  Marcus-Levich-Jortner (MLJ) theory is one of the most commonly used methods
for including nuclear quantum effects into the calculation of electron-transfer
rates and for interpreting experimental data. It divides the molecular problem
into a subsystem treated quantum-mechanically by Fermi's golden rule and a
solvent bath treated by classical Marcus theory. As an extension of this idea,
we here present a "reduced" semiclassical instanton theory, which is a
multiscale method for simulating quantum tunnelling of the subsystem in
molecular detail in the presence of a harmonic bath. We demonstrate that
instanton theory is typically significantly more accurate than the cumulant
expansion or the semiclassical Franck-Condon sum, which can give
orders-of-magnitude errors and in general do not obey detailed balance. As
opposed to MLJ theory, which is based on wavefunctions, instanton theory is
based on path integrals and thus does not require solutions of the
Schr\"odinger equation, nor even global knowledge of the ground- and
excited-state potentials within the subsystem. It can thus be efficiently
applied to complex, anharmonic multidimensional subsystems without making
further approximations. In addition to predicting accurate rates, instanton
theory gives a high level of insight into the reaction mechanism by locating
the dominant tunnelling pathway as well as providing information on the
reactant and product vibrational states involved in the reaction and the
activation energy in the bath similarly to what would be found with MLJ theory.
