A new method is developed for integrating coupled differential equations arising in bound state and scattering problems in quantum mechanics. The wavefunctions are easily constructed in piecewise analytic form, to any prescribed accuracy.

New Method for Constructing Wavefunctions for Bound States and Scattering

An explicit method for the exact calculation of unimolecular linewidths and line shapes, using an adaptation of known numerical techniques, is presented. The method renders feasible the calculation of decay from one composite ``bound'' subspace, into another composite ``continuum'' subspace. The eigenvalues of the Hamiltonian for the whole ``bound'' subspace are also obtained to a high degree of accuracy. Two specific examples are studied: (a) A single closed channel (containing four bound states) coupled to a single open channel, and (b) three closed channels corresponding to eight bound states, interacting with a single open channel. The exact line shapes for the first of these cases are computed. Interference effects, due to overlapping resonances, are clearly demonstrated and discussed in terms of Fano's theory. Line‐widths and resonance energies are calculated for the second case. The results are contrasted to those of the related scattering problem and discussed in terms of the different boundary conditions.

Dynamics of Dissociation. I. Computational Investigation of Unimolecular Breakdown Processes

A practical method is presented for computing the T‐matrix elements for a rotating, vibrating oscillator. A simple model is used which approximates the features of the He–H2 system. It is found that the rigidrotor approximation is in error even at energies well below the threshold for vibrational excitation. For computation of rotational transitions from the ground to the first accessible excited rotational state, many of the excited rotational transitions may be neglected but some of the excited vibrational transitions must be included. At high energies, it is shown that for any particular transition many of the states not strongly coupled to the states involved in the transition may be neglected. It is found that the computation of T‐matrix elements for vibrational transitions in the presence of rotational transitions is not prohibitively time consuming. In computing the total cross section, it is shown that a calculation including only the ground state gives remarkably good results.

Calculation of Rotational and Vibrational Transitions for the Collision of an Atom with a Rotating Vibrating Diatomic Oscillator

A new method (homogeneous integral solution) for solving collision problems utilizing an integral equation formalism is presented. The approach is noniterative and applies equally well to purely local interactions or problems with combined local and nonlocal interactions. The method involves transforming the integral equation of scattering into a Volterra integral equation of the second kind. It follows that even for nonlocal interactions, the wavefunction may be determined noniteratively from knowledge of its value at a single point (the origin). A simple numerical procedure is proposed for solution of the Volterra equation which allows one to avoid matrix inversions completely. At the end of the calculation, the scattering T or R matrix is obtained directly from the solutions. The method is illustrated by a calculation of the singlet and triplet s‐wave Hartree–Fock phase shifts for electron–H‐atom scattering.

Noniterative solutions of integral equations for scattering. i. single channels.

Substantial effort has been directed toward developing methods for describing molecular collision processes that are electronically adiabatic, i.e., for which it can be assumed that nuclear motion evolves on a single potential energy hypersurface. A number of recent reviews are devoted to this subject.(1–8) Considerably less attention has been paid to processes that are nonadiabatic,i.e., that involve electronic transitions between potential energy surfaces. This is in spite of the fact that nonadiabatic behavior is both common and important, even in thermal energy collisions.

Nonadiabatic Processes in Molecular Collisions

The method of ``amplitude density functions'' is a new formalism which allows a scattering problem to be broken up into problems with weaker interaction potentials. These simpler problems may be solved separately and be ``added'' together to give the total solution. A numerical method is discussed which takes advantage of this property. The formulas are given for the use of this method in the solution of the one‐dimensional atom‐molecule collision and the e+−H collision. A numerical example is discussed.

The Solution of the Nonrelativistic Quantum Scattering Problem without Exchange

The problem of finding linear combinations of Slater determinants which are eigenfunctions of both the S2 and Sz operators is considered. The method of projection operators is used in forming eigenfunctions of S2. A recursion relation is given for the coefficients in the linear combination. Rules are given for selecting a complete linearly independent set of eigenfunctions and an explicit formula is given for the members of the set.

Don Secrest

Papers

The Solution of the Nonrelativistic Quantum Scattering Problem without Exchange

Eigenfunctions of the Electron Spin Operators