Theory of Molecular Inelastic Scattering

When molecules collide with atoms or other molecules, the vibrational and rotational quantum state may change. In simple cases, it is possible to solve the Schrödinger equation on a computer to calculate these changes, if one knows the full intermolecular potential. The computation, however, is long and usually gives little insight on the details of the collision -- a block of numbers emerges from the computer after a few hours of grinding. We have devised several methods to get approximate solutions of modest to high accuracy in much less time. The approximations give a simple direct connection between the potential and the final scattering matrix (S-matrix).

We start by expanding the complete wave function in terms of the complete set of basis functions for the vibrations, rotations, and the orbital angular momentum. Schrödinger's equation then becomes a set of coupled differential equations,

If we neglect the coupling terms, we can define an easily calculable reference solution which we can remove from the exact solution to get a set of coupled differential equations for the perturbation. If the resulting equations are solved by perturbation theory, we can get a remarkably accurate and very fast approximation. We can continue the process. Suppose we have a second aproximate solution. We can remove this from the coupled equations for the perturbation to get a new set of equations. At this point we can again resort to perturbation theory, but, having removed more of the potential, the result is more accurate.

Another trick is to find an r-dependent transformation matrix that diagonalizes the potential. Applying this transformation gives us an adiabatic representation where much of the coupling between states is removed. In a diabatic representation, the off-diagonal elements of the potential matrix couple the various states. In the adiabatic representation this coupling is provided by the derivative of the transformation matrix. Calculations show that the adiabatic perturbation is quite accurate and only slightly slower than the diabatic case. By using semiclassical approximations to the wave functions, a simple set of coupled equations can be obtained with interesting properties. The incoming and outgoing parts of the scattering are completley separated, and the equations can be rapidly solved by a simple extension of the perturbation theory. The approximation agrees with the exact results to better than 1%. The theory can be easily extended to include energetically closed channels that contribute only during the collision.

A summary of recent work can be found in:

• The Onion Method for Multiple Perturbation Theory, R. J. Cross, J. Chem. Phys. 88, 4871 (1988).
• Perturbation Treatment of Nonadiabatic Collisions, R. J. Cross, J. Chem. Phys. 89, 4700 (1988).
• Semiclassical Perturbation Treatment of Molecular Charge Exchange, R. J. Cross, J. Chem. Phys. 95, 1900 (1991).
• Use of Approxiamation Theories as Interpolation Guides, R. J. Cross, J. Chem. Phys. 97, 3166 (1992).
• Tests of an Adiabatic Perturbation Theory for Symmetric Charge Exchange, R. J. Cross, J. Phys. Chem. 97, 2092 (1993).
• An Adiabatic Exponential Perturbation Theory for Rotationally Inelastic Scattering, E. Curotto and R. J. Cross, J. Chem. Phys. 106, 2225 (1997).

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Last updated: 11/29/07 .