Spin-orbit pseudopotentials

There is a variety of approaches to the development of pseudopotential spin-orbit operators. The spin-orbit potential is mostly expressed in the semi-local form... 

Naveh, D., Kronik, L., Tiago, M.L. and Chelikowsky J.R. (2007) Real-space pseudopotential method for spin-orbit coupling within density functional theory. Physical Review B - Condensed Matter, 76, 153407-1-153407-4. 

Vaara, J., Malkina, O.L., Stoll, H., Malkin, V.G. and Kaupp, M. (2001) Study of relativistic effects on nuclear shieldings using density-functional theory and spin-orbit pseudopotentials. Journal of Chemical Physics, 114, 61-71. 

Table I. Four-fold ionization energy —EfpU) of Si obtained from a rigid core full potential Dirac equation calculation. Arix is the reduction of Epp (j) when the core relaxes. —E(j) is the ionization energy obtained from a pseudopotential calculation and Aps = Epp (j) — E(j). Ago is the spin-orbit splitting obtained from a pseudopotential calculation. —. E(Tt) is the ionization energy obtained from a pseudopotential calculation for parallel spin...

Teichteil et al.41 fit a spin-orbit pseudo-operator such that its action on a pseudo-orbital optimally reproduces the effect of the true spin-orbit operator on the corresponding all-electron orbital. Ermler, Ross, Christiansen, and co-workers42 9 and Titov and Mosyagin50 define a spin-orbit operator as the difference between the and j dependent relativistic effective pseudopotentials (REPs)51... 

We discuss now the choice of the spin orbitals. The spin-orbitals are conceptually more important than the pseudopotential because they provide the nodal structure of the trial function. With the fixed node approximation in RQMC, the projected ground state has the same nodal surfaces of the trial function, while the other details of the trial function are automatically optimized for increasing projection time. It is thus important that the nodes provided by given spin-orbitals be accurate. Moreover, the optimization of nodal parameters (see below) is, in general, more difficult and unstable than for the pseudopotential parameters [6]. 

The simplest form of spin-orbitals for a system with translational invariance are plane waves (PW) 9k r,a) = exp[ fe r]. This form was used in the first QMC study of metallic hydrogen [33]. It is particularly appealing for its simplicity and still qualitatively correct since electron-electron and electron-proton correlations are considered through the pseudopotential . The plane waves orbitals are expected to reasonably describe the nodal structure for metallic atomic hydrogen, but no information about the presence of protons appears in the nodes with PW orbitals. 

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... 

Shape-consistent pseudopotentials including spin-orbit operators based on Dirac-Hartree-Fock AE calculations using the Dirac-Coulomb Hamiltonian have been generated by Christiansen, Ermler and coworkers [161-170]. The potentials and corresponding valence basis sets are also available on the internet under http //www.clarkson.edu/ pac/reps.html. A similar, quite popular set for main group and transition elements based on scalar-relativistic Cowan-Griffin AE calculations was published by Hay and Wadt [171-175]. 

Parameters of energy-consistent ab initio pseudopotentials and corresponding valence basis sets are available for almost all elements of the periodic table [93,94,117,190-192,194-201]. A compilation of parameters for the use within the MOLPRO program system also exists on the internet under the address http //www.theochem.uni-stuttgart.de. Special care has to be taken when spin-orbit coupling is included in calculations with small-core PPs some SO operators are constructed (similar to the large-core case) for a fully variational two-component treatment, whereas in some cases effective valence SO operators are defined. The latter have to be applied in SO-CI calculations for the valence electrons, in which the semi-core shells (outside the PP core) are frozen in their scalar-relativistic form. 

NMR chemical shifts can be calculated for light elements, treated at allelectron level, while the heavier elements in the system are treated using pseudopotentials. Examples exist for H,233 13C,233-238 l70,235 236 239,240 27A1,241 29Si,242 31p 243,244 Ti245 and Nb 245 jn 1 the Spin-Orbit-... 

Other SC RECPs start from different reference data, mostly omitting spin-orbit interactions, but all approaches use, as reference data, the shape of the valence orbitals in the spatial valence region and their corresponding one-particle energies. We omit details concerning energy-adjusted and other types of relativistic pseudopotentials or effective core potentials. At this point, we summarize some characteristics of valence-only methods such as RECPs. 

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-... 

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