Second-order Dirac equations

The most popular way to derive the non-relativistic limit (nrl) of the Dirac equation is known as the method of the elimination of the small component [38, 39]. One starts by writing the DE as 

Otherwise Eq. (60) is of little use, since the operator on the l.h.s. is energy-dependent, and hence for every eigenstate a different Hamiltonian has to be used. 

The derivation of the nrl depends, of course, on the assumption that the eigenvalue W = E + mc is close to mc, i.e. that we consider an electronic state. For positronic states the limit (40) is obtained, and for states for which E is not small compared to mc, the nrl does not exist. 

Somewhat simpler second order Dirac equations (we always consider W = mc + E) involve 

---

By substituting this expression for tfi into (10-389) one finds that must satisfy the following second-order Dirac equation... 

An overview of the development of the finite difference Hartree-Fock method is presented. Some examples of it axe given construction of sequences of highly accurate basis sets, generation of exact solutions of diatomic states, Cl with numerical molecular orbitals, Dirac-Hartree-Fock method based on a second-order Dirac equation. 

The finite difference HF scheme can also be used to solve the Schrodinger equation of a one-electron diatomic system with an arbitrary potential. Thus the approach can be applied, for example, to the construction of exchange-correlation potentials employed by the density functional methods. The eigenvalues of several GaF39+ states have been reported and the Th 79+ system has been used to search for the influence of the finite charge distribution on the potential energy curve. It has been also indicated that the machinery of the finite difference HF method could be used to find exact solutions of the Dirac-Hartree-Fock equations based on a second-order Dirac equation. 

D. Sundhohn, P. Pyykko, L. Laaksonen. Two-Dunensional PuUy Numerical Solutions of Second-order Dirac Equations for Diatomic Molecules, Part 3. Phys. Scr., 36 (1987) 400-402. 

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]... 

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] ... 

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. 

As a first-order differential equation, the radial Dirac equation, which is the central equation to be solved in the case of atoms, requires a discretization scheme, and several options are at hand of which some should be presented here for the sake of completeness. The first one is analogous to the Nu-merov procedure for second-order differential equations without first derivatives [491,492]. The derivation in terms of Taylor series expansions provides a derivation which is easier to understand. However, using operator techniques is the most elegant way for this particular task. 

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]. 

It is well known that the Dirac equation with scalar and vector potentials can be reduced to a second-order differential equation and manipulated into the form of the Schrodinger equation [Cl 85]. 

Here (following ref. [Lu87]) we reduce the single particle Dirac equation to a second-order differential equation which yields an equivalent upper component for the projectile wave function and consequently the same scattering observables. A more detailed account of the derivation of this so-called Schrodinger equivalent potential for local scalar, vector and tensor interactions is given in ref. [Cl 85],... 

The Schrodinger equation and the Klein-Gordon equation both involve second order partial derivatives, and to recover such an equation from the Dirac equation we can operate on equation 18.12 with the operator... 

Actually Schrddinger s original paper on quantum mechanics already contained a relativistic wave equation, which, however, gave the wrong answer for the spectrum of the hydrogen atom. Due to this fact, and because of problems connected with the physical interpretation of this equation, which is of second order in the spaoe and time variables, it was temporarily discarded. Dirac took seriously the notion of first... 

The second-order equation (10-393) is more useful than the first-order equation (10-389) because (10-393) involves only one ef the Dirac matrices, r, and can, therefore, be solved more easily than... 

We then turn to the question of how to eliminate the spin-orbit interaction in four-component relativistic calculations. This allows the assessment of spin-orbit effects on molecular properties within the framework of a single theory. In a previous publication [13], we have shown how the spin-orbit interaction can be eliminated in four-component relativistic calculations of spectroscopic properties by deleting the quaternion imaginary parts of matrix representations of the quaternion modified Dirac equation. We show in this chapter how the application of the same procedure to second-order electric properties takes out spin-forbidden transitions in the spectrum of the mercury atom. Second-order magnetic properties require more care since the straightforward application of the above procedure will extinguish all spin interactions. After careful analysis on how to proceed we... 

Here we have used the natural expansion (33), with spin-orbitals written in the form (29). The second term in (41), absent in a Pauli-type approximation, contains the correction arising from the use of a 4roomponent formulation it is of order (2tmoc) and is usually negligible except at singularities in the potential. As expected, for AT = 1, (41) reproduces the density obtained from a standard treatment of the Dirac equation but now there is no restriction on the particle number. 

If p 7 0 Eqs. (21) and (22) are coupled, but the relations between components of the wavefunction are much simpler than in the standard Dirac-Pauli representation. By the elimination of and respectively from Eq. (21) and from Eq. (22), we get two decoupled second-order equations for and ... 

It is also convenient to convert the Dirac equation into a second-order partial differential equation, by multiplying both sides of Eq. (3.6.22) by E/ =i ( /i /() + im0c]. After some travail, using quantities that are more familiar, the result is... 

This important equation is known as the Klein-Gordon equation, and was proposed by various authors [6, 7, 8, 9] at much the same time. It is, however, an inconvenient equation to use, primarily because it involves a second-order differential operator with respect to time. Dirac therefore sought an equation linear in the momentum operator, whose solutions were also solutions of the Klein-Gordon equation. Dirac also required an equation which could more easily be generalised to take account of electromagnetic fields. The wave equation proposed by Dirac was [10]... 

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. 

---