Relativistic Contributions

Up to this point, we have considered the nonrelativistic Schrodinger equation. However, to calculate AEs to an accuracy of a few kJ/mol, it is necessary to account for relativistic effects, even for molecules containing only hydrogen and first-row atoms. Fortunately, the major relativistic contributions to the AEs of such molecules - the mass-velocity (MV), one-electron Darwin (ID), and first-order spin-orbit (SO) terms - are easily obtained [58]. 

Whereas the SO corrections are accurately known from atomic measurements, the MV and ID corrections must be calculated as the expectation values of the operators 

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The experimentally observed isomer shift, (5exp, includes a relativistic contribution, which is called second-order Doppler shift, sod> and which adds to the genuine isomer shift d. 

The electron density i/ (0)p at the nucleus primarily originates from the ability of s-electrons to penetrate the nucleus. The core-shell Is and 2s electrons make by far the major contributions. Valence orbitals of p-, d-, or/-character, in contrast, have nodes at r = 0 and cannot contribute to iA(0)p except for minor relativistic contributions of p-electrons. Nevertheless, the isomer shift is found to depend on various chemical parameters, of which the oxidation state as given by the number of valence electrons in p-, or d-, or /-orbitals of the Mossbauer atom is most important. In general, the effect is explained by the contraction of inner 5-orbitals due to shielding of the nuclear potential by the electron charge in the valence shell. In addition to this indirect effect, a direct contribution to the isomer shift arises from valence 5-orbitals due to their participation in the formation of molecular orbitals (MOs). It will be shown in Chap. 5 that the latter issue plays a decisive role. In the following section, an overview of experimental observations will be presented. 

In fact, for 5d transition-metals relativistic contributions, and in particular spin-orbit coupling, can be of the same order of magnitude as chemical bonding. 

Table 1.10 Harmonic and anharmonic ZPVE contributions, and first-order relativistic contributions to AEs (kJ/mol).

In Table 1.11, the AEs are listed for twenty small molecules. The AEs are obtained by adding vibrational and relativistic corrections to the nonrelativistic CCSD(T)/cc-pcV(56)Z equilibrium AEs. The ZPVEs have been taken from the compilation of Helgaker, Jprgensen, and Olsen [12] the relativistic contributions contain the MV and ID scalar corrections calculated at the CCSD(T)/cc-pCVQZ level, in addition to first-order SO corrections from atomic measurements [9], Table 1.11 also contains experimental AEs. 

The scalar relativistic contribution is computed as the first-order Darwin and mass-velocity corrections from the ACPF/MTsmall wave function, including inner-shell correlation. 

Perhaps the simplest and most cost-effective way of treating relativistic contributions in an all-electron framework is the first-order perturbation theory of the one-electron Darwin and mass-velocity operators [46, 47]. For variational wavefunctions, these contributions can be evaluated very efficiently as expectation values of one-electron operators. 

It has been found repeatedly [1, 43, 45] that scalar relativistic contributions are overestimated by about 20 - 25 % in absolute value at the SCF level. Hence inclusion of electron correlation is essential we found the ACPF method (which is both variational and approximately size extensive) to be an excellent compromise between quality and cost. It is reasonable to suppose that for a property that becomes more important as one approaches the nucleus, one wants maximum flexibility of the wavefunction near the nucleus as well as correlation of all electrons thus we finally opted for ACPF/MTsmall as our approach of choice. Typically the cost of the scalar relativistic step is a fairly small fraction of that of the core correlation step, since only n2N4 scaling is involved in the ACPF calculations. 

The model (which requires essentially no CPU time) was found to work very satisfactorily its performance for the W2-1 set can be seen in Table 2.3. Somewhat to our surprise, we found that the same model performs reasonably well when applied to the scalar relativistic contributions, albeit with larger individual deviations. 

On the other hand, high-level computational methods are limited, for obvious reasons, to very simple systems.122 Calculations are likely to have limited accuracy due to basis set effects, relativistic contributions, and spin orbit corrections, especially in the case of tin hydrides, but these concerns can be addressed. Given the computational economy of density functional theories and the excellent behavior of the hybrid-DFT B3LYP123 already demonstrated for calculations of radical energies,124 we anticipate good progress in the theoretical approach. We hope that this collection serves as a reference for computational work that we are certain will be forthcoming. 

