the features of the vibrational circular dichroism rotational strengths. Combined with earlier demonstrations that such Hamiltonians can also recover qualitatively correct electronic momenta with electronic densities that approximately satisfy a continuity equation, the data would suggest that, if one looks closely enough, chemistry in fact occurs on potential energy surfaces parameterized by both X and P, EPS(X, P). While the dynamical implications of such a phase-space electronic Hamiltonian are not yet known, we hypothesize that, by offering classical trajectories that explicitly offer nonzero electronic momentum while also conserving the total angular momentum (unlike Born-Oppenheimer theory), this new phase-space electronic structure Hamiltonian may well explain some fraction of the chiral-induced spin selectivity effect.

Toward a Correct Description of Initial Electronic Coherence in Nonadiabatic Dynamics Simulations

The recent improvement in experimental capabilities for interrogating and controlling molecular systems with ultrafast coherent light sources calls for the development of theoretical approaches that can accurately and efficiently treat electronic coherence. However, the most popular and practical nonadiabatic molecular dynamics techniques, Tully's fewest-switches surface hopping and Ehrenfest mean-field dynamics, are unable to describe the dynamics proceeding from an initial electronic coherence. While such issues are not encountered with the analogous coupled-trajectory algorithms or numerically exact quantum dynamics methods, applying such techniques necessarily comes with a higher computational cost. Here we show that a correct description of initial electronic coherence can indeed be achieved using independent-trajectory methods derived from the semiclassical mapping formalism. The key is the introduction of an initial sampling over the electronic phase space and a means of incorporating phase interference between trajectories, both of which are naturally achieved when working within the semiclassical mapping framework.

Extracting the Geometric Phase from the Ensemble of Trajectories

Traditionally, methods designed to investigate the effects of the geometric phase in reaction dynamics, such as including a vector potential in the nuclear Hamiltonian, necessitate the explicit manipulation of geometric phase-related terms in the adiabatic representation. In contrast, the diabatic representation provides an alternative approach that implicitly addresses the geometric phase and nonadiabatic issues. In this study, we present a method to directly extract the phase information on the geometric phase from the ensemble of interdependent trajectories utilizing the diabatic representation. This approach presents a direct means of quantitatively examining the geometric phase effects in dynamics and has the potential to yield observables suitable for experimental measurement.

Simulations of photoinduced processes with the exact factorization: state of the art and perspectives

This perspective offers an overview of the applications of the exact factorization of the electron-nuclear wavefunction to the domain of theoretical photochemistry, where the aim is to gain insights into the ultrafast dynamics of molecular systems via simulations of their excited-state dynamics beyond the Born-Oppenheimer approximation. The exact factorization offers an alternative viewpoint to the Born-Huang representation for the interpretation of dynamical processes involving the electronic ground and excited states as well as their coupling through the nuclear motion. Therefore, the formalism has been used to derive algorithms for quantum molecular-dynamics simulations where the nuclear motion is treated using trajectories and the electrons are treated quantum mechanically. These algorithms have the characteristic features of being based on coupled and on auxiliary trajectories, and have shown excellent performance in describing a variety of excited-state processes, as this perspective illustrates. We conclude with a discussion on the authors' point of view on the future of the exact factorization.

Emergence of the Molecular Geometric Phase from Exact Electron-Nuclear Dynamics

Geometric phases play a crucial role in diverse fields. In molecules, they appear when a reaction path encircles an intersection between adiabatic potential energy surfaces and the molecular wave function experiences quantum-mechanical interference effects. This intriguing effect, closely resembling the magnetic Aharonov-Bohm effect, crucially relies on the adiabatic description of the dynamics, and it is an open issue whether and how it persists in an exact quantum dynamical framework. Recent works suggest that the molecular geometric phase is an artifact of the adiabatic approximation, thereby challenging the entire concept. Here, building upon a recent investigation (Martinazzo, R.; Burghardt, I. Phys. Rev. Lett. 2024, 132, 243002), we address this issue using the exact factorization of the total wave function. We introduce instantaneous gauge-invariant phases separately for the electrons and nuclei and use them to monitor the phase difference between the trailing edges of a wavepacket encircling a conical intersection between adiabatic surfaces. The transition from the time-dependent open-path phase differences to the closed-path limit is examined, revealing how the phase differences in the electronic and nuclear subspaces compensate for each other upon path closure. In this way, we unambiguously demonstrate the role of the geometric phase in the interference process and shed light on its persistence beyond the adiabatic approximation.

