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We derive the non-adiabatic ring polymer molecular dynamics (RPMD) approach in the phase space of the SU(N) Lie Group. This method, which we refer to as the spin mapping non-adiabatic RPMD (SM-NRPMD), is based on the spin-mapping formalism for the electronic degrees of freedom (DOFs) and ring polymer path-integral description for the nuclear DOFs. Using the Stratonovich–Weyl transform for the electronic DOFs and the Wigner transform for the nuclear DOFs, we derived an exact expression of the Kubo-transformed time-correlation function (TCF). We further derive the spin mapping non-adiabatic Matsubara dynamics using the Matsubara approximation that removes the high frequency nuclear normal modes in the TCF and derive the SM-NRPMD approach from the non-adiabatic Matsubara dynamics by discarding the imaginary part of the Liouvillian. The SM-NRPMD method has numerical advantages compared to the original NRPMD method based on the Meyer–Miller–Stock–Thoss (MMST) mapping formalism due to a more natural mapping using the SU(N) Lie Group that preserves the symmetry of the original system. We numerically compute the Kubo-transformed position auto-correlation function and electronic population correlation function for three-state model systems. The numerical results demonstrate the accuracy of the SM-NRPMD method, which outperforms the original MMST-based NRPMD. We envision that the SM-NRPMD method will be a powerful approach to simulate electronic non-adiabatic dynamics and nuclear quantum effects accurately. [ABSTRACT FROM AUTHOR]

We perform on-the-fly non-adiabatic molecular dynamics simulations using the symmetrical quasi-classical (SQC) approach with the recently suggested molecular Tully models: ethylene and fulvene. We attempt to provide benchmarks of the SQC methods using both the square and triangle windowing schemes as well as the recently proposed electronic zero-point-energy correction scheme (the so-called γ correction). We use the quasi-diabatic propagation scheme to directly interface the diabatic SQC methods with adiabatic electronic structure calculations. Our results showcase the drastic improvement of the accuracy by using the trajectory-adjusted γ-corrections, which outperform the widely used trajectory surface hopping method with decoherence corrections. These calculations provide useful and non-trivial tests to systematically investigate the numerical performance of various diabatic quantum dynamics approaches, going beyond simple diabatic model systems that have been used as the major workhorse in the quantum dynamics field. At the same time, these available benchmark studies will also likely foster the development of new quantum dynamics approaches based on these techniques. [ABSTRACT FROM AUTHOR]

We present a new non-adiabatic ring polymer molecular dynamics (NRPMD) method based on the spin mapping formalism, which we refer to as the spin mapping NRPMD (SM-NRPMD) approach. We derive the path-integral partition function expression using the spin coherent state basis for the electronic states and the ring polymer formalism for the nuclear degrees of freedom. This partition function provides an efficient sampling of the quantum statistics. Using the basic properties of the Stratonovich–Weyl transformation, we further justify a Hamiltonian that we propose for the dynamical propagation of the coupled spin mapping variables and the nuclear ring polymer. The accuracy of the SM-NRPMD method is numerically demonstrated by computing the nuclear position and population auto-correlation functions of non-adiabatic model systems. The results obtained using the SM-NRPMD method agree very well with the numerically exact results. The main advantage of using the spin mapping variables over the harmonic oscillator mapping variables is numerically demonstrated, where the former provides nearly time-independent expectation values of physical observables for systems under thermal equilibrium. We also explicitly demonstrate that SM-NRPMD provides invariant dynamics upon various ways of partitioning the state-dependent and state-independent potentials. [ABSTRACT FROM AUTHOR]