The high quality of the CC3 model and the convergence in one-electron space have allowed us to determine the Cauchy moments for Ne which have the accuracy comparable to the experimental results. To arrive at the similar conclusion for the Ar and Kr atoms a more detailed investigation of the convergence in one-electron and A-electron space is required. The relativistic contribution to the Cauchy moments may also need to be taken into account for these two atoms. 

As the MSFT (38) additivity model predicts only very weak core correlation and scalar relativistic contributions to the proton affinity, we have not attempted their explicit (and very expensive) calculation. 

Table II compares these empirical estimates with those obtained from the DSW calculation. Relativistic contributions have little effect on the spin-dipolar interactions, and both calculations are in reasonably good agreement with the empirical estimates. The spin-orbit contributions are also in moderately good agreement with the empirical estimates, showing that electron currents about the -axis are considerably more important than those about axes in the plane of the ligand. Indeed, in view of the approximations that enter into Equation 7, (, ), one might have as much confidence in the DSW result as in the empirical estimate given in the final column.

The recoil correction in (4.19) is the leading order (Za) relativistic contribution to the energy levels generated by the Braun formula. All other contributions to the energy levels produced by the remaining terms in the Braun formula start at least with the term of order (Za) [17]. The expression in (4.19) exactly reproduces all contributions linear in the mass ratio in (3.5). This is just what should be expected since it is exactly Coulomb and Breit potentials which were taken in account in the construction of the effective Dirac equation which produced (3.5). The exact mass dependence of the terms of order Za) m/M)m and Za) m/M)m is contained in (3.5), and, hence,... 

In the atomic context the need for relativistic corrections to Exc[n] is obvious and has led to the development of the relativistic LDA (RLDA) [5,6,24]. On the basis of RLDA calculations for metallic Au and Pt, MacDonald et al. [25,26] have concluded that in solids relativistic contributions to Exc[n] can produce small but significant modifications of measurable quantities, as eg. the Fermi surface area. On the other hand, it has been shown [7] that the RLDA suffers from several shortcomings, eg. from a drastic overestimation of transverse exchange contributions, thus making the RLDA a less reliable tool than its nonrelativistic counterpart. As relativistic corrections are clearly misrepresented by the RLDA, it seems worthwhile to reinvestigate the role of relativistic arc-effects in solids on the basis of a more accurate form for Exc[n. ... 

In summary, there can be little doubt that for the cohesive properties of solids the relativistic contributions to Exc[n are much less important than the nonlocal contributions to Jxc[n] not contained in the LDA. It should be pointed out, however, that the RGGA is as efficiently applied as the GGA, so that there seems to be no reason to rely on error cancellation in the calculation of E h Moreover, at least for some systems (as Au) the relativistic rc-corrections are visible in the band structure, indicating that a complete description of these systems requires a relativistic form for Exc[n], Finally, if relativistic ax-corrections are to be included at all this should be done on the GGA- rather than the LDA-level. 

By comparison of one quarter of the IS1 — 25 transition frequency with the 25 — 45 and 25 — 4D transition frequency, the main energy contributions described by the simple Rydberg formula are eliminated. The remaining difference frequency (about 5 GHz) is determined by well known relativistic contributions, the hyperfine interaction, and a combination of Lamb shifts. Since quantum electrodynamic contributions scale roughly as 1/n3 with the principal quantum number, the Lamb shift of the 15 level is the largest. 

Table 8. Details of QED and higher-order relativistic contributions to the ionization energies of helium. Units axe MHz...

Precise measurements on g factors of electrons bound in atomic Hydrogen and the Helium ion 4He+ were carried out by Robinson and coworkers. The accuracies of 3 x 10-8 for the Hydrogen atom [5] and of 6 x 10-7 for the Helium ion [6] were sensitive to relativistic effects. Other measurements of the magnetic moment of the electron in Hydrogen-like ions were performed at GSI by Seelig et al. for Lead (207Pb81+) [7] and by Winter et al. for Bismuth (209Bi82+) [8] with precisions of about 10-3 via lifetime measurements of hyperfine transitions. These measurements were also only sensitive to the relativistic contributions. 

In the case of normal hydrogen [14] the difference is mainly determined by a relativistic contribution of order (Za)2Ep (so-called Breit term [15]). In muonic atoms the leading effect is due to vacuum polarization and it is of order aEp. 

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