  We explore the relation between the quantum and semiclassical instanton
approximations for the reaction rate constant. From the quantum instanton
expression, we analyze the contributions to the rate constant in terms of
minimum-action paths and find that two such paths dominate the expression. For
symmetric barriers, these two paths join together to describe the semiclassical
instanton periodic orbit. However, for asymmetric barriers, one of the two
paths takes an unphysically low energy and dominates the expression, leading to
order-of-magnitude errors in the rate predictions. Nevertheless, semiclassical
instanton theory remains accurate. We conclude that semiclassical instanton
theory can only be obtained directly from the semiclassical limit of the
quantum instanton for symmetric systems. We suggest a modification of the
quantum instanton approach which avoids sampling the spurious path and thus has
a stronger connection to semiclassical instanton theory, giving numerically
accurate predictions even for very asymmetric systems in the low temperature
limit.
