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.

Improved Local-Mode Zero-Point-Energy Conservation Scheme for Quasiclassical Trajectories.

We present an improvement of the local-pair zero-point-energy (LP-ZPE) scheme of Mukherjee and Barbatti. The new approximation is called the improved LP-ZPE scheme or iLP-ZPE. This scheme can produce trajectories that do not have unphysical leaking of zero-point energy from high-frequency spectator modes into low-frequency modes. We illustrate the method with a successful direct dynamics application to the Ne···HF van der Waals molecule. The method is well suited for direct dynamics calculations because it does not require costly evaluations of local Hessians or instantaneous normal modes along the trajectories.

Dissociation Time, Quantum Yield, and Dynamic Reaction Pathways in the Thermolysis of trans-3,4-Dimethyl-1,2-dioxetane.

The thermolysis of trans-3,4-dimethyl-1,2-dioxetane is studied by trajectory surface hopping. The significant difference between long and short dissociation times is rationalized by frustrated dissociations and the time spent in triplet states. If the C-C bond breaks through an excited state channel, then the trajectory passes over a ridge of the potential energy surface of that state. The calculated triplet quantum yields match the experimental results. The dissociation half-times and quantum yields follow the same ascending order as per the product states, justifying the conjecture that the longer dissociation time leads to a higher quantum yield, proposed in the context of the methylation effect. The populations of the molecular Coulomb Hamiltonian and diagonal states reach equilibrium, but the triplet populations with different Sz components fluctuate indefinitely. Certain initial velocities, leading the trajectories to given product states, can be identified as the most characteristic features for sorting trajectories according to their product states.

Beyond MD17: the reactive xxMD dataset.

System specific neural force fields (NFFs) have gained popularity in computational chemistry. One of the most popular datasets as a bencharmk to develop NFF models is the MD17 dataset and its subsequent extension. These datasets comprise geometries from the equilibrium region of the ground electronic state potential energy surface, sampled from direct adiabatic dynamics. However, many chemical reactions involve significant molecular geometrical deformations, for example, bond breaking. Therefore, MD17 is inadequate to represent a chemical reaction. To address this limitation in MD17, we introduce a new dataset, called Extended Excited-state Molecular Dynamics (xxMD) dataset. The xxMD dataset involves geometries sampled from direct nonadiabatic dynamics, and the energies are computed at both multireference wavefunction theory and density functional theory. We show that the xxMD dataset involves diverse geometries which represent chemical reactions. Assessment of NFF models on xxMD dataset reveals significantly higher predictive errors than those reported for MD17 and its variants. This work underscores the challenges faced in crafting a generalizable NFF model with extrapolation capability.

Parametrically Managed Activation Functions for Improved Global Potential Energy Surfaces for Six Coupled 5A' States and Fourteen Coupled 3A' States of O + O2.

We report new potential energy surfaces for six coupled 5A' states and 14 coupled 3A' states of O3. The new surfaces are created by parametrically managed diabatization by deep neural network (PM-DDNN). The PM-DDNN method uses calculated adiabatic potential energy surfaces to discover and fit an underlying adiabatic-equivalent set of diabatic surfaces and their couplings and obtains the fit to the adiabatic surfaces by diagonalization of the diabatic potential energy matrix (DPEM). The procedure yields the adiabatic surfaces and their gradients, as well as the DPEM and its gradient. If desired one can also compute the nonadiabatic coupling due to the transformation. The present work improves on previous work by using a new coordinate to guide the decay of the neural network contribution to the many-body fit to the whole DPEM. The main objective was to obtain smoother potentials than the previous ones with better suitability for dynamics calculations, and this was achieved. Furthermore, we obtained suitably small deviations from the input reference data. For the six coupled 5A' surfaces, the 60,366 data below 10 eV are fit with a mean unsigned error (MUE) of 49 meV, and for the 14 coupled 3A' surfaces, the 76,733 data below 10 eV are fit with an MUE of 28 meV. The data below 5 eV fit even more accurately with MUEs of 37 meV (5A') and 20 meV (3A').

ChemPotPy: A Python Library for Analytic Representations of Potential Energy Surfaces and Diabatic Potential Energy Matrices.

