On- and off-resonance radiation-atom-coupling matrix elements involving extended atomic wave functions

Abstract

In continuation of our earlier works, we present results concerning the computation of matrix elements of the multipolar Hamiltonian (MPH) between extended wave functions that are obtained numerically. The choice of the MPH is discussed in connection with the broader issue of the form of radiation-atom (or -molecule) interaction that is appropriate for the systematic solution of various problems of matter-radiation interaction. We derive analytic formulas, in terms of the sine-integral function and spherical Bessel functions of various orders, for the cumulative radial integrals that were obtained and calculated by Komninos, Mercouris, and Nicolaides [Phys. Rev. A 71, 023410 (2005)]. This development allows the much faster and more accurate computation of such matrix elements, a fact that enhances the efficiency with which the time-dependent Schr\odinger equation is solved nonperturbatively, in the framework of the state-specific expansion approach. The formulas are applicable to the general case where a pair of orbitals with angular parts $|{\ensuremath{\ell}}_{1},\phantom{\rule{1.0pt}{0ex}}\phantom{\rule{1.0pt}{0ex}}{m}_{1}\ensuremath{\rangle}$ and $|{\ensuremath{\ell}}_{2},\phantom{\rule{1.0pt}{0ex}}\phantom{\rule{1.0pt}{0ex}}{m}_{2}\ensuremath{\rangle}$ are coupled radiatively. As a test case, we calculate the matrix elements of the electric field and of the paramagnetic operators for on- and off-resonance transitions, between hydrogenic circular states of high angular momentum, whose quantum numbers are chosen so as to satisfy electric dipole and electric quadrupole selection rules. Because of the nature of their wave function (they are nodeless and the large centrifugal barrier keeps their overwhelming part at large distances from the nucleus), the validity of the electric dipole approximation in various applications where the off-resonance couplings must be considered becomes precarious. For example, for the transition from the circular state with $n$ = 20 to that with $n$ = 21, for which $\ensuremath{\langle}r\ensuremath{\rangle}\ensuremath{\approx}400$ a.u., the dipole approximation starts to fail already at XUV wavelengths ($\ensuremath{\lambda}l125\phantom{\rule{0.28em}{0ex}}\mathrm{nm}$).