Non-adiabatic ring polymer molecular dynamics with spin mapping variables

Abstract

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.

Non-adiabatic Dynamics using the Generators of the su(N) Lie Algebra

“…Enforcing this separation in the MMST mapping formalism can significantly improve the stability and accuracy of the dynamics. 21,23,38 In the su(N ) mapping formalism, this is intrinsically achieved. Using the generators defined in Eqs.…”

“…as a basis to describe the quantum dynamics, 24,38 where θ and ϕ are the angles defining the Bloch vector, with the radius of the Bloch sphere being fixed. The spin coherent states can be generalized for a N -level system as follows 32,39,40…”

“…In order to obtain equivalent EOMs in terms of the 2N − 2 underlying {θ n } and {ϕ n } variables for n ∈ [1, N − 1], we want to find a set of canonical variables. We recognize that in the two-state special case, 38,74 the conjugate momentum of ϕ 1 is r s cos θ 1 ≡ r s Ω γ2 , where γ 2 is the index of the diagonal generator (see Eq. 5c).…”

General Formulas of the Structure Constants in the $\mathfrak{su}(N)$ Lie Algebra

“…For example, the su(6) Lie algebra in the quark model [6]; in the search of a Grand Unification Theory (GUT), su(5) has been proposed as the simplest possible version of GUT by Georgi and Glashow [7]. In atomic and optical physics [8], as well as in physical chemistry [9,10], spin analogy is used to map the electronic-nuclear dynamics of open quantum systems involving non-adiabaticity [9][10][11][12][13]. The su(N ) Lie algebra is also proposed as general mapping between a multi-state Hamiltonian and a classical-like Hamiltonian [12].…”

“…Mapping Hamiltonian using the su(N ) Lie Algebra. The SU(2) representation of the Lie algebra (spin-1 2 analogy) is often used in quantum dynamics to study systems with two states [9,10,13]. For a two level system with the Hamiltonian Ĥ = H…”