Donald J. Kouri

On the factorization and fitting of molecular scattering information

The factorization of cross sections of various kinds resulting from the infinite order sudden approximation is considered in detail. Unlike the earlier study of Goldflam, Green, and Kouri, we base the present analysis on the factored IOS T‐matrix rather than on the S‐matrix. This enables us to obtain somewhat simpler expressions. For example, we show that the factored IOS approximation to the Arthurs–Dalgarno T‐matrix involves products of dynamical coefficients TLl and Percival–Seaton coefficients fL(jl‖j0l0‖J). It is shown that an optical theorem exists for the TlL dynamical coefficients of the T‐matrix. The differential scattering amplitudes are shown to factor into dynamical coefficients qL(χ) times spectroscopic factors that are independent of the dynamics (potential). Then a generalized form of the Parker–Pack result for Σj(dσ/d?)(j0→j) is derived. It is also shown that the IOS approximation for (dσ/d?)(j0→j) factors into sums of spectroscopic coefficients times the differential cross sections out of...

Close‐coupling wave packet approach to numerically exact molecule–surface scattering calculations

In this paper we describe the theory and application of the recently developed close‐coupling wave packet (CCWP) method to the study of transition probabilities of H2 scattered from flat and corrugated surfaces. We present an improved method of analyzing the final wave function which permits S matrices and transition probabilities to be obtained over a wide range of energies from the propagation of a single wave packet. Transition probabilities obtained using the CCWP method are in excellent agreement with those obtained from time‐independent close‐coupling (CC) calculations. For the present three‐dimensional H2‐corrugated surface scattering study, the CCWP calculations require roughly one‐tenth of the computation time of the CC method. Our results indicate that the CCWP method should be a very efficient method of obtaining highly accurate scattering results both for standard collision problems and for collision problems which cannot be readily treated using standard CC methods.

The application of time-dependent wavepacket methods to reactive scattering

In this article, we review several methods for performing numerically-exact reactive scattering calculations using time-dependent wavepackets. The basic idea we imply is to take the multi-arrangement reactive problem and reformulate it as one or more inelastic ones. In the simplest method, we extract total reaction probabilities by calculating the flux of the wavepacket as it leaves the interaction region in the direction of the reactive arrangement. To make this practical, we use complex potentials that absorb the wavepacket before it reaches the numerical grid boundary. We describe methods that generate observables ranging from total, energy-averaged reaction probabilities up to energy- and state-resolved S-matrix elements. We also review techniques for efficiently performing the necessary inelastic wavepacket propagation.

L2 amplitude density method for multichannel inelastic and rearrangement collisions

A new method for quantum mechanical calculations of cross sections for molecular energy transfer and chemical reactions is presented, and it is applied to inelastic and reactive collisions of I, H, and D with H2. The method involves the expansion in a square‐integrable basis set of the amplitude density due to the difference between the true interaction potential and a distortion potential and the solution of a large set of coupled equations for the basis function coefficients. The transition probabilities, which correspond to integrals over the amplitude density, are related straightforwardly to these coefficients.

A time‐dependent wave packet approach to atom–diatom reactive collision probabilities: Theory and application to the H+H2 (J=0) system

This paper describes a new approach to the study of atom–diatom reactive collisions in three dimensions employing wave packets and the time‐dependent Schrodinger equation. The method uses a projection operator approach to couple the inelastic and reactive portions of the total wave function and optical potentials to circumvent the necessity of using product arrangement coordinates. Reactive transition probabilities are calculated from the state resolved flux of the wave packet as it leaves the interaction region in the direction of the reactive arrangement channel. The wave packet does not need to be propagated into the asymptotic reactive region in order to determine accurate vibrationally resolved, but rotationally summed reaction probabilities. The present approach is used to obtain such vibrationally resolved reaction probabilities for the three‐dimensional H+H2 (J=0) hydrogen exchange reaction, using a body‐fixed system of coordinates.

Vibrational deactivation of diatomic molecules by collisions with solid surfaces

A model is proposed for vibrational deexcitation of diatomic molecules by collisions with a solid surface. The expressions obtained are analyzed to yield insight into the collision dynamics and used to predict the rotational and translational energy distributions, and other properties of interest. The method is developed in the approximation of a stationary surface, and is closely related to a recent model for vibrational relaxation in atom–molecule collisions. From considerations based on the scales of the relevant energy spacings and coupling strengths applied to the vibrational, rotational, and diffraction states involved, the scattering equations are greatly simplified by several approximations. For a simple but realistic class of potentials, analytical expressions are obtained for the deactivation probabilities pertaining to all final translational–rotational channels. Using the expressions of the model, a detailed study is made of: (i) The rotational–translational energy distribution produced by the...

General, energy‐separable Faber polynomial representation of operator functions: Theory and application in quantum scattering

A general, uniformly convergent series representation of operator‐valued functions in terms of Faber polynomials is presented. The method can be used to evaluate the action of any operator‐valued function which is analytic in a simply connected region enclosed by a curve, Lγ. The three most important examples include the time‐independent Green’s operator, G+(E)=1/[E−(H−ie)], where H may be Hermitian or may also contain a negative imaginary absorbing potential, the time‐dependent Green’s or evolution operator, exp(−iHt/ℏ), and the generalized collision operator from nonequilibrium statistical mechanics, 1/[E−(L−ie)], where L is the Liouvillian operator for the Hamiltonian. The particular uniformly convergent Faber polynomial expansion employed is determined by the conformal mapping between the simply connected region external to the curve Lγ, which encloses the spectrum of H−ie (or L−ie), and the region external to a disk of radius γ. A locally smoothed conformal mapping is introduced containing a finite n...

A general time-to-energy transform of wavepackets. Time-independent wavepacket-Schrödinger and wavepacket-Lippmann—Schwinger equations

Abstract Recently, new time-independent wavepacket-Schrodinger and wavepacket-Lippmann—Schwinger equations have been derived making use of absorbing potentials. We show that these equations, which are characterized by the occurrence of an initial ¢L2-wavepacket source of scattered waves, can be gotten without introducing the absorbing potential. We also show that a powerful method of solving the equations can be based on a Chebychev representation of the causal full Green function, combined with the distributed approximating function representation of the system Hamiltonian, and we present two example applications illustrating approach.

Orthogonal polynomial expansion of the spectral density operator and the calculation of bound state energies and eigenfunctions

Abstract An orthogonal polynomial expansion method is presented, and illustrated with calculations, for calculating δ( E – H ), the spectral density operator (SDO), the projection operator that projects out of any L 2 wavepacket the eigenstate (s) of H having energy E . If applied to an L 2 wavepacket which overlaps the interaction, it yields either scattering-type (improper) eigenstates or proper bound eigenstates. For negative energies, the exact SDO yields zero away from an eigenvalue, and yields the energy eigenstate (times a constant) when E equals an eigenvalue. The finite orthogonal polynomial expansion of the SDO, acting on an L 2 wavepacket, yields approximately zero for E not equal to an eigenvalue, and becomes nonzero in the neighborhood of an eigenvalue.

Infinite order sudden approximation for reactive scattering. I. Basic l‐labeled formulation

An infinite order sudden (IOS) treatment of reactive scattering is developed taking into account recent results of nonreactive collision studies on the importance of l‐labeling, nonconservation of helicity, and transformation properties of sudden approximation wave functions. The present IOS method should be sufficiently simple to apply to a number of chemically interesting atom–diatom reactions. Such applications are currently in progress.