  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 to describe dynamics
in both adiabatic and nonadiabatic regimes. We show that the MF-RPI for
symmetric systems are in good agreement with previous literature, and we show
that our numerical techniques are robust for systems with non-zero driving
force. The nuclear MF-RPI and the nuclear MV-RPI are similar, but the MV-RPI
uniquely reports on the changes in the electronic state populations along the
instanton path. In both cases, we analytically demonstrate the existence of a
zero-mode and we numerically and that these solutions are true instantons with
a single unstable mode as expected for a first order saddle point. Finally, we
use the MF-RPI to accurately calculate rate constants for adiabatic and
nonadiabatic model systems with the coupling strength varying over three orders
of magnitude.
