This work identifies geometric effects on dynamics due to nonadiabatic couplings in Born Oppenheimer systems and provides a systematic method for deriving corrections to mixed quantum-classical methods. Specifically, an exact path integral formulation of the quantum nonadiabatic dynamics of Born Oppenheimer systems is described. Stationary phase approximations to the propagator for full quantum dynamics are derived. It is shown that quantum corrections to mixed quantum classical methods can be obtained through stationary phase approximations to the full quantum dynamics. A rigorous description of the quantum corrections due to electronic nonadiabatic coupling on the nuclear dynamics within the Ehrenfest framework is obtained. The fewest switches surface hopping method is shown to be obtained as a quasiclassical approximation to the dynamics and natural semiclassical extensions to include classically forbidden nonadiabatic transitions are suggested.
Nonadiabatic dynamical processes are one of the most important quantum mechanical phenomena in chemical, materials, biological, and environmental molecular systems, where the coupling between different electronic states is either inherent in the molecular structure or induced by the (intense) external field. The curse of dimensionality indicates the intractable exponential scaling of calculation effort with system size and restricts the implementation of numerically exact approaches for realistic large systems. The phase space formulation of quantum mechanics offers an important theoretical fr…
We have developed a numerical differentiation scheme which eliminates evaluation of overlap determinants in calculating the time-derivative non-adiabatic couplings (TDNACs). Evaluation of these determinants was the bottleneck in previous implementations of mixed quantum-classical methods using numerical differentiation of electronic wave functions in the Slater-determinant representation. The central idea of our approach is, first, to reduce the analytic time derivatives of Slater determinants to time derivatives of molecular orbitals, and then to apply a finite-difference formula. Benchmark c…
An important physical process unique to neutral-ion systems is the charge-transfer (CT) reaction. Here, we present measurements of and models for CT processes between co-trapped ultracold Ca atoms and Yb ions under well-controlled conditions. The theoretical analysis reveals the existence of three reaction mechanisms when lasers from a magneto-optical trap (MOT) and an additional catalyst laser are present. Besides the direct CT involving existent excited Ca population in the MOT, the second pathway is controlled by MOT-induced CT, whereas the third one mostly involves the additional red-detun…
We extend the Mixed Quantum-Classical Initial Value Representation (MQC-IVR), a semiclassical method for computing real-time correlation functions, to electronically nonadiabatic systems using the Meyer-Miller-Stock-Thoss (MMST) Hamiltonian to treat electronic and nuclear degrees of freedom (dofs) within a consistent dynamic framework. We introduce an efficient symplectic integration scheme, the MInt algorithm, for numerical time-evolution of the nuclear and electronic phase space variables as well as the Monodromy matrix, under the non-separable MMST Hamiltonian. We then calculate the probabi…
We present semiclassical approximations to Green's functions of multidimensional systems, extending Gutzwiller's work to the classically forbidden region. Based on steepest-descent integrals over these functions, we derive an instanton method for computing the rate of nonadiabatic reactions, such as electron transfer, in the weak-coupling limit, where Fermi's golden-rule can be employed. This generalizes Marcus theory to systems for which the environment free-energy curves are not harmonic and where nuclear tunnelling plays a role. The derivation avoids using the Im F method or short-time appr…
The mapping approach addresses the mismatch between the continuous nuclear phase space and discrete electronic states by creating an extended, fully continuous phase space using a set of harmonic oscillators to encode the populations and coherences of the electronic states. Existing quasiclassical dynamics methods based on mapping, such as the linearised semiclassical initial value representation (LSC-IVR) and Poisson bracket mapping equation (PBME) approaches, have been shown to fail in predicting the correct relaxation of electronic-state populations following an initial excitation. Here we …
We show that a novel, general phase space mapping Hamiltonian for nonadiabatic systems, which is reminiscent of the renowned Meyer-Miller mapping Hamiltonian, involves a commutator variable matrix rather than the conventional zero-point-energy parameter. In the exact mapping formulation on constraint space for phase space approaches for nonadiabatic dynamics, the general mapping Hamiltonian with commutator variables can be employed to generate approximate trajectory-based dynamics. Various benchmark model tests, which range from gas phase to condensed phase systems, suggest that the overall pe…
