Complex energies and the polyelectronic Starkproblem

Abstract

The problem of computing the energy shifts and widths of groundor excited N-electron atomic states perturbed by weak or strongstatic electric fields is dealt with by formulating astate-specific complex eigenvalue Schrödinger equation (CESE),where the complex energy contains the field-induced shift andwidth. The CESE is solved to all orders nonperturbatively, byusing separately optimized N-electron function spaces, composedof real and complex one-electron functions, the latter beingfunctions of a complex coordinate. The use of such spaces is asalient characteristic of the theory, leading to economy andmanageability of calculation in terms of a two-stepcomputational procedure. The first step involves only Hermitianmatrices. The second adds complex functions and the overallcomputation becomes non-Hermitian. Aspects of the formalism andof computational strategy are compared with those of thecomplex absorption potential (CAP) method, which was recentlyapplied for the calculation of field-induced complex energiesin H and Li. Alsocompared are the numerical results of the two methods, and thequestions of accuracy and convergence that were posed by Sahooand Ho (Sahoo S and Ho Y K 2000 J. Phys. B: At. Mol. Opt. Phys. 33 2195)are explored further. We draw attention to the fact that,because in the region where the field strength is weak thetunnelling rate (imaginary part of the complex eigenvalue)diminishes exponentially, it is possible for even large-scalenonperturbative complex eigenvalue calculations either to failcompletely or to produce seemingly stable results which,however, are wrong. It is in this context that the discrepancyin the width of Li 1s22s 2S between results obtainedby the CAP method and those obtained by the CESE method isinterpreted. We suggest that the very-weak-field regime must becomputed by the golden rule, provided the continuum isrepresented accurately. In this respect, existing one-particlesemiclassical formulae seem to be sufficient. In addition tothe aforementioned comparisons and conclusions, we present anumber of new results from the application of the state-specificCESE theory to the calculation of field-induced shifts andwidths of the H n = 3 levels and of the prototypical Be1s22s2 1S state, for a range of fieldstrengths. Using the H n = 3 manifold as the example, it isshown how errors may occur for small values of the field,unless the function spaces are optimized carefully for eachlevel.