Estimation of radii of convergence of Rayleigh-Schrödinger perturbation expansions: Application to the1Zexpansions of two- through ten-electron atomic isoelectronic sequences

An almost a priori method based on a simple theoretical model is developed for obtaining good estimates of the radius of convergence of Rayleigh-Schrödinger (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 $\frac{1}{Z}$ expansions of $N$-electron atomic isoelectronic sequences. The only system- and state-dependent information required is the norm of the first-order eigenfunction $\parallel {\psi }_1\parallel$. In those cases where $\parallel {\psi }_1\parallel$ is inaccessible or unavailable, it is shown how adequate perturbational-variational (PV) approximations can be simply obtained. The procedure has been applied to the $\frac{1}{Z}$ expansions of the ground states and several low-lying states of the $2\le N\le 10$ 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 $1s^2{}_{}{}^1{}_{}{}^{}S$ state of the helium isoelectronic sequence, convergence is found for $Z\ge 1$, hence for the first time predicting convergence for ${\mathrm{H}}^{-}$. 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 $3\le N\le 10$ isoelectronic sequences, the predicted region of convergence can be represented approximately by $Z\ge 3N-7$. 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 $Z\ge 3$ for the beryllium isoelectronic sequence, for $Z\ge N$ for the boron through oxygen isoelectronic sequences, and for $Z\ge N+1\mathrm{}$ for the fluorine and neon isoelectronic sequences, thus extending the application of such expansions to at least $N=10\mathrm{}$. Optimal screening is quantitatively tested for the $\frac{1}{Z}$ eigenvalue expansion of the $1s^{2}2s^2{}_{}{}^1{}_{}{}^{}S$ state of the beryllium isoelectronic sequence and the results are found to be in accord with predictions.