Dirac electron density

By the functional derivative respecting the Fock-Dirac electron density one gets ... 

In many materials, the relaxations between the layers oscillate. For example, if the first-to-second layer spacing is reduced by a few percent, the second-to-third layer spacing would be increased, but by a smaller amount, as illustrated in figure Al,7,31b). These oscillatory relaxations have been measured with FEED [4, 5] and ion scattering [6, 7] to extend to at least the fifth atomic layer into the material. The oscillatory nature of the relaxations results from oscillations in the electron density perpendicular to the surface, which are called Eriedel oscillations [8]. The Eriedel oscillations arise from Eenni-Dirac statistics and impart oscillatory forces to the ion cores. 

Similarides Between Potential Ruid Dynamics and Quantum Mechanics Electrons in the Dirac Theory The Nearly Nonrelativistic Limit The Lagrangean-Density Correction Term Topological Phase for Dirac Electrons What Have We Learned About Spinor Phases ... 

Density functional theory-based methods ultimately derive from quantum mechanics research from the 1920 s, especially the Thomas-Fermi-Dirac model, and from Slater s fundamental work in quantum chemistry in the 1950 s. The DFT approach is based upon a strategy of modeling electron correlation via general functionals of the electron density. 

This result was rediscovered by Slater (1951) with a slightly different numerical coefficient of C. Authors often refer to a term Vx which is proportional to the one-third power of the electron density as a Slater-Dirac exchange potential. 

Just to remind you, the electron density and therefore the exchange potential are both scalar fields they vary depending on the position in space r. We often refer to models that make use of such exchange potentials as local density models. The disagreement between Slater s and Dirac s numerical coefficients was quickly resolved, and authors began to write the exchange potential as... 

With a = 2/3 this is identical to the Dirac expression. The original method used a = 1, but a value of 3/4 has been shown to give better agreement for atomic and molecular systems. The name Slater is often used as a synonym for the L(S)DA exchange energy involving die electron density raised to the 4/3 power (1/3 power for the energy density). 

When wave mechanical calculations are made according to the Schrodinger equation, the probability of finding the electron in a node is zero, but this treatment ignores relativistic considerations. When such considerations are applied, Dirac has shown that nodes do have a very small electron density Powell, R.E. J. Chem. Educ., 1968,45,558. See also Ellison, F.O. and Hollingsworth, C.A. J. Chem. Educ., 1976, 53, 767 McKelvey, D.R. J. Chem. Educ., 1983, 60, 112 Nelson, P.G. J. Chem. Educ., 1990, 67, 643. For a review of relativistic effects on chemical structures in general, see Pyykko, P. Chem. Rev., 1988, 88, 563. 

Fig. 7.57 Plot of the isomer shift 5 of the 36.2 keV Mdssbauer transition of Os versus Dirac-Fock values for the electron density differences at the Os nuclei in free ion 5d configurations. The numbers of the data points refer to the numbering of the compounds in Table 7.9 (from [258])...

Fig. 7.78 Linear relation of the quadmpole splitting A q = ( jl)eqQ (1 + j /3)l/2 and the isomer shift b for aurous (a) and auric (b) compounds. Also included is a correlation with the relative change in electron density at the gold nucleus, Ali/r(o)P, as derived from Dirac-Fock atomic structure calculations for several electron configurations of gold. An approximate scale of the EFG (in the principal axes system) is given on the right-hand ordinate (from [341])...

The expressions (4.22)-(4.23) found in chap. 4 for the isomer shift 5 in nonrelativ-istic form may be applied to lighter elements up to iron without causing too much of an error. In heavier elements, however, the wave function j/ is subject to considerable modification by relativistic effects, particularly near the nucleus (remember that the spin-orbit coupling coefficient increases with Z ). Therefore, the electron density at the nucleus l /(o)P will be modified as well and the aforementioned equations for the isomer shift require relativistic correction. This has been considered [1] in a somewhat restricted approach by using Dirac wave functions and first-order perturbation theory in this approximation the relativistic correction simply consists of a dimensionless factor S (Z), which is introduced in the above equations for S,... 

Dirac electrons, 266-268 topological phase, 270-272 Lagrangean-density correction term, 269-... 

With the purpose of evaluate not only the energy but also the electron density itself, Ashby and Holzman [15] performed calculations in which the relativistic TF density was replaced at short distancies from the nucleus from the one obtained for the 1 s Dirac orbital for an hydrogenic atom, matched continuously to the semiclassical density at a switching radius rg where the kinetic energy density of both descriptions also match. 

In these expressions, e and N refer to electron and nucleus, respectively, Lg is the orbital angular moment operator, rg is the distance between the electron and nnclens. In and Sg are the corresponding spins, and reN) is the Dirac delta fnnction (eqnal to 1 at rgN = 0 and 0 otherwise). The other constants are well known in NMR. It is worth mentioning that eqs. 3.8 and 3.9 show the interaction of nnclear spins with orbital and dipole electron moments. It is important that they not reqnire the presence of electron density directly on the nuclei, in contrast to Fermi contact interaction, where it is necessary. 

After the discovery of the relativistic wave equation for the electron by Dirac in 1928, it seems that all the problems in condensed-matter physics become a matter of mathematics. However, the theoretical calculations for surfaces were not practical until the discovery of the density-functional formalism by Hohenberg and Kohn (1964). Although it is already simpler than the Hartree-Fock formalism, the form of the exchange and correlation interactions in it is still too complicated for practical problems. Kohn and Sham (1965) then proposed the local density approximation, which assumes that the exchange and correlation interaction at a point is a universal function of the total electron density at the same point, and uses a semiempirical analytical formula to represent such universal interactions. The resulting equations, the Kohn-Sham equations, are much easier to handle, especially by using modern computers. This method has been the standard approach for first-principles calculations for solid surfaces. 

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