On the equivalence of two commonly used forms of semiclassical instanton theory

Semiclassical instanton theory gives an approximate description of deep tunneling by means of periodic orbits on the inverted potential energy surface. There are two versions of the theory, one derived by taking a semiclassical limit of the exact flux-side time-correlation function and the other by starting from the "Im F" premise, in which the partition function is analytically continued into the complex plane. Here, we provide a derivation showing that the two versions of the theory are exactly equivalent. Unlike a previous derivation (which was restricted to a system-bath model), our derivation is completely general, and thus establishes that the "Im F" premise, which is behind such methods as quantum transition-state theory and ring polymer molecular dynamics rate-theory, is correct in the steepest-descent limit.