Constructing analytic representations of global and semiglobal potential energy surfaces is difficult and can be laborious, and it is even harder when one needs coupled potential energy surfaces and their electronically nonadiabatic couplings. When accomplished, however, the resulting potential functions are a valuable resource. To facilitate the convenient use of potentials that have been developed, we provide a collection of existing surfaces in a library with consistent units and formats. A potential energy surface library of this type, namely PotLib, was built more than 20 years ago. However, that library only provided pristine Fortran subroutines for each potential energy surface, and therefore, it is not as user-friendly as would be desirable. Here, we report the creation of ChemPotPy, a CHEMical library of POTential energy surfaces in PYthon. ChemPotPy is a user-friendly library for analytic representation of single-state and multistate potential energy surfaces and couplings. A given entry in the library contains an analytic potential energy function or analytic functions for a set of coupled potential energy surfaces, and depending on the case, it may also include analytic or numerical gradients, nonadiabatic coupling vectors, and/or diabatic potential energy matrices and their gradients. Only three inputs, namely, the chemical formula of the system, the name of the potential energy surface or surface set, and the Cartesian geometry, are required. ChemPotPy uses the same units for input and output quantities of all surfaces and surface sets to facilitate general interfaces with the dynamics programs. The initial version of the library contains 338 entries, and we anticipate that more will be added in the future.

Nonadiabatic Coupling in Trajectory Surface Hopping: Accurate Time Derivative Couplings by the Curvature-Driven Approximation.

Trajectory surface hopping (TSH) is a widely used mixed quantum-classical dynamics method that is used to simulate molecular dynamics with multiple electronic states. In TSH, time-derivative coupling is employed to propagate the electronic coefficients and in that way to determine when the electronic state on which the nuclear trajectory is propagated switches. In this work, we discuss nonadiabatic TSH dynamics algorithms employing the curvature-driven approximation and overlap-based time derivative couplings, and we report test calculations on six photochemical reactions where we compare the results to one another and to calculations employing analytic nonadiabatic coupling vectors. We correct previous published results thanks to a bug found in the software. We also provide additional, more detailed studies of the time-derivative couplings. Our results show good agreement between curvature-driven algorithms and overlap-based algorithms.

Semiclassical Multistate Dynamics for Six Coupled 5A' States of O + O2.

Dynamics simulations of high-energy O2-O collisions play an important role in simulating thermal energy content and heat flux in flows around hypersonic vehicles. To carry out such dynamics simulations efficiently requires accurate global potential energy surfaces and (in most algorithms) state couplings for many energetically accessible electronic states. The ability to treat collisions involving many coupled electronic states has been a challenge for decades. Very recently, a new diabatization method, the parametrically managed diabatization by deep neural network (PM-DDNN), has been developed. The PM-DDNN method uses a deep neural network architecture with an activation function parametrically dependent on input data to discover and fit the diabatic potential energy matrix (DPEM) as a function of geometry, and the adiabatic potential energy surfaces are obtained by diagonalization of a small matrix with analytic matrix elements. Here, we applied the PM-DDNN method to the six lowest-energy potential energy surfaces in the 5A' manifold of O3 to perform simultaneous diabatization and fitting; the data are obtained by extended multistate complete-active-space second-order perturbation theory. We then used the adiabatic surfaces for dynamics calculations with three methods: coherent switching with decay of mixing (CSDM), curvature-driven CSDM (κCSDM), and electronically curvature-driven CSDM (eκCSDM). The κCSDM calculations require only adiabatic potential energies and gradients. The three dynamical methods are in good agreement. We then calculated electronically nonadiabatic, electronically inelastic, and dissociative cross sections for seven initial collision energies, five initial vibrational levels, and four initial rotational levels. Trends in the electronically inelastic cross sections as functions of the initial collision energy and vibrational level were rationalized in terms of the coordinate ranges where the gaps between the second and third potential energy surfaces are small.

Parametrically Managed Activation Function for Fitting a Neural Network Potential with Physical Behavior Enforced by a Low-Dimensional Potential.

