Scalar-relativistic calculations

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... 

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. 

We note from Figure 1 that scalar relativistic calculations (7) are entirely unable to reproduce the experimental trend for X = Br and I. Indeed, scalar relativistic and non-relativistic calculations gave almost identical results in this case (7). However, with the inclusion of spin-orbit/Fermi contact operators, the experimental trend is reproduced nicely (9). 

Figure 1. JH absolute shielding in HX, X = F, Cl, Br, I. The figure illustrates the importance of spin-orbit/Fermi contact effects in these systems (9) scalar relativistic calculation (7) are unable to reproduce the experimental trend.

Autschbach and Ziegler presented relativistic spin-spin coupling constants based on the two-component ZORA formulation. They published four papers. In the first paper of their series, only the scalar relativistic part was included, and a full inclusion of the ZORA effects was implemented in the second paper. They used the density functional theory (DFT) approach. The first paper showed that scalar relativistic calculations are able to reproduce major parts of the relativistic effects on the one-bond metal-ligand couplings of systems containing Pt, Hg and Pb. It was found that the... 

If spin-orbit effects are considered in ECP calculations, additional complications for the choice of the valence basis sets arise, especially when the radial shape of the / -f-1/2- and / — 1/2-spinors differs significantly. A noticeable influence of spin-orbit interaction on the radial shape may even be present in medium-heavy elements as 53I, as it is seen from Fig. 21. In many computational schemes the orbitals used in correlated calculations are generated in scalar-relativistic calculations, spin-orbit terms being included at the Cl step [244] or even after the Cl step [245,246]. It therefore appears reasonable to determine also the basis set contraction coefficients in scalar-relativistic calculations. Table 9 probes the performance of such basis sets for the fine structure splitting of the 531 P ground state in Kramers-restricted Hartree-Fock [247] and subsequent MRCI calculations [248-250], which allow the largest flexibility of... 

Table 3. Four-component CCSD(T) EFGs at the halogen center for the HX molecules [162] compared to the scalar relativistic calculations by Kello and Sadlej [163]. All values are given in a.u.

A small spin-orbit Cl using configurations from scalar-relativistic calculations cannot describe the large relaxation from the scalar-relativistic to the spin-orbit coupled orbitals. Methods that use two-component orbitals from the start show much faster convergence of the Cl expansion. 

Spin-polarized LDA all-electron scalar-relativistic calculations were performed... 

After averaging over spin-orbit interaction the following form for one-component, i.e. scalar relativistic, calculations is obtained ... 

The computation times for the evaluation of relativistic Hamiltonians and for one SCF iteration step are presented in Table 14.5. Note that only the nuclear external potential has been considered in the construction of the unitary transformation [cf. Eq. (14.59)]. We see that the computation of the X2C Hamiltonian is slightly faster than that of the BSS Hamiltonian since three additional matrix multiplications are required for the BSS approach, as is evident from Table 14.2. For scalar calculations, the computation time of the X2C Hamiltonian is very close to that of DKH8. The fastest DKH2 approach is about five times faster than the X2C approach (for the setup of the one-electron Hamiltonian). Compared with the computation time of SCF iterations, one Hartree-Fock iteration is about twice as expensive as the X2C transformation. Because several tens of iterations are usually required to obtain converged results, the SCF iterations dominate the total computation time in a Hartree-Fock calculation. The DLU approximation dramatically reduces the computation time. Point-group symmetry can be exploited in scalar-relativistic calculations. As... 

M. Kaupp. Scalar relativistic calculations of hyperfine coupling tensors using the Douglas-KroU-Hess method. Chem. Phys. Lett, 396 (2004) 268-276. 

The all-electron DFT LCAO approach with Gaussian basis sets was extended to scalar-relativistic calculations of periodic systems [561]. The approach is based on a third-order DKH approximation, and similar to the molecular case, requires only a modification of the one-electron Hamiltonian. The effective core Hamiltonian is obtained by applying the DKH transformation to the nuclear-electron potential Vn-Considering that relativistic effects are dominated by the short-range part of the Coulomb interaction, it is proposed to replace the nuclear-electron Coulomb operator used to build the DKH Hamiltonian by a short-range Coulomb operator... 

The benchmark scalar-relativistic calculations [561] were made for the bulk metals (Pd, Ag, Pt and Au) and the large bandgap semiconductors AgF and AgCl. It was shown that scalar-relativistic effects reduce the lattice constant by 0.06-0.10 A for the 4d metals (Pd and Ag), and by 0.14-0.22 A for the 5d metals (Pt and Au). For the 4d metals, scalar-relativistic effects increase the calculated bulk moduli by 20-40 GPa, while for the 5d metals this increase is between 60 GPa and 100 GPa. For both AgF and AgCl crystals scalar-relativistic effects decrease the energy gap - by 1.0 eV for AgF and 0.9 eV for AgCl. 

To be entirely consistent with the one-particle terms, a second transformation should be applied to this operator. However, it has been found that this first transformation of the two-electron operators is about as important in scalar relativistic calculations as the transformations of the one-electron operators up to fifth order (Wolf et al. 2002). The second transformation of the two-electron operators is therefore unlikely to be of great significance. The reason is the strength of the nuclear potential, which is a factor of Z larger than the electron-electron interaction and is attractive rather than repulsive. 

Fig. 3.26 Calculated and experimental lattice constant for actinide nitrides. Solid curve scalar relativistic calculations with spin polarisation. Dashed curve fully relativistic calculations. Dash-and-dot curve scalar relativistic calculations without spin polarisation. Dots experimental lattice constant.

Magnetization can be calculated from first principles by band theory. Calculation methods are usually separated into (i) scalar-relativistic calculations and (ii) full-relativistic calculations. The scalar-relativistic calculations can predict the occurrence of ferromagnetism, the value of the spin moment, and several nonmagnetic properties. The fuU-relativistic calculations are additionally able to determine the magnetocrystalline anisotropy and the induced orbital moment. The most common approach uses the formalism of local density approximation (LDA) to the density functional theory (DFT) [10]. 

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