Chebyshev method

In most cases, this Lanczos-based technique proves to be superior to the Chebyshev method introduced above. It is the method of choice for the application problems of class 2b of Sec. 2. The Chebyshev method is superior only in the case that nearly all eigenstates of the Hamiltonian are substantially occupied. 

The tower characteristic KaV/L can be determined by integration. Normally used is the Chebyshev method for numerically ev uating the integral, whereby... 

Chebyshev Method for Calculating State-to-State Reaction Probabilities from the Time-Independent Wavepacket Reactant-Product Decoupling Equations. 

Panovsky and Richardson, A family of implicit Chebyshev methods for the numerical integration of second-order differential equations, J. Comput. Appl. Math., 1988, 23, 35-51. 

The electron dynamics has been integrated with the Chebyshev method. [430] The time propagation of multi-state coupled nuclear wavefunction has been performed with use of the Chebyshev expansion method. [430] Since the ground electronic state jl) involves molecular dissociation, we... 

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. 

Althorpe S, Kouri D, Hoffman D (1997) A chebyshev method for calculating state-to-state reaction probabilities from the time-independent wavepacket reactant-product decoupling... 

The combined AO/MO method takes advantage of the simple form that the equations for the time-propagation take up in the adiabatic MO basis set within the time slice, whereas the transfer of the wavepacket coefficients between consecutive time slices is still carried out in the non-adiabatic AO basis set. The procedure, illustrated in Fig. 3, is robust and produces results in complete agreement with the Chebyshev method. 

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