Virial Relations for the Dirac Equation and Their Applications to Calculations of Hydrogen-Like Atoms

Abstract

Virial relations for the Dirac equation in a central field and their applications to calculations of H-like atoms are considered. It is demonstrated that using these relations allows one to evaluate various average values for a hydrogenlike atom. The corresponding relations for non-diagonal matrix elements provide an effective method for analytical evaluations of infinite sums that occur in calculations based on using the reduced Coulomb-Green function. In particular, this method can be used for calculations of higher-order corrections to the hyperfine splitting and to the g factor in hydrogenlike atoms.

Accurate solution of the Dirac equation on Lagrange meshes

“…[21] or from Ref. [27]. If the Gauss quadrature is performed on block (1,2) rather than on block (2,1), the mean values are closer to the exact ones for 2p 1/2 and 2p 3/2 but they are slightly less good for 1s 1/2 and 2s 1/2 .…”

“…Let us introduce expansions (26) and (27) in the coupled radial Dirac equations (5). A projection on the Lagrange functions leads to the 2N × 2N algebraic system of equations…”

Algebraic and Algorithmic Aspects of Differential and Integral Operators

“…We want to stress that the overall computing time for the integrals (7)-(9) does not exceed a few minutes on a standard laptop. Our next challenge is to study the off-diagonal matrix elements that are important in applications [12][13][14][15]17] (see also the references therein). For the radial functions F n,κ (r) and G n,κ (r) given by (1) in terms of the Laguerre polynomials, one needs to investigate the following four integrals: ∞ 0 r p+2 F n1,κ1 F n2,κ2 ± G n1,κ1 G n2,κ2 dr, ∞ 0 r p+2 F n1,κ1 G n2,κ2 ± G n1,κ1 F n2,κ2 dr as off-diagonal extensions of (7)- (9).…”

Theory of Bound-Electron g Factor in Highly Charged Ions

“…The expectation value of H rad is evaluated with the Dirac wave function of the electron that accounts for the interaction with the homogeneous magnetic field to first order in H. The first-order correction to the electronic wave function due to the interaction with the magnetic field is easily obtained using the generalized virial relations for the Dirac equation 18,19 .…”

“…The low-order term can be evaluated analytically employing the generalized virial relations 18,19 . This yields 39…”