An alternative implementation of the Lanczos algorithm for wave function propagation

We reformulate the Lanczos algorithm for quantum wave function propagation in terms of variational principle. By including some basis states of previous time steps into the variational subspace, the resultant accuracy increases by several orders. Numerical errors of the alternative method accumulate much slower than that of the original Lanczos method. There is almost no extra numeric cost for the gaining of the accuracy, i.e., the accuracy increase needs no extra operations of the Hamiltonian acting on state vectors, which are the major numeric cost for wave function propagation. A wave packet moving in a 2-dimensional H\'enon-Heiles model serves as an illustration. This method is suitable for small time step propagation of quantum wave functions in large scale time dependent calculations where the operations of the Hamiltonian acting on state vectors are expensive.