Approximate solution of the Dirac equation

Let us introduce the total angular momentum as a sum of the orbital and spin contribution 

For such an operator the commutators of [la, HD] and [6a, HD] just cancel so that this operator commutes with the Dirac Hamiltonian 

It must be mentioned again that the Dirac equation was not derived—it was postulated. However, it is useful in describing the correct behaviour of electrons. One example is the magnetogyric factor for the electron, defined as the ratio of the intrinsic magnetic moment to the intrinsic angular momentum, expressed in units of (q/2m) 

This value cannot be properly derived from the non-relativistic theory because the application of the classical expression for the magnetic moment of a charged particle associated with some angular momentum / 

Although the Dirac equation in its four-component form can be solved exactly for a few systems (the hydrogen-like atom, electron in a uniform magnetic field), normally a decoupling to the two-component form has to be done [2-6], For this purpose two techniques were developed the Foldy-Wouthuysen transformation and the partitioning method (for a small component). We will follow the second approach, which is based on these steps  

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Morrison, J., and Moss, R., 1980, Approximate solution of the Dirac equation using the Foldy-Wouthuysen Hamiltonian , A/o/. Phys. 41 491. 

It has been demonstrated in a step-by-step calculation how to obtain an approximate solution of the Dirac equation for the hydrogen-like atom. One of the results is that the relativistic orbitals are contracted compared to the non-relativistic ones. 

Fia. 2-19.—Curves representing values of electron energies, as a function of atomic number. These curves were obtained by approximate solution of the wave equation by the Thomas-Fermi-Dirac method. 

Obviously, the solution of this equation for X is as complex as the solution of the Dirac equation itself, and approximations have to be employed. 

This so-called Hellmann-Feynman force (HFF) represents the electrostatic interaction between the negatively charged electrons and the nuclei as well as the interaction among the nuclei. Equation (4.58) would describe the forces correctly for an exact solution of the Dirac equation. However, in practical calculations we have to introduce approximations which have a rather large influence on the forces. There are two such artificial forces resulting from the following. 

Abstract. This chapter concerns a presentation of the Darwin solutions of the Dirac equation, in the Hestenes form of this equation, for the central potential problem. The passage from this presentation to that of complex spinor is entirely explicited. The nonrelativistic Pauli and Schrodinger theories are deduced as approximations of the Dirac theory. 

The approximation of the nucleus as an infinitely heavy point charge makes possible analytical solution of the Dirac equation for the hydrogen-like problem. The resulting orbitals are, however, too tightly bound and clearly unphysical within the nucleus. A homogeneously charged nucleus is a significant improvement and is sufficient for many applications. For more detailed studies of nuclear properties, it is, however, desirable to use a more physical nuclear distribution, such as the Fermi and the Fourier-Bessel distributions described below. 

In many quantum-mechanical calculations, use is made of the wave functions obtained by the Dirac—Slater and the Hartree—Fock methods for the approximate solution of the Schrddinger equation for free atoms. It woiild be very interesting to determine whether these functions could be refined specifically for crystals and whether the problem could be solved using relatively simple analytic approximations to the calculated functions. In particular, the approximation by Gaussian functions demands attention. 

Several authors have considered a number of approximate solutions to the Dirac equation. One such method is the use of the Foldy-Wouthuysen transformation (see, for example, Morrison and Moss ). Upon application of a unitary transformation of the form... 

Over the past few years has been increasing interest in the use of the algebraic approximation in estimating the solutions of the Dirac equation and... 

Spatial forms of the electron density and the potential are determined in the muffin-tin approximation. Then, the wave function is expanded by the solutions of the Dirac equation inside the APW spheres and by the plane waves in the interstitial region. To determine the expansion coefficients and the eigenvalues, a variational principle is derived from eq. (9)... 

For the calculations of relativistic density functions we used a multi-configuration Dirac-Fock approach (MCDF), which can be thought of as a relativistic version of the MCHF method. The MCDF approach implemented in the MDF/GME program [4, 27] calculates approximate solutions to the Dirac equation with the effective Dirac-Breit Hamiltonian [27]... 

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 inclusion of relativistic effects is essential in quantum chemical studies of molecules containing heavy elements. A full relativistic calculation, i.e. based upon Quantum Electro Dynamics, is only feasible for the smallest systems. In the SCF approximation it involves the solution of the Dirac Fock equation. Due to the four component complex wave functions and the large number of basis functions needed to describe the small component Dirac spinors, these computations are much more demanding than the corresponding non-relativistic ones. This limits Dirac Fock calculations, which can be performed using e.g. the MOLFDIR package [1], to small molecular systems, UFe being a typical example, see e.g. [2]. 

Transformed Dirac equations are convenient starting points for the derivation of quasi-relativistic Hamiltonians. The transformed Dirac equations can be obtained by using approximate solutions for the small components as ansatze for the wave function. The ansatz can be deduced from the lower half of the Dirac equation by an approximate elimination of the small component. 

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