Relativistic scalar

As expected, Ap vanishes if the strength of the spin-orbit coupling is reduced to 0 by reducing (co/c) or respectively. Both sets "f model calculations give nearly the same results indicating that the so-called scalar relativistic effects due to the mass-velocity and Darwin-term, are of minor importance for the absolute value of Ap. 

For both alloy systems the theoretical results for p obtained in a fully relativistic are found in very satisfying agreement with the corresponding experimental data. In addition to these calculations a second set of calculations has been done making use of the two-current model. This means the partial resistivities p have been calculated by performing scalar relativistic calculations for every spin subsystem separately. As can be seen, the resulting total isotropic resistivity p is reasonably close to the fully relativistic result. Furthermore, one notes that the relative deviation of both sets of theoretical data is more pronounced for Co2,Pdi 2, than for Co2,Pti 2,. This has to be... 

We use s, p, and d partial waves, 16 energy points on a semi circular contour, 135 special k-points in the l/12th section of the 2D Brillouin zone and 13 plane waves for the inter-layer scattering. The atomic wave functions were determined from the scalar relativistic Schrodinger equation, as described by D. D. Koelling and B. N. Harmon in J. Phys. C 10, 3107 (1977). 

Assuming that substituted Sb at the surface may work as catalytic active site as well as W, First-principles density functional theory (DFT) calculations were performed with Becke-Perdew [7, 9] functional to evaluate the binding energy between p-xylene and catalyst. Scalar relativistic effects were treated with the energy-consistent pseudo-potentials for W and Sb. However, the binding strength with p-xylene is much weaker for Sb (0.6 eV) than for W (2.4 eV), as shown in Fig. 4. 

If not otherwise stated the four-component Dirac method was used. The Hartree-Fock (HF) calculations are numerical and contain Breit and QED corrections (self-energy and vacuum polarization). For Au and Rg, the Fock-space coupled cluster (CC) results are taken from Kaldor and co-workers [4, 90], which contains the Breit term in the low-frequency limit. For Cu and Ag, Douglas-Kroll scalar relativistic CCSD(T) results are used from Sadlej and co-workers [6]. Experimental values are from Refs. [91, 92]. 

Ag+ with 1.2 A ) [99]. Spin-orbit coupling is neglected in our analysis because the results shown in Table 4.2 are from scalar relativistic Douglas-Kroll calculations. Because of the additional shell expansion of the 5ds/2 orbital due to spin-orbit coupling, we expect a further increase of the polarizability of Au. Table 4.3 also... 

All calculations are scalar relativistic calculations using the Douglas-Kroll Hamiltonian except for the CC calculations for the neutral atoms Ag and Au, where QCISD(T) within the pseudopotential approach was used [99], CCSD(T) results for Ag and Au are from Sadlej and co-workers, and Cu and Cu from our own work, using an uncontracted (21sl9plld6f4g) basis set for Cu [6,102] and a full active orbital space. 

ForAu(CN)2 the dissociation energy is defined asAu(CN)2 - AuCN + CN. A scalar relativistic ... 

Haberlen, O.D. and Rdsch, N. (1992) A scalar-relativistic extension of the linear combination of Gaussian-type orbitals local density functional method application to AuFl, AuCl and Au2. Chemical Physics Letters, 199, 491-496. 

Suzumura, T., Nakajima, T. and Hirao, K. (1999) Ground-state properties of MH, MCI, and M2 (M—Cu, Ag, and Au) calculated by a scalar relativistic density functional theory International Journal of Quantum Chemistry, 75, lVJ-1. ... 

However, there also exists a third possibility. By using a famous relation due to Dirac, the relativistic effects can be (in a nonunique way) divided into spin-independent and spin-dependent terms. The former are collectively called scalar relativistic effects and the latter are subsumed under the name spin-orbit coupling (SOC). The scalar relativistic effects can be straightforwardly included in the one-electron Hamiltonian operator h. Unless the investigated elements are very heavy, this recovers the major part of the distortion of the orbitals due to relativity. The SOC terms may be treated in a second step by perturbation theory. This is the preferred way of approaching molecular properties and only breaks down in the presence of very heavy elements or near degeneracy of the investigated electronic state. 

Table 5.1 Effect of relativity on Hartree-Eock orbital energies (in eV) for the neutral Hg and Fe atoms. Scalar relativistic effects were treated with the DKH2 approximation...

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%. 

Schreckenbach, G., Ziegler, T., 1997a, Calculation of NMR Shielding Tensors Based on Density Functional Theory and a Scalar Relativistic Pauli-Type Hamiltonian. Application to Transition Metal Complexes , Int. J. 

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. 

The importance of scalar relativistic effects for compounds of transition metals and/or heavy main group elements is well established by now [44], Somewhat surprisingly (at first sight), they may have nontrivial contributions to the TAE of first-row and second-row systems as well, in particular if several polar bonds to a group VI or VII element are involved. For instance, in BF3, S03) and SiF4, scalar relativistic effects reduce TAE by 0.7, 1.2, and 1.9kcal/mol, respectively - quantities which clearly matter even if only chemical accuracy is sought. Likewise, in a benchmark study on the electron affinities of the first-and second-row atoms [45] - where we were able to reproduce the experimental values to... 

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. 

Bauschlicher [48] compared a number of approximate approaches for scalar relativistic effects to Douglas-Kroll quasirelativistic CCSD(T) calculations. He found that the ACPF/MTsmall level of theory faithfully reproduces his more rigorous calculations, while the use of non-size extensive approaches like CISD leads to serious errors. For third-row main group systems, studies by the same author [49] indicate that more rigorous approaches may be in order. 

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