Charge density Dirac

There are two approaches to map crystal charge density from the measured structure factors by inverse Fourier transform or by the multipole method [32]. Direct Fourier transform of experimental structure factors was not useful due to the missing reflections in the collected data set, so a multipole refinement is a better approach to map charge density from the measured structure factors. In the multipole method, the crystal charge density is expanded as a sum of non-spherical pseudo-atomic densities. These consist of a spherical-atom (or ion) charge density obtained from multi-configuration Dirac-Fock (MCDF) calculations [33] with variable orbital occupation factors to allow for charge transfer, and a small non-spherical part in which local symmetry-adapted spherical harmonic functions were used. 

From the symmetric set, an extended set of Maxwell equations was exhibited in Section V.E. This set contains currents and sources for both fields E, B. The old conjecture of Dirac s is vindicated, but the origin of charge density is always electric (i.e., no magnetic monopole). Standard Maxwell s equations are a limiting case in far field. 

This expression involves no reference to a wave function, and electronic information is provided by the electronic density p(r), described in the Dirac delta formalism as given in Eq. (303). The computation of the expectation value force operator requires only a simple sum of a classical contribution from the electronic charge density and the inter-nuclear repulsion. 

To put the definition of this property into direct correspondence with the definition of other atomic properties, as one for which the property density at r is determined by the effect of the field over the entire molecule, we express the perturbed density in terms of the first-order corrections to the state function. This is done in a succinct manner by using the concept of a transition density (Longuet-Higgins 1956). The operator whose expectation value yields the total electronic charge density at the position r may be expressed in terms of the Dirac delta function as... 

The potential surrouding each atom in a molecule is not the same as that for the free atom, because electron transfer occurs between atoms in the molecule. This means that atomic orbitals in the molecule are distinct from those in the free atom. Accordingly, it is necessary to use atomic orbitals optimized for each atomic potential in the molecule, as basis functions. In the present methods, the molecular wave functions were expressed as linear combinations of atomic orbitals obtained by numerically solving the Dirac-Slater or Hartree-Fock-Slater equations in the atomic-like potential derived from the spherical average of the molecular charge density around the nuclei [15]. Thus the atomic orbitals used as basis functions were automatically optimized for the molecule and thus the minimum size of the present basis set has enough flexibility to form accurate molecular orbitals. 

In all the calculations for the electronic and geometric structures of the system, the density functional method (6-9) was used. The total energy, E, in the Dirac-Fock-Slater approximation is expressed as a functional of charge density... 

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). 

This is the charge density distribution for the point-like nucleus case (PNC), which we include for completeness and because of the importance of this model as a reference for any work with an extended model of the atomic nucleus (finite nucleus case, FNC). The charge density distribution can be given in terms of the Dirac delta distribution as... 

A uniform distribution of charge over the surface of a sphere of radius R can be represented as charge density distribution in terms of the Dirac delta distribution as follows ... 

The use of extended nuclear charge density distributions, instead of the simple point-like Dirac delta distribution, is almost a standard in present-... 

The MPIB and VIB [35] models attempt to improve the aeeuraey of the earlier models and to overcome some of the difficulties associated with the use of Hartree-Fock wave functions. We have already stated some of the advantages of using the density functional approach to obtain ionic wave-functions they were amply demonstrated by the PIB model which replaced the Hartree-Fock equation with a density functional implementation of the Dirac equation [21]. The MPIB is so called because it also adopts the density functional approach to obtain ionic charge densities (specifically anon-relativistic version derived from the Herman-Skillman [48],but replaces the potential inside the Watson shell with the spherical average of the potential due to the rest of the material, IF (r)[36] ... 

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