Scalar relativistic correction

A comparison of anisotropic Fe HFCs with the experimental results shows good agreement between theory and experiment for the ferryl complexes and reasonable agreement for ferrous and ferric complexes. Inspection reveals that the ZORA corrections are mostly small ( 0.1 MHz) but can approach 2 MHz and improve the agreement with the experiment. The SOC contributions are distinctly larger than the scalar-relativistic corrections for the majority of the investigated iron complexes. They can easily exceed 20%. 

Reduced-Cost Approaches to the Scalar Relativistic Correction... 

In systems where a large number of inner-shell electrons makes the inner-shell correlation (and, to a lesser extent, scalar relativistic) steps in W1 and W2 theory unfeasible, the use of a bond equivalent model for the inner-shell correlation and scaled B3LYP/cc-pVTZuc+l scalar relativistic corrections offers an alternative under the name of Wlc and Wlch theories. 

All-electron DFT calculations were performed using the DMOL [24] code. These incorporated scalar relativistic corrections and employed the non-local exchange and correlation functional Perdew-Wang91 [25] denoted GGA in the rest of the paper, which is generally found to be superior to the local density approximation (EDA)... 

Scalar relativistic corrections, 209 Spin-spin interaction, 211 Turnover rule, 124 Woodward-Hoffmann forbidden and allowed... 

Spin-orbit (SO) coupling corrections were calculated for the Pt atom since the relativistic effects are essential for species containing heavy elements. Other scalar relativistic corrections like the Darwin and mass-velocity terms are supposed to be implicitly included in (quasi)relativistic pseudopotentials because they mostly affect the core region of the considered heavy element. Their secondary influence can be seen in the contraction of the outer s-orbitals and the expansion of the d-orbitals. This is considered in the construction of the pseudoorbitals. The effective SO operator can be written within pseudopotential (PS) treatment in the form71 75... 

We have already discussed in chapters 12 and 13 that low-order scalar-relativistic operators such as DKH2 or ZORA provide very efficient variational schemes, which comprise all effects for which the (non-variational) Pauli Hamiltonian could account for (as is clear from the derivations in chapters 11 and 13). It is for this reason that historically important scalar relativistic corrections which can only be considered perturbatively (such as the mass-velocity and Darwin terms in the Pauli approximation in section 13.1), are no longer needed and their significance fades away. There is also no further need to develop new pseudo-relativistic one- and two-electron operators. This is very beneficial in view of the desired comparability of computational studies. In other words, if there were very many pseudo-relativistic Hamiltonians available, computational studies with different operators of this sort on similar molecular systems would hardly be comparable. 

Nonetheless, care needs to be exercised in calculations of the one-electron scalar relativistic correction by perturbation theory using these operators. It has been found that the magnitude of the correction is quite sensitive to the contraction of the basis set. This follows from the fact that the operators weight the region near the nucleus. If a basis function is taken out of the contraction the nonrelativistic wave funetion may not change very much, but a small change in the coefficient of the eore function can have a big effect on the relativistic correction. 

This is the correction to the Hartree-Fock energy. Obviously, only the terms that are diagonal in the spin and are spatially totally symmetric will contribute. These terms are the scalar relativistic corrections the mass-velocity term, the one- and two-electron Darwin terms, and the orbit-orbit term. Other terms such as the spin-spin term and the z component of the spin-orbit interaction contribute for open-shell systems, where the spin is nonzero. 

Third, since this term mixes relativity and correlation, a nonzero value means that relativity and correlation are not additive. If the term is small, the degree of nonadditivity will be small, and it may be negligible for the purposes of the calculation in hand. The fact that the lowest cross-term is zero for the one-electron scalar relativistic corrections implies that at least for light elements the neglect of the cross-term is reasonably well justified. 

Apart from the expectation of the relativistic correction to the property operator, we get a mixed expression, like the mixed relativistic-correlation correction above. The scalar relativistic correction to the property operator is zero for external electric fields but nonzero othemise, as we saw in chapter 16. If the property is a first-order property, that is, I Wool I does not vanish, the mixed correction will also be nonzero... 

Scalar relativistic corrections at the levels of Douglas-Kroll, ZORA, and the infinite-order relativistic approximation (lORA) (Dyall and van Lenthe 1999). 

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