Relativistic Theory of the Electron

Having introduced the principles of special relativity in classical mechanics and electrodynamics as well as the foundations of quantum theory, we now discuss their unification in the relativistic, quantum mechanical description of the motion of a free electron. One might start right away with an appropriate ansatz for the basic equation of motion with arbitrary parameters to be chosen to fulfill boundary conditions posed by special relativity, which would lead us to the Dirac equation in standard notation. However, we proceed stepwise and derive the Klein-Gordon equation first so that the subsequent steps leading to Dirac s equation for a freely moving electron can be better understood. 

One postulate that has not explicitly been formulated as a basic axiom of quantum mechanics in the last chapter, because this postulate is valid for any physical theory, is that the equations of quantum mechanics have to be valid and invariant in form in all intertial reference frames. In this chapter, we take the first step toward a relativistic electronic structure theory and start to derive the basic quantum mechanical equation of motion for a single, freely moving electron, which shall obey the principles of relativity outlined in chapter 3. We are looking for a Hamiltonian which keeps Eq. (4.16) invariant in form under Lorentz transformations. 

Classical Energy Expression and First Hints from the Correspondence Principle 

We first consider the option to set up a quantum mechanical equation of motion which obeys the correspondence principle. If we apply the correspondence principle to the classical nonrelativistic kinetic energy expression E = (2m) we arrive at the time-dependent Schrodinger equation, in which 

Relativistic Quantum Chemistry. Markus Reiher and Alexander Wolf 

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In Eq. (15), 8(rik) is the Dirac delta function which, when integrated with the wave function, gives the value of the wave function at rik = 0. The two terms in Eq. (15) are in reality two limiting forms of the same interaction. The first term is the ordinary dipole-dipole interaction for two dipoles that are not too close to each other. It is the proper form of M S1 to be applied to p, d, and / electrons which are not found near the nucleus. For s electrons, which have a finite probability of being at the nucleus, the first term is clearly inappropriate, since it gives zero contribution at large values of rik and does not hold for small values of rik. From Dirac s relativistic theory of the electron, it is found (4) that the second term in Eq. (15) is the correct form for Si when the electron is close to the nucleus. Thus the contribution toJT S] from s electrons is proportional to the wave function squared at the site of the nucleus and the second term in Eq. (15) is often called the contact term in the hyperfine interaction. 

Just as orbital angular momentum L gives rise to a magnetic dipole moment pL, spin angular momentum S gives rise to a spin magnetic dipole moment fis. Dirac s relativistic theory of the electron showed that... 

The topics of the individual chapters are well separated and the division of the book into five major parts emphasizes this structure. Part I contains all material, which is essential for understanding the physical ideas behind the merging of classical mechanics, principles of special relativity, and quantum mechanics to the complex field of relativistic quantum chemistry. However, one or all of these three chapters may be skipped by the experienced reader. As is good practice in theoretical physics (and even in textbooks on physical chemistry), exact treatments of the relativistic theory of the electron as well as analytically solvable problems such as the Dirac electron in a central field (i.e., the Dirac hydrogen atom) are contained in part 11. 

The 2 comes from Dirac s relativistic theory of the electron the 0.0023 comes from additional correction terms. 

Section VI shows the power of the modulus-phase formalism and is included in this chapter partly for methodological purposes. In this formalism, the equations of continuity and the Hamilton-Jacobi equations can be naturally derived in both the nonrelativistic and the relativistic (Dirac) theories of the electron. It is shown that in the four-component (spinor) theory of electrons, the two exha components in the spinor wave function will have only a minor effect on the topological phase, provided certain conditions are met (nearly nonrelativistic velocities and external fields that are not excessively large). 

We demonstrated that by the selection of a representation of the Dirac Hamiltonian in the spinor space one may strongly influence the performance of the variational principle. In a vast majority of implementations the standard Pauli representation has been used. Consequently, computational algorithms developed in relativistic theory of many-electron systems have been constructed so that they are applicable in this representation only. The conditions, under which the results of these implementations are reliable, are very well understood and efficient numerical codes are available for both atomic and molecular calculations (see e.g. [16]). However, the representation of Weyl, if the external potential is non-spherical, or the representation of Biedenharn, in spherically-symmetric cases, seem to be attractive and, so far, hardly explored options. 

The prediction, and subsequent discovery, of the existence of the positron, e+, constitutes one of the great successes of the theory of relativistic quantum mechanics and of twentieth century physics. When Dirac (1930) developed his theory of the electron, he realized that the negative energy solutions of the relativistically invariant wave equation, in which the total energy E of a particle with rest mass m is related to its linear momentum V by... 

J. Sucher, Foundations of the relativistic theory of many-electron atoms, Phys. Rev. A 22 (1980) 348. 

So far we have three quantum numbers. However, we know from the relativistic treatment of the electron due to Dirac, which we described in chapter 3, that there is a fourth quantum number, called electron spin. In the first instance the need for a fourth quantum number became evident from experiment in the Dirac theory it is a consequence of introducing time as the fourth dimension. The spin angular momentum % s has the value (1/2)15, so that the magnitude of the spin angular momentum is [(1 /2)(3/2) l/ 2 ti. It can be oriented in two possible directions, with the fourth quantum number ms taking the values +1/2 or —1/2. Conventionally, these two orientations are described as a or f) respectively. 

SO coupling is a relativistic effect. The theory of the interaction of the magnetic moments of the electron spin and the orbital motion in one- and two-electron atoms has been formulated independently by Heisenberg and Pauli [12,13], shortly before the advent of the four-component Dirac theory of the electron [14]. Breit later has added the retardation correction [15]. The resulting Breit-Pauli SO operator, which can more elegantly be derived from the Dirac equation via a Foldy-Wouthuysen transformation [16], was thus well known for atoms since the early 1930s [17]. 

A proper framework for a relativistic description of the various scenarios indicated above is based on Dirac s theory of the electron and QED as the quantum field theory of leptons and photons. 

Independently of the approximations used for the representation of the spinors (numerical or basis expansion), matrix equations are obtained for Equations (2.4) that must be solved iteratively, as the potential v(r) depends on the solution spinors. The quality of the resulting solutions can be assessed as in the nonrelativistic case by the use of the relativistic virial theorem (Kim 1967 Rutkowski et al. 1993), which has been generalized to allow for finite nuclear models (Matsuoka and Koga 2001). The extensive contributions by I. P. Grant to the development of the relativistic theory of many-electron systems has been paid tribute to recently (Karwowski 2001). The higher-order QED corrections, which need to be considered for heavy atoms in addition to the four-component Dirac description, have been reviewed in great detail (Mohr et al. 1998) and in Chapter 1 of this book. 

The aim of this volume is twofold. First, it is an attempt to simplify and clarify the relativistic theory of the hydrogen-like atoms. For this purpose we have used the mathematical formalism, introduced in the Dirac theory of the electron by David Hestenes, based on the use of the real Clifford algebra Cl(M) associated with the Minkwoski space-time M, that is, the euclidean R4 space of signature (1,3). This algebra may be considered as the extension to this space of the theory of the Hamilton quaternions (which occupies an important place in the resolution of the Dirac equation for the central potential problem). 

Once we know the solution of the Dirac equation for a single electron moving in the external potential it is tempting to build the relativistic theory of many-electron systems in a similar way as the non-relativistic theory is built, i.e., by combining the one-electron Dirac Hamiltonian for each electron with the interaction between electrons. 

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