Electronic Vector Potential from the Exact Factorization of a Complex Wavefunction

We generalize the definitions of local scalar potentials namedυ kin ${\upsilon _{{\rm{kin}}} }$ andυ N - 1 ${\upsilon _{N - 1} }$ , which are relevant to properly describe phenomena such as molecular dissociation with density-functional theory, to the case in which the electronic wavefunction corresponds to a complex current-carrying state. In such a case, an extra term in the form of a vector potential appears which cannot be gauged away. Both scalar and vector potentials are introduced via the exact factorization formalism which allows us to express the given Schrödinger equation as two coupled equations, one for the marginal and one for the conditional amplitude. The electronic vector potential is directly related to the paramagnetic current density carried by the total wavefunction and to the diamagnetic current density in the equation for the marginal amplitude. An explicit example of this vector potential in a triplet state of two non-interacting electrons is showcased together with its associated circulation, giving rise to a non-vanishing geometric phase. Some connections with the exact factorization for the full molecular wavefunction beyond the Born-Oppenheimer approximation are also discussed.

A Phase-Space Electronic Hamiltonian For Vibrational Circular Dichroism

We show empirically that a phase-space non-Born-Oppenheimer electronic Hamiltonian approach to quantum chemistry (where the electronic Hamiltonian is parametrized by both nuclear position and momentum, ĤPS(R,P)) is both a practical and accurate means to recover vibrational circular dichroism spectra. We further hypothesize that such a phase-space approach may lead to very new dynamical physics beyond spectroscopic circular dichroism, with potential implications for understanding chiral induced spin selectivity (CISS), noting that classical phase-space approaches conserve the total nuclear plus electronic momentum, whereas classical Born-Oppenheimer approaches do not (they conserve only the nuclear momentum).

Multiconfigurational Surface Hopping: a Time-Dependent Variational Approach with Momentum-Jump Trajectories

The Ehrenfest mean field dynamics and trajectory surface hopping have been widely used in nonadiabatic dynamics simulations. Based on the time-dependent variational principle (TDVP), the multiconfigurational Ehrenfest (MCE) method has also been developed and can be regarded as a multiconfigurational extension of the traditional Ehrenfest dynamics. However, it is not straightforward to apply the TDVP to surface hopping trajectories because there exists momentum jump during surface hops. To solve this problem, we here propose a multiconfigurational surface hopping (MCSH) method, where continuous momenta are obtained by linear interpolation and the interpolated trajectories are used to construct the basis functions for TDVP in a postprocessing manner. As demonstrated in a series of representative spin-boson models, MCSH achieves high accuracy with only several hundred trajectory bases and can uniformly improve the performance of surface hopping. In principle, MCSH can be combined with all kinds of mixed quantum-classical trajectories and thus has the potential to properly describe general nonadiabatic dynamics.

Exact factorization of the photon-electron-nuclear wavefunction: Formulation and coupled-trajectory dynamics

We employ the exact-factorization formalism to study the coupled dynamics of photons, electrons, and nuclei at the quantum mechanical level, proposing illustrative examples of model situations of nonadiabatic dynamics and spontaneous emission of electron-nuclear systems in the regime of strong light-matter coupling. We make a particular choice of factorization for such a multi-component system, where the full wavefunction is factored as a conditional electronic amplitude and a marginal photon-nuclear amplitude. Then, we apply the coupled-trajectory mixed quantum-classical (CTMQC) algorithm to perform trajectory-based simulations, by treating photonic and nuclear degrees of freedom on equal footing in terms of classical-like trajectories. The analysis of the time-dependent potentials of the theory along with the assessment of the performance of CTMQC allows us to point out some limitations of the current approximations used in CTMQC. Meanwhile, comparing CTMQC with other trajectory-based algorithms, namely multi-trajectory Ehrenfest and Tully surface hopping, demonstrates the better quality of CTMQC predictions.

Nonadiabatic Dynamics with Exact Factorization: Implementation and Assessment

In this work, we report our implementation of several independent-trajectory mixed-quantum-classical (ITMQC) nonadiabatic dynamics methods based on exact factorization (XF) in the Libra package for nonadiabatic and excited-state dynamics. Namely, the exact factorization surface hopping (SHXF), mixed quantum-classical dynamics (MQCXF), and mean-field (MFXF) are introduced. Performance of these methods is compared to that of several traditional surface hopping schemes, such as the fewest-switches surface hopping (FSSH), branching-corrected surface hopping (BCSH), and the simplified decay of mixing (SDM), as well as conventional Ehrenfest (mean-field, MF) method. Based on a comprehensive set of 1D model Hamiltonians, we find the ranking SHXF ≈ MQCXF > BCSH > SDM > FSSH ≫ MF, with the BCSH sometimes outperforming the XF methods in terms of describing coherences. Although the MFXF method can yield reasonable populations and coherences for some cases, it does not conserve the total energy and is therefore not recommended. We also find that the branching correction for auxiliary trajectories is important for the XF methods to yield accurate populations and coherences. However, the branching correction can worsen the quality of the energy conservation in the MQCXF. Finally, we find that using the time-dependent Gaussian width approximation used in the XF methods for computing decoherence correction can improve the quality of energy conservation in the MQCXF dynamics. The parameter-free scheme of Subotnik for computing the Gaussian widths is found to deliver the best performance in situations where such widths are not known a priori.

