Dirac radial equations

Eq. (36) may also be expressed as a system of two first-order equations, i.e. as the radial Dirac equation in the representation of Biedenharn. Let us rewrite the radial Dirac-Pauli equation (18) with V = —Zjr in the form... 

For numerical evaluation (to sum over the entire spectrum of Dirac equation) B-splines are used [28], in particular the version developed by I.A. Goidenko [29]. Earlier the full QED calculations were carried out only for the ground (lsi/2)2 state He-like ions for the various nuclear charges Z. At that ones used either B-splines or the technique of discretization of radial Dirac equations [27]. As well as in [27] we used the Coulomb gauge. For control we reproduced the results of the calculation of (lsi/2)2 state and compared them with ones of [27]. Coulomb-Coulomb interaction is reproduced for every Z with the accuracy, on average, 0.01 %, Coulomb-Breit is with the accuracy 0.05 % and Breit-Breit (with disregarding retardation) is with the accuracy 0.1%. The small discrepancy is explained by the difference in the numerical procedures applied in [27] and in this work. 

In the previous part of the chapter we expressed the problem of an electron in a local, central potential in terms of radial equations and eigenstates of orbital angular momentum. In generalising to the case where the electron obeys the Dirac equation (3.154) we remember that spin and orbital angular momentum are coupled. 

The LDA radial Schrodinger equation is solved by matching the outward numerical finite-difference solution to sin inward-going solution (which vanishes at infinity) of the same energy, near the classical turning point. Continuity of P t(r) = rAn/(r) and its derivative determines the eigenvalue /. The second order differential equation is actually solved as a pair of simultaneous first-order equations, so that the nonrelativistic and relativistic (Dirac equation) procedures appear similar. 

The investigations of Asada et al. and Christensen - were carried out with linear-muffin-tin orbitals within the atomic sphere approximation (LMTO- AS A) Within the muffin-tin model suitable s, p and d basis functions (muffin-tin orbitals, MTO) are chosen. In contrast to the APW procedure the radial wave functions chosen in the linear MTO approach are not exact solutions of the radial Schrodinger (or Dirac) equation. Furthermore, in the atomic sphere approximation (ASA) the radii of the atomic spheres are of the Wigner-Seitz type (for metals the spheres have the volume of the Wigner-Seitz cell) and therefore the atomic spheres overlap. The ASA procedure is less accurate than the APW method. However, the advantage of the ASA-LMTO method is the drastic reduction of computer time compared to the APW procedure. 

In the Darwin solution of the Dirac equation for a hydrogenic atom, the radial functions g(r) (great) and /(r) (fine) satisfy the system... 

Contrary to the mass of the nucleus, its size influences the binding energy considerably in heavy ions (Fig. 10). In studying nuclear size effects nowadays always a spherically symmetric charge distribution of the nucleus is assumed which allows a separation of the Dirac equation and corresponding wave function into an angular part and a radial part similar to the point nucleus case. The radial Dirac equation then reads [45]... 

If we remember that the eigenvalues of the Schrodinger equation are directly proportional to the mass m of a particle while the expectation mean value scales as Mm, we expect that the eigenvalues of the Dirac equation will show both an increase in the binding energy of an electron and a contraction of the radial wave functions towards the origin because of the relativistic increase in the mass given by ... 

In nonrelativistic quantum mechanics, the angular momentum barrier prevents the collaps to the center. The angular momentum barrier is an effective potential of the form + l)/r that appears if one writes the kinetic energy in polar coordinates. For the Dirac equation the role of the angular momentum barrier is obviously played by the term /c/r in the radial Dirac operator. This term is effectively repulsive for both signs of k, because it appears only off-diagonal. The point is, that the repulsive angular momentum barrier k jr cannot balance the attractive Coulomb potential 7/r for r —> 0, as soon as I7I > k. ... 

The corresponding square-integrable solution of the radial Dirac equation... 

Here P and Q are the radial large and small components of the wavefunction, the angular functions are 2-component spinors, the quantum number k = 2 - j) j + 1/2), -j < rrij < j, and the phase factor i is introduced for convenience in some atomic applications because it makes the radial Dirac equation real. 

Eq. 20 can be solved iteratively and yields the same one-particle energies as the corresponding Dirac-equation. The radial functions P K r) correspond to the large components. In the many-electron case the correct nonlocal Hartree-Fock potential is used in Eq. 21, but a local approximation to it in Eqs. 22. Averaging over the relativistic quantum number k leads to a scalar-relativistic scheme. 

The third step is to generate radial wave functions and the corresponding potential parameters. To this end, the programme solves the Dirac equation without the spin-orbit interaction (Sect.9.6.1) using the trial potential. Hence, the programme includes the important relativistic mass-velocity and Darwin shifts. The potential parameters are calculated from (3.33-35) and then converted to standard parameters by the formulae in Sect. 4.6. The energy derivatives are calculated from the solutions of the Dirac equation at two energies, E + e and E - e, where e is some small fraction of the relevant bandwidth. 

The radial Dirac equations without spin-orbit interaction (9.35) are in a form which, except for the term -(k + l)v /(rv )P, is identical to the equations (4-90,4-91) solved by Louaks [9.10]. In the subroutine WAVEFC a technique similar to that used by Louoks is therefore applied to solve the case without spin-orbit interaction. 

In this subsection, we discuss analytical solutions to the radial Dirac equations (17) and (18) for the special case V r) = —Zfr. (We adopt atomic units where h = e = m= l in this section and in the sequel.) As a first step in our analysis, we examine these equations at large values of r. Retaining only dominant terms as i—> oo, we find... 

Next, we expand the solutions to the radial Dirac equation in a finite basis. This basis is chosen to be a set of n B-splines of order k. Following deBoor [33], we divide the interval [0, R] into segments. The end points of these segments are given by the knot sequence tj, i = 1,2, , n + k. The B-splincs of order k, Bi k r), on this knot sequence are defined recursively by the relations. 

The set of B-splines of order k on the knot sequence f< forms a complete basis for piecewise polynomials of degree fc — 1 on the interval spanned by the knot sequence. We represent the solution to the radial Dirac equation as a linear combination of these B-splines and work with the B-spline representation of the wave functions rather than the wave functions themselves. 

We now consider the corresponding variational treatment of the Dirac equation. The radially reduced equation for the the ground state is [11]... 

In systems with heavier elements, relativistic effects must be included. In the medium range of atomic numbers (up to about 54) the so called scalar relativistic scheme is often used [21], It describes the main contraction or expansion of various orbitals (due to the Darwin s-shift or the mass-velocity term), but omits spin-orbit interaction. The latter becomes important for the heavy elements or when orbital magnetism plays a significant role. In the present version of WIEN2k the core states always are treated fully relativistically by numerically solving the radial Dirac equation. For all other states, the scalar relativistic approximation is used by default, but spin-orbit interaction (computed in a second-variational treatment [22]) can be included if needed [23]. 

The radial Dirac equation for a hydrogen-like atom with nuclear charge Z may be written, in atomic units, in the form... 

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