A comparative study of two-electron systems with screened Coulomb potentials

Abstract

The relative ordering of energy levels is investigated for bound two-electron systems with potentials of the form V(r1,r2,r12)=Z(v(r1)+v(r2))−v(r12). Given the two one-body binding potentials v(1)(r) and v(2)(r), it is argued that if f(r)≡v(1)(r)−v(2)(r) is positive and monotonically decreasing upon increasing r then the corresponding eigenvalues of the two-electron Hamiltonians Hi=−[Formula presented]∇12+∇22+Z(v(i)(r1)+v(i)(r2))−v(i)(r12),i=1,2are highly likely to be pairwise ordered, i.e., En(1)≥En(2), n=1,2,…, where E1(i)≤E2(i)≤⋯≤Ek(i)≤⋯, for both i=1 and i=2. This conjecture certainly holds at sufficiently large Z. The range of n may be finite or infinite, depending on the nature of the potentials. In fact, the range of values of n for v(1)(r) may be shorter than that of v(2)(r) (which is more binding). The one-electron potentials specifically considered are: v(r)=vC(r)=−[Formula presented]CoulombvD(r)=−[Formula presented]exp(−λr)Debye (Yukawa)vHu(r)=−[Formula presented]HulthénvECSC(r)=−[Formula presented]exp(−λr)cos(λr)ECSC∗,∗Exponential-Cosine-Screened-Coulombwhere λ is the screening parameter. For each of the λ-dependent potentials we compare one- and two-electron spectra corresponding to distinct values of λ. This is followed by pairwise comparison of distinct potentials.