Dirac equations and electron spin

A detailed study of the Dirac equation and its solutions will not be required it will simply be assumed, as already indicated, that the S3rstem of N electrons above the negative-energy sea may be described using a wavefunction constructed from antisymraetrized products of (positive energy) spin-orbitals of type (29). It is, however, necessary to know the basic properties of the operators Q/i, which appear in the Dirac equation... 

G. Breit, Dirac s equation and the spin-spin interactions of two electrons, Phys. Rev. 39 (1932) 616. 

One can gain some insight into the nature of the Dirac wave equation and the spin angular momentum of the electron by considering the Foldy Wouthuysen transformation for a free particle. In the absence of electric and magnetic interactions, the Dirac Hamiltonian is... 

In writing equation (4.6), we have assumed that the nuclei can be treated as Dirac particles, that is, particles which are described by the Dirac equation and behave in the same way as electrons. This is a fairly desperate assumption because it suggests, for example, that all nuclei have a spin of 1 /2. This is clearly not correct a wide range of values, integral and half-integral, is observed in practice. Furthermore, nuclei with integral spins are bosons and do not even obey Fermi Dirac statistics. Despite this, if we proceed on the basis that the nuclei are Dirac particles but that most of them have anomalous spins, the resultant theory is not in disagreement with experiment. If the problem is treated by quantum electrodynamics, the approach can be shown to be justified provided that only terms of order (nuclear mass) 1 are retained. 

All the terms of the operator (164) are the order Z (ctZy in r.u., i.e. Z (aZ) o where So is the characteristic binding energy. The terms in the first line of Eq(164) describe the relativistic orbit - orbit interaction, the terms in the second line describe the spin - other orbit interaction (unlike the spin-orbit interaction that is included in the one-electron Dirac equation) and the terms in the third line describe the spin-spin interaction. 

In the Dirac relativistic equation, spin is naturally included. It is possible to identify the energy terms in the Dirac equation and include them in the ordinary Hamiltonian as magnetic terms. In particular, the spin-orbit coupling term is important. This term is physically due to interactions between the electron spin and its motional spin around the atomic nucleus. Spin-orbit coupling increases in importance for heavy atoms. Transitions or curve crossings are no longer spin forbidden. 

In the next section, we will examine some properties of the solutions of the spin-free modified Dirac equation, and we then proceed to a closer inspection of the one-electron operators in the modified formalism before treating the two-electron terms. 

Secondly, the exposition will be restricted to non-relativistic quantum mechanics, i.e. to the Schrodinger equation. This approach is justified, if we restrict ourselves to atoms of the first three rows of the periodic table, for which relativistic effects are generally unimportant. However, if we are interested in discussing properties, which include interactions with the spin of the electrons such as NMR and ESR couphng constants, the Schrodinger equation is not sufficient alone, because it is in principle a spin-free theory contrary to the Dirac equation. The necessary operators for the interaction with the electron spin are therefore derived from the Dirac equation and then added to the Schrodinger Hamiltonian in an ad hoc fashion. 

We could continue now with the Dirac equation and derive expressions for the molecular properties using standard perturbation theory. However, as stated earlier, the exposition in these notes is restricted basically to a non-relativistic treatment with the exception that we want to include also interactions with the spin of the electrons. The appropriate operator can be found by reduction of the Dirac equation to a non-relativistic two-component form, which can be achieved by several approaches. Here, we want to discuss only the simplest approach, the so-called elimination of the small component. 

Dirac showed in 1928 dial a fourth quantum number associated with intrinsic angidar momentum appears in a relativistic treatment of the free electron, it is customary to treat spin heiiristically. In general, the wavefimction of an electron is written as the product of the usual spatial part (which corresponds to a solution of the non-relativistic Sclnodinger equation and involves oidy the Cartesian coordinates of the particle) and a spin part a, where a is either a or p. A connnon shorthand notation is often used, whereby... 

The most common description of relativistic quantum mechanics for Fermion systems, such as molecules, is the Dirac equation. The Dirac equation is a one-electron equation. In formulating this equation, the terms that arise are intrinsic electron spin, mass defect, spin couplings, and the Darwin term. The Darwin term can be viewed as the effect of an electron making a high-frequency oscillation around its mean position. 

The Dirac equation automatically includes effects due to electron spin, while this must be introduced in a more or less ad hoc fashion in the Schrodinger equation (the Pauli principle). Furthermore, once the spin-orbit interaction is included, the total electron spin is no longer a good quantum number, an orbital no longer contains an integer number of a and /) spin functions. The proper quantum number is now the total angular momentum obtained by vector addition of the orbital and spin moments. 

Following the hypothesis of electron spin by Uhlenbeck and Goudsmit, P. A. M. Dirac (1928) developed a quantum mechanics based on the theory of relativity rather than on Newtonian mechanics and applied it to the electron. He found that the spin angular momentum and the spin magnetic moment of the electron are obtained automatically from the solution of his relativistic wave equation without any further postulates. Thus, spin angular momentum is an intrinsic property of an electron (and of other elementary particles as well) just as are the charge and rest mass. 

Spin-orbit coupling problems are of a genuine quantum nature since a priori spin is a quantity that only occurs in quantum mechanics. However, already Thomas (Thomas, 1927) had introduced a classical model for spin precession. Later, Rubinow and Keller (Rubinow and Keller, 1963) derived the Thomas precession from a WKB-like approach to the Dirac equation. They found that although the spin motion only occurs in the first semiclassical correction to the relativistic classical electron motion, it can be expressed in merely classical terms. 

The second term on the right-hand side of the equation gives for point nuclei directly the one-electron spin-orhit operator (2) of the Breit-Pauli Hamiltonian and can he eliminated to give a spin-free equation that becomes equivalent to the Schrddinger equation in the non-relativistic limit. In a quaternion formulation of the Dirac equation the elimination becomes particularly simple. The algebra of the quaternion units is that of the Pauli spin matrices... 

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... 

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