Coupled-channels expansion

Although the Sclirodinger equation associated witii the A + BC reactive collision has the same fonn as for the nonreactive scattering problem that we considered previously, it cannot he. solved by the coupled-channel expansion used then, as the reagent vibrational basis functions caimot directly describe the product region (for an expansion in a finite number of tenns). So instead we need to use alternative schemes of which there are many. 

A powerful way of achieving this goal uses the coupled-channels expansion, a method widely used in calculations of scattering cross sections [6]. In the context of quantized matter-radiation problems, the coupled-channels method amounts to expanding E, n, N ) in number states. Concentrating on the expansion in the /th mode, we write E, n, N ) as... 

The coupled channels expansion can be further simplified by introducing the (number state) rotating-wave approximation (RWA), valid only when the field is jjsfif moderate intensity and the system is near resonance. As pointed out above, igtyen an initial photon number state [JVf), the components of E, n, N — 1") of. . greatest interest for a one-photon transition are (JV, , n",JV, —1 ) and y (Nj dt l[Ji, n, Nj — 1 ). If [ , ) is the ground material state, then the (Nj+m,... 

The final stage in the adiabatic reduction is the solution of Eq. (4.24). Given the adiabatic potential of Eq. (4.26) this cannot be done analytically, but the resulting ordinary differential equation may be solved numerically using the finite difference method. As an example, we show in Fig. 20 a comparison between the even-parity adiabatic eigenvalues and the exact ones, obtained by solving the full coupled channels expansion, using the artificial channel method.69... 

QM1 and QM2 indicates that the coupled channel expansion may not have converged for these vibrationally inelastic processes 78 within this level of uncertainty, however, there is good agreement between the semiclassical and quantum calculations. Also shown in Fig. 12 is a phase space distribution... 

Instead of proceeding with the coupled channel expansion of in terms of eigenfunctions of consider an adiabatic approximation to the y-motion. Thus, we shall express as... 

In the application of the ESA to the entrance channel a, the operator in the Hamiltonian of equation (1) is replaced by the average value 3 (T+1) [27,28]. If 3T is the projection of along z, then the problem reduces to obtaining solutions for each separate M value. We have the coupled channel expansion... 

CSA as the coupled-channel expansion in the entrance channel contains the orbital spherical harmonics, the sum over which in equation (15) achieves the partial-wave expansion. For light-heavy-light reactions at room temperature energies, the maximum value of J required is about 20 and the ESA-CSA computations will then be 20 times cheaper than the CSA... 

This is an area that has benefited enormously from the use of supercomputers and mini-supercomputers. The theories described in this review are almost all based on a coupled-channel expansion of the wavefiinction and the numerical woik involved in the calculation is largely concerned with matrix manipulations and the computations of multidimensional integrds. These are all operations that are made efficient through the use of computers with vector processing capabilities. Furthermore, the power of these computers has enabled more accurate calculations to be performed on more compUcated systems which have provided benchmark results that have allowed the accuracy of the many approximate methods to be properly calibrated. Also, the new computers have now made it possible to perform approximate computations on quite large molecules of real chemical interest such as aromatic molecules. The impact of these types of calculations on experimental chemistry is already being realised and will become even more significant in the years to come. 

This equation may be solved by the same methods as used with the nonreactive coupled-channel equations (discussed later in section A3.11.4.2). Flowever, because F(p, p) changes rapidly with p, it is desirable to periodically change the expansion basis set ip. To do this we divide the range of p to be integrated into sectors and within each sector choose a (usually the midpoint) to define local eigenfimctions. The coiipled-chaimel equations just given then apply withm each sector, but at sector boundaries we change basis sets. Let y and 2 be the associated with adjacent sectors. Then, at the sector boundary p we require... 

Following the well-established coupled-channel theory [17], we represent the eigenstates of the full Hamiltonian by an expansion in terms of some orthonormal states. 

The main conclusion of this section is that the matrix elements of all terms in the collision Hamiltonian in the fully uncoupled space-fixed representation can be reduced to simple products of integrals of the type (8.46). Such matrix elements are very easy to evaluate numerically. The fiilly uncoupled representation is therefore very convenient for the development of the coupled channel codes for collision problems involving open-shell molecules with many angular momenta that need to be accounted for. The price for simplicity is a very large number of basis states that need to be included in the expansion of the eigenstates of the full Hamiltonian to achieve full basis set convergence (see Section 8.3.4). 

When elastic scattering is the only open channel, k is positive but all other values of kf, and all values of nj, are negative. Consequently, all the functions F)(ri) and Gj(p), except for Fi(ri), decay exponentially for large values of r and p. The resulting equation for Fj(ri) is similar in form to equation (3.20), in which the optical potential Vopt was introduced indeed a truncated coupled-state expansion essentially defines an approximation to the optical potential which satisfies the conditions for the phase shifts to be lower bounds on the exact values. 

The present coupled-channel calculations also allow for the inclusion of projectile-centered states according to following expansion... 

The following restrictions have been found to the application of coupled-channel calculations for the computation of pulsed-laser ionization. The dipole approximation restricts the photon energy to < 1 keV in the current treatment. This, however, does not pose a strict condition since a partial-wave expansion of the laser field may be used, similar to as in the case of screened Coulomb potentials. In comparison to ion/atom collisions, typical photon/atom interaction times are extremely long. An upper limit of the pulse width AZp = 100 fs at intermediate laser-power densities follows from the numerically restricted density of continuum states. 

During the past few years we have observed an intensive development of many-channel approaches to the collision problem. In particular, the coupled-channels method is based on an expansion of the total wave fmiction in internal states of reactants and products and a numerical solution of the coupled-channels equations.This method was applied in the usual way to the atom-diatom reaction A + BC by MOR-TENSBN and GUCWA /86/, MILLER /102/, WOLKEN and KARPLUS /103/, and EL-KOWITZ and WYATT /101b/. Operator techniques based on the Lippmann-Schwinger equation (46.II) or on the transition operator (38 II) has also been used, for instance, by BAER and KIJORI /104/ The effective Hamiltonian approach( opacity and optical-potential models) and the statistical approach (phase space models, transition state models, information theory) provide other relatively simple ways for a solution of the collision problem in the framework of the many-channel method /89/<. 

An alternative approach to solving the coupled equations is to use a basis set expansion for the R coordinate as well as for the angular variables. The angular basis sets used in such calculations are generally the same as in coupled channel calculations. This approach was pioneered by Le Roy and Van Kranendonk, who used numerical basis sets for the radial (R) functions. Such basis sets are adequate for the rare gas-H2 systems, but converge very poorly for more strongly anisotropic systems. An alternative basis set, based on Morse-oscillator-like functions, has been used extensively by Tennyson and coworker... 

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