FURTHER GENERALIZATION AND NUMERICAL IMPLEMENTATION OF PSEUDO-TIME SCHRÖDINGER EQUATIONS FOR QUANTUM SCATTERING CALCULATIONS

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) = ϕ T Ut 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 energy-dependent 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.