Mean-Field Ring Polymer Molecular Dynamics (MF-RPMD) is a powerful, efficient, and accurate method for approximate quantum dynamic simulations of multi-level system dynamics. Initial efforts to compute nonadiabatic reaction rates using MF-RPMD were not successful; recent work showed that this can be remedied by including a simple, if adhoc, correction term that accounts for the formation of `kinked' or mixed electronic state ring polymer configurations. Here, we build on this idea, introducing a electronic state population based reaction coordinate and novel skew dividing surface that constrai…
We present an accurate theoretical determination of rovibrational energy levels of the hydrogen molecule and its isotopologues in its electronic ground state. We consider all significant corrections to the Born-Oppenheimer approximation, obtained within nonadiabatic perturbation theory, including the mixed nonadiabatic-relativistic effects. Quantum electrodynamic corrections in the leading $α^5\,m$ and the next-to-leading $α^6\,m$ orders, as well as finite nuclear size effect, are also taken into account but within the Born-Oppenheimer approximation only. Final results for the transition wavel…
We derive an exact quantum propagator for nonadiabatic dynamics in multi-state systems using the mapping variable representation, where classical-like Cartesian variables are used to represent both continuous nuclear degrees of freedom and discrete electronic states. The resulting expression is a Moyal series that, when suitably approximated, can allow for the use of classical dynamics to efficiently model large systems. We demonstrate that different truncations of the exact propagator lead to existing approximate semiclassical and mixed quantum-classical methods and we derive an associated er…
First-order nonadiabatic coupling matrix elements (fo-NACMEs) are the basic quantities in theoretical descriptions of electronically nonadiabatic processes that are ubiquitous in molecular physics and chemistry. Given the large size of systems of chemical interests, time-dependent density functional theory (TDDFT) is usually the first choice. However, the lack of wave functions in TDDFT renders the formulation of NAC-TDDFT for fo-NACMEs conceptually difficult. The present account aims to analyze the available variants of NAC-TDDFT in a critical but concise manner and meanwhile point out the pr…
Nuclear Berry curvature effects emerge from electronic spin degeneracy and canlead to non-trivial spin-dependent (nonadiabatic) nuclear dynamics. However, such effects are completely neglected in all current mixed quantum-classical methods such as fewest switches surface-hopping. In this work, we present a phase-space surface-hopping (PSSH) approach to simulate singlet-triplet intersystem crossing dynamics. We show that with a simple pseudo-diabatic ansatz, a PSSH algorithm can capture the relevant Berry curvature effects and make predictions in agreement with exact quantum dynamics for a simp…
Evaluation of the importance of spin-orbit couplings in the nonadiabatic quantum dynamics with quantum fidelity and with its efficient "on-the-fly" ab initio semiclassical approximation
We develop a trajectory-based approach for excited-state molecular dynamics simulations of systems subject to an external periodic drive. We combine the exact-factorization formalism, allowing to treat electron-nuclear systems in nonadiabatic regimes, with the Floquet formalism for time-periodic processes. The theory is developed starting with the molecular time-dependent Schroedinger equation with inclusion of an external periodic drive that couples to the system dipole moment. With the support of the Floquet formalism, quantum dynamics is approximated by combining classical-like, trajectory-…
Accurate and efficient methods to simulate nonadiabatic and quantum nuclear effects in high-dimensional and dissipative systems are crucial for the prediction of chemical dynamics in condensed phase. To facilitate effective development, code sharing and uptake of newly developed dynamics methods, it is important that software implementations can be easily accessed and built upon. Using the Julia programming language, we have developed the NQCDynamics.jl package which provides a framework for established and emerging methods for performing semiclassical and mixed quantum-classical dynamics in c…
The celebrated Meyer-Miller mapping model has been a useful approach for generating practical trajectory-based nonadiabatic dynamics methods. It is generally assumed that the zero-point-energy (ZPE) parameter is positive. The constraint implied in the conventional Meyer-Miller mapping Hamiltonian for an F-electronic-state system actually requires that parameter γis larger than -1/F for the ZPE parameter for each electronic degree of freedom. Both negative and positive values are possible for such a parameter. We first establish a rigorous formulation to construct exact mapping models in the Ca…
We present an approximate scheme for analytical gradients and nonadiabatic couplings for calculating state-average density matrix renormalization group self-consistent-field wavefunction. Our formalism follows closely the state-average complete active space self-consistent-field (SA-CASSCF) \emph{ansatz}, which employs a Lagrangian, and the corresponding Lagrange multipliers are obtained from a solution of the coupled-perturbed CASSCF (CP-CASSCF) equations. We introduce a definition of the matrix product state (MPS) Lagrange multipliers based on a single-site tensor in a mixed-canonical form o…
