Single Lanczos propagation

Transition Amplitudes with a Single Lanczos Propagation. 

A1 A" —> X A Resonance Emission Spectra of HCN/DCN Using Single Lanczos Propagation... 

A commonly used approach for computing the transition amplitudes is to approximate the propagator in the Krylov subspace, in a similar spirit to the time-dependent wave packet approach.7 For example, the Lanczos-based QMR has been used for U(H) = (E — H)-1 when calculating S-matrix elements from an initial channel (%m )-93 97 The transition amplitudes to all final channels (Xm) can be computed from the cross-correlation functions, namely their overlaps with the recurring vectors. Since the initial vector is given by xmo only a column of the S-matrix can be obtained from a single Lanczos recursion. 

The above linear system of differential equations can be solved by various propagation methods [2], as the time split [28, 29], Chebyshev [30], or Lanczos [31, 32] method, provided an efficient and accurate method for computing the matrix elements Hij is given (see, e.g., Appendix B of ref. 3). The basic limitation of the method stays in its bad scaling properties, since the number of equations goes as rf, where n is the dimension of the single-mode basis set (supposed here to be identical for all the modes). 

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