Chebyshev propagation

The RWP method also has features in common with several other accurate, iterative approaches to quantum dynamics, most notably Mandelshtam and Taylor s damped Chebyshev expansion of the time-independent Green s operator [4], Kouri and co-workers time-independent wave packet method [5], and Chen and Guo s Chebyshev propagator [6]. Kroes and Neuhauser also implemented damped Chebyshev iterations in the time-independent wave packet context for a challenging surface scattering calculation [7]. The main strength of the RWP method is that it is derived explicitly within the framework of time-dependent quantum mechanics and allows one to make connections or interpretations that might not be as evident with the other approaches. For example, as will be shown in Section IIB, it is possible to relate the basic iteration step to an actual physical time step. 

The propagator nature of the Chebyshev operator is not merely a formality it has several important numerical implications.136 Because of the similarities between the exponential and cosine propagators, any formulation based on time propagation can be readily transplanted to one that is based on the Chebyshev propagation. In addition, the Chebyshev propagation can be implemented easily and exactly with no interpolation errors using Eq. [56], whereas in contrast the time propagator has to be approximated. 

Like the time propagation, the major computational task in Chebyshev propagation is repetitive matrix-vector multiplication, a task that is amenable to sparse matrix techniques with favorable scaling laws. The memory request is minimal because the Hamiltonian matrix need not be stored and its action on the recurring vector can be generated on the fly. Finally, the Chebyshev propagation can be performed in real space as long as a real initial wave packet and real-symmetric Hamiltonian are used. 

The cosine form of the Chebyshev propagator also affords symmetry in the effective time domain, which allows for doubling of the autocorrelation function. In particular, 2K values of autocorrelation function can be obtained from a E-step propagation 147... 

H. Guo and R. Chen,/. Chem. Phys., 110, 6626 (1999). Short-Time Chebyshev Propagator... 

Dordrecht, The Netherlands, 2004, pp. 217-229. Chebyshev Propagation and Applications to Scattering Problems. 

Z.G. Sun, S.Y. Lee, H. Guo, D.H. Zhang, Comparison of second-order split operator and Chebyshev propagator in wave packet based state-to-state reactive scattering calculations, J. Chem. Phys. 130 (2009) 174102. 

Abstract. The Chebyshev operator is a diserete eosine-type propagator that bears many formal similarities with the time propagator. It has some unique and desirable numerical properties that distinguish it as an optimal propagator for a wide variety of quantum mechanical studies of molecular systems. In this contribution, we discuss some recent applications of the Chebyshev propagator to scattering problems, including the calculation of resonances, cumulative reaction probabilities, S-matrix elements, cross-sections, and reaction rates. 

We note here in passing that Chebyshev propagation is related to several other recursive methods based on the Krylov subspace = span i//(, Hy/(, ...,H Wq) ... 

This differenee stems from the abandonment in the damped Chebyshev propagation of the doubling relation commonly used for the conventional Chebyshev autocorrelation function, [3 5] which can again be derived from trigonometry ... 

Sueh doubling relations obviously do not hold for damped Chebyshev propagation prescribed by Eq. (8), but the real question is whether they will allow the extraction of the narrow resonanees using the Chebyshev-based LSFD. 

To answer this question, we eomputed the resonance positions and widths of HCO 41] and HN2,[42] using both doubled and undoubled autocorrelation funetions obtained from the damped Chebyshev propagation. The results indieated tiiat the enforeed doubling of the autocorrelation function yields no appreeiable differenees in both positions and widths of the narrow resonances when compared with those obtained from a directly calculated autocorrelation function. The differences are plotted in Fig. 1 for the low-lying resonances of HN2. The largest differences are for resonanees with widths on the order of a few hundred wave numbers.[42]... 

The advantage of the correlation function approach is that only the storage of scalar quantities, rather than wave packets, is needed. Thus, the memory requirement is significantly reduced, an issue that may become more important for large systems. The implementation with the Chebyshev propagator takes further advantage of its numerical properties discussed above. In cases where resonances are dominant, the LSFD approach can be used to further reduce computational costs. We note in passing this approach can be extended to the calculation of thermal rate constants. 

We have recently studied rotational and vibrational inelastic scattering between two H2 molecules using the damped Chebyshev propagation with a full-dimensional Hamiltonian.[63-65] The corresponding S-matrix elements were obtained as Fourier... 

The total reaction probability is typically obtained fiom the reactive flux calculated at the dividing surface placed at a point-of-no-retum.[70,71] This surface is often located in the product channel, but not necessarily at the asymptote where the S-matrix elements are completely converged. Consequently, such calculations can be conveniently carried out in reactant Jacobi coordinates and the computational costs are no more expensive than that for inelastic scattering. Implemented for the Chebyshev propagation, the reaction probability is given as below [72]... 

Chebyshev propagation and apphcations to scattering problems Harding L.B. 

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