Unitary quantum time evolution by iterative Lanczos reduction

TLDR

The paper develops a general unitary time‑evolution method for wave packets on a fixed ℒ2 basis. The method reduces the full Hamiltonian to a p‑dimensional Krylov subspace via Lanczos iterations, evolves the state unitarily with exp(−iH_pt) over an interval τ determined from the reduced Hamiltonian, and repeats this process for successive intervals, demonstrated on one‑ and two‑dimensional potentials with a Gaussian basis and applied to compute a thermal rate constant for the Eckart barrier. The approach yields accurate long‑time results but is most efficient for large systems over short time intervals.

Abstract

A general unitary time evolution method for wave packets defined on a fixed ℒ2 basis is developed. It is based on the Lanczos reduction of the full N×N Hamiltonian to a p-dimensional subspace defined by the application of H p−1 times to the initial vector. Unitary time evolution in the subspace is determined by exp{−iHpt}, retaining accuracy for a time interval τ, which can be estimated from the Lanczos reduced Hamiltonian Hp. The process is then iterated for additional time intervals. Although accurate results over long times can be obtained, the process is most efficient for large systems over short times. Time evolution employing this method in one- (unbounded) and two-dimensional (bounded) potentials are done as examples using a distributed Gaussian basis. The one-dimensional application is to direct evaluation of a thermal rate constant for the one-dimensional Eckart barrier.