We present the non-adiabatic Matsubara dynamics, a general framework for computing the time-correlation function (TCF) of electronically non-adiabatic systems. This new formalism is derived based on the generalized Kubo-transformed TCF using the Wigner representation for both the nuclear degrees of freedom and the electronic mapping variables. By dropping the non-Matsubara nuclear normal modes in the quantum Liouvillian and explicitly integrating these modes out from the expression of the TCF, we derived the non-adiabatic Matsubara dynamics approach. Further making the approximation to drop the imaginary part of the Matsubara Liouvillian and enforce the nuclear momentum integral to be real, we arrived at the non-adiabatic ring-polymer molecular dynamics (NRPMD) approach. We have further justified the capability of NRPMD for simulating the non-equilibrium TCF. This work provides the rigorous theoretical foundation for several recently proposed state-dependent RPMD approaches and offers a general framework for developing new non-adiabatic quantum dynamics methods in the future. [ABSTRACT FROM AUTHOR]

A recently proposed nonadiabatic ring polymer molecular dynamics (NRPMD) approach has shown to provide accurate quantum dynamics by incorporating explicit state descriptions and nuclear quantizations. Here, we present a rigorous derivation of the NRPMD Hamiltonian and investigate its performance on simulating excited state nonadiabatic dynamics. Our derivation is based on the Meyer-Miller-Stock-Thoss mapping representation for electronic states and the ring-polymer path-integral description for nuclei, resulting in the same Hamiltonian proposed in the original NRPMD approach. In addition, we investigate the accuracy of using NRPMD to simulate the photoinduced nonadiabatic dynamics in simple model systems. These model calculations suggest that NRPMD can alleviate the zero-point energy leakage problem that is commonly encountered in the classical Wigner dynamics and provide accurate excited state nonadiabatic dynamics. This work provides a solid theoretical foundation of the promising NRPMD Hamiltonian and demonstrates the possibility of using the state-dependent RPMD approach to accurately simulate electronic nonadiabatic dynamics while explicitly quantizing nuclei. [ABSTRACT FROM AUTHOR]

A new partially linearized approximate approach to non-adiabatic quantum dynamics is derived based on linearizing the path difference for nuclear degrees of freedom (DOF) in the classical mapping Hamiltonian while keeping quantum interference effects inherent in the forward and backward propagators for the electronic DOF. With this new approach, the non-adiabatic force that acts on the nuclear DOF is a mean force rather than a state dependent force as found in some alternative approaches. Various benchmark examples are explored to test the accuracy of this new approach, and compare its performance with other approaches for a wide range of physical phenomena including: non-adiabatic scattering, excited state conical intersection dynamics, excited state photoisomerization, and excitation energy transfer in realistic condensed phase model systems. Results indicate that, even though the method is based on a “mean trajectory”-like scheme, it can accurately capture electronic population branching through multiple avoided crossing regions and that the approach offers a robust and reliable way to treat quantum dynamical phenomena in a wide range of condensed phase applications. [ABSTRACT FROM AUTHOR]

Powerful approximate methods for propagating the density matrix of complex systems that are conveniently described in terms of electronic subsystem states and nuclear degrees of freedom have recently been developed that involve linearizing the density matrix propagator in the difference between the forward and backward paths of the nuclear degrees of freedom while keeping the interference effects between the different forward and backward paths of the electronic subsystem described in terms of the mapping Hamiltonian formalism and semi-classical mechanics. Here we demonstrate that different approaches to developing the linearized approximation to the density matrix propagator can yield a mean-field like approximate propagator in which the nuclear variables evolve classically subject to Ehrenfest-like forces that involve an average over quantum subsystem states, and by adopting an alternative approach to linearizing we obtain an algorithm that involves classical like nuclear dynamics influenced by a quantum subsystem state dependent force reminiscent of trajectory surface hopping methods. We show how these different short time approximations can be implemented iteratively to achieve accurate, stable long time propagation and explore their implementation in different representations. The merits of the different approximate quantum dynamics methods that are thus consistently derived from the density matrix propagator starting point and different partial linearization approximations are explored in various model system studies of multi-state scattering problems and dissipative non-adiabatic relaxation in condensed phase environments that demonstrate the capabilities of these different types of approximations for treating non-adiabatic electronic relaxation, bifurcation of nuclear distributions, and the passage from nonequilibrium coherent dynamics at short times to long time thermal equilibration in the presence of a model dissipative environment. [ABSTRACT FROM AUTHOR]