  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 (DOFs). This
partition function provides an efficient sampling of the quantum statistics.
Using the basic property of the Stratonovich-Weyl transformation, we derive a
Hamiltonian which 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 nuclear position and population
auto-correlation functions of non-adiabatic model systems. The results from
SM-NRPMD 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, the latter can not preserve the initial quantum Boltzmann
distribution. We also explicitly demonstrate that SM-NRPMD provides invariant
dynamics upon various ways of partitioning the state-dependent and
state-independent potentials.
