CONVERGENCE OF THE NONRELATIVISTIC AND RELATIVISTIC R-MATRIX EXPANSIONS AT THE REACTION VOLUME BOUNDARY

The convergence of the non-relativistic and relativistic fixed-boundar y-condition R-matrix expansions at the reaction volume boundary is dis cussed. It is shown that in the non-relativistic case the expansion of the wavefunction converges to this function at the boundary. In the r elativistic case, however, the expansion does not generally converge t o the wavefunction at the reaction surface because the set of relativi stic basis functions spanning the interior of the reaction volume is i ncomplete on the surface. With one exception, this fact has not been r ecognized before and has been a source of errors in previous presentat ions of the relativistic R-matrix method. We present a corrected deriv ation of the R-matrix theory for the Dirac equation and generalize it to the multi-channel case. We also explain why, in spite of being base d on an incorrect theory, the results of electron-atom and electron-io n scattering calculations performed so far within the framework of the relativistic R-matrix method were not afflicted by these errors.