# An R-Matrix Approach to Electron-Molecule Collisions

## Abstract

The R-matrix formalism has a long and venerable history. The method was introduced into nuclear physics by Wigner^{1} and Wigner and Eisenbud^{2} in the late 1940’s to enable a unified treatment of nuclear reactions dominated by compound state formation. However, there are earlier sources,^{3–4} which developed quite similar approaches to resonant nuclear reactions. All of these theories utilize the short-range character of the nuclear force to define a reaction zone of finite radius but differ in the mathematical details of the treatment of the wavefunction within that reaction zone. By enclosing the scattering partners within this sphere of radius *r* = *a* (the R-matrix surface), where a is chosen to be the range of the nuclear force, it should be possible to characterize the system using energies and wavefunctions computed within the sphere. By matching to the known asymptotic solutions, which in the nuclear problem are simply free waves, one can easily extract the relevant scattering parameters. The connection between the internal and external solutions is provided by the R-matrix, which is a sum over quantities related to the overlap integrals (level widths) of the internal and external wavefunctions evaluated on the surface of the sphere, and the energies of the internal states.^{5–8} Since the low-energy nuclear scattering problem is dominated by the formation of resonances which can be identified fairly easily with the internal states, the method is a natural one for the parametrization of nuclear cross sections. Thus the R-matrix method becomes a systematic framework for understanding and characterizing large amounts of data in terms of energy levels and widths obtained from experimental measurement. In addition, once these energies and level widths are obtained, the R-matrix provides a vehicle for predicting new results which may be difficult or impossible to obtain from experiment.

## Keywords

External Region Schrodinger Equation Hamiltonian Matrix Fixed Boundary Condition Basis Ofthe## Preview

Unable to display preview. Download preview PDF.

