coupledstates

Coupled States Approximation

Introduction

In the Coupled States approximation [1-5] the diagonal centrifugal matrix of the centrifugal barrier in the close-coupled formulation

(l²)_ii = (h²/2 mR²) l_i (l_i + 1) , (1)

is replaced by a constant diagonal matrix

(l²)_ii = (h²/2 mR²)

(

+ 1) , (2)

where (designated L-bar) is some average orbital angular momentum, then the CC equations become block-diagonal in the index which is the body-frame projection of J. Equivalently, the block diagonalization can be achieved by neglecting all Coriolis coupling in a body-frame expansion of the total scattering wavefunction, and, in addition, approximating the diagonal terms in the body-frame expansion of the orbital angular momentum

l = J_tot - J

by Eq. (2).

Because the computer time required for solution of the CC equations scales as N_ch³, this block diagonalization leads to a substantial reduction in the total time required in a scattering calculation. Moreover, for problems in which the potential is not long-ranged, many tests have shown the the CS approximation is remarkably accurate.

Cross Sections

Within the CS formulation, the integral cross section for a transition from an initial state i to a final state f is given by

(3)

where k_i is the wavevector of the initial state and the sum runs over:

all values of the projection quantum number such that

min

J_i, J_f

Note that in the CS formulation, the S matrix is indexed in both

and

. Since the S matrix is independent of the sign of

, the summation in Eq. (3) can be further restricted to positive, definite values of

At present the Hibridon code contains no provision for the determination of differential cross sections with the CS approximation.

References

1. P. McGuire and D. J. Kouri, J. Chem. Phys. 60, 2488 (1974).
2. P. McGuire, Chem. Phys. 13, 81 (1976).
3. R. T. Pack, J. Chem. Phys. 60, 633 (1974).
4. D. J. Kouri in Atom-Molecule Collision Theory: A Guide for the Experimentalist, edited by R. B. Bernstein (Plenum, New York, l979) p. 301.
5. A. E. DePristo and M. H. Alexander, Chem. Phys. 19, 181 (1977).

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