Relativistic effective pseudo-potentials

The relativistic effects (Rl) and (R2) can be simulated by adjusting the sizes of basis functions used in a standard variational treatment. This adjustment is usually combined with an effective-core-potential [ECP] approximation in which inner-shell electrons are replaced by an effective [pseudo] potential of chosen radius. The calculations of this chapter were carried out with such ECP basis sets in order to achieve approximate incorporation of the leading relativistic effects.)... 

At B3LYP/6-311G(2d,p), pseudo-relativistic effective core potential and a (31/31/1) valence basis set were used for Si, Ge, Sn, Pb, from Ref 40. 

Most common among the approximate spin-orbit Hamiltonians are those derived from relativistic effective core potentials (RECPs).35-38 Spin-orbit coupling operators for pseudo-potentials were developed in the 1970s.39 40 In the meantime, different schools have devised different procedures for tailoring such operators. All these procedures to parameterize the spin-orbit interaction for pseudo-potentials have one thing in common The predominant action of the spin-orbit operator has to be transferred from... 

In the calculations based on effective potentials the core electrons are replaced by an effective potential that is fitted to the solution of atomic relativistic calculations and only valence electrons are explicitly handled in the quantum chemical calculation. This approach is in line with the chemist s view that mainly valence electrons of an element determine its chemical behaviour. Several libraries of relativistic Effective Core Potentials (ECP) using the frozen-core approximation with associated optimised valence basis sets are available nowadays to perform efficient electronic structure calculations on large molecular systems. Among them the pseudo-potential methods [13-20] handling valence node less pseudo-orbitals and the model potentials such as AIMP (ab initio Model Potential) [21-24] dealing with node-showing valence orbitals are very popular for transition metal calculations. This economical method is very efficient for the study of electronic spectroscopy in transition metal complexes [25, 26], especially in third-row transition metal complexes. 

An ab initio effective core potential method derived from the relativistic all-electron Dirac-Fock solution of the atom, which we call the relativistic effective core potential (RECP) method, has been widely used by several investigators to study the electronic structure of polyatomics including the lanthanide- and actinide-containing molecules. This RECP method was formulated by Christiansen et al. (1979). It differs from the conventional Phillips-Kleinman method in the representation of the nodeless pseudo-orbital in the inner region. The one-electron valence equation in an effective potential of the core electron can be written as... 

Desclaux has developed a numerical Dirac-Fock code for atoms which can be used to obtain relativistic numerical allelectron four-component spinor wavefunctions for any atom in the periodic table. The relativistic four-component wave-functions for all the atomic orbitals could then be used for the construction of pseudo-orbitals and relativistic effective core potentials. The resulting relativistic potentials would also have four component spinor forms. 

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. 

Hafner, P. and Schwarz, W.H.E. (1978) Pseudo-potential approach including relativistic effects. Journal of Physics B, 11, 217-233. 

The rotational barriers obtained from pseudo-potential and all-electron calculations generally agree within ca 0.15 kcal mol-1, even when X, Y = Pb. Hence, relativistic effects do not appear to influence the barriers. 

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. 

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. 

In order to overcome these problems, the core electrons are often excluded from the calculation (frozen-core approximation), and their effect on the valence electrons is parameterized in the form of a pseudo potential based on a relativistic atomic calculation [12]. In connection with GTO basis sets, the most common form of pseudo potential is the effective core potential (ECP) using Gaussian-type radial functions to describe the potential [13-16]. 

The hyperpolarizability of tin derivatives can alternatively be computed within the framework of the density functional theory (DFT) approach (e.g. at the B3PW91/6-31+G /LANL2DZ(Sn) level), using the time-consuming finite field procedure. " The use of a pseudo-potential is required to allow the description of relativistic effects for tin. In this approach, p is obtained as the numerical partial derivative of the energy (W) with respect to the electric field (E), evaluated at zero field, according to the following equation ... 

