Generalized spherical functions in the theory of many‐electron atoms

Abstract

AbslractsA new method for finding non-relativistic and relativistic wave-functions of an electron moving in the field of a nuclear charge in the j j 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 j j coupling scheme; only their radial integrals and some phase multipliers depend on orbital quantum numbers.On propose une nouvelle mtthode pour construire des fonctions d'onde d'un tlectron dans le champ d'une charge nucltaire, dans le schtma de couplage jj. Elle est baste sur I'utilisation des functions spheriques gtntralisees. On dtveloppe le formalisme mathtmatique ntcessaire pour calculer les expressions des Cldments des matrices de I'tnergie non-relativiste et relativiste et des optrateurs de transitions Clectroniques. Ces formules conviennent mieux que celles qu'on utilise ordinairement dans le schtma de couplage j j ; les inttgrales radiales ainsi que certains multiplicateurs de phase dtpendent des nombres quantiques secondaires.Eine neue Methode wird vorgeschlagen um nicht-relativistische und relativistische Wellenfunktionen fur ein im Feld einer Kernladung bewegenden Elektron in jj-Kopplung zu finden. Diese Methode basiert auf die Anwendung von verallgerneinerten Kugelflachenfunktionen. Der fur die Berechnung der Matrixelemente der Energie-und Ubergangsoperatoren gebrauchte Formalismus wird entwickelt. Die fur diese Matrixelemente erhaltenen Formeln sind einfacher als die ublichen Formeln. Nur die Radial-integrale und einige Phasenfaktoren hangen von den azimutalen Quantenzahlen ah.

An efficient approach for spin - angular integrations in atomic structure calculations

“…All the above-mentioned approaches were applied in the coordinate representation. The second-quantization formalism (Judd 1967, Rudzikas and Kaniauskas 1984, Rudzikas 1991and Rudzikas 1997) has a number of advantages compared with the coordinate representation. First, it is much easier to find algebraic expressions for complex operators and their matrix elements, when relying on second-quantization formalism.…”

“…Uylings (1992) suggested a fairly simple approach for dealing with separate cases of three-particle operators. Moreover, in the second quantization approach the quasispin formalism was efficiently developed by Rudzikas and Kaniauskas (1984). The main advantage of this approach is that applying the quasispin method for calculating the matrix elements of any operator, we can use the reduced coefficients of fractional parentage whose matrix elements are independent of the occupation number of the shell.…”

The Transformation of Irreducible Tensor Operators Under Spherical Functions

“…. Expressions over D of other operators, such as spin operator S 1 , angular momentum operator L 1 , can be found, for instance, in [8]. All these mentioned operators also transform among themselves according to Eq.…”

“…It is noticeable, obtained expression, by applying technique of coordinate transformations, coincides with Eq. ( 41) in [8], if reducing given Kronecker products l × k → L ′ , L ′ × l → L ′′ . This would signify L ′ = l and L ′′ = 0.…”

“…The eigenstates φ λ m of central-field atomic Hamiltonian are usually enunciated by spherical functions on SO(3)/SO (2) quotient group, multiplied by 2 × 1 spin matrices. However, in [8] it was showed that φ x ξ = φ θ ξ , ϕ ξ can be expressed by Wigner D Ω ξ functions on SU (2), whereΩ ξ = Φ ξ ,Θ ξ , 0 = ϕ ξ + π/2, θ ξ , 0 . Furthermore, Bhatia et al in [9] constructed two-electron wave function Φ (x 1 ,x 2 ) in terms of the spherical functions D (Ω ), where Ω , as usually, denotes Euler angles (Φ,Θ ,Ψ ), i.e., rotation on S 2 fromx 1 tox 2 and vice verse.…”