Machine-learned representations of potential energy surfaces generated in the output layer of a feedforward neural network are becoming increasingly popular. One difficulty with neural network output is that it is often unreliable in regions where training data is missing or sparse. Human-designed potentials often build in proper extrapolation behavior by choice of functional form. Because machine learning is very efficient, it is desirable to learn how to add human intelligence to machine-learned potentials in a convenient way. One example is the well-understood feature of interaction potentials that they vanish when subsystems are too far separated to interact. In this article, we present a way to add a new kind of activation function to a neural network to enforce low-dimensional constraints. In particular, the activation function depends parametrically on all of the input variables. We illustrate the use of this step by showing how it can force an interaction potential to go to zero at large subsystem separations without either inputting a specific functional form for the potential or adding data to the training set in the asymptotic region of geometries where the subsystems are separated. In the process of illustrating this, we present an improved set of potential energy surfaces for the 14 lowest 3A' states of O3. The method is more general than this example, and it may be used to add other low-dimensional knowledge or lower-level knowledge to machine-learned potentials. In addition to the O3 example, we present a greater-generality method called parametrically managed diabatization by deep neural network (PM-DDNN) that is an improvement on our previously presented permutationally restrained diabatization by deep neural network (PR-DDNN).

New Gradient Correction Scheme for Electronically Nonadiabatic Dynamics Involving Multiple Spin States.

It has been recommended that the best representation to use for trajectory surface hopping (TSH) calculations is the fully adiabatic basis in which the Hamiltonian is diagonal. Simulations of intersystem crossing processes with conventional TSH methods require an explicit computation of nonadiabatic coupling vectors (NACs) in the molecular-Coulomb-Hamiltonian (MCH) basis, also called the spin-orbit-free basis, in order to compute the gradient in the fully adiabatic basis (also called the diagonal representation). This explicit requirement destroys some of the advantages of the overlap-based algorithms and curvature-driven algorithms that can be used for the most efficient TSH calculations. Therefore, although these algorithms allow one to perform NAC-free simulations for internal conversion processes, one still requires NACs for intersystem crossing. Here, we show that how the NAC requirement is circumvented by a new computation scheme called the time-derivative-matrix scheme.

Direct Nonadiabatic Dynamics of Ammonia with Curvature-Driven Coherent Switching with Decay of Mixing and with Fewest Switches with Time Uncertainty: An Illustration of Population Leaking in Trajectory Surface Hopping Due to Frustrated Hops.

Mixed quantum-classical nonadiabatic dynamics is a widely used approach to simulate molecular dynamics involving multiple electronic states. There are two main categories of mixed quantum-classical nonadiabatic dynamics algorithms, namely, trajectory surface hopping (TSH) in which the trajectory propagates on a single potential energy surface, interrupted by hops, and self-consistent-potential (SCP) methods, such as semiclassical Ehrenfest, in which propagation occurs on a mean-field surface without hops. In this work, we will illustrate an example of severe population leaking in TSH. We emphasize that such leaking is a combined effect of frustrated hops and long-time simulations that drive the final excited-state population toward zero as a function of time. We further show that such leaking can be alleviated-but not eliminated-by the fewest switches with time uncertainty TSH algorithm (here implemented in the SHARC program); the time uncertainty algorithm slows down the leaking process by a factor of 4.1. The population leaking is not present in coherent switching with decay of mixing (CSDM), which is an SCP method with non-Markovian decoherence included. Another result in this paper is that we find very similar results with the original CSDM algorithm, with time-derivative CSDM (tCSDM), and with curvature-driven CSDM (κCSDM). Not only do we find good agreement for electronically nonadiabatic transition probabilities but also we find good agreement of the norms of the effective nonadiabatic couplings (NACs) that are derived from the curvature-driven time-derivative couplings as implemented in κCSDM with the time-dependent norms of the nonadiabatic coupling vectors computed by state-averaged complete-active-space self-consistent field theory.

Decoherence and Its Role in Electronically Nonadiabatic Dynamics.

Decoherence is the tendency of a time-evolved reduced density matrix for a subsystem to assume a form corresponding to a statistical ensemble of states rather than a coherent combination of pure-state wave functions. When a molecular process involves changes in the electronic state and the coordinates of the nuclei, as in ultraviolet or visible light photochemistry or electronically inelastic collisions, the reduced density matrix of the electronic subsystem suffers decoherence, due to its interaction with the nuclear subsystem. We present the background necessary to conceptualize this decoherence; in particular, we discuss the density matrix description of pure states and mixed states, and we discuss pointer states and decoherence time. We then discuss how decoherence is treated in the coherent switching with decay of mixing algorithm and the trajectory surface hopping method for semiclassical calculations of electronically nonadiabatic processes.

Nonadiabatic Dynamics of 1,3-Cyclohexadiene by Curvature-Driven Coherent Switching with Decay of Mixing.

