Dirac one-electron equation

For studies of the Fermi surface in the lanthanide compounds, it is necessary to develop a reliable theoretical method in which hybridization of the 4f electrons with other electrons as well as the relativistic effect can be taken into account quantitatively. For that purpose, the relativistic APW method proposed by Loucks (1967) provides a good starting basis. Loucks derived his original method from the Dirac one-electron equation, which is a natural extension of Slater s non-relativistic APW method (Slater 1937). It proved to be a powerful method comparable to a relativistic KKR method (Onodera and Okazaki 1966, Takada 1966). Loucks method does not accocunt for the symmetrization of the wave functions by group theory, nor it is a self-consistent method. These shortcomings are serious limitations for calculations of the energy band structure in the lanthanide compounds. 

Yamagami and Hasegawa carried out a self-consistent calculation of the energy band structure by solving the Kohn-Sham-Dirac one-electron equation by the density-functional theory in a local-density approximation (LDA). This self-consistent, symmetrized relativistic APW approach was applied to many lanthanide compounds and proved to give quite accurate results for the Fermi surface. 

Relativity affects the kinetic term and the exchange-correlation potential in the Kohn-Sham equation. As investigated in detail for the uranium atom and the cerium atom, the relativistic effect on the exchange correlation potential is rather small and therefore we use /Ac[ ( )] in n relativistic band structure calculation. The relativistic effect on the kinetic term is appreciably large and can be taken into account by adopting the Kohn-Sham-Dirac one-electron equation instead of eq. (3) as follows ... 

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 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... 

Dirac, the discoverer of the relativistic one-electron equation, thought that relativity would be unimportant in chemistry (P. A. M. Dirac, Quantum Mechanics of Many-Electron Systems , Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences, 1929, 123(192), 714). Why was he mistaken ... 

For atoms with high atomic numbers it is important that relativistic effects are included in the band-structure calculations. In SCFC we therefore solve the Dirac rather than the Schrodinger equation, but leave out the spin-orbit interaction. In doing so we obtain an effective one-electron equation which is essentially the Schrodinger equation with the important mass-velocity and Darwin corrections included. The present technique is based on unpublished work by O.K. Andersen and U.K. Poulsen. KoelUng and Harmon [9.8] have taken a related approach. 

The procedure not to apply the (a p)-operator onto the basis functions but to consider it explicitly and rewrite the one-electron equation is a major trick which has been dicsussed by many authors and which is intimately connected with the modified Dirac equation and the so-called exact-decoupling methods discussed in detail in section 14.1. 

For this purpose, we focus on one-electron operators only, because even the first-quantized many-electron theory reduces to Dirac-like one-electron equations, i.e., to the self-consistent field equations. The Dirac Hamiltonian is then substituted by the four-dimensional Fock operator... 

Before moving on to this more complicated two body problem, let us look at a few more properties of the simple one-electron equation. If we expand the central field Dirac equation roughly in powers of Za, we find a number of familiar interactions. 

The Dirac equation can be readily adapted to the description of one electron in the held of the other electrons (Hartree-Fock theory). This is called a Dirac-Fock or Dirac-Hartree-Fock (DHF) calculation. 

Dirac equation one-electron relativistic quantum mechanics formulation direct integral evaluation algorithm that recomputes integrals when needed distance geometry an optimization algorithm in which some distances are held fixed... 

Intrinsic Semiconductors. For semiconductors in thermal equiHbrium, (Ai( )), the average number of electrons occupying a state with energy E is governed by the Fermi-Dirac distribution. Because, by the Pauli exclusion principle, at most one electron (fermion) can occupy a state, this average number is also the probabiHty, P E), that this state is occupied (see Fig. 2c). In equation 2, K... 

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... 

First, we consider the NRF and NRL of the one-electron external field Dirac equation. Before we continue any further, let us rewrite the equation (11), for a fermion in an external radiation field... 

The Xa multiple scattering method generates approximate singledeterminant wavefunctions, in which the non-local exchange interaction of the Hartree-Fock method has been replaced by a local term, as in the Thomas-Fermi-Dirac model. The orbitals are solutions of the one-electron differential equation (in atomic units)... 

The relativistic coupled cluster method starts from the four-component solutions of the Drrac-Fock or Dirac-Fock-Breit equations, and correlates them by the coupled-cluster approach. The Fock-space coupled-cluster method yields atomic transition energies in good agreement (usually better than 0.1 eV) with known experimental values. This is demonstrated here by the electron affinities of group-13 atoms. Properties of superheavy atoms which are not known experimentally can be predicted. Here we show that the rare gas eka-radon (element 118) will have a positive electron affinity. One-, two-, and four-components methods are described and applied to several states of CdH and its ions. Methods for calculating properties other than energy are discussed, and the electric field gradients of Cl, Br, and I, required to extract nuclear quadrupoles from experimental data, are calculated. 

Heavy atoms exhibit large relativistic effects, often too large to be treated perturba-tively. The Schrodinger equation must be supplanted by an appropriate relativistic wave equation such as Dirac-Coulomb or Dirac-Coulomb-Breit. Approximate one-electron solutions to these equations may be obtained by the self-consistent-field procedure. The resulting Dirac-Fock or Dirac-Fock-Breit functions are conceptually similar to the familiar Hartree-Fock functions the Hartree-Fock orbitals are replaced, however, by four-component spinors. Correlation is no less important in the relativistic regime than it is for the lighter elements, and may be included in a similar manner. 

The relativistic many-electron Hamiltonian cannot be written in closed form it may be derived perturbatively from quantum electrodynamics [1]. The simplest form is the Dirac-Coulomb (DC) Hamiltonian, where the nonrelativistic one-electron terms in the Schrodinger equation are replaced by the one-electron Dirac operator hj). 

It is known (Chap. A) that Koopmans theorem is not vahd for the wavefunctions and eigenvalues of strongly bound states in an atom or in the cores of a solid, i.e. for those states which are a solution of the Schrodinger (or Dirac) equation in a central potential. In them the ejection (or the emission) of one-electron in the electron system means a strong change in Coulomb and exchange interactions, with the consequent modification of the energy scheme as well as of the electronic wavefunction, in contradiction to Koopmans theorem. 

Consider the Dirac-Fock equations for a three-electron system Is nlj. Formally they fall into one-electron Dirac equations for the orbitals l5 and nlj with the potential ... 

Band theory is a one-electron, independent particle theory, which assumes that the electrons are distributed amongst a set of available stationary states following the Fermi-Dirac statistics. The states are given by solutions of the Schrodinger equation... 

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