Solutions of the Radial Dirac Equation

We now turn to the radial functions and the radial Dirac equation. Using (7.22) in (7.14) and (7.15) we arrive at the pair of coupled radial equations  

We omit the mass in this section, and work strictly in Hartree atomic units. 

For the purposes of comparison with the nonrelativistic limit, we eliminate the small component from these equations, using (7.27b) to express Q in terms of P and substituting into (7.27a), with the result 

The solutions of these equations depend on the choice of the nuclear potential. For a point nucleus V = —Zjr, analytic solutions have been derived (Darwin 1928, Gordon 1928). We will not go into the details of the derivation, but we reproduce the essential features of the solutions for the purposes of comparison between nonrelativistic and relativistic solutions and the examination of the effects of relativity. 

The solutions of the point nucleus radial Dirac equation can be written as a product of an exponential function, a power of r, and a polynomial  

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

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 turn to the discretization of the coupled first-order differential equations as they occur in the solution of the Dirac radial equation for atoms (see chapter 9). While the Numerov scheme is well established for second-order differential equations — and, hence, for the solution of the radial Schrodinger equation for atoms — this is not the case for first-order differential equations. Indeed, it was long believed that the Numerov scheme cannot be used at all in this context [1165]. 

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... 

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. 

The relativistic version (RQDO) of the quantum defect orbital formalism has been employed to obtain the wavefunctions required to calculate the radial transition integral. The relativistic quantum defect orbitals corresponding to a state characterized by its experimental energy are the analytical solutions of the quasirelativistic second-order Dirac-like equation [8]... 

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

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. 

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. 

For the solution of the Dirac equation for hydrogen-like atoms, Eq. (6.7), we have succeeded in showing that the angular (and spin) variables can be separated from the radial-dependent part and that the eigenstates of total angular... 

This sum clearly truncates after a finite number of terms if a is a negative integer. The two functions Fi and F2 appearing in the radial solution of the point nucleus Dirac equation are... 

At small distances from the nucleus, the solutions of the Dirac equation are determined largely by the nuclear potential. In section 7.3, the solutions for a point nuclear potential were presented. Here, we adopt a more general approach, to determine the behavior of the solutions for an arbitrary (but realistic) nuclear potential. The radial functions are expanded in a power series. 

Relativistic effects have to be taken into account for compounds containing transition elements with higher atomic numbers the 5d transition elements (Hf, Ta, W) are of particular concern in the present review. A fiiUy relativistic treatment requires the solution of the Dirac equation instead of the Schrodinger equation. However, in many cases, it is sufficient to use a scalar relativistic scheme (48) as an approximation. In this technique, the mass-velocity term and the Darwin 5-shift are considered. The spin-orbit splitting, however, is neglected. In this approximation a different procedure must be used to calculate the radial wave functions, but the nonrelativistic formalism, which is computationally much simpler than solving Dirac s equation, is retained. 

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] ... 

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