The photoinduced ring-opening reaction of 1,3-cyclohexadiene to produce 1,3,5-hexatriene is a classic electrocyclic reaction and is also a prototype for many reactions of biological and synthetic importance. Here, we simulate the ultrafast nonadiabatic dynamics of the reaction in the manifold of the three lowest valence electronic states by using extended multistate complete-active-space second-order perturbation theory (XMS-CASPT2) combined with the curvature-driven coherent switching with decay of mixing (κCSDM) dynamical method. We obtain an excited-state lifetime of 79 fs, and a product quantum yield of 40% from the 500 trajectories initiated in the S1 excited state. The obtained lifetime and quantum yield values are very close to previously reported experimental and computed values, showing the capability of performing a reasonable nonadiabatic ring-opening dynamics with the κCSDM method that does not require nonadiabatic coupling vectors, time derivatives, or diabatization. In addition, we study the ring-opening reaction by initiating the trajectories in the dark state S2. We also optimize the S0/S1 and S1/S2 minimum-energy conical intersections (MECIs) by XMS-CASPT2; for S1/S2, we optimized both an inner and an outer local-minimum-energy conical intersections (LMECIs). We provide the potential energy profile along the ring-opening coordinate by joining selected critical points via linear synchronous transit paths. We find the inner S1/S2 LMECI to be more crucial than the outer one.

Recommendation of Orbitals for G0W0 Calculations on Molecules and Crystals.

The many-body GW approximation, especially the G0W0 method, has been widely used for condensed matter and molecules to calculate quasiparticle energies for ionization, electron attachment, and band gaps. Because G0W0 calculations are well-known to have a strong dependence on the orbitals, the goal of the present work is to provide guidance on the choice of density functional used to generate orbitals and to recommend a choice that gives the most broadly accurate results. We have systematically investigated the dependence of G0W0 calculations on the orbitals for 100 molecules and 8 crystals by considering orbitals obtained with a diverse set of Kohn-Sham (KS) and generalized KS (GKS) functionals (63 functionals plus Hartree-Fock). The percentage of Hartree-Fock exchange employed in density functionals has been found to have strong influence on the predicted molecular ionization energy and crystal fundamental band gaps (with optimum values between 40 and 56%), but to have less effect on predicting molecular electron affinities. The low cost of the Gaussian implementation, even with hybrid functionals in periodic calculations, the better performance of global hybrids as compared to range-separated hybrids of either than screened exchange or long-range-corrected type, and the relatively low cost of global-hybrid-functional periodic calculations using Gaussians means that one can employ global-hybrid functionals at a very reasonable cost and obtain more accurate band gaps of semiconductors than are obtained by the methods currently widely employed, namely local gradient approximations. We single out three global-hybrid functionals that give especially good results for both molecules (100 in the test set) and crystals (8 in the test set, for all of which our benchmark data are the proper band gap rather than an optical band gap uncorrected for exciton effects).

Diabatic States of Molecules.

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

Nonadiabatic Dynamics Algorithms with Only Potential Energies and Gradients: Curvature-Driven Coherent Switching with Decay of Mixing and Curvature-Driven Trajectory Surface Hopping.

Direct dynamics by mixed quantum-classical nonadiabatic methods is an important tool for understanding processes involving multiple electronic states. Very often, the computational bottleneck of such direct simulation comes from electronic structure theory. For example, at every time step of a trajectory, nonadiabatic dynamics requires potential energy surfaces, their gradients, and the matrix elements coupling the surfaces. The need for the couplings can be alleviated by employing the time derivatives of the wave functions, which can be evaluated from overlaps of electronic wave functions at successive time steps. However, evaluation of overlap integrals is still expensive for large systems. In addition, for electronic structure methods for which the wave functions or the coupling matrix elements are not available, nonadiabatic dynamics algorithms become inapplicable. In this work, building on recent work by Baeck and An, we propose new nonadiabatic dynamics algorithms that only require adiabatic potential energies and their gradients. The new methods are named curvature-driven coherent switching with decay of mixing (κCSDM) and curvature-driven trajectory surface hopping (κTSH). We show how powerful these new methods are in terms of computation time and accuracy as compared to previous mixed quantum-classical nonadiabatic dynamics algorithms. The lowering of the computational cost will allow longer nonadiabatic trajectories and greater ensemble averaging to be affordable, and the ability to calculate the dynamics without electronic structure coupling matrix elements extends the dynamics capability to new classes of electronic structure methods.

