Dirac equation density

Again, the summation convention is used, unless we state otherwise. As will appear below, the same strategy can be used upon tbe Dirac Lagrangean density to obtain the continuity equation and Hamilton-Jacobi equation in the modulus-phase representation. 

By following [323], we substitute in the Lagrangean density, Eq. (149), from the Dirac equations [322], namely, from... 

We thus obtain a Lagrangean density, whieh is equivalent to Eq. (149) for all solutions of the Dirac equation, and has the structure of the nonrelativistic Lagrangian density, Eq. (140). Its variational derivations with respect to v / and v / lead to the solutions shown in Eq. (152), as well as to other solutions. 

Relativistic density functional theory can be used for all electron calculations. Relativistic DFT can be formulated using the Pauli formula or the zero-order regular approximation (ZORA). ZORA calculations include only the zero-order term in a power series expansion of the Dirac equation. ZORA is generally regarded as the superior method. The Pauli method is known to be unreliable for very heavy elements, such as actinides. 

Here we have used the natural expansion (33), with spin-orbitals written in the form (29). The second term in (41), absent in a Pauli-type approximation, contains the correction arising from the use of a 4roomponent formulation it is of order (2tmoc) and is usually negligible except at singularities in the potential. As expected, for AT = 1, (41) reproduces the density obtained from a standard treatment of the Dirac equation but now there is no restriction on the particle number. 

Accurate simulation of radical g tensors has only recently become possible, courtesy of accurate solvent models, ZORA approximation to Dirac equation, and the arrival of an accurate treatment of spin-orbit coupling within the density functional theory. All these methods are implemented in the ADF 2007 package, which was used for the g tensor calculations reported in Table 10.2. 

Accounting for relativistic effects in computational organotin studies becomes complicated, because Hartree-Fock (HF), density functional theory (DFT), and post-HF methods such as n-th order Mpller-Plesset perturbation (MPn), coupled cluster (CC), and quadratic configuration interaction (QCI) methods are non-relativistic. Relativistic effects can be incorporated in quantum chemical methods with Dirac-Hartree-Fock theory, which is based on the four-component Dirac equation. " Unformnately the four-component Flamiltonian in the all-electron relativistic Dirac-Fock method makes calculations time consuming, with calculations becoming 100 times more expensive. The four-component Dirac equation can be approximated by a two-component form, as seen in the Douglas-Kroll (DK) Hamiltonian or by the zero-order regular approximation To address the electron cor-... 

The finite difference HF scheme can also be used to solve the Schrodinger equation of a one-electron diatomic system with an arbitrary potential. Thus the approach can be applied, for example, to the construction of exchange-correlation potentials employed by the density functional methods. The eigenvalues of several GaF39+ states have been reported and the Th 79+ system has been used to search for the influence of the finite charge distribution on the potential energy curve. It has been also indicated that the machinery of the finite difference HF method could be used to find exact solutions of the Dirac-Hartree-Fock equations based on a second-order Dirac equation. 

The term Lamb shift of a single atomic level usually refers to the difference between the Dirac energy for point-like nuclei and its observable value shifted by nuclear and QED effects. Nuclear effects include energy shifts due to static nuclear properties such as the size and shape of the nuclear charge density distribution and due to nuclear dynamics, i.e. recoil correction and nuclear polarization. To a zeroth approximation, the energy levels of a hydrogen-like atom are determined by the Dirac equation. For point-like nuclei the eigenvalues of the Dirac equation can be found analytically. In the case of extended nuclei, this equation can be solved either numerically or by means of successive analytical approximation (see Rose 1961 Shabaev 1993). 

Methods for solid-state calculations have been devised on the basis of the Dirac equation (bei der Kellen and Freeman 1996 Shick et al. 1999 Wang et al. 1992). Very recent progress has been achieved in the framework of four-component density functional theory for solids (Theileis and Bross 2000) (compare also the review on the... 

Note, ft is to emphasize that the above entities are independent of the probability density p, and so are relevant of the part Di (four real scalar equations) of the Dirac equation D which is independent of p. Let Du (four real scalar equations) be the part of D which depends on p. About the role of the density p with respect to these entities, we have established in [11] the following theorem. 

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