Two Lanczos subspace propagation techniques are discussed in this work and compared with the Chebyshev method applied to the original Hamiltonian matrix. Both procedures involve the use of a reduced propagator in the Lanczos subspace to calculate the solution to the time-dependent Schrodinger equation but differ in the Ε½ . way the propagator is evaluated. The LSC Lanczos subspace Chebyshev expresses the propagator in terms of Chebyshev polynomials that are functions of the tridiagonal Ε½ Hamiltonian matrix in the Lanczos space. In contrast, the LSV Lanczos subspace . variational is implemented by solving the eigenproblem in the Lanczos subspace and then performing a variational expansion of the propagator in the M-dimensional eigenvector space. Although the LSV is the same as the reduced propagator scheme proposed by Park and Light, in the present study the LSV is implemented as a one-step long-time propagator. As a numerical example, the interaction of a molecule with a strong laser pulse is investigated. The Hamiltonian is explicitly time dependent in this case, and thus the stationary formalism is employed in this work to solve the timedependent Schrodinger equation. Application of either the LSC or LSV yields a wave function in the M-dimensional Lanczos subspace. Nonetheless, the transition amplitudes computed from this wave function are in excellent agreement with those calculated by direct application of the Chebyshev method in the original space used to define the Hamiltonian matrix.