Semiclassical Trajectory Studies of Reactive and Nonreactive Scattering of OH(A 2 Σ+ ) by H2 Based on an Improved Full-Dimensional Ab Initio Diabatic Potential Energy Matrix.

We present a new full-dimensional diabatic potential energy matrix (DPEM) for electronically nonadiabatic collisions of OH(A 2 Σ+ ) with H2 , and we calculate the probabilities of electronically adiabatic inelastic collisions, nonreactive quenching, and reactive quenching to form H2 O+H. The DPEM was fitted using a many-body expansion with permutationally invariant polynomials in bond-order functions to represent the many-body part. The dynamics calculations were carried out with the fewest-switches with time uncertainty and stochastic decoherence (FSTU/SD) semiclassical trajectory method. We present results both for head-on collisions (impact parameter b equal to zero) and for a full range of impact parameters. The results are compared to experiment and to earlier FSTU/SD and quantum dynamics calculations with a previously published DPEM. The various theoretical results all agree that nonreactive quenching dominates reactive quenching, but there are quantitative differences between the two DPEMs and between the b=0 results and the all-b results, especially for the probability of reactive quenching.

Direct coherent switching with decay of mixing for intersystem crossing dynamics of thioformaldehyde: The effect of decoherence.

We evaluate the effect of electronic decoherence on intersystem crossing in the photodynamics of thioformaldehyde. First, we show that the state-averaged complete-active-space self-consistent field electronic structure calculations with a properly chosen active space of 12 active electrons in 10 active orbitals can predict the potential energy surfaces and the singlet-triplet spin-orbit couplings quite well for CH2S, and we use this method for direct dynamics by coherent switching with decay of mixing (CSDM). We obtain similar dynamical results with CSDM or by adding energy-based decoherence to trajectory surface hopping, with the population of triplet states tending to a small steady-state value over 500 fs. Without decoherence, the state populations calculated by the conventional trajectory surface hopping method or the semiclassical Ehrenfest method gradually increase. This difference shows that decoherence changes the nature of the results not just quantitatively but qualitatively.

CAS without SCF-Why to use CASCI and where to get the orbitals.

The complete active space self-consistent field (CASSCF) method has seen broad adoption due to its ability to describe the electronic structure of both the ground and excited states of molecules over a broader swath of the potential energy surface than is possible with the simpler Hartree-Fock approximation. However, it also has a reputation for being unwieldy, computationally costly, and un-black-box. Here, we discuss a class of alternatives, complete active space configuration interaction (CASCI) methods, paying particular attention to their application to electronic excited states. The goal of this Perspective is fourfold. First, we argue that CASCI is not merely an approximation to CASSCF, in that it can be designed to have important qualitative advantages over CASSCF. Second, we present several insights drawn from our experience experimenting with different schemes for computing orbitals to be employed in CASCI. Third, we argue that CASCI is well suited for application to nanomaterials. Finally, we reason that, with the rise in new low-scaling approaches for describing multireference systems, there is a greater need than ever to develop new methods for defining orbitals that provide an efficient and accurate description of both static correlation and electronic excitations in a limited active space.

Permutationally Restrained Diabatization by Machine Intelligence.

Simulations of electronically nonadiabatic processes may employ either the adiabatic or diabatic representation. Direct dynamics calculations are usually carried out in the adiabatic basis because the energy, force, and state coupling can be evaluated directly by many electronic structure methods. However, although its straightforwardness is appealing, direct dynamics is expensive when combined with quantitatively accurate electronic structure theories. This generates interest in analytically fitted surfaces to cut the expense, but the cuspidal ridges of the potentials and the singularities and vector nature of the couplings at high-dimensional, nonsymmetry-determined intersections in the adiabatic representation make accurate fitting almost impossible. This motivates using diabatic representations, where the surfaces are smooth and the couplings are also smooth and-importantly-scalar. In a recent previous work, we have developed a method called diabatization by deep neural network (DDNN) that takes advantage of the smoothness and nonuniqueness of diabatic bases to obtain them by machine learning. The diabatic potential energy matrices (DPEMs) learned by the DDNN method yield not only diabatic potential energy surfaces (PESs) and couplings in an analytic form useful for dynamics calculations, but also adiabatic surfaces and couplings in the adiabatic representation can be calculated inexpensively from the transformation. In the present work, we show how to extend the DDNN method to produce good approximations to global permutationally invariant adiabatic PESs simultaneously with DPEMs. The extended method is called permutationally restrained DDNN.