Relativistic central fields

For heavy atoms, it is often necessary to go beyond the Hartree-Fock approach, although the decision whether relativistic corrections are important depends on the problem in hand. For example, if it is desired to study atomic effects down a column of the periodic table, consistency may require that all the calculations are performed with the same code, 

2 The special significance of the ground state emerges as follows let ip be any wavefunction (not necessarily normalised) in the space of the exact solutions % of eigenenergies en. Expand ip = fTiinCn- Then the expectation value 

The correction B(i,j) to the Coulomb potential is treated as a perturbation of the zero-order Hamiltonian, and may include relaxation effects, correlations, quantum electrodynamic corrections and the relativistic retardation of the two-electron potential. 

For open inner shells, the Breit correction is important and can be included perturbatively. QED effects can be allowed for by interpolation of tabulated data [14]. 

In relativistic calculations, L and S cease to be good quantum numbers, because the spin must be included within the single particle Hamiltonians hn(i), and so the results must be expressed in jj coupling. The constants of the motion become n, j and k, where 

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In the 5 d series however it is possible to derive additional information bearing upon the problem of the relative extent of central field and symmetry restricted covalency. For many 5 d complexes reasonable estimates of the effective spin-orbit coupling constant can be derived from the spectra, and thence the relativistic ratio, / (= complex/ gas). When both f) and / are known for a given system, Jorgensen (74) has suggested how estimates of both covalencv contributions may be made. 

Although Dirac s equation does not directly admit of a completely self-consistent single-particle interpretation, such an interpretation is physically acceptable and of practical use, provided the potential varies little over distances of the order of the Compton wavelength (h/mc) of the particle in question. It allows, for instance, first-order relativistic corrections to the spectrum of the hydrogen atom and to the core-level densities of many-electron atoms. The latter aspect is of special chemical importance. The required calculations are invariably numerical in nature and this eliminates the need to investigate central-field solutions in the same detail as for Schrodinger s equation. A brief outline suffices. 

In non-relativistic Schrodinger theory every component of the orbital angular momentum L = r x p, as well as L2, commutes with the Hamiltonian H = p2/2m + V of a spinless particle in a central field. As a result, simultaneous eigenstates of the operators H, L2 and Lz exist in Schrodinger theory, with respective eigenvalues of E, l(l + l)h2 and mh. In Dirac s theory, however, neither the components of L, nor L2, commute with the Hamiltonian 10. 

Despite the complication due to the interdependence of orbital and spin angular momenta, the Dirac equation for a central field can be separated in spherical polar coordinates [63]. The energy eigenvalues for the hydrogen atom (V(r) = e2/r, in electrostatic units), are equivalent to the relativistic terms of the old quantum theory [64]... 

One-particle wave-functions in a central field are obtained as solutions of the relativistic Dirac equation, which can be written in the two-component form ... 

In order to improve the theoretical description of a many-body system one has to take into consideration the so-called correlation effects, i.e. to deal with the problem of accounting for the departures from the simple independent particle model, in which the electrons are assumed to move independently of each other in an average field due to the atomic nucleus and the other electrons. Making an additional assumption that this average potential is spherically symmetric we arrive at the central field concept (Hartree-Fock model), which forms the basis of the atomic shell structure and the chemical regularity of the elements. Of course, relativistic effects must also be accounted for as corrections, if they are small, or already at the very beginning starting with the relativistic Hamiltonian and relativistic wave functions. 

Two coupled first order differential equations derived for the atomic central field problem within the relativistic framework are transformed to integral equations through the use of approximate Wentzel-Kramers-Brillouin solutions. It is shown that a finite charge density can be derived for a relativistic form of the Fermi-Thomas atomic model by appropriate attention to the boundary conditions. A numerical solution for the effective nuclear charge in the Xenon atom is calculated and fitted to a rational expression. 

Condon and Shortley wrote the book(4) on atomic theory and I will use their equations for the radial components of the relativistic wave function for an electron in a central field(5). Thus 1 quote... 

Abstract. This chapter is a recall of the properties of the long-range part of the electromagnetic fields created by time-periodic currents, as they may be observed in particular in the Zeeman effect. The aim of this part is also to place the vector frame of these observations, that is, one of the spherical coordinates, which is in the center of the presentation in the real formalism of the relativistic central potential problem. This frame is the one in which are expressed the Dirac probability current, associated with a state and with the transition between two states. But it is to notice that, as a specificity of the real formalism, the form given by Hestenes to the wave function of the electron, strictly equivalent to the Dirac spinor, may be presented, in the case of central potential, as a combination of the vectors of this frame. 

Magnetism is a central field in condensed matter research, basic as well as applied. Several physical effects such as, for example, the magnetooptic Kerr effect, are caused by the simultaneous occurrence of spin polarization and spin-orbit coupling. It is therefore necessary to include spin polarization in the (fully) relativistic band structure formalisms. Feder et al. 

Later, in Sec. 4, we will give a detailed discussion of the need for the no-pair Hamiltonian in relativistic calculations, its limitations, and its relation to QED. To establish a foundation for our studies of few-electron systems, we start in Sec. 2 with a discussion of the one-electron central-field Dirac equation and radiative corrections to one-electron atoms. In Sec. 3 we describe many-body perturbation theory (MBPT) calculations of few-electron atoms, and finally, in Sec. 4 we turn to relativistic configuration-interaction (RCI) calculations. 

For heavy atoms, the instantaneous velocities of the electrons near the nuclei cannot be neglected with respect to the velocity of light. These electrons must be described within the Dirac relativistic theory. For the sake of simplicity, let us consider a one-electron system in a central field. The Dirac Hamiltonian, shifted for the energy by e, can be written in atomic units (a.u.) as... 

An accurate calculation in heavy atoms requires a thoroughgoing relativistic treatment. The central field-independent particle approximation gives a first-order answer, but corrections involving electron exchange, noncentral effects, and other electron-electron correlations are needed in many cases. Details of the calculations have been reviewed elsewhere ... 

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. 

All aspects of Newtonian mechanics can equally well be formulated within the more general Lagrangian framework based on a single scalar function, the Lagrangian. These formal developments are essential prerequisites for the later discussion of relativistic mechanics and relativistic quantum field theories. As a matter of fact the importance of the Lagrangian formalism for contemporary physics cannot be overestimated as it has strongly contributed to the development of every branch of modem theoretical physics. We will thus briefly discuss its most central formal aspects within the framework of classical Newtonian mechanics. 

The most significant difference of Dirac s results from those of the non-relativistic Pauli equation is that the orbital angular momentum and spin of an electron in a central field are no longer separate constants of the motion. Only the components of J = L - - S and J, which commute with the Hamiltonian, emerge as conserved quantities [1]. Dirac s equation, extended to general relativity by the method of projective relativity [2], automatically ensures invariance with respect to gauge, coordinate and spinor transformations, but has never been solved in this form. 

In general, we shall assume hat relativistic calculations are based on the Dirac equation. Therefore, before discussing the many-body situation, it is worthwhile to briefly review the relativistic effects contained in the one electron Dirac equation for an electron moving in a central field ... 

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