Scalar relativistic pseudo-potential

In subsection 3.1, we will present GGA and LDA calculations for Au clusters with 6first principles method outlined in section 2, which employs the same scalar-relativistic pseudo-potential for LDA and GGA (see Fig 1). These calculations show the crucial relevance of the level of density functional theory (DFT), namely the quality of the exchange-correlation functional, to predict the correct structures of Au clusters. Another, even more critical, example is presented in subsection 3.2, where we show that both approaches, LDA and GGA, predict the cage-like tetrahedral structure of Au2o as having lower energy than amorphous-like isomers, whereas for other Au clusters, namely Auig, Au ... 

By analyzing the density matrix composition of planar and 3D structures of seven atom clusters (II and IV of Fig 1), calculated using scalar relativistic pseudo-potential at the GGA theory level, Fernandez and coworkers conclude that the planarity of An clusters is driven by the hybridization of the half-filled 6s orbital with the fully occupied 5d 2 orbital, which is favored by relativistic effects. Thus, the three valence electrons in the orbitals 6s and 5d 2, form a sticky-waist cylinder , where the cylinder is due to the almost filled s + d 2 hybrid, and the sticky-waist is due to the nearly half-filled s — d 2 hybrid orbital. 

Figure 2. Left equilibrium geometries of the two lowest energy isomeric states of Au clusters obtained using LDA or GGA scalar relativistic pseudo-potentials. The ground state is Au for GGA and Auj for LDA (except for n=6, which LDA structure is also Aue). Right difference in the binding energy per atom of the planar and 3D structures given in the left panel for neutral gold clusters with 6

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. 

In this work we recalculate the structures of Au clusters with 6scalar relativistic Troullier-Martins pseudo-potentials , respectively, and within the SIESTA code" . In Fig 2 we present our results for the structures and relative binding energies. We see that GGA leads to planar structures whereas LDA favors 3D structures for n>7 clusters. Thus, in addition to relativistic effects, the observed planarity of Au clusters is accounted for using only the GGA level of theory. 

The scalar-relativistic effects can be easily absorbed into the effective potential by taking the all-electron (AE) calculation results of the same order of relativistic approximation as the references to parametrize the potentials. Taking the two-component (or even four-component) form of the pseudo-valence orbitals, the spin-orbit coupling effect can also be absorbed into the ECR Because the pseudovalence orbitals are energetically the lowest-eigenvalue eigenvectors of the Fock... 

In the MCP, or more advanced AIMP, approximations [72, 73], is represented by an adjustable local potential and a projection operator. This potential is constructed so that the inner nodal stmcture of the pseudo-valence orbitals is conserved, thus closely approximating all-electron valence AOs. Scalar relativistic effects are directly taken into account by relativistic operators such as Douglas-Kroll (DK) one. SO effects can be included with the use of the SO operator,... 

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