Further generalization and numerical implementation of pseudo-time Schrödinger equations for quantum scattering calculations
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چکیده
We review and further develop the recently introduced numerical approach [Phys. Rev. Lett. 86, 5031, (2001)] for scattering calculations based on a so called pseudo-time Schrödinger equation, which is in turn a modification of the damped Chebyshev polynomial expansion scheme [J. Chem. Phys. 103, 2903, (1995)]. The method utilizes a special energy-dependent form for the absorbing potential in the time-independent Schrödinger equation, in which the complex energy spectrum is mapped inside the unit disk Ek → uk, where uk are the eigenvalues of some explicitly known sparse matrix U . Most importantly for the numerical implementation, all the physical eigenvalues uk are the extreme eigenvalues of U (i.e., uk ≈ 1 for resonances and uk = 1 for the bound states), which allows one to extract these eigenvalues very efficiently by harmonic inversion of a pseudo-time autocorrelation function y(t) = φU φ using the filter diagonalization method. The computation of y(t) up to time t = 2T requires only T sparse real matrix-vector multiplications. We describe and compare different schemes, effectively corresponding to different choices of the energydependent absorbing potential, and test them numerically by calculating resonances of the HCO molecule. Our numerical tests suggest an optimal scheme that provide accurate estimates for most resonance states using a single autocorrelation function.
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تاریخ انتشار 2002