  Fermi's golden rule defines the transition rate between weakly coupled states
and can thus be used to describe a multitude of molecular processes including
electron-transfer reactions and light-matter interaction. However, it can only
be calculated if the wave functions of all internal states are known, which is
typically not the case in molecular systems. Marcus theory provides a
closed-form expression for the rate constant, which is a classical limit of the
golden rule, and indicates the existence of a normal regime and an inverted
regime. Semiclassical instanton theory presents a more accurate approximation
to the golden-rule rate including nuclear quantum effects such as tunnelling,
which has so far been applicable to complex anharmonic systems in the normal
regime only. In this paper we extend the instanton method to the inverted
regime and study the properties of the periodic orbit, which describes the
tunnelling mechanism via two imaginary-time trajectories, one of which now
travels in negative imaginary time. It is known that tunnelling is particularly
prevalent in the inverted regime, even at room temperature, and thus this
method is expected to be useful in studying a wide range of molecular
transitions occurring in this regime.
