Omar A. Sharafeddin

Time-dependent treatment of scattering: integral equation approaches using the time-dependent amplitude density

The time‐dependent form of the Lippmann–Schwinger integral equation is used as the basis of several new wave packet propagation schemes. These can be formulated in terms of either the time‐dependent wave function or a time‐dependent amplitude density. The latter is nonzero only in the region of configuration space for which the potential is nonzero, thereby in principle obviating the necessity of large grids or the use of complex absorbing potentials when resonances cause long collision times (leading, consequently, to long propagation times). Transition amplitudes are obtained in terms of Fourier transforms of the amplitude density from the time to the energy domain. The approach is illustrated by an application to a standard potential scattering model problem where, as in previous studies, the action of the kinetic energy operator is evaluated by fast Fourier transform (FFT) techniques.

Spectroscopic analysis of transition state energy levels - Bending-rotational spectrum and lifetime analysis of H3 quasibound states

We report converged quantum mechanical calculations of scattering matrices and transition probabilities for the reaction of H with H2 with total angular momentum 0, 1, and 4 as functions of total energy in the range 0.85–1.15 eV on an accurate potential energy surface. These calculations show energy dependences that may be attributed to dynamical resonances with vibrational quantum numbers (100 0) and (111 0). The resonance structure is illustrated with Argand diagrams, and we present state‐to‐state reactive collision delay times and lifetimes. For J=0, 1, and 4, we found the lowest‐energy H3 resonance at total energies of 0.983, 0.985, and 1.01 eV, respectively, with lifetimes of about 16–17 fs. For J=1 and 4 there is a higher‐energy resonance at 1.10–1.11 eV; for J=1 the lifetime is about 4 fs and for J=4 it is about 1 fs.

A comparison of three time‐dependent wave packet methods for calculating electron–atom elastic scattering cross sections

We compare three time‐dependent wave packet methods for performing elastic scattering calculations from screened Coulomb potentials. The three methods are the time‐dependent amplitude density method (TDADM), what we term a Cayley‐transform method (CTM), and the Chebyshev propagation method of Tal‐Ezer and Kosloff. Both the TDADM and the CTM are based on a time‐dependent integral equation for the wave function. In the first, we propagate the time‐dependent amplitude density, ‖ζ(t)〉=U‖ψ(t)〉, where U is the interaction potential and ‖ψ(t)〉 is the usual time‐dependent wave function. In the other two, the wave function is propagated. As a numerical example, we calculate phase shifts and cross sections using a screened Coulomb, Yukawa type potential over the range 200–1000 eV. One of the major advantages of time‐dependent methods such as these is that we get scattering information over this entire range of energies from one propagation. We find that in most cases, all three methods yield comparable accuracy and...

Time-dependent treatment of scattering. II - Novel integral equation approach to quantum wave packets

The time‐dependent form of the Lippmann–Schwinger integral equation is used as the basis for a novel wave‐packet propagation scheme. The method has the advantage over a previous integral equation treatment in that it does not require extensive matrix inversions involving the potential. This feature will be important when applications are made to systems where in some degrees of freedom the potential is expressed in a basis expansion. As was the case for the previous treatment, noniterated and iterated versions of the equations are given; the iterated equations, which are much simpler in the present new scheme than in the old, eliminate a matrix inversion that is required for solving the earlier noniterated equations. In the present noniterated equations, the matrix to be inverted is a function of the kinetic energy operator and thus is diagonal in a Bessel function basis set (or a sine basis set, if the centrifugal potential operator is incorporated into an effective potential). Transition amplitudes for ...

Numerical evaluation of spherical Bessel transforms via fast Fourier transforms

The purpose of this article is to describe a new fast Fourier transform (FFT) method for calculating spherical Bessel transforms. The method is based on an expansion representation of the spherical Bessel functions in terms of sine and cosine trigonometric functions, multiplied by polynomials in inverse powers of the argument of the Bessel function [ 11. The method should be of value in algorithms in which frequent calculations of spherical Bessel transforms are required at many values of the transform variables, which is frequently the situation in time-dependent scattering calculations. The method makes use of FFTs for which the computing time for each transform scales as Nlog,(N), where N is the number of quadrature points, rather than the N2 scaling of ordinary numerical quadratures. Also, the explicit evaluation of the spherical Bessel functions is not required in the method. The method differs from that given in the work of J. D. Talman [2, 33 and A. E. Siegmon [4]. In the work of Talman [2, 33 and Siegman [4], a change to logarithmic variables is required to recast the integral transform as a convolution integral, which is then evaluated by FFT procedures. However, as pointed out by Talman, this makes the step size Ar increase proportionally with r and renders the method unsuitable for functions of an oscillatory nature. In particular, the oscillatory nature of the function is lost in the increasing mesh intervals. In essence, the Talman-Siegman method is excellent for bound state wave functions, but it is not

Quadrature-based, coarse-grained treatment of the coordinate representation free particle real-time evolution operator

In this paper we report a quadrature evaluation of the coordinate representation, short‐time free particle propagator, 〈R‖exp(−iH0τ)‖R’〉. The result is the elimination of most of the highly oscillatory behavior in this quantity yielding in its stead a much smoother function, strongly peaked at R=R’. We view this as a numerical coarse graining of the propagator which leads to the intuitively reasonable result that for short times τ or large mass, the particle should not have a significant amplitude for R points that are far from R’. This leads to an interesting, and potentially useful, banded structure for 〈R‖exp(−iH0τ)‖R’〉. Calculations have been carried out both for zero and nonzero orbital angular momenta, for which we also give the exact analytic results, and the same behavior is found. The quadrature‐coarse graining procedure still appears to retain the important quantum effects as demonstrated by subsequent use of the coarse‐grained free propagator to calculate the scattering of an electron by a simp...

On the role of parallel architecture supercomputers in time-dependent approaches to quantum scattering

SummaryResults of our initial study of the use of parallel architecture super-computers in solving time-dependent quantum scattering equations are reported. The specific equations solved are obtained from the time-dependent Lippmann-Schwinger integral equation by means of a quadrature approximation to the time integral. This leads to a modified Cayley transform algorithm in which the primary computational step is a matrix-vector multiplication. Implementation has been carried out both for the MasPar MP-1 and the NCUBE 6400 parallel machines. The codes are written in a modular form that greatly facilitates porting from one machine architecture to another. Both parallel machines prove to be more powerful for this application than the serial architecture VAX 8650. Specific analysis of machine performance is given.

Time dependent integral equation approaches to quantum scattering: Comparative application to atom–rigid rotor multichannel scattering

In this paper we generalize earlier work on potential scattering to atom–rigid rotor scattering. We compare six approaches including the interaction picture, modified Cayley, amplitude density, and symmetric split operator methods. All methods derive from the integral equation form of the time‐dependent Schrodinger equation. The methods were tested using the standard Lester–Bernstein model potential. All methods were found to perform well with the same parameters. Fast Fourier transforms were not used in these methods, and an average execution time for a 16 channel problem on CRAY YMP supercomputer was about 45 s. This single calculation yields results at any energy significantly contained in the initial packet. In the present study, the S matrix was computed at a total of 42 energies, but results could have been obtained at many more energies without a large increase in computing time. Timing results for one of the methods are reported for 25, 64, 144, and 256 coupled channels.