Coupled channel T-operator equations with connected kernels: (II). N coupled two-body channels

A time-independent theory of rearrangement collisions involving transitions between two-body states is presented. It is assumed that the system of interest consists of particles that may be partitioned into two-body systems in N ways, including interchanges of particle labels without changing the kind of channel. An infinite family of sets of N coupled T-operator equations is derived by use of the channel coupling array, as in previous work on the three-body problem. Specialization to the channel-permuting arrays guaranteeing connected (N?1)th iterates of the kernel of the coupled equations is made in the N-channel case (N > 3) and the nature of the solutions to the coupled equations is discussed. Various approximation schemes to be used with numerical calculations are suggested. Since the transition operators for all rearrangement channels are coupled together, no problems concerning non-orthogonality of the eigenstates of different channel Hamiltonians are encountered; also the presence of the outgoing wave boundary condition in all channels is made explicit. The close resemblance of the equations in matrix form to those of one-channel scattering is exploited by introducing Møller wave operators and associated channel scattering states, an optical potential formalism that leads to rearrangement channel optical potential operators, and a variational formulation of the coupled equations using a Schwinger-like variational principle. A brief comparison with other many-body formalisms is also given.     

Computational tests of the factorization of cross sections in the sudden approximation

The factorization expressions for cross sections reported by Goldflam, Green and Kouri and independently by Khare are tested using accurate close coupling input for e^? + H₂, H + H₂, He + (HF, DF, HCl, DCl) and Ar + N₂. The results at the degeneracy averaged cross section level are used to illustrate accuracy criteria given for the factorization. Effects studied include variation of the projectile reduced mass, influence of initial and final rotor state, collision energy and potential anisotropy.     

Qualitative properties of modified equations

Suppose that a consistent one-step numerical method of orderr is applied to a smooth system of ordinary differential equations.Given any integer m 1, the method may be shown to be of orderr + m as an approximation to a certain modified equation. Ifthe method and the system have a particular qualitative propertythen it is important to determine whether the modified equationsinherit this property. In this article, a technique is introducedfor proving that the modified equations inherit qualitativeproperties from the method and the underlying system. The techniqueuses a straightforward contradiction argument applicable toarbitrary one-step methods and does not rely on the detailedstructure of associated power series expansions. Hence the conclusionsapply, but are not restricted, to the case of Runge-Kutte methods.The new approach unifies and extends results of this type thathave been derived by other means: results are presented forintegral preservation, reversibility, inheritance of fixed points.Hamiltonian problems and volume preservation. The techniquealso applies when the system has an integral that the methodpreserves not exactly, but to order greater than r. Finally,a negative result is obtained by considering a gradient systemand gradient numerical method possessing a global property thatis not shared by the associated modified equations.     

Diatom-diatom collisions: jz-conserving coupled states calculations for para-H2 + para-H2 scattering

The first calculations of diatom-diatom collision cross sections using the j_z-conserving coupled states approximation are reported. Results are given at 806.6 cm⁻¹ and are compared with the accurate close coupling results of Green and the effective potential results of Rabitz for the transitions 0,0 → 0, 0, 0, 0 → 2, 0 and 0, 0 → 2, 2. The j_z CCS method is found to be quantitative for these transtitions, with a level of accuracy the same as that found by McGuire and Kouri for the He + H₂ system. This is true even for the 0, 0 → 2, 2 transition, which is dominated by the long range quadrupole-quadrupole interaction. These preliminary results indicate that the j_z CCS method works as well for diatom-diatom collisions as for atom-molecule systems. Additional topics currently under study are indicated.     

Comparison of multichannel and two-state calculations for H-atom transfer between two nearly degenerate states

Multichannel and two-state calculations are made for a collinear H-atom transfer between two heavy masses. The particular reaction studied is on where the initial vibrational state of the reactants is nearly degenerate with a vibrational state of the products. The two sets of results for the reaction probability are in good agreement with each other.     

On the decoupling of angular momenta in molecular collisions

It is pointed out that the more exact treatment of the j_z-conserving coupled states (R?-helicity) approximation is quite closely related to the P?-helicity method of Tamir and Shapiro, and the two methods lead to approximate equations identical in form. The difference in interpretation of the magnetic quantum numbers in the two methods is used to understand the accuracy of the crudest approximations obtained by neglecting the transformation from rotating to space-fixed coordinate frames. Further it is expected that the two approaches lead to similar results in the case where the exact diagonal elements of the l² operator are retained and the correct transformations to non-helicity frames are used. Finally, the two methods should lead to identical results for integral cross sections and opacities.