  We analyse the golden rule limit of the recently proposed isomorphic ring
polymer (iso-RP) method. This method aims to combine an exact expression for
the quantum mechanical partition function of a system with multiple electronic
states with a pre-existing mixed quantum-classical (MQC) dynamics
approximation, such as fewest switches surface hopping. Since the choice of the
MQC method adds a degree of flexibility, we simplify the analysis by assuming
that the dynamics used correctly reproduces the exact golden rule rate for a
non-adiabatic (e.g., electron transfer) reaction in the high temperature limit.
Having made this assumption, we obtain an expression for the iso-RP rate in the
golden rule limit that is valid at any temperature. We then compare this rate
with the exact rate for a series of simple spin-boson models. We find that the
iso-RP method does not correctly predict how nuclear quantum effects affect the
reaction rate in the golden rule limit. Most notably, it does not capture the
quantum asymmetry in a conventional (Marcus) plot of the logarithm of the
reaction rate against the thermodynamic driving force, and it also
significantly overestimates the correct quantum mechanical golden rule rate for
activationless electron transfer reactions. These results are analysed and
their implications discussed for the applicability of the iso-RP method to more
general non-adiabatic reactions.
