Time propagation and spectral filters in quantum dynamics: A Hermite polynomial perspective

We present an investigation of Hermite polynomials as a basic paradigm for quantum dynamics, and make a thorough comparison with the well-known Chebys hev method. The motivation of the present study is to develop a compact and numerically efficient formulation of the spectral filter problem. In parti cular, we expand the time evolution operator in a Hermite series and obtain thereby an exponentially convergent propagation scheme. The basic features of the present formulation vis a vis Chebyshev scheme are as follows: (i) Contrary to the Chebyshev scheme Hamiltonian renormalization is not needed. However, an arbitrary time scaling may be necessary in order to avoid nume rical hazards, and this time scaling also provides a leverage to accelerate the convergence of the Hermite series. We emphasize the final result is in dependent of the arbitrary scaling. (ii) As with the Chebyshev scheme the m ethod is of high accuracy but not unitary by definition, and thus any devia tion from unitarity may be used as a guideline for accuracy. The calculatio n of expansion coefficients in the present scheme is extremely simple. To c ontrast the convergence property of present method with that of the Chebysh ev one for finite time propagation, we have introduced a time-energy scalin g concept, and this has given rise to a unified picture of the overall conv ergence behavior. To test the efficacy of the present method, we have compu ted the transmission probability for a one-dimensional symmetric Eckart bar rier, as a function of energy, and shown that the present method, by suitab le time-energy scaling, can be very efficient for numerical simulation. Tim e-energy scaling analysis also suggests that it may be possible to achieve a faster convergence with the Hermite based method for finite time propagat ion, by a proper choice of scaling parameter. We have further extended the present formulation directed toward the spectral filter problem. In particu lar, we have utilized the Gaussian damping function for the purpose. The He rmite propagation scheme has allowed all the time integrals to be done full y analytically, a feature not completely shared by the Chebyshev based sche me. As a result, we have obtained a very compact and numerically efficient scheme for the spectral filters to compute the interior eigenspectra of a l arge rank eigensystem. The present formulation also allows us to obtain a c losed form expression to estimate the error of the energies and spectral in tensities. As a test, we have utilized the present spectral filter method t o compute the highly excited vibrational states for the two-dimensional LiC N (J=0) system and compared with the exact diagonalization result. (C) 1999 American Institute of Physics. [S0021- 9606(99)00647-9].