The development of the QM/MM interface and its application for the on-the-fly QM/MM nonadiabatic dynamics in JADE package: Theory, implementation, and applications

Understanding the nonadiabatic dynamics of complex systems is a challenging task in computational photochemistry. Herein, we present an efficient and user-friendly quantum mechanics/molecular mechanics (QM/MM) interface to run on-the-fly nonadiabatic dynamics. Currently, this interface consists of an independent set of codes designed for general-purpose use. Herein, we demonstrate the ability and feasibility of the QM/MM interface by integrating it with our long-term developed JADE package. Tailored to handle nonadiabatic processes in various complex systems, especially condensed phases and protein environments, we delve into the theories, implementations, and applications of on-the-fly QM/MM nonadiabatic dynamics. The QM/MM approach is established within the framework of the additive QM/MM scheme, employing electrostatic embedding, link-atom inclusion, and charge-redistribution schemes to treat the QM/MM boundary. Trajectory surface-hopping dynamics are facilitated using the fewest switches algorithm, encompassing classical and quantum treatments for nuclear and electronic motions, respectively. Finally, we report simulations of nonadiabatic dynamics for two typical systems: azomethane in water and the retinal chromophore PSB3 in a protein environment. Our results not only illustrate the power of the QM/MM program but also reveal the important roles of environmental factors in nonadiabatic processes.

Dynamics of the Molecular Geometric Phase

The fate of the molecular geometric phase in an exact dynamical framework is investigated with the help of the exact factorization of the wave function and a recently proposed quantum hydrodynamical description of its dynamics. An instantaneous, gauge-invariant phase is introduced for arbitrary paths in nuclear configuration space in terms of hydrodynamical variables, and shown to reduce to the adiabatic geometric phase when the state is adiabatic and the path is closed. The evolution of the closed-path phase over time is shown to adhere to a Maxwell-Faraday induction law, with nonconservative forces arising from the electron dynamics that play the role of electromotive forces. We identify the pivotal forces that are able to change the value of the phase, thereby challenging any topological argument. Nonetheless, negligible changes in the phase occur when the local dynamics along the probe loop is approximately adiabatic. That is, the geometric phase effects that arise in an adiabatic limiting situation remain suitable to effectively describe certain dynamic observables.

Generalized Semiclassical Ehrenfest Method: A Route to Wave Function-Free Photochemistry and Nonadiabatic Dynamics with Only Potential Energies and Gradients

We reconsider recent methods by which direct dynamics calculations of electronically nonadiabatic processes can be carried out while requiring only adiabatic potential energies and their gradients. We show that these methods can be understood in terms of a new generalization of the well-known semiclassical Ehrenfest method. This is convenient because it eliminates the need to evaluate electronic wave functions and their matrix elements along the mixed quantum-classical trajectories. The new approximations and procedures enabling this advance are the curvature-driven approximation to the time-derivative coupling, the generalized semiclassical Ehrenfest method, and a new gradient correction scheme called the time-derivative matrix (TDM) scheme. When spin-orbit coupling is present, one can carry out dynamics calculations in the fully adiabatic basis using potential energies and gradients calculated without spin-orbit coupling plus the spin-orbit coupling matrix elements. Even when spin-orbit coupling is neglected, the method is useful because it allows calculations by electronic structure methods for which nonadiabatic coupling vectors are unavailable. In order to place the new considerations in context, the article starts out with a review of background material on trajectory surface hopping, the semiclassical Ehrenfest scheme, and methods for incorporating decoherence. We consider both internal conversion and intersystem crossing. We also review several examples from our group of successful applications of the curvature-driven approximation.

Exploring Exact-Factorization-Based Trajectories for Low-Energy Dynamics near a Conical Intersection

We study low-energy dynamics generated by a two-dimensional two-state Jahn-Teller Hamiltonian in the vicinity of a conical intersection using quantum wave packet and trajectory dynamics. Recently, these dynamics were studied by comparing the adiabatic representation and the exact factorization, with the purpose to highlight the different nature of topological-phase and geometric-phase effects arising in the two theoretical representations of the same problem. Here, we employ the exact factorization to understand how to accurately model low-energy dynamics in the vicinity of a conical intersection using an approximate description of the nuclear motion that uses trajectories. We find that since nonadiabatic effects are weak but non-negligible, the trajectory-based description that invokes the classical approximation struggles to capture the correct behavior.

