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• W2078652561 endingPage "454" @default.
• W2078652561 startingPage "441" @default.
• W2078652561 abstract "An almost a priori method based on a simple theoretical model is developed for obtaining good estimates of the radius of convergence of Rayleigh-Schrodinger (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 $ensuremath{parallel}{ensuremath{psi}}_{1}ensuremath{parallel}$. In those cases where $ensuremath{parallel}{ensuremath{psi}}_{1}ensuremath{parallel}$ 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 $2ensuremath{le}Nensuremath{le}10$ 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 $1{s}^{2}^{1}S$ state of the helium isoelectronic sequence, convergence is found for $Zensuremath{ge}1$, hence for the first time predicting convergence for ${mathrm{H}}^{ensuremath{-}}$. 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 $3ensuremath{le}Nensuremath{le}10$ isoelectronic sequences, the predicted region of convergence can be represented approximately by $Zensuremath{ge}3Nensuremath{-}7$. 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 $Zensuremath{ge}3$ for the beryllium isoelectronic sequence, for $Zensuremath{ge}N$ for the boron through oxygen isoelectronic sequences, and for $Zensuremath{ge}N+1$ 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 $frac{1}{Z}$ eigenvalue expansion of the $1{s}^{2}2{s}^{2}^{1}S$ state of the beryllium isoelectronic sequence and the results are found to be in accord with predictions." @default.
• W2078652561 created "2016-06-24" @default.
• W2078652561 creator A5052915782 @default.
• W2078652561 date "1981-02-01" @default.
• W2078652561 modified "2023-10-16" @default.
• W2078652561 title "Estimation of radii of convergence of Rayleigh-Schrödinger perturbation expansions: Application to the<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML display=inline><mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mi>Z</mml:mi></mml:mrow></mml:mfrac></mml:math>expansions of two- through ten-electron atomic isoelectronic sequences" @default.
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