Relativistic potential

The orbital distortions due to relativity can be pictured in a qualitative way as follows the relativistic potential appears as a short-range attractive term that, above all, stabilizes s-orbitals and makes them more compact as well as steep close to the nucleus. The same happens to a lesser extent for the p-orbitals. Since the low-lying s- and /7-orbitals are now closer to the nucleus they do shield the nuclear charge better than in the nonrelativistic case. Consequently, d- and /-orbitals are... 

If we subtract this zeroth order solution, fourier transform the x coordinates, convert the time coordinate to conformal time, r), defined by dr) = dt/a, and ignore vector and tensor perturbations (discussed in the lectures by J. Bartlett on polarization at this school), the Liouville operator becomes a first-order partial differential operator for /( (k, p, rj), depending also on the general-relativistic potentials, (I> and T. We further define the temperature fluctuation at a point, 0(jfc, p) = f( lj i lodf 0 1 /<9To) 1 where To is the average temperature and )i = cos 6 in the polar coordinates for wavevector k. 

Av represents the percentage deviation of the selfconsistent relativistic potential I x ([ ] ) from the corresponding selfconsistent nonrelativistic potential yJ0 ([n ] r). The selfconsistent Dx([n ] f) is calculated by insertion of the self-consistent n (r) into the functional derivative (3.28) for that [n] which has been used to determine n (r). In particular, the ROPM x-only potential can, in principle, be obtained by insertion of the exact x-only density "(r) into the exact Dx([n] r) = r J ([ ] r) and thus can be used as a comparative standard ... 

The integral equations for relativistic potential scattering are conveniently written in terms of a four-dimensional notation for the four-component spinor vp). 

This figure shows the orbital moments as calculated from Equation (5.37) with tabulated spin-orbit parameters (Mackintosh and Andersen 1980) and the LDOS obtained from a scalar relativistic treatment. These data are compared with the results of a nonself-consistent relativistic calculation, i.e. the Dirac equation has been solved using converged scalar relativistic potentials as input. Obviously, the trends are very well described by the above model. The self-consistent fully relativistic calculation... 

The method works as follows. The mass velocity, Darwin and spin-orbit coupling operators are applied as a perturbation on the non-relativistic molecular wave-functions. The redistribution of charge is then used to compute revised Coulomb and exchange potentials. The corrections to the non-relativistic potentials are then included as part of the relativistic perturbation. This correction is split into a core correction, and a valence electron correction. The former is taken from atomic calculations, and a frozen core approximation is applied, while the latter is determined self-consistently. In this way the valence electrons are subject to the direct influence of the relativistic Hamiltonian and the indirect effects arising from the potential correction terms, which of course mainly arise from the core contraction. 

The details of implementation of scalar relativity in GTOFF were presented in [41] and reviewed in [75], so we summarize the essential assumptions and methodological features here. First, all practical DFT implementations of relativistic corrections of which we are aware assume the validity (either explicitly or implicitly) of an underlying Dirac-Kohn-Sham four-component equation. We do also. The Hamiltonian is therefore a relativistic free particle Hamiltonian augmented by the usual non-relativistic potentials... 

The relativistic Hamiltonian may be defined by adding Hso to Hel. The eigenfunctions of this new Hamiltonian are the relativistic wavefunctions, i,n, which define the relativistic potential curves... 

Le Roy, et al, (2002) have reviewed all of the different types of experimental observations and theoretical calculations for HI. By an empirical analysis, they have shown that, because in HI the spin-orbit interaction is especially important, the adiabatic relativistic potential curves can explain all of the experimental data without introducing residual nonadiabatic coupling. For the lighter halogen hydrides, the J = 1/2 J = 3/2 branching ratio can be obtained from the solution of inhomogeneous coupled equations with a source term representing the initial vibrational wavefunction multiplied by the electronic transition moment (Band, et al., 1981). These calculations are based on adiabatic electronic (or diabatic relativistic) potential curves (see, for example, for HC1, Alexander, et al., 1993 and for HBr, Peoux, et al., 1997). 

The spin-orbit operator derived this way can be ab initio if it is derived from relativistic ab initio potentials. It can be introduced in molecular calculations. Pacios and Christiansen have published Gaussian analytic fits of averaged relativistic effective potentials and spin-orbit operators for Li through Ar. The relativistic potentials of other elements are also being tabulated. ... 

Since V has the same decoupled block-diagonal structure as in Eq. (14.14), only the scalar term K VK is needed to construct V. For the special relativistic potential matrix W, four real matrices,... 

The basis-set representation of the antihermitean matrix operators Wjt is denoted as Wfc. To keep the convention for the antihermitean matrix consistent with chapter 12 and with the research literature, the letter W is used although it could be confused with the relativistic potential-energy integral defined in Eq. (14.17). 

If the scalar approximation (neglecting spin-orbit-coupling terms) is activated, the computational demands turn out to be rather different. The operation count for the scalar-relativistic variants is also given in Table 14.2. Most important, only real matrix operations are required since one can employ real basis functions and the Hamiltonian operators are also real. Then, spin is a good quantum number and the spin symmetry can be exploited so that dimensions of all two- and four-component matrix operators are reduced by half. Finally, since the spin-orbit components of the relativistic potential matrix, i.e., W, Wy, and W, are neglected, the number of matrix multiplications required for the orthonormal basis transformation is decreased from ten to four. 

While the non-relativistic potential remains the most successful and simplest approach for calculating and predicting energy levels mid decay rates, other, more sophisticated, methods have been devised including bag models, QCD sum rules (Shifman et al., 1978 Reinders et al., 1985) and lattice calculations (Rebbi, 1983 Creutz, 1983 Creutz et al., 1983). The properties of quarkonia test some aspects of both perturbative mid non-perturbative QCD the heavier the quarkonium system, the less important are both relativistic effects and higher order perturbative QCD corrections. For a more recent approach to the problem of heavy flavours, see Isgur and Wise (1989, 1990) and for a comprehensive review, Neubert (1992). 

In this section we consider cc and bb bound states treated within the non-relativistic potential picture. Light quarks are excluded from our present considerations. 

In section 3.8 we discussed the contribution of the target nudeon spin-ffip amplitudes to the second-order NR optical potential. Analogous terms exist for the relativistic potential. Calculations of these quantities based on the PV form of the RIA amplitude or the IA2 interaction remain to be done. 

L, A, Lajohn, P. A. Christiansen, R. B. Ross, T. Atashroo, and W. C. Ermler,/. Chem. Phys., 87, 2812 (1987). Ab Initio Relativistic Potentials with Spin-Orbit Operators. III. Rb Through Xe. 

From this we can deduce the relativistic potential set up by a charge q moving at velocity u. We let S be the observer frame, and place the x axis in the direction of u. In a charge-centered frame S, the potential is the same as from a stationary charge,... 

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