Formulation of transition dipole gradients for non-adiabatic dynamics with polaritonic states

A general formulation of the strong coupling between photons confined in a cavity and molecular electronic states is developed for the state-interaction state-average spin-restricted ensemble-referenced Kohn-Sham method. The light-matter interaction is included in the Jaynes-Cummings model, which requires the derivation and implementation of the analytical derivatives of the transition dipole moments between the molecular electronic states. The developed formalism is tested in the simulations of the nonadiabatic dynamics in the polaritonic states resulting from the strong coupling between the cavity photon mode and the ground and excited states of the penta-2,4-dieniminium cation, also known as PSB3. Comparison with the field-free simulations of the excited-state decay dynamics in PSB3 reveals that the light-matter coupling can considerably alter the decay dynamics by increasing the excited state lifetime and hindering photochemically induced torsion about the C=C double bonds of PSB3. The necessity of obtaining analytical transition dipole gradients for the accurate propagation of the dynamics is underlined.

A Many-Body Perspective of Nuclear Quantum Effects in Aqueous Clusters

Nuclear quantum effects play an important role in the structure and thermodynamics of aqueous systems. By performing a many-body expansion with nuclear-electronic orbital (NEO) theory, we show that proton quantization can give rise to significant energetic contributions for many-body interactions spanning several molecules in single-point energy calculations of water clusters. Although zero-point motion produces a large increase in energy at the one-body level, nuclear quantum effects serve to stabilize higher-order molecular interactions. These results are significant because they demonstrate that nuclear quantum effects play a nontrivial role in many-body interactions of aqueous systems. Our approach also provides a pathway for incorporating nuclear quantum effects into water potential energy surfaces. The NEO approach is advantageous for many-body expansion analyses because it includes nuclear quantum effects directly in the energies.

A simple one-electron expression for electron rotational factors

Within the context of fewest-switch surface hopping (FSSH) dynamics, one often wishes to remove the angular component of the derivative coupling between states J and K. In a previous set of papers, Shu et al. [J. Phys. Chem. Lett. 11, 1135-1140 (2020)] posited one approach for such a removal based on direct projection, while we isolated a second approach by constructing and differentiating a rotationally invariant basis. Unfortunately, neither approach was able to demonstrate a one-electron operatorÔ whose matrix element JÔK was the angular component of the derivative coupling. Here, we show that a one-electron operator can, in fact, be constructed efficiently in a semi-local fashion. The present results yield physical insight into designing new surface hopping algorithms and are of immediate use for FSSH calculations.

Practical phase-space electronic Hamiltonians for ab initio dynamics

Modern electronic structure theory is built around the Born-Oppenheimer approximation and the construction of an electronic Hamiltonian Ĥel(X) that depends on the nuclear position X (and not the nuclear momentum P). In this article, using the well-known theory of electron translation (Γ') and rotational (Γ″) factors to couple electronic transitions to nuclear motion, we construct a practical phase-space electronic Hamiltonian that depends on both nuclear position and momentum, ĤPS(X,P). While classical Born-Oppenheimer dynamics that run along the eigensurfaces of the operator Ĥel(X) can recover many nuclear properties correctly, we present some evidence that motion along the eigensurfaces of ĤPS(X,P) can better capture both nuclear and electronic properties (including the elusive electronic momentum studied by Nafie). Moreover, only the latter (as opposed to the former) conserves the total linear and angular momentum in general.

Consistent Construction of the Density Matrix from Surface Hopping Trajectories

Proper construction of the density matrix based on surface hopping trajectories remains a difficult problem. Due to the well-known overcoherence in traditional surface hopping simulations, the electronic wave function cannot be used directly. In this work, we propose a consistent density matrix construction method, which takes the advantage of occupation of active states to rescale the coherence calculated by wave functions and ensures the intrinsic consistency of the density matrix. This new trajectory analysis method can be used for both Tully's fewest switches surface hopping (FSSH) and our recently proposed branching corrected surface hopping (BCSH). As benchmarked in both one- and two-dimensional standard scattering models, the new approach combined with BCSH trajectories achieves highly accurate time-dependent spatial distributions of adiabatic populations and coherence compared to exact quantum results.

Floquet Engineering of a Diatomic Molecule through a Bichromatic Radiation Field

We report on a theoretical study of a Cs2 molecule illuminated by two lasers and show how this can result in novel quantum dynamics. We reveal that these interactions facilitate the bypass of the non-crossing rule, forming light-induced conical intersections and modifiable avoided crossings. Our findings show how laser field orientation and strength, along with initial phase differences, can control molecular-state transitions, especially on the micromotion scale. We also extensively discuss how the interaction of radiation with matter gives rise to the emergence of potential energy surfaces of hybrids of radiation and molecular states. This research advances a technique for manipulating photoassociation processes in Cs2 molecules, offering potential new avenues in quantum control.

