Angular Fock coefficients: Refinement and further development

Abstract

The angular coefficients ψ k,p (α, θ) of the Fock expansion characterizing the S-state wave function of the two-electron atomic system, are calculated in hyperspherical angular coordinates α and θ. To solve the problem the Fock recurrence relations separated into the independent individual equations associated with definite power j of the nucleus charge Z, are applied. The "pure" j-components of the angular Fock coefficients, orthogonal to of the hyperspherical harmonics Y kl , are found for even values of k. To this end, the specific coupling equation is proposed and applied. Effective techniques for solving the individual equations with simplest nonseparable and separable righthand sides are proposed. Some mistakes/misprints made earlier in representations of ψ 2,0 , were noted and corrected. All j-components of ψ 4,1 and the majority of components and subcomponents of ψ 3,0 are calculated and presented for the first time. All calculations were carried out with the help of the Wolfram Mathematica.

The Green’s function approach to the Fock expansion calculations of two-electron atoms

“…[12]. Then, it should be noted that the numerical calculations of the subcomponents ψ [15] show that the results of the Green's function approach and the analytic results presented in Ref. [15,16] are not coincident.…”

“…(93) in Ref. [15]). We shall also obtain a similar discrepancy, for example, in case of solving the IFFR…”

“…It was shown in the works [15][16][17] that any component of the AFC can be represented by a single series of the form…”

Two-particle atomic coalescences: Boundary conditions for the Fock coefficient components

“…The subcomponent ψ (2d) 4,1 was represented by single series of the form (7) with function ω (4,1;2d) l (α) ≡ τ l (ρ) defined by Eqs. (94)-(96) [19] for l = 0, 1, 2. The explicit analytic expression for τ l with l ≥ 3 was represented in [20].…”

Averaged electron densities of the helium-like atomic systems

“…The standard coordinate system {r 1 , r 2 , r 12 } of the interparticle distances, as well as the hyperspherical coordinates {R, α, θ} and the coordinates {r 1 , r 2 , cos θ} were applied.The averaged wave functions Ψ av (R) of the hyperspherical radius R were calculated numerically by averaging Ψ(R, α, θ) over the hyperspherical angles α and θ. The exact analytic representations for the relative derivatives Ψ av (0)/Ψ av (0) and Ψ av (0)/Ψ av (0) were derived by the use of the angular coefficients[3] of the Fock expansion[6]. The analytic approximations very close to the actual Ψ av (R) were obtained on the base of the specific form of the Schrödinger equation for the two-electron atoms.…”