Dirac Equation Radial Equations

The no-pair DCB Hamiltonian (6) is used as a starting point for variational or many-body relativistic calculations [9], The procedure is similar to the nonrelativistic case, with the Hartree-Fock orbitals replaced by the four-component Dirac-Fock-Breit (DFB) functions. The spherical symmetry of atoms leads to the separation of the one-electron equation into radial and spin-angular parts [10], The radial four-spinor has the so-called large component the upper two places and the small component Q, in the lower two. The quantum number k (with k =j+ 1/2) comes from the spin-angular equation, and n is the principal quantum number, which counts the solutions of the radial equation with the same k. Defining... 

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... 

The RQDO radial, scalar, equation derives from a non-unitary decoupling of Dirac s second order radial equation. The analytical solutions, RQDO orbitals, are linear combinations of the large and small components of Dirac radial function [6,7] ... 

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. 

For w = 1 or 2 they have the general form of a radial eigenvalue problem arising from some Hamiltonian. In fact, the radial parts of the nonrelativistic hydrogenic Hamiltonian, Klein-Gordon, and second-order iterated Dirac Hamiltonians with 1/r potential can all be expressed in this form for w = 1 and suitable choices of the parameters , rj, x. Similarly, the three-dimensional isotropic harmonic oscillator radial equation has this form for w = 2. 

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... 

The relativistic theory and computation of atomic structures and processes has therefore attained some sort of maturity and the various codes now available are widely used. Those mentioned so far were based on ideas originating from Hartree and his students [28], and have been developed in much the same way as the non-relativistic self-consistent field theory recorded in [28-30]. All these methods rely on the numerical solution, using finite differences, of the coupled differential equations for radial orbital wave-functions of the self-consistent field. This makes them unsuitable for the study of molecules, for which it is preferable to expand the radial amplitudes in a suitably chosen set of analytic functions. This nonrelativistic matrix Hartree-Fock method, as it is often termed, was pioneered by Hall and Lennard-Jones [31], Hall [32,33] and Roothaan [34,35], and it was Roothaan s students, Synek [36] and Kim [37] who were the first to attempt to solve the corresponding matrix Dirac-Hartree-Fock equations. Kim was able to obtain solutions for the ground state of neon in 1967, but at the expense of some numerical instability, and it seemed at the time that the matrix Dirac-Hartree-Fock scheme would not be a serious competitor to the finite difference codes. 

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. 

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