Relativistic Dirac equation

Spatial extension, as expressed by the expectation value (r), is roughly comparable for 4 f and 5 f wave functions (Figs. 7 and 8). However, the many-electron wave functions resulting from the solution of the relativistic Dirac equation may also be used to calculate a number of physically interesting quantities, i.e. expectation values of observable... 

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 ... 

This experimental development was matched by rapid theoretical progress, and the comparison and interplay between theory and experiment has been important in the field of metrology, leading to higher precision in the determination of the fundamental constants. We feel that now is a good time to review modern bound state theory. The theory of hydrogenic bound states is widely described in the literature. The basics of nonrelativistic theory are contained in any textbook on quantum mechanics, and the relativistic Dirac equation and the Lamb shift are discussed in any textbook on quantum electrodynamics and quantum field theory. An excellent source for the early results is the classic book by Bethe and Salpeter [6]. A number of excellent reviews contain more recent theoretical results, and a representative, but far from exhaustive, list of these reviews includes [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17]. 

For a system which contains elements with a large atomic number, the error due to the non-relativistic approximation become very serious in the electronic state calculation. Besides, a large energy shift of core level takes place even for rather light elements. Thus, relativistic Dirac equation should be... 

The LDA radial Schrodinger equation is solved by matching the outward numerical finite-difference solution to sin inward-going solution (which vanishes at infinity) of the same energy, near the classical turning point. Continuity of P t(r) = rAn/(r) and its derivative determines the eigenvalue /. The second order differential equation is actually solved as a pair of simultaneous first-order equations, so that the nonrelativistic and relativistic (Dirac equation) procedures appear similar. 

In the preceeding sections we have discussed the DV scheme as it has been applied in nonrelativistic DF theory here we extend the discussion to solutions of the relativistic Dirac equation,... 

The zero-order regular approximation (ZORA), a two-component form of the fully-relativistic Dirac equation, is currently used for organotin computational calculations using basis sets specifically designed for ZORA. It should be noted that while all-electron calculations, whether non-relativistic or relativistic, can be used for organotin systems, the 6-3IG Pople basis set is not available for tin and therefore, most all-electron calculations involving tin employ the smaller 3-21G basis set. 

We solve the dme-dependent relativistic Dirac-equation for complicated ion-atom systems iti the Diiac-Fock-Slater approximation including the many-electron effects by means of an inclusive probability description. Various gas-gas and gas-solid target collision systems are discussed. We find the dominant effect in the observed excitation probability of inner shells to be a reladvisdc dynamic coupling of various levels which in a non-reladvisdc descripdon is zero. The effect of excitation and transfer during the passage within the solid allows us to understand the gas-solid target systems. An ab-inido calculation of K MO X-rays is presented for the system Cl" on Ar. 

In order to derive an expression for the energy of interaction between an intrinsic magnetic moment (as for a particle with nonzero spin angular momentum) and the electromagnetic field, one cannot proceed via classical mechanics as for the charged particle. Rather one proceeds via the relativistic Dirac equation and seeks the nonrelativistic limit... 

The spin moment operator has been introduced arbitrarily here, but appears naturally if the equation is derived by reducing the relativistic Dirac equation. With the spin operator included the wavefimction must be treated as a two component spin function and the operators as 2 X 2 matrices. 

As a result, quasi-particles in graphene exhibit the linear dispersion relation E = hvp, where vp is the Fermi velocity ( 10 m/s), as if they were massless relativistic particles. Thus, graphene s quasiparticles behave differently from those in conventional metals and semiconductors, where the energy spectrum can be approximated by a parabolic dispersion relation. Electron transport in all known condensed-matter systems is described by the (non-relativistic) Schrodinger equation and relativistic effects are usually negligible. In contrast, the electrons of graphene are described by the (relativistic) Dirac equation, i.e. they mimic relativistic charged particles with zero rest mass and constant velocity [10]. 

At this point, we are flirting with the more difficult relativistic Dirac equation for the F atom. That method involves representing the spin function, as 2 x 2 Pauh matrices. Not only is that method... 

The above equations can be solved explicitly for a single charged particle in a coulombic field, and exactly in the case of the hydrogen atom. The main difference between the ordinary noiurelativistic solution to the Schrddinger equation of the hydrogen atom and the relativistic Dirac equation is that the solution is a column vector of length four, known as the four-component spinor. Thus we have four-component spinor orbitals instead of simple orbitals in the fully relativistic description of atoms. 

In this way the relativistic effects are taken into account in an effective way. At the same time the 5 — L coupling, natural for the non-relativistic Schrodinger equation and the standard theory of tri-positive lanthanide ions, is preserved, instead of the j — j basis of a relativistic Dirac equation. As a result of such replacements of all the operators, new angular terms appear and the radial integrals in (10.33) are defined by the small and large... 

---