Scalable kernel polynomial method for calculating transition rates

We present an efficient method for calculating the prefactors of harmonic transition state theory rates. We reformulate the prefactors in terms of the density of states (DOS) of the Hessian matrices at the basin minimum and the saddle point. The DOS is then approximated using the kernel polynomial method as an expansion in terms of Chebyshev polynomials. The cost of the calculation scales linearly with the number of atoms, in contrast with the cubic scaling of the direct method. This approach hence greatly facilitates the investigations of kinetic processes in very large systems. We demonstrate the method by calculating the prefactors of the transition rates for two processes in bulk silver: vacancy hopping and Frenkel pair formation.