Lanczos propagation scheme

Usually, the propagator (7(r, to) is approximated by various schemes [55,60,137], and there are plenty of wonderful articles that have explained each in detail, such as the split operator method and higher order split operator methods [11, 36, 130], Chebyshev polynomial expansion [131], Faber polynomial expansion [51, 146], short iterative Lanczos propagation method [95], Crank-Nicholson second-order differencing [10,56,57], symplectic method [14,45], recently proposed real Chebyshev method [24,44,125], and Multi-configuration Time-Dependent Hartree (MCTDH) Method [ 12,73,81-83]. For details, one may refer to the corresponding references. 

Since the potential matrix V is diagonal in the grid representation, the multiplication with the exponential operators involving the potential is straightforward. The operation with the exponential kinetic energy operator is, as we have seen, equally as straightforward if we use the FFT method. Another time-propagation method is based upon the Lanczos iterative scheme where each time-step requires a number of operations with the hamiltonian operator on the wavefunction... 

Fig. 3. Stepsize r used in the simulation of the collinear photo dissociation of ArHCl the adaptive Verlet-baaed exponential integrator using the Lanczos iteration (dash-dotted line) for the quantum propagation, and a stepsize controlling scheme based on PICKABACK (solid line). For a better understanding we have added horizontal lines marking the collisions (same tolerance TOL). We observe that the quantal H-Cl collision does not lead to any significant stepsize restrictions.

Pig. 4. Photo dissociation of ArHCl. Left hand side the number of force field evaluations per unit time. Right hand side the number of Fast-Fourier-transforms per unit time. Dotted line adaptive Verlet with the Chebyshev approximation for the quantum propagation. Dash-dotted line with the Lanczos iteration. Solid line stepsize controlling scheme based on PICKABACK. If the FFTs are the most expensive operations, PiCKABACK-like schemes are competitive, and the Lanczos iteration is significantly cheaper than the Chebyshev approximation. 

---