This paper provides an overview of some recent developments in quantum dynamic and spectroscopic calculations using the Chebyshev propagator. It is shown that the Chebyshev operator (Tk(H)) can be considered as a discrete cosine type propagator (cos(kΘ)), in which the angle operator (Θ = arccos H) is a single-valued mapping of the scaled Hamiltonian (H) and the order (k) is an effective time. Properties in the angle domain (thus in energy domain) can be obtained from those in the order domain via a cosine Fourier transform. The equivalence and parallelism between the order-angle formulation and the time-energy formulation allow one to transplant existing time-dependent quantum mechanical methods. Compared with the time propagator, the Chebyshev propagator has a number of numerical advantages. Because of its polynomial form, the action of the Chebyshev propagator can be evaluated exactly except for roundoff errors. The three-term recursion formula for Chebyshev polynomials is stable and requires minimal memory. The discretization error in time propagation can be avoided because the Chebyshev propagator is naturally discrete. Additional computational savings over the time propagation are possible because the Chebyshev wave packet can be propagated in the real space. Various applications of the Chebyshev propagator are discussed.
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