Diagonalizing the Born-Oppenheimer Hamiltonian via Moyal perturbation theory, nonadiabatic corrections, and translational degrees of freedom

This article describes a method for calculating higher order or nonadiabatic corrections in Born-Oppenheimer theory and its interaction with the translational degrees of freedom. The method uses the Wigner-Weyl correspondence to map nuclear operators into functions on the classical phase space and the Moyal star product to represent operator multiplication on those functions. These are explained in the body of the paper. The result is a power series in κ2, where κ = (m/M)1/4 is the usual Born-Oppenheimer parameter. The lowest order term is the usual Born-Oppenheimer approximation, while higher order terms are nonadiabatic corrections. These are needed in calculations of electronic currents, momenta, and densities. The separation of nuclear and electronic degrees of freedom takes place in the context of the exact symmetries (for an isolated molecule) of translations and rotations, and these, especially translations, are explicitly incorporated into our discussion. This article presents an independent derivation of the Moyal expansion in molecular Born-Oppenheimer theory. We show how electronic currents and momenta can be calculated within the framework of Moyal perturbation theory; we derive the transformation laws of the electronic Hamiltonian, the electronic eigenstates, and the derivative couplings under translations; we discuss in detail the rectilinear motion of the molecular center of mass in the Born-Oppenheimer representation; and we show how the elimination of the translational components of the derivative couplings leads to a unitary transformation that has the effect of exactly separating the translational degrees of freedom.

Exact-Factorization-Based Surface Hopping without Velocity Adjustment

While surface hopping has emerged as a powerful method for simulating non-adiabatic dynamics in large molecules, the ad hoc nature of the necessary velocity adjustments and decoherence corrections in the algorithm somewhat reduces its reliability. Here we propose a new scheme that eliminates these aspects by combining the nuclear equation from the quantum-trajectory surface-hopping approach with the electronic equation derived from the exact-factorization approach. The resulting method, denoted QTSH-XF, yields a surface-hopping method on firmer ground than previous and is shown to successfully capture dynamics in Tully models and in a linear vibronic coupling model of the photoexcited uracil cation.

Real-Time Time-Dependent Density Functional Theory for Simulating Nonequilibrium Electron Dynamics

The explicit real-time propagation approach for time-dependent density functional theory (RT-TDDFT) has increasingly become a popular first-principles computational method for modeling various time-dependent electronic properties of complex chemical systems. In this Perspective, we provide a nontechnical discussion of how this first-principles simulation approach has been used to gain novel physical insights into nonequilibrium electron dynamics phenomena in recent years. Following a concise overview of the RT-TDDFT methodology from a practical standpoint, we discuss our recent studies on the electronic stopping of DNA in water and the Floquet topological phase as examples. Our discussion focuses on how RT-TDDFT simulations played a unique role in deriving new scientific understandings. We then discuss existing challenges and some new advances at the frontier of RT-TDDFT method development for studying increasingly complex dynamic phenomena and systems.

Nonadiabatic dynamics with classical trajectories: The problem of an initial coherent superposition of electronic states

Advances in coherent light sources and development of pump-probe techniques in recent decades have opened the way to study electronic motion in its natural time scale. When an ultrashort laser pulse interacts with a molecular target, a coherent superposition of electronic states is created and the triggered electron dynamics is coupled to the nuclear motion. A natural and computationally efficient choice to simulate this correlated dynamics is a trajectory-based method where the quantum-mechanical electronic evolution is coupled to a classical-like nuclear dynamics. These methods must approximate the initial correlated electron-nuclear state by associating an initial electronic wavefunction to each classical trajectory in the ensemble. Different possibilities exist that reproduce the initial populations of the exact molecular wavefunction when represented in a basis. We show that different choices yield different dynamics and explore the effect of this choice in Ehrenfest, surface hopping, and exact-factorization-based coupled-trajectory schemes in a one-dimensional two-electronic-state model system that can be solved numerically exactly. This work aims to clarify the problems that standard trajectory-based techniques might have when a coherent superposition of electronic states is created to initialize the dynamics, to discuss what properties and observables are affected by different choices of electronic initial conditions and to point out the importance of quantum-momentum-induced electronic transitions in coupled-trajectory schemes.

Total angular momentum conservation in Ehrenfest dynamics with a truncated basis of adiabatic states

We show that standard Ehrenfest dynamics does not conserve linear and angular momentum when using a basis of truncated adiabatic states. However, we also show that previously proposed effective Ehrenfest equations of motion [M. Amano and K. Takatsuka, "Quantum fluctuation of electronic wave-packet dynamics coupled with classical nuclear motions," J. Chem. Phys. 122, 084113 (2005) and V. Krishna, "Path integral formulation for quantum nonadiabatic dynamics and the mixed quantum classical limit," J. Chem. Phys. 126, 134107 (2007)] involving the non-Abelian Berry force do maintain momentum conservation. As a numerical example, we investigate the Kramers doublet of the methoxy radical using generalized Hartree-Fock with spin-orbit coupling and confirm that angular momentum is conserved with the proper equations of motion. Our work makes clear some of the limitations of the Born-Oppenheimer approximation when using ab initio electronic structure theory to treat systems with unpaired electronic spin degrees of freedom, and we demonstrate that Ehrenfest dynamics can offer much improved, qualitatively correct results.