Making the best of mixed-field orientation of polar molecules: A recipe for achieving adiabatic dynamics in an electrostatic field combined with laser pulses
Fewest-switches surface hopping is studied in the context of quantum-classical Liouville dynamics. Both approaches are mixed quantum-classical theories that provide a way to describe and simulate the nonadiabatic quantum dynamics of many-body systems. Starting from a surface-hopping solution of the quantum-classical Liouville equation, it is shown how fewest-switches dynamics can be obtained by dropping terms that are responsible for decoherence and restricting the nuclear momentum changes that accompany electronic transitions to those events that occur between population states. The analysis …
Nonadiabatic Dynamics of Molecules Interacting with Metal Surfaces: Extending the Hierarchical Equations of Motion and Langevin Dynamics Approach to Position-Dependent Metal-Molecule Couplings
We present two multistate ring polymer instanton (RPI) formulations, both obtained from an exact path integral representation of the quantum canonical partition function for multistate systems. The two RPIs differ in their treatment of the electronic degrees of freedom; whereas the Mean-Field (MF)-RPI averages over the electronic state contributions, the Mapping Variable (MV)-RPI employs explicit continuous Cartesian variables to represent the electronic states. We compute both RPIs for a series of model two-state systems coupled to a single nuclear mode with electronic coupling values chosen …
Trajectory-based methods that propagate classical nuclei on multiple quantum electronic states are often used to simulate nonadiabatic processes in the condensed phase. A long-standing problem of these methods is their lack of detailed balance, meaning that they do not conserve the equilibrium distribution. In this article, we investigate ideas for how to restore detailed balance in mixed quantum--classical systems by tailoring the previously proposed spin-mapping approach to thermal equilibrium. We find that adapting the spin magnitude can recover the correct long-time populations but is insu…
The diagonal Born-Oppenheimer correction (DBOC) stems from the diagonal second derivative coupling term in the adiabatic representation, and it can have an arbitrary large magnitude when a gap between neighbouring Born-Oppenheimer (BO) potential energy surfaces (PESs) is closing. Nevertheless, DBOC is typically neglected in mixed quantum-classical methods of simulating nonadiabatic dynamics (e.g., fewest-switch surface hopping (FSSH) method). A straightforward addition of DBOC to BO PESs in the FSSH method, FSSH+D, has been shown to lead to numerically much inferior results for models containi…
We review techniques for simulating fully quantum nonadiabatic dynamics using the frozen-width moving Gaussian basis functions to represent the nuclear wavefunction. A choice of these basis functions is primarily motivated by the idea of the on-the-fly dynamics that will involve electronic structure calculations done locally in the vicinity of each Gaussian center and thus avoiding the "curse of dimensionality" appearing in large systems. For quantum dynamics involving multiple electronic states there are several aspects that need to be addressed. First, the choice of the electronic state repr…
We derive a formulation of mixed quantum-classical dynamics for describing electronic carriers interacting with phonons in reciprocal space. For dispersionless phonons, we start by expressing the real-space classical coordinates in terms of complex variables. A Fourier series over these coordinates then yields the reciprocal-space coordinates. Evaluating the electron-phonon interaction term through Ehrenfest's theorem, we arrive at a reciprocal-space formalism that is equivalent to mean-field mixed quantum-classical dynamics in real space. This equivalence is numerically verified for the Holst…
An accurate description of nonadiabatic dynamics of molecular species on metallic surfaces poses a serious computational challenge associated with a multitude of closely-spaced electronic states. We propose a mixed quantum-classical scheme that addresses this challenge by introducing collective electronic variables. These variables are defined through analytic block-diagonalization applied to the time-dependent Hamiltonian matrix governing the electronic dynamics. We compare our scheme with the Ehrenfest approach and with a full-memory electronic friction model on a one-dimensional "adatom + a…
We derive a Hierarchical Equations of Motion (HEOM) description of nonadiabatic Herzberg-Teller type coupling effects and of non-Condon effects in a system of electronic transitions interacting with intra- and inter-molecular vibrational modes. We point out analogies between the auxiliary density operators (ADOs) of the hierarchy and the terms arising from explicit description of the vibrational modes in the Hamiltonian. In comparison with the standard formulation of HEOM, our equations contain additional connections between ADOs in the hierarchy scheme, which account for the dependence of the…
