Any order imaginary time propagation method for solving the Schrِdinger equation

The eigenvalue-function pair of the 3D Schrِdinger equation can be efficiently computed by use of high order, imaginary time propagators. Due to the diffusion character of the kinetic energy operator in imaginary time, algorithms developed so far are at most 4th order. In this work, we show that for a grid based algorithm, imaginary time propagation of any even order can be devised on the basis of multi-product splitting. The effectiveness of these algorithms, up to the 12 th order, is demonstrated by computing all 120 eigenstates of a model C 60 molecule to very high precisions. The algorithms are particularly useful when implemented on parallel computer architectures.