Lanczos iteration

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. 

Every Lanczos iteration with P, however, requires two operations of the Green s function onto a vector,... 

Fig. 2. Calculated GT-strength function for the 130Cd ground-state (ON decays to the low-lying 1+ (T=16) states in 130In as a sequence of the number of Lanczos iterations. The abscissa is the excitation energy in 130In. For convenience of drawing, the minimum width is chosen to be 100 keV.

Yu, H.-G. Two-layer Lanczos iteration approach to molecular spectroscopic calculation, J. Chem. Phys. 2002,117,8190-6. 

C. Lanczos, Iteration method for the solution of the eigenvalue problem of linear differential and integral operators, J. Res. Nat. Bur. Standards vol. 45 (1950) RP 2133. 

Here H may be any hermitean operator of interest, but in our case is identified with the vibronic Hamiltonian. Given these states, one may start the following three-term recurrence relations, also called Lanczos iteration ... 

We note that a very efficient version of this method based on the Lanczos iterative eigensolver has been implemented for the specific case of vibronically coupled systems described by the vibronic coupling model Hamiltonian, allowing for the computation of absorption or photoelectron spectra for systems with bases containing up to 10 basis functions. This method is described in details in the Chap. 7 of Ref. [64]. 

The calculation of the molecular eigenstates with the MVCM model, necessary in traditional time-independent methods, can prove to be very cumbersome or even unfeasible. However, time-independent effective solutions, practicable for reduced-dimensionality models (in practice when the number of relevant normal coordinates is less than 10), may be obtained by taking advantage of the Lanczos iterative tridiagonalization of the Hamiltonian matrix [130]. The Lanczos algorithm proves to be very suitable for the computation of low-resolution spectra however, its effectiveness is better highlighted in a time-dependent framework. In fact, it can be easily realized that Lanczos states are only sequentially coupled, and it is therefore clear that only a limited number of states is necessary to describe short-time dynamics since the latter is the only relevant information for low-resolution spectra (see Chapter 10). 

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... 

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