Abstract
To use the short iterative Lanczos algorithm to solve the time-dependent Schroedinger equation, one must choose, for a given Lanczos space size, a time step. We compare the derivation of the well-known Lubich and Hochbruck time step from SIAM J. Numer. Anal. 34 (1997) 1911 with the a priori time step we proposed in Mohankumar and Carrington (MC) Comput. Phys. Commun., 181 (2010) 1859 and demonstrate that the MC time step is somewhat larger, i.e., that the MC error bound is tighter. In addition, we use the MC approach to derive an error bound and time step for imaginary time propagation. The error bound we derive is much tighter than the error bound of Stewart and Leyk.
Highlights
The short iterative Lanczos (SIL) algorithm [1] for solving the time-dependent Schroedinger equation (TDSE), i ∂ψ( x, t) = Ĥψ( x, t), ∂t (1)is widely used in chemical physics and other fields
In typical calculations, ∆t is small enough that 1 − α ∼ 1 and e−my ∼ 1 and in this case the time step ∆t Mohankumar and Carrington (MC) is larger than ∆t L by a factor of (8πm)1/m . ∆t MC is larger than ∆t L whenever α < 0.8
Once an error bound has been found, it is straightforward to derive an equation for the best time step to use
Summary
The short iterative Lanczos (SIL) algorithm [1] for solving the time-dependent Schroedinger equation (TDSE) (in atomic units), i. M is small enough that loss of orthogonality of the Lanczos vectors is unimportant. We shall assume that a(t = 0) is the Lanczos starting vector. To use the SIL algorithm, one chooses a value of m, propagates from t = 0 to t = ∆t, and uses a(∆t) as the starting vector and propagates again from t = ∆t to t = 2∆t, etc. Small enough that loss of orthogonality is not a problem, one must choose ∆t. We compare the derivations of the Lubich and MC equations and show that. We derive and test a new equation for ∆t for propagating in imaginary time
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