Scalar Relativistic Scheme

Eq. 20 can be solved iteratively and yields the same one-particle energies as the corresponding Dirac-equation. The radial functions P K r) correspond to the large components. In the many-electron case the correct nonlocal Hartree-Fock potential is used in Eq. 21, but a local approximation to it in Eqs. 22. Averaging over the relativistic quantum number k leads to a scalar-relativistic scheme. 

Other methods of formulation a scalar relativistic scheme based on a differential equation where SO coupling is eliminated have been developed. [74,75]... 

For basis sets derived from a cellular decomposition of space, e.g. the muflSn-tin decomposition used in LAPW and LMTO methods, the difficulty can be sidestepped by constructing the basis in terms of fully relativistic atomic-like functions determined by the miiflfin-tin potential. By restricting the selection to the electron states, the desired decomposition is automatically enforced. This is the essence of the Koelling-Harmon scalar relativistic scheme widely used in cellular-based modified plane-wave basis methods [45]. 

In systems with heavier elements, relativistic effects must be included. In the medium range of atomic numbers (up to about 54) the so called scalar relativistic scheme is often used [21], It describes the main contraction or expansion of various orbitals (due to the Darwin s-shift or the mass-velocity term), but omits spin-orbit interaction. The latter becomes important for the heavy elements or when orbital magnetism plays a significant role. In the present version of WIEN2k the core states always are treated fully relativistically by numerically solving the radial Dirac equation. For all other states, the scalar relativistic approximation is used by default, but spin-orbit interaction (computed in a second-variational treatment [22]) can be included if needed [23]. 

Relativistic effects have to be taken into account for compounds containing transition elements with higher atomic numbers the 5d transition elements (Hf, Ta, W) are of particular concern in the present review. A fiiUy relativistic treatment requires the solution of the Dirac equation instead of the Schrodinger equation. However, in many cases, it is sufficient to use a scalar relativistic scheme (48) as an approximation. In this technique, the mass-velocity term and the Darwin 5-shift are considered. The spin-orbit splitting, however, is neglected. In this approximation a different procedure must be used to calculate the radial wave functions, but the nonrelativistic formalism, which is computationally much simpler than solving Dirac s equation, is retained. 

A different approach is chosen when the screening of nuclear potential due to the electrons is incorporated in /z . Transformation to the eigenspinor basis is then only possible after the DHF equation is solved which makes it more difficult to isolate the spin-orbit coupling parts of the Hamiltonian. Still, it is also in this case possible to define a scalar relativistic formalism if the so-called restricted kinetic balance scheme is used to relate the upper and lower component expansion sets. The modified Dirac formalism of Dyall [24] formalizes this procedure and makes it possible to identify and eliminate the spin-orbit coupling terms in the selfconsistent field calculations. The resulting 4-spinors remain complex functions, but the matrix elements of the DCB Hamiltonian exhibit the non-relativistic symmetry and algebra. 

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... 

Pseudopotentials have been the subject of considerable attention in the last two decades, and they have been developed by a number of different groups. They are also the most widely used effective core potentials in chemical applications either for the study of chemical reactions or spectroscopy. A large variety of pseudopotentials are now available, and all the coupling schemes at the SCF step have been implemented four-component, two-component, and scalar relativistic along with spin-orbit pseudopotentials. However, it is well known that four-component calculations can (in the worst cases) be 64 times more expensive than in the non relativistic case. In addition, the small component of the Dirac wave function has little density in the valence region, and pseudopoten-... 

Another pertinent question is related to the accuracy of the common approximation to describe relativistic effects at the pseudopotential level. Our AE scalar relativistic DKH scheme allows to evaluate the precision of the latter scheme. A relativistic pseudopotential [196] was utilized to treat the heavy element Pd in the Pd-0 complexes employing extended EPE-embedded cluster models of the quality comparable to that for the AE cluster model. This resulted in the adsorption energy value 123 kJ/mol and the Pd-0 bond length 213 pm. For the Pd-0 complexes under scrutiny the deviations from the corresponding scalar relativistic values, by 3 kJ/mol and 2 pm respectively, are rather small. Clearly, relativistic pseudopotentials for heavier atoms have to be constructed with due care [8]. The AE scalar relativistic DKH approach certainly provides an attractive alternative. 

The field of relativistic electronic structure theory is generally not part of theoretical chemistry education, and is therefore not covered in most quantum chemistry textbooks. This is due to the fact that only in the last two decades have we learned about the importance of relativistic effects in the chemistry of heavy and super-heavy elements. Developments in computer hardware together with sophisticated computer algorithms make it now possible to perform four-component relativistic calculations for larger molecules. Two-component and scalar all-electron relativistic schemes are also becoming part of standard ab-initio and density functional program packages for molecules and the solid state. The second volume of this two-part book series is therefore devoted to applications in this area of quantum chemistry and physics of atoms, molecules and the solid state. Part 1 was devoted to fundamental aspects of relativistic electronic structure theory. Both books are in honour of Pekka Pyykko on his 60 birthday - one of the pioneers in the area of relativistic quantum chemistry. 

Tables 6.1 through 6.4 summarize the spectroscopic parameters for HF, HCl, HBr, and HI molecules calculated at CASSCF and SF-SSMRPT2 levels using the non-relativistic and SFX2c-le schemes. Dynamical correlation effects are demonstrated as the differences between CASSCF and SSMRPT2 results, while scalar relativistic effects can be obtained by taking differences between SFX2c-le and nonrelativistic computations.

Reference [223] Cl scheme MT scalar-relativistic 13e pseudopotentials with s, p, d radii given in the paper spin-polarized LDA and B-LYP GC plane-wave cutoff of 60 Ry Hockney s boundary conditions. The effect of a 5e-pseudopotential was studied, with and without nonlinear core corrections. 

Reference [226] C2 scheme with scalar-relativistic 1 le s, p, d pseudopotentials (given in the paper) also given are results with le pseudopotentials from [266] for the sake of comparison LDA. 

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