›1mLARGE ORDERS OF THE 1/n-EXPANSION IN ATOMIC PHYSICS›m

POPOV V.S., ›4mSERGEEV A.V.›m

Institute of Theoretical and Experimental Physics,

Bol'shaya Cheriomushkinskaya 25, Moscow, 117259 Russian Federation

S.I.Vavilov State Optical Institute, Tuchkov per. 1, Saint

Petersburg, 199034 Russian Federation, e-mail sergeev@soi.spb.su

The energy of atoms is expanded as a power series in 1/n, where

n is the principal quantum number. We assume that

while . In an essence, such an expansion is

equivalent to the recently developed dimensional expansion [1]. It

is a semiclassical method, which is rather similar to the methods

of molecular vibration analysis. The problem reduces to Rayleigh -

Schrödinger perturbation theory for anharmonic oscillator. Some

difficulties arise from the divergence of the 1/n-expansion. To

sum the divergent series, various methods are used, such as Padé -

Borel approximants. To take full advantage of them, one should

take into account the divergent large-order behaviour of the

expansion responsible for the nearest singularity to the origin in

the Borel function [1].

Typically, the coefficients in the expansion grow as

factorials, , with . We found that the

parameter coincides with the reciprocal of the action

integral, the constants and are also calculable.

Here we extend our recent results [2] to multidimensional

effective potentials for treating nonspherically symmetric and two

-electron atoms. In this case, the determination of the most

probable escape path which minimizes the classical action becomes

a nontrivial task. We examine the parameter for Stark and

Zeeman effects in a hydrogen atom, ion, and helium

isoelectronic sequence. The results may be incorporated into

summation schemes in order to yield highly accurate energies.

1. GOODSON D.Z.,LÓPEZ-CABRERA M.et al, J.Chem.Phys.,›4m97›m,8481,1992.

2. POPOV V.S., SERGEEV A.V., Phys.Lett.A, ›4m172›m, 193, 1993.›z›"z

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