Short-iterative Lanczos method

Equation (4) is a three-term recursion for propagating a wave packet, and, assuming one starts out with some 4>(0) and (r) consistent with Eq. (1), then the iterations of Eq. (4) will generate the correct wave packet. The difficulty, of course, is that the action of the cosine operator in Eq. (4) is of the same difficulty as evaluating the action of the exponential operator in Eq. (1), requiring many evaluations of H on the current wave packet. Gray [8], for example, employed a short iterative Lanczos method [9] to evaluate the cosine operator. However, there is a numerical simplification if the representation of H is real. In this case, if we decompose the wave packet into real and imaginary parts. 

The Lanczos algorithm can also be used to approximate a short-time propagator. The so-called short-iterative Lanczos (SIL) method of Park and Light constructs a small set of Lanczos vectors,226 which can be summarized by Eq. [96] ... 

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. 

EOMCC = equation-of-motion coupled-clu.ster FOD = fourth-order differencing MBGF = many-body Green s function RR = resonance Raman RRGM = recursive-residue-generation method SIL = short-time iterative Lanczos SOD = second-order differencing. 

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). 

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