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A new method for finding non-relativistic and relativistic wave-functions of an electron moving in the field of a nuclear charge in the jj coupling scheme is proposed. It is based on the usage of generalized spherical functions. The mathematical apparatus necessary to find the expressions for matrix elements of the non-relativistic and relativistic energy or electron transition operators is developed. The formulas obtained for these matrix elements are more convenient than those usually used in jj coupling scheme; only their radial integrals and some phase multipliers depend on orbital quantum numbers.

It is shown that the isospin formalism is convenient for discussing the properties of the wavefunctions of the configurations n1lN1n2lN2 with respect to the orthogonal transformations of the radial orbitals. The main point of this formalism consists of the statement that the existence of an electron in one of the two shells n1lN1 or n2lN2 is considered as the additional degree of freedom in some abstract 'isospin' space. The corresponding basis is defined and the properties of the wavefunctions and operators are studied in this basis. The authors analyse the conditions under which the isospin T may be a fairly exact quantum number for complex electronic configurations.

The isospin basis is put into operation for investigation of atomic configurations, having two shells of equivalent electrons, characterized by the same orbital (LS coupling) or total (jj coupling) angular momenta of each electron. Tensorial properties of both the operators and the wave functions are studied in this basis. The two-particle operator is expressed in terms of the tensors irreducible in the isospin space. The problem of the additional classification of the levels is considered. The accuracy of the quantum numbers of the isospin basis in jj coupling scheme is discussed.

The relativistic electric multipole transition operators are considered, not specifying the gauge condition of the electromagnetic field potential. The nonrelativistic limit of the relativistic expressions obtained is found. The relativistic corrections to various forms of the electric dipole transition operators are investigated. The electric multipole transition operator contains an arbitrary parameter, the gauge constant, dependence on which can serve as one of the criteria for the accuracy of the wavefunctions used. Calculated examples of the theory, starting with the general expressions found, are presented. The methods described and the formulae presented allow the authors to investigate in an effective way the electric multipole transitions for atoms and their isoelectronic sequences, including very highly ionised atoms.

The mirror reflection symmetry which consists in the correlation of the substitution j −j −1 with the reflection of the coordinate system and of m −m with the reflection of the space with respect to the xy-plane has been applied to the quasi-spin formalism. The phases of the wave functions of the shell of equivalent atomic electrons treated as partially filled, as they relate to those when the same shell is treated as almost filled, have been established. The mirror reflection symmetry has been generalized for the supplementary quantum numbers used for distinguishing the terms of the same kind. The phases of the coefficients of fractional parentage of almost filled shell enabling us to determine them from those of partially filled shell have been specified. Corresponding phase relations for the matrix elements of operators have also been considered.

At present the number of papers in which many-electron atoms and ions are studied both theoretically and experimentally is increasing rapidly. The greatest attention is paid to the multiple-charged ions, observed both in astrophysical and laboratory high-temperature plasma. They are particularly important in the investigation of the solar spectra in the short wavelength region as well as in the study of the thermonuclear fusion.