Modelling atomic core potentials

In the case when there is no appreciable overlap between atomic cores there is every reason to believe that the simulation of the Pauli principle in the region of each atomic core would be approximated to a high degree of accuracy by the simple sum of the separate-atom core potentials. This statement can be made more precise by saying that, if the core-core overlap is negligible, then the product of the corresponding core projectors will be zero, which is equivalent to saying that the electrons of one core have no Pauli-principle driven interaction with those of a remote core. In this case the molecular case is just the same as the above many-electron atomic case with the model potential replaced by a sum of model potentials for each atomic core ... 

The combination of this pseudopotential with the Coulomb and exchange potential to form a model potential for each atomic core is possible, and a sensible form for this combined atomic core potential is the semi-local form ... 

It is probably prudent to use a single-determinant model of the atomic cores since all our introductory theory assumed this model. However since we are now modelling the core potential not transforming an exact equation for the electronic structure, there is no a priori reason to stay with the HF model. 

After some experience with MNDO, it became clear that there were certain systematic en ors. For example the repulsion between two atoms which are 2-3 A apart is too high. This has as a consequence that activation energies in general are too large. The source was traced to too repulsive an interaction in the core-core potential. To remedy this, the core-core function was modified by adding Gaussian functions, and the whole model was reparameterized. The result was called Austin Model 1 (AMl) in honour of Dewar s move to the University of Austin. The core-core repulsion of AMI has the form ... 

The shape-consistent (or norm-conserving ) RECP approaches are most widely employed in calculations of heavy-atom molecules though ener-gy-adjusted/consistent pseudopotentials [58] by Stuttgart team are also actively used as well as the Huzinaga-type ab initio model potentials [66]. In plane wave calculations of many-atom systems and in molecular dynamics, the separable pseudopotentials [61, 62, 63] are more popular now because they provide linear scaling of computational effort with the basis set size in contrast to the radially-local RECPs. The nonrelativistic shape-consistent effective core potential was first proposed by Durand Barthelat [71] and then a modified scheme of the pseudoorbital construction was suggested by Christiansen et al. [72] and by Hamann et al. [73]. 

An early method of describing electrons in crystals was the method of nearly free electrons we shall refer to it as the NFE model. In this the potential energy V(x, y, z) in (6) is treated as small compared with the electron s total energy . This is, of course, never the case in real crystals the potential energy near the atomic core is always large enough to produce major deviations from the free-electron form. Therefore, until the introduction of the concept of a pseudopotential , it was thought that the NFE model was not relevant to real crystalline solids. 

The complexity of the parameter-fitting procedure in the MINDO models can only be appreciated by a detailed study of the inherent assumptions. It is perhaps indicative to say the only molecular integral that is calculated exactly from the basis atomic orbitals is the overlap integral, all others being approximated or given empirical values. The repulsion potential of the atomic cores is, for example, one of the critical functions in the theory. There has to the present time been four distinct versions of the MINDO parameterization, and the latest (MINDO/3) (102) is said to remove certain deficiencies in the earlier versions (such as the prediction that HjO was linear and the underestimation of the strain energies in small ring hydrocarbons). 

Model potential methods and their utilization in atomic structure calculations are reviewed in [139], main attention being paid to analytic effective model potentials in the Coulomb and non-Coulomb approximations, to effective model potentials based on the Thomas-Fermi statistical model of the atom, as well as employing a self-consistent field core potential. Relativistic effects in model potential calculations are discussed there, too. Paper [140] has examples of numerous model potential calculations of various atomic spectroscopic properties. 

All of the measurements employed the technique described above that involves the analysis of the isotope composition of 02 released from the carrier complexes in preequilibrated solutions. In addition, an established DFT method (mPWPW91)34 with the atomic orbital basis functions, Co, Fe, and Cl (the compact relativistic effective core potential basis CEP-31G),35 N and O (6-311G ), P (6-311G ), C(6-31G), and H (STO-3G),36 were used to calculate the 180 EIE in terms of actual and model structures. The latter approach has also been employed for hypothetical intermediates in enzymes as described below. 

In the one-electron transition model it is assumed that only one core electron is excited to an unfilled state present in the initial, unperturbed solid. The remaining electrons are assumed to be unaffected, remaining frozen in their original states. In essence the one-electron model describes the potential seen by the final-state electrons as nonoverlapping spherically symmetric spin-independent potentials (muffin-tin scatterers) centered around an atom from which the X-ray cross section from a deep core level of an atom to final states above the Fermi level can in principle be calculated for any energy above threshold. 

The well-known muffin-tin model of the electron atom interaction potential has in most cases proved to be an adequate compromise between accuracy and computational efficiency, at least for electron kinetic energies exceeding 20 eV./27,28,29/ The muffin-tin potential is spherical inside the muffin-tin spheres. This ion core usually provides the dominant contribution to electron scattering and emission, to be discussed shortly. 

To model the copper (100) surface a two-layer cluster of C4V symmetry, with 5 copper atoms in one layer and 4 copper atoms in the other layer, has been used. In this cluster, all the 9 metal atoms were described by the LANL2DZ basis set. The LANL2DZ basis set treats the 3s 3p 3d 4s Cu valence shell with a double zeta basis set and treats all the remainder inner shell electrons with the effective core potential of Hay and Wadt [33]. The non-metallic atoms (C and H) were described by the 6-3IG basis set of double zeta quality with p polarization functions in... 

In the present work, the interaction of the ethylene molecule with the (100) surfaces of platinum, palladium and nickel is studied using the cluster model approach. All these metals have a face centered cubic crystal structure. The three metal surfaces are modelled by a two-layer M9(5,4) cluster of C4V symmetry, as shown in Fig. 6, where the numbers inside brackets indicate the number of metal atoms in the first and second layer respectively. In the three metal clusters, all the metal atoms are described by the large LANL2DZ basis set. This basis set treats the outer 18 electrons of platinum, palladium and nickel atoms with a double zeta basis set and treats all the remainder electrons with the effective core potential of Hay and Wadt... 

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