Squeezed Protons and Infrared Plasmonic Resonance Energy Transfer

Unusual nuclear quantum effects may emerge near noble metal nanostructures such as squeezed vibrational states in molecular junctions and plasmonic resonance energy transfer in the infrared domain. Herein, nuclear quantum effects near heavy metals are studied by nuclear-electronic orbital density functional theory (NEO-DFT) with an effective core potential. For a quantum proton sandwiched between a pair of gold tips modeled by two Au6 clusters, NEO-DFT calculations suggest that the quantum proton density can be squeezed as the tip distance decreases. For an HF molecule placed near a one-dimensional Au nanowire composed of up to 34 Au atoms, real-time NEO time-dependent density functional theory (RT-NEO-TDDFT) shows that the infrared plasmonic motion within the Au nanowire may resonantly transfer electronic energy to the HF proton vibrational stretch mode. Overall, these calculations illustrate the advantages of the NEO approach for probing nuclear quantum effects, such as squeezed proton vibrational states and infrared plasmonic resonance energy transfer.

Investigating the Photodynamics of trans-Azobenzene with Coupled Trajectories

In this work, we present the first implementation of coupled-trajectory Tully surface hopping (CT-TSH) suitable for applications to molecular systems. We combine CT-TSH with the semiempirical floating occupation molecular orbital-configuration interaction electronic structure method to investigate the photoisomerization dynamics of trans-azobenzene. Our study shows that CT-TSH can capture correctly decoherence effects in this system, yielding consistent electronic and nuclear dynamics in agreement with (standard) decoherence-corrected TSH. Specifically, CT-TSH is derived from the exact factorization and the electronic coefficients' evolution is directly influenced by the coupling of trajectories, resulting in the improvement of internal consistency if compared to standard TSH.

Resolving the Experimental Photoelectron Spectra of CAl3Si. J Phys Chem A 2024;128:355-369. [PMID: 38189257 DOI: 10.1021/acs.jpca.3c06295] [Citation(s) in RCA: 0] [Impact Index Per Article: 0]

The experimental photoelectron spectra concerning the six electronic states of CAl3Si- are resolved through electronic structure calculations and quantum nuclear dynamics in this study. It incorporates a model diabatic Hamiltonian to evaluate the coupling parameters and fit the potential energy curves (PECs). The analysis of these PECs showed us that there are sufficient nonadiabatic effects in the photoelectron spectra through the presence of various conical intersections. Poisson intensity distributions (PIDs) and the wave packet density plots are utilized for assigning the fundamental and first overtone excitations. The nuclear dynamics study is accomplished by employing time-dependent (TD) and time-independent (TI) quantum chemistry methods. Ultimately, our theoretical results concurred well with the experimental findings exhibiting vibronic coupling amidst the nearly positioned electronic states.

Photoelectron spectra of benzene: Can path dependent diabatic surfaces provide unique observables?

While carrying out Beyond Born-Oppenheimer theory based diabatization, the solutions of adiabatic-to-diabatic transformation equations depend on the paths of integration over two-dimensional cross-sections of multi-dimensional space of nuclear degrees of freedom. It is shown that such path-dependent solutions leading to diabatic potential energy surface matrices computed along any two different paths are related through an orthogonal matrix, and thereby, those surface matrices should provide unique observables. While exploring the numerical validity of the theoretical framework, we construct diabatic Hamiltonians for the five low-lying electronic states (X̃2E1g, B̃2E2g, and C̃2A2u) of benzene radical cation (C6H6+) along three different approaches of contour integration over two dimensional nuclear planes constituted by seven non-adiabatically active normal modes. Three different diabatic surface matrices are further employed to generate the photoelectron spectra of the benzene molecule (C6H6). It is interesting to note that the spectral peak positions and intensity patterns for all three cases are almost close to each other and also exhibit very good agreement with the experimental results.

On the Nature of Geometric and Topological Phases in the Presence of Conical Intersections

The observable nature of topological phases related to conical intersections in molecules is studied. Topological phases should be ubiquitous in molecular processes, but their elusive character has often made them a topic of discussion. To shed some light on this issue, we simulate the dynamics governed by a Jahn-Teller Hamiltonian and analyze it employing two theoretical representations of the molecular wave function: the adiabatic and the exact factorization. We find fundamental differences between effects related to topological phases arising exclusively in the adiabatic representation, and thus not related to any physical observable, and geometric phases within the exact factorization that can be connected to an observable quantity. We stress that while the topological phase of the adiabatic representation is an intrinsic property of the Hamiltonian, the geometric phase of the exact factorization depends on the dynamics that the system undergoes and is connected to the circulation of the nuclear momentum field.

