Abstract

Introduction. In the theoretical description of the behavior of a hydrogen-like atom in external electric and magnetic fields, it is convenient to use Coulomb spheroidal wave functions. Generally, they are defined in two limiting cases of large and small distances R between the foci of the spheroidal coordinate system. The Coulomb spheroidal wave functions are presented in the form of a linear combination of the Coulomb parabolic functions in the first case and the spherical functions in the second case. In order to find the expansions, which connect these bases, the standard perturbation theory as well as the additional integrals of motion were used.
Purpose. Determine the spheroidal corrections to the spherical and parabolic bases of the hydrogen atom at large and small intercenter distances R. 
Results. The asymptotical expressions for the eigenvalues and eigenfunctions (the Coulomb spheroidal functions ψсфрnqm) of the hydrogen atom system in the form of series in R for small (R<<1) and in R^-1 for large (R>>1) internuclear distances were obtained. For this purpose, the additional integrals of motion and the standard Raeleigh-Schrödinger perturbation theory scheme within the terms of 3^rd order were used. It is shown that in each order of perturbation theory the corrections to the Coulomb spheroidal functions are expressed in a finite number of the basic functions of the corresponding representation. 
Conclusion. In this paper, a brief analysis of the fundamental bases of the hydrogen atom, which are the eigenfunctions of its Hamiltonian and of the one of the generators of the hidden symmetry group SO (4) is carried out. The expansion of the one of the fundamental bases with respect to another one was analyzed in terms of additional integrals of motion. The information about additional integrals of motion allowed to calculate the spheroidal corrections to the spherical and parabolic bases of the hydrogen atom at small and large intercenter distances R using a purely algebraic scheme of the Rayleigh-Schrödinger perturbation theory

Keywords: hydrogen atom, spherical basis, parabolic basis, perturbation theory, two-center corrections, additional integral of motion

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