  Ring-polymer instanton theory has been developed to simulate the quantum
dynamics of molecular systems at low temperatures. Chemical reaction rates can
be obtained by locating the dominant tunneling pathway and analyzing
fluctuations around it. In the standard method, calculating the fluctuation
terms involves the diagonalization of a large matrix, which can be unfeasible
for large systems with a high number of ring-polymer beads. Here we present a
method for computing the instanton fluctuations with a large reduction in
computational scaling. This method is applied to three reactions described by
fitted, analytic and on-the-fly ab initio potential-energy surfaces and is
shown to be numerically stable for the calculation of thermal reaction rates
even at very low temperature.