Toward Accurate Post-Born-Oppenheimer Molecular Simulations on Quantum Computers: An Adaptive Variational Eigensolver with Nuclear-Electronic Frozen Natural Orbitals

Nuclear quantum effects such as zero-point energy and hydrogen tunneling play a central role in many biological and chemical processes. The nuclear-electronic orbital (NEO) approach captures these effects by treating selected nuclei quantum mechanically on the same footing as electrons. On classical computers, the resources required for an exact solution of NEO-based models grow exponentially with system size. By contrast, quantum computers offer a means of solving this problem with polynomial scaling. However, due to the limitations of current quantum devices, NEO simulations are confined to the smallest systems described by minimal basis sets, whereas realistic simulations beyond the Born-Oppenheimer approximation require more sophisticated basis sets. For this purpose, we herein extend a hardware-efficient ADAPT-VQE method to the NEO framework in the frozen natural orbital (FNO) basis. We demonstrate on H2 and D2 molecules that the NEO-FNO-ADAPT-VQE method reduces the CNOT count by several orders of magnitude relative to the NEO unitary coupled cluster method with singles and doubles while maintaining the desired accuracy. This extreme reduction in the CNOT gate count is sufficient to permit practical computations employing the NEO method─an important step toward accurate simulations involving nonclassical nuclei and non-Born-Oppenheimer effects on near-term quantum devices. We further show that the method can capture isotope effects, and we demonstrate that inclusion of correlation energy systematically improves the prediction of difference in the zero-point energy (ΔZPE) between isotopes.

Stabilized coupled trajectory mixed quantum-classical algorithm with improved energy conservation: CTMQC-EDI

Coupled trajectory mixed quantum-classical (CTMQC) dynamics is a rigorous approach to trajectory-based non-adiabatic dynamics, which has recently seen an improvement to energy conservation via the introduction of the CTMQC-E algorithm. Despite this, the method's two key quantities distinguishing it from Ehrenfest dynamics, the modified Born-Oppenheimer momentum and the quantum momentum, require regularization procedures in certain circumstances. Such procedures in the latter can cause instabilities, leading to undesirable effects, such as energy drift and spurious population transfer, which is expected to become increasingly prevalent when the system gets larger as such events would happen more frequently. We propose a further modification to CTMQC-E, which includes a redefinition of the quantum momentum, CTMQC-EDI (double intercept), such that it has no formal divergences. We then show for Tully models I-III and the double arch model that the algorithm has greatly improved total energy conservation and negligible spurious population transfer at all times, in particular in regions of strong non-adiabatic coupling. CTMQC-EDI, therefore, shows promise as a numerically robust non-adiabatic dynamics technique that accounts for decoherence from first principles and that is scalable to large molecular systems and materials.

First-Principles Approach for Coupled Quantum Dynamics of Electrons and Protons in Heterogeneous Systems

The coupled quantum dynamics of electrons and protons is ubiquitous in many dynamical processes involving light-matter interaction, such as solar energy conversion in chemical systems and photosynthesis. A first-principles description of such nuclear-electronic quantum dynamics requires not only the time-dependent treatment of nonequilibrium electron dynamics but also that of quantum protons. Quantum mechanical correlation between electrons and protons adds further complexity to such coupled dynamics. Here we extend real-time nuclear-electronic orbital time-dependent density functional theory (RT-NEO-TDDFT) to periodic systems and perform first-principles simulations of coupled quantum dynamics of electrons and protons in complex heterogeneous systems. The process studied is an electronically excited-state intramolecular proton transfer of o-hydroxybenzaldehyde in water and at a silicon (111) semiconductor-molecule interface. These simulations illustrate how environments such as hydrogen-bonding water molecules and an extended material surface impact the dynamical process on the atomistic level. Depending on how the molecule is chemisorbed on the surface, excited-state electron transfer from the molecule to the semiconductor surface can inhibit ultrafast proton transfer within the molecule. This Letter elucidates how heterogeneous environments influence the balance between the quantum mechanical proton transfer and excited electron dynamics. The periodic RT-NEO-TDDFT approach is applicable to a wide range of other photoinduced heterogeneous processes.

Zombie cats on the quantum-classical frontier: Wigner-Moyal and semiclassical limit dynamics of quantum coherence in molecules

In this paper, we investigate the time evolution of quantum coherence-the off-diagonal elements of the density matrix of a multistate quantum system-from the perspective of the Wigner-Moyal formalism. This approach provides an exact phase space representation of quantum mechanics. We consider the coherent evolution of nuclear wavepackets in a molecule with two electronic states. For harmonic potentials, the problem is analytically soluble for both a fully quantum mechanical description and a semiclassical description. We highlight the serious deficiencies of the semiclassical treatment of coherence for general systems and illustrate how even qualitative accuracy requires higher order terms in the Moyal expansion to be included. The model provides an experimentally relevant example of a molecular Schrödinger's cat state. The alive and dead cats of the exact two-state quantum evolution collapse into a "zombie" cat in the semiclassical limit-an averaged behavior, neither alive nor dead, leading to significant errors. The inclusion of the Moyal correction restores a faithful simultaneously alive and dead representation of the cat that is experimentally observable.