A further reduction of the computational effort in investigations of electronic structure can be achieved by the restriction of the actual quantum chemical calculations to the valence electron system and the implicit inclusion of the influence of the chemically inert atomic cores by means of suitable parametrized effective (core) potentials (ECPs) and, if necessary, effective core polarization potentials (CPPs). Initiated by the pioneering work of Hellmann and Gombas around 1935, the ECP approach developed into two successful branches, i.e. the model potential (MP) and the pseudopotential (PP) techniques. Whereas the former method attempts to maintain the correct radial nodal structure of the atomic valence orbitals, the latter is formally based on the so-called pseudo-orbital transformation and uses valence orbitals with a simplified radial nodal structure, i.e. pseudovalence orbitals. Besides the computational savings due to the elimination of the core electrons, the main interest in standard ECP techniques results from the fact that they offer an efficient and accurate, albeit approximate, way of including implicitly, i.e. via parametrization of the ECPs, the major relativistic effects in formally nonrelativistic valence-only calculations. A number of reviews on ECPs has been published and the reader is referred to them for details (Bala-subramanian 1998 Bardsley 1974 Chelikowsky and Cohen 1992 Christiansen et... 

Another option that reduces the number of functions, particularly when heavy atoms are involved, is the replacement of inner shell electrons by effective (or pseudo) potentials. Such procedures have been incorporated into many ab initio program systems including ACES II. Since the core electrons are not explicitly considered, effective potentials can drastically reduce the computational effort demanded by the integral evaluation. However, because the step is an inexpensive part of a correlated calculation, the role of effective potentials in correlated calculations is less important, due to the fact that dropping orbitals is tantamount to excluding them via effective potentials. An exception occurs when relativistic effects are important, as they would be in a description of heavy atom systems. Most such chemically relevant effects are due to inner shell elearons their important physical effects, like expanding the Pt valence shell, can be introduced via effective potentials that are extracted from Dirac-Fock or other relativistic calculations on atoms. ° Similarly, some effeaive potentials introduce some spin-orbital effects as well. Thus, besides simplifying the computation, effective potential calculations could include important physical effects absent from the ordinary nonrelativistic methods routinely applied. 

Another group of methods successfully used for calculations of the electronic structures of the heaviest element molecules are effective core potentials (ECP) (see the Chapters of M. Dolg and Y.-S. Lee in these issues). The relativistic ECPs (RECP) were applied to calculations of the electronic structures of halides and oxyhalides of Rf and Sg and of some simple compounds (mostly hydrides and fluorides) of elements 113 through 118 [126-131]. Using energy-adjusted pseudo-potentials (PP) [132] electronic structures and properties, and the influence of relativistic effects were studied for a number of compounds of elements at the end of the 6d series (elements 111 and 112), as well as at the beginning of the 7p series (elements 113 and 114) (see Refs. 26 and 133 for reviews and references therein). Some other methods, like the Douglas-Kroll-Hess (DKH) [134], were also used for calculations of small heaviest-element species (e.g. IIIH [95]). 

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... 

If one defines the core projector and the pseudo-orbital in the same way as done in the Phillips-Kleinman method, for relativistic spinor wavefunctions one obtains relativistic pseudo-orbitals and relativistic effective potentials... 

Effective core potential (ECP) or pseudo-potential approximation, has been proved to be very useful for modeling of heavy atoms in the ab initio methods (Hay and Wadt 1985). In this approximation, core electrons are replaced by an effective potential, thereby reducing the number of electrons to be considered and hence requiring fewer basis functions. The ECP method takes into account the relativistic effect on valence electrons, thus making it applicable to heavy atoms (e.g., second- and third-row transition metals, lanthanides and actinides). It is relatively cheap, works very well, and has very little loss in reliability. 

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,... 

In the effective core potential (ECP) approximation, is represented by a semi-local potential [74]. Unlike in the MCP methods, there are no core functions and the pseudo-valence orbitals are nodeless for the radial part, which is an essential approximation. The semi-local ansatz gives rise to rather complicated integrals over the Gaussian functions compared to the MCP methods, though efficient algorithms were developed for their solution. Relativistic and SO effects are treated by relativistic one-electron PPs (RPP) [76]... 

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