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We present two multistate ring polymer instanton (RPI) formulations, bothobtained from an exact path integral representation of the quantum canonicalpartition function for multistate systems. The two RPIs differ in theirtreatment of the electronic degrees of freedom; whereas the Mean-Field (MF)-RPIaverages over the electronic state contributions, the Mapping Variable (MV)-RPIemploys explicit continuous Cartesian variables to represent the electronicstates.  We compute both RPIs for a series of model two-state systems coupled to asingle nuclear mode with electronic coupling values chosen to describe dynamicsin both adiabatic and nonadiabatic regimes. We show that the MF-RPI forsymmetric systems are in good agreement with previous literature, and we showthat our numerical techniques are robust for systems with non-zero drivingforce. The nuclear MF-RPI and the nuclear MV-RPI are similar, but the MV-RPIuniquely reports on the changes in the electronic state populations along theinstanton path. In both cases, we analytically demonstrate the existence of azero-mode and we numerically and that these solutions are true instantons witha single unstable mode as expected for a first order saddle point. Finally, weuse the MF-RPI to accurately calculate rate constants for adiabatic andnonadiabatic model systems with the coupling strength varying over three ordersof magnitude.

Multistate ring polymer instantons and nonadiabatic reaction rates