Significance of Energy Conservation in Coupled-Trajectory Approaches to Nonadiabatic Dynamics

Through approximating electron-nuclear correlation terms in the exact factorization approach, trajectory-based methods have been derived and successfully applied to the dynamics of a variety of light-induced molecular processes, capturing quantum (de)coherence effects rigorously. These terms account for the coupling among the trajectories, recovering the nonlocal nature of quantum nuclear dynamics that is completely overlooked in traditional independent-trajectory algorithms. Nevertheless, some of the approximations introduced in the derivation of some of these methods do not conserve the total energy. We analyze energy conservation in the coupled-trajectory mixed quantum-classical (CTMQC) algorithm and explore the performance of a modified algorithm, CTMQC-E, where some of the terms are redefined to restore energy conservation. A set of molecular models is used as a test, namely, 2-cis-penta-2,4-dienimium cation, bis(methylene) adamantyl radical cation, butatriene cation, uracil radical cation, and neutral pyrazine.

Orbital-Free Density Functional Theory: An Attractive Electronic Structure Method for Large-Scale First-Principles Simulations

Kohn-Sham Density Functional Theory (KSDFT) is the most widely used electronic structure method in chemistry, physics, and materials science, with thousands of calculations cited annually. This ubiquity is rooted in the favorable accuracy vs cost balance of KSDFT. Nonetheless, the ambitions and expectations of researchers for use of KSDFT in predictive simulations of large, complicated molecular systems are confronted with an intrinsic computational cost-scaling challenge. Particularly evident in the context of first-principles molecular dynamics, the challenge is the high cost-scaling associated with the computation of the Kohn-Sham orbitals. Orbital-free DFT (OFDFT), as the name suggests, circumvents entirely the explicit use of those orbitals. Without them, the structural and algorithmic complexity of KSDFT simplifies dramatically and near-linear scaling with system size irrespective of system state is achievable. Thus, much larger system sizes and longer simulation time scales (compared to conventional KSDFT) become accessible; hence, new chemical phenomena and new materials can be explored. In this review, we introduce the historical contexts of OFDFT, its theoretical basis, and the challenge of realizing its promise via approximate kinetic energy density functionals (KEDFs). We review recent progress on that challenge for an array of KEDFs, such as one-point, two-point, and machine-learnt, as well as some less explored forms. We emphasize use of exact constraints and the inevitability of design choices. Then, we survey the associated numerical techniques and implemented algorithms specific to OFDFT. We conclude with an illustrative sample of applications to showcase the power of OFDFT in materials science, chemistry, and physics.

A Discrete-Variable Local Diabatic Representation of Conical Intersection Dynamics

Conical intersections (CIs) are ubiquitous in polyatomic molecules and are responsible for a wide range of phenomena in photochemistry and photophysics. Modeling the conical intersection dynamics with adiabatic electronic states is hindered by the divergence of the first- and second-order derivative couplings at CIs due to electronic degeneracy. We introduce and implement a novel diabatic representation for exact correlated electron-nuclear wave packet dynamics through conical intersections. It directly employs the adiabatic electronic states but avoids the singular first- and second-order derivative couplings and is robust to different gauge choices of the electronic wave function phases. The reference nuclear geometries defining the adiabatic electronic states are determined by a discrete-variable representation of the nuclear coordinates. The nonadiabatic effects are accounted for by the electronic overlap matrix instead of derivative couplings as in the adiabatic representation. Illustrated by a two-mode conical intersection model, this representation captures all nonadiabatic effects, including electronic transitions, electronic coherence, and geometric phases. Thus, this representation provides a singularity-free framework for modeling ab initio conical intersection wave packet dynamics.

Different flavors of exact-factorization-based mixed quantum-classical methods for multistate dynamics

The exact factorization approach has led to the development of new mixed quantum-classical methods for simulating coupled electron-ion dynamics. We compare their performance for dynamics when more than two electronic states are occupied at a given time, and analyze: (1) the use of coupled versus auxiliary trajectories in evaluating the electron-nuclear correlation terms, (2) the approximation of using these terms within surface-hopping and Ehrenfest frameworks, and (3) the relevance of the exact conditions of zero population transfer away from nonadiabatic coupling regions and total energy conservation. Dynamics through the three-state conical intersection in the uracil radical cation as well as polaritonic models in one dimension are studied.