The combination of R-matrix and complex coordinate methods: application to resonances in the diamagnetic Rydberg spectrum of Li

Abstract

The R-matrix and complex coordinate techniques are combined for the first time and applied to the problem of the positive energy spectrum of a non-hydrogenic atom in a laboratory strength magnetic field. Quantum defect theory is extended to calculate the appropriate irregular Coulomb function at complex energy and complex radial distance. Calculations are carried out for lithium to compare with previous experimental measurements and with a corresponding calculation for hydrogen.

Features of the Theory of Resonant Electron Scattering on Atoms

“…If we assume that at there is an equation (10) for whose solution at ( ) the continuity conditions (11) (12) are satisfied, then, using (9), we can restore the value at the border of the inner and outer regions according to the known energy value. In particular, this approach was used when combining the methods of the R-matrix and complex scaling [27,28], as well as in the construction of the theory of the CAP method [10].…”

“…This feature can be eliminated by the method of stitching independent solutions on the border of the inner and outer regions using, for example, the formalism of the R-matrix method [23]. Such approaches have been successfully used, in particular, to solve the problems of reaction scattering [22,[24][25][26] and to calculate the parameters of scattering resonances by the method of complex scaling (complex coordinate) [27,28].…”

Calculation of the Lowest Resonant States of H− and Li by the Complex Absorbing Potential Method

“…Thus, solving Equation ( 11) with the given boundary condition b −1 = R(E) r=R , it is possible to find a value E that satisfies the radial Equation (1). Since the eigenfunction in Equation ( 1) is assumed to be continuous at the point r = R, the solution of Equation (11) for the given boundary condition can be considered as one of the options for combining the R-matrix with the approximate solution (1) in the basis of functions that are squareintegrated over the range r ≥ R. The possibility of using the simplified form (11) to combine R-matrix and the complex scaling method was discussed earlier, but the practical side of the issue was reduced to solving the generalized eigenvalue problem [61,62]. Case (11) was also used to solve the model problem by CAP [7].…”

“…This feature can be removed by stitching the solutions at the boundary of the overlapping region of the absorbing potential using the R-matrix formalism [59,60]. Similar approaches were previously proposed to calculate the parameters of scattering resonances by the complex coordinate method [61,62] and solving problems of reaction scattering [25,[63][64][65]. The combination of the R-matrix with the CAP method was also used in the construction of the theory of the CAP [7].…”

Complex absorbing potentials for stark resonances

“…Let us mention several methods that have been used for calculating Stark resonances: uniform complex scaling of the coordinates as first presented by Maquet, Chu, and Reinhardt (see also Chapter 5 in ref. ); exterior complex scaling; complex basis functions; complex spectral decomposition as developed by Reichl, Nickel and Emmanouilidou; Weyl's theory as implemented by Brandas and Hehenherger for Stark resonances; Korsch's procedure for Stark resonances of periodic systems; and the combination of the R‐matrix method and the complex coordinate methods as developed by Halley and his coworkers …”