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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 present the rigorous theoretical framework of the generalized spin mapping representation for non-adiabatic dynamics. Our work is based upon a new mapping formalism recently introduced by Runeson and Richardson [J. Chem. Phys. 152, 084110 (2020)], which uses the generators of the s u (N) Lie algebra to represent N discrete electronic states, thus preserving the size of the original Hilbert space. Following this interesting idea, the Stratonovich–Weyl transform is used to map an operator in the Hilbert space to a continuous function on the SU(N) Lie group, i.e., a smooth manifold which is a phase space of continuous variables. We further use the Wigner representation to describe the nuclear degrees of freedom and derive an exact expression of the time-correlation function as well as the exact quantum Liouvillian for the non-adiabatic system. Making the linearization approximation, this exact Liouvillian is reduced to the Liouvillian of several recently proposed methods, and the performance of this linearized method is tested using non-adiabatic models. We envision that the theoretical work presented here provides a rigorous and unified framework to formally derive non-adiabatic quantum dynamics approaches with continuous variables and connects the previous methods in a clear and concise manner. [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]

We use the ring polymer (RP) representation to quantize the radiation field inside an optical cavity to investigate polariton quantum dynamics. Using a charge transfer model coupled to an optical cavity, we demonstrate that the RP quantization of the photon field provides accurate rate constants of the polariton mediated electron transfer reaction compared to Fermi's golden rule. Because RP quantization uses extended phase space to describe the photon field, it significantly reduces the computational costs compared to the commonly used Fock state description of the radiation field. Compared to the other quasi-classical descriptions of the photon field, such as the classical Wigner based mean-field Ehrenfest model, the RP representation provides a much more accurate description of the polaritonic quantum dynamics because it alleviates the potential quantum distribution leakage problem associated with the photonic degrees of freedom (DOF). This work demonstrates the possibility of using the ring polymer description to treat the quantized radiation field in polariton chemistry, offering an accurate and efficient approach for future investigations in cavity quantum electrodynamics. [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]