1981/02/01 by Jeremiah N. Silverman · 1 voice · 1 citation
Physics and Astronomy · #Advanced Chemical Physics Studies #Atomic and Molecular Physics #Cold Atom Physics and Bose-Einstein Condensates
paper · doi:10.1103/physreva.23.441
openalex publication_date 1981/02/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
An almost a priori method based on a simple theoretical model is developed for obtaining good estimates of the radius of convergence of Rayleigh-Schr"odinger (RS) perturbation expansions. The procedure is applicable to the RS expansions of all stationary states of any system described by a Hamiltonian linear in a real perturbing parameter, e.g., the (1)/(Z) expansions of N-electron atomic isoelectronic sequences. The only system- and state-dependent information required is the norm of the first-order eigenfunction \ensuremath∥\ensuremathψ1\ensuremath∥. In those cases where \ensuremath∥\ensuremathψ1\ensuremath∥ is inaccessible or unavailable, it is shown how adequate perturbational-variational (PV) approximations can be simply obtained. The procedure has been applied to the (1)/(Z) expansions of the ground states and several low-lying states of the 2\ensuremath≤N\ensuremath≤10 isoelectronic sequences. Where comparison is possible, the estimates are in close agreement with numerically obtained accurate convergence data and are greatly improved over the weak Kato-type bounds. For example, for the 1s21S state of the helium isoelectronic sequence, convergence is found for Z\ensuremath≥1, hence for the first time predicting convergence for H^\ensuremath-. Further, in harmony with physical expectations, our findings indicate that the effect of increasing N on radii of convergence is drastic; thus, for the ground states of the 3\ensuremath≤N\ensuremath≤10 isoelectronic sequences, the predicted region of convergence can be represented approximately by Z\ensuremath≥3N\ensuremath-7. The influence of screening the nucleus in compensating for the effect of increasing N is investigated and it is shown how the radius of convergence can be maximized by optimal screening. A PV method is introduced for obtaining estimates of the optimal screening parameter for arbitrary N and states. It is predicted that for the ground states, the optimally screened expansions will converge for Z\ensuremath≥3 for the beryllium isoelectronic sequence, for Z\ensuremath≥N for the boron through oxygen isoelectronic sequences, and for Z\ensuremath≥N+1 for the fluorine and neon isoelectronic sequences, thus extending the application of such expansions to at least N=10. Optimal screening is quantitatively tested for the (1)/(Z) eigenvalue expansion of the 1s22s21S state of the beryllium isoelectronic sequence and the results are found to be in accord with predictions.