PHYSICAL REVIEW A 97, 022503 (2018) Critical screening in the one- and two-electron Yukawa atoms H. E. Montgomery, Jr. * Chemistry Program, Centre College, Danville, Kentucky 40422, USA K. D. Sen † School of Chemistry, University of Hyderabad, Hyderabad 500 046, India Jacob Katriel ‡ Department of Chemistry, Technion—Israel Institute of Technology, Haifa 32000, Israel (Received 15 December 2017; published 9 February 2018) The one- and two-electron Yukawa atoms, also referred to as the Debye-Hückel or screened Coulomb atoms, have been topics of considerable interest both for intrinsic reasons and because of their relevance to terrestrial and astrophysical plasmas. At sufﬁciently high screening the one-electron Yukawa atom ceases to be bound. Some calculations appeared to suggest that as the screening increases in the ground state of the two-electron Yukawa atom (in which both the one-particle attraction and the interparticle repulsion are screened) the two electrons are detached simultaneously, at the same screening constant at which the one-electron atom becomes unbound. Our results rule this scenario out, offering an alternative that is not less interesting. In particular, it is found that for Z< 1 a mild amount of screening actually increases the binding energy of the second electron. At the nuclear charge Z c ≈ 0.911028 ..., at which the bare Coulomb two-electron atom becomes unbound, and even over a range of lower nuclear charges, an appropriate amount of screening gives rise to a bound two-electron system. DOI: 10.1103/PhysRevA.97.022503 I. INTRODUCTION In the present paper we consider the behaviors of the nonrelativistic one- and two-electron Yukawa atoms, in which both the one-particle attractive and the interparticle repulsive Coulomb interactions are multiplied by exponentially decay- ing screening factors. Such potentials are also referred to as the Debye-Hückel [1] or screened Coulomb potentials. Thus, the screened hydrogenlike atom is speciﬁed by the Hamiltonian h Yu =− 1 2 ∇ 2 − Z exp(−λr ) r , (1) and the screened heliumlike atom is speciﬁed by H Yu =− 1 2 ( ∇ 2 1 +∇ 2 2 ) − Z  exp(−λr 1 ) r 1 + exp(−λr 2 ) r 2  + exp(−λr 12 ) r 12 . (2) The extensive work done on the one-electron case is reviewed in Sec. II. In particular, the value of λ at which the ground state ceases to be bound and its (trivial) Z dependence have been determined (by several authors). There is little that we can add on this matter. In the two-electron case we seek the value of λ at which the binding energy of the second electron vanishes. Although this issue was also studied by several authors, whose work is reviewed in Sec. III, it turns out that * ed.montgomery@centre.edu † kds77@uohyd.ac.in ‡ jkatriel@technion.ac.il some interesting features have not been properly dealt with. The claim that motivated our curiosity is that in the ground state of the two-electron Yukawa atom, upon raising the nonlinear screening constant, the two electrons simultaneously cease being bound, at the same critical screening constant at which the one-electron Yukawa atom becomes unbound [2–5]. This is sharply different from the He-isoelectronic sequence, where the critical charge below which only one electron remains bound is Z c ≈ 0.91102822407725573 [6], but this remaining electron remains bound for all Z> 0. Hence, one would wish to understand how the transition from the unscreened to the screened behavior takes place. The two electrons would trivially unbind simultaneously if no interelectronic repulsion existed. They could unbind simultaneously if the expectation value of the interelectronic repulsion vanished more rapidly than the one-electron components of the energy, upon approaching the critical charge. In any case, a more careful examination of this issue appears worthwhile, and the results reported below, that refute the claim cited above and extrapolate in an interesting manner to the bare Coulomb scenario, clearly justify this effort. A rigorous study of the behavior of the spectra of short- range one-particle systems bound by potentials that depend linearly on the real parameter μ, i.e., h =− 1 2 ∇ 2 + μV, was presented by Klaus and Simon [7]. Upon lowering the parameter μ, a threshold, μ (c) , is often observed at which an eigenvalue vanishes. In three dimensions the approach to this 2469-9926/2018/97(2)/022503(8) 022503-1 ©2018 American Physical Society