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

The relative ordering of energy levels is investigated for bound two-electron systems with potentials of the formV(r1,r2,r12)=Z(v(r1)+v(r2))−v(r12). Given the two one-body binding potentialsv(1)(r) andv(2)(r), it is argued that iff(r)≡v(1)(r)−v(2)(r) is positive and monotonically decreasing upon increasingr then the corresponding eigenvalues of the two-electron Hamiltonians Hi=−12∇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,…, whereE1(i)≤E2(i)≤⋯≤Ek(i)≤⋯, for bothi=1 andi=2. This conjecture certainly holds at sufficiently largeZ. The range ofn may be finite or infinite, depending on the nature of the potentials. In fact, the range of values ofn forv(1)(r) may be shorter than that ofv(2)(r) (which is more binding). The one-electron potentials specifically considered are: v(r)=vC(r)=−1rCoulombvD(r)=−1rexp(−λr)Debye (Yukawa)vHu(r)=−λexp(λr)−1HulthénvECSC(r)=−1rexp(−λr)cos(λr)ECSC∗,∗Exponential-Cosine-Screened-Coulombwhereλ is the screening parameter. For each of theλ-dependent potentials we compare one- and two-electron spectra corresponding to distinct values ofλ. This is followed by pairwise comparison of distinct potentials. •A comparison criterion for screened potentials is conjectured for two-electron atoms.•The limits of validity of the correlation hypothesis are discussed.•Independent Hylleraas and Lagrange mesh calculations verify the comparison criterion.