## References

- 1.
- 2.Wigner, E. P. and Eisenbud, L., Phys. Rev.
**72**, 29 (1947)ADSCrossRefGoogle Scholar - 3.Kapur, P. L. and Peierls, R. E., Proc. Roy. Soc. (London), A
**166**, 277 (1938)ADSCrossRefGoogle Scholar - 4.Siegert, A. J. F., Phys. Rev.
**56**, 750 (1939)ADSCrossRefGoogle Scholar - 5.Thomas, R. G., Phys. Rev.
**88**, 1109 (1952)ADSzbMATHCrossRefGoogle Scholar - 6.Bloch, C, Nucl. Phys.
**4**, 5039 (1957); An elegant treatment which unifies all of the reaction theories in one formalismGoogle Scholar - 7.Lane, A. N. and Thomas, R. G., Rev. Mod. Phys.
**30**, 257 (1958); A very important early reference, although the notation is cumbersomeMathSciNetADSCrossRefGoogle Scholar - 8.Lane, A. N. and Robson, D., Phys. Rev.
**3**, 774 (1966); Shows how the framework of reference 6 can used to systematize the theories developed after 1957.ADSCrossRefGoogle Scholar - 9.Buttle, P. J. A., Phys. Rev.
**160**, 719 (1967)ADSCrossRefGoogle Scholar - 10.Lippmann, B. A. and Schwinger, J., Phys. Rev.
**79**, 469 (1950)MathSciNetADSzbMATHCrossRefGoogle Scholar - 11.Burke, P. G., Hibbert, A. and Robb, W. D., J. Phys. B
**4**, 1153 (1971)CrossRefGoogle Scholar - 12.Burke, P. G. and Seaton, M. J., Methods Comput. Phys.
**10**, 1 (1971)Google Scholar - 13.Burke, P. G., Comput. Phys. Comm.
**6**, 288 (1973)ADSCrossRefGoogle Scholar - 14.Burke, P. G. and Robb W. D., J. Phys. B
**5**, 44 (1972)ADSCrossRefGoogle Scholar - 15.Burke, P. G. and Robb W. D., Adv. At. Mol. Phys.
**11**, 143 (1975)ADSCrossRefGoogle Scholar - 16.Burke, P.G. and Berrington, K. A., “R-matrix Theory of Atomic and Molecular Processes” (IOP Publishing, Bristol, 1993)Google Scholar
- 17.Schneider, B. I., Chem. Phys. Lett.
**2**, 237 (1975)ADSCrossRefGoogle Scholar - 18.Schneider, B. I., Phys. Rev.
**A11**, 1957 (1975)ADSGoogle Scholar - 19.Schneider, B. I., Invited Paper. Proceedings of X ICPEAC, Paris, France (1977)Google Scholar
- 20.Schneider, B. I. and Hay, P. J., Phys. Rev.
**A13**, 2049 (1976)ADSGoogle Scholar - 21.Schneider, B. I. and Morrison, M. A., Phys. Rev.
**A16**, 1003 (1977)ADSGoogle Scholar - 22.Burke, P. G., Mackey, I. and Shimamura, I., J. Phys. B
**10**, 2497 (1977)MathSciNetADSCrossRefGoogle Scholar - 23.Buckley, B. D., Burke, P. G., and Vo Ky Lan, Comput. Phys. Commun.
**17**, 175 (1979)ADSCrossRefGoogle Scholar - 24.Noble, C. J., Burke, P. G., and Salvini, S., J. Phys. B
**15**, 3779 (1979)ADSCrossRefGoogle Scholar - 25.An example of this situation can be found in the dissociation of a number of molecules by electron impact. The electron dynamics must be computed at each geometry to high precision but once the adiabatic nuclei wavefunctions are known a semiclassical or reflection approach to the nuclear breakup will often suffice.Google Scholar
- 26.Schneider, B. L, Phys. Rev.
**A24**, 1 (1981)ADSGoogle Scholar - 27.Nesbet, R. K., “Variational Methods in Electron-Atom Scattering Theory”, P. G. Burke and H. Kleinpoppen Eds., (Plenum Press, New York and London, 1980)CrossRefGoogle Scholar
- pointed out that very early work on variational approaches to scattering such as Kohn, W., Phys. Rev.
**74**, 1763 (1948)ADSzbMATHCrossRefGoogle Scholar - and Jackson, J. L., Phys. Rev.
**83**, 301 (1951) had implicity recognized that it was not necessary to impose a fixed boundary condition on the trial scattering wavefunction. However, neither of these authors couched their discussion in terms of the usual R-matrix theory and more importantly did not propose the diagonalization of a modified Hamiltonian as in reference 6 to simplify the computational effort over many energiesADSzbMATHCrossRefGoogle Scholar - 28.A simple application of Greens’s theorem to the volume integral involving the co-ordinate of the scattered electron illustrates the situation quite simplyGoogle Scholar
- 29.Light, J. C. and Walker, R. B., J. Chem. Phys.
**65**, 4272 (1976)ADSCrossRefGoogle Scholar - 30.Noble, C. J. and Nesbet, R. K., Comput. Phys. Comm.
**33**, 399 (1984)ADSCrossRefGoogle Scholar - 31.Schneider, B. I. and Walker, R. B., J. Chem. Phys.
**70**, 2466 (1979)ADSCrossRefGoogle Scholar - 32.Greene, C. H and Longhuan, K., Phys. Rev.
**A38**, 5953 (1988)ADSGoogle Scholar - 33.Le Rouzo, H. and Raseev, G., Phys. Rev.
**A29**, 1214 (1984)ADSGoogle Scholar - 34.Robicheaux, F. To Be PublishedGoogle Scholar
- 35.V. R. Saunders of Daresbury Laboratory has programmed and used such a procedure with Gaussian orbitals to compute photoionization cross sectionsGoogle Scholar
- 36.The word open channel is here meant to mean any channel with non-negligible amplitude on the R-matrix surface. In practice it may be convenient to include certain closed channels in this definition, especially if they are Rydberg in character, in order to keep the size of the internal region at a manageable levelGoogle Scholar
- 37.Temkin, A. and Vasavada, K. V., Phys. Rev.
**160**, 109 (1967)ADSCrossRefGoogle Scholar - 38.Shugard, M. and Hazi, A. U., Phys. Rev.
**A12**, 1895 (1975)ADSGoogle Scholar - 39.Morrison, M. A., J. Phys. B
**19**, L707, (1986)ADSCrossRefGoogle Scholar - Morrison, M. A., Abdolsalami, M. and Elza, B. K., Phys. Rev.
**A43**, 3440 (1991)ADSGoogle Scholar - 40.See the part on the Complex Kohn Variational Method in this bookGoogle Scholar
- 41.Schneider, B. I., LeDourneuf, M. and Burke, P. G., J. Phys. B
**12**, L365 (1979)CrossRefGoogle Scholar - 42.Schneider, B. I., LeDourneuf, M. and Vo Ky Lan, Phys. Rev. Lett.
**43**, 1926 (1979)ADSCrossRefGoogle Scholar - 43 There are earlier as well as later approaches to resonant scattering which recognize the importance of an internal, fixed nuclei, intermediate electronic level. A representative sample includes: Birtwistle, D. T. and Herzenberg, A., J. Phys. B
**4**, 53 (1972)ADSCrossRefGoogle Scholar - Dube, L. and Herzenberg, A., Phys. Rev.
**A20**, 195 (1979)ADSGoogle Scholar - Domcke, W. and Cederbaum, L. S., Phys. Rev.
**A16**, 1465 (1977)ADSGoogle Scholar - Berman, M., Estrada, H., Cederbaum, L. S. and Domcke, W., Phys. Rev.
**A28**, 1363 (1983)ADSGoogle Scholar - Greene, C. H and Jungen, Ch., Adv. Atom. Mol. Phys.
**21**, 51 (1985)ADSCrossRefGoogle Scholar - 44.Chang, E. S. and Fano, U., Phys. Rev.
**A6**, 173 (1972)ADSGoogle Scholar