Modelling atomic pseudopotentials

Fig. 10.15 Distribution of electron density within a Ag lattice at the (Iff) surface on the basis of the jellium model with pseudopotentials located at the metal atom cores as indicated by the arrows [26]. The broken vertical line shows the position of the metal surface.

The description of the metal is improved considerably if metallic structure is introduced by accounting for the local attractive force of the metal atoms on the free electron gas. This corresponds to the jellium model with pseudopotentials. Each metal atom in the lattice is pictured as being surrounded by a spherical volume Fc in which electrostatic effects may be ignored. Outside of the sphere the metal atom behaves as a point charge of charge number n. Thus, it has a pseudopotential < )(., where... 

Now, a decision about the symmetry type ( -value) of an AO is, colloquially speaking, essentially a non-local type of act the -value of a function cannot be deduced from its properties in an infinitesimal neighbourhood of a point. We are therefore forced to incorporate some form of non-local operator if we are to develop any sort of realistic modelling of the atomic pseudopotential. 

This evidence is by no means exhaustive or convincing in itself, but there is a vast body of such evidence to support the idea that the most convenient form for an atomic model potential which will simulate the effect of the true atomic pseudopotential is... 

Pseudopotentials describe the interaction of a valence electron with the core of the atoms. They are known in the literature under various names, such as model potentials, effective core potentials,. Model potentials are generally parametrized from atomic spectroscopic data whereas effective core potentials and pseudopotentials are most often derived from ab initio calculations. There is a huge literature on the subject and several review articles. " The recent paper by Krauss and Stevens is recommended for an overall survey of the subject with applications and comparisons with all-electron calculations. The recent review paper of Pelissier et al is devoted to transition elements. In the following we shall only review the main characteristics of the determination of atomic pseudopotentials by the ab initio simulation techniques of Section II.B. 

The method provides valence orbitals in (168) with internal nodes, which closely resemble the original valence Hartree-Fock orbitals of (166). This method has been developed mainly by Huzinaga and colleagues, who determine atomic pseudopotentials (model potentials in their terminology) of the form ... 

The simple electronic structure of sodium also renders the application of other types of models relatively easy and extendable to relatively large sizes. See e.g. Hiickel calculations and MC structural search (R. Poteau and F. Spiegelmann, Phys. Rev. B 45, 1878 (1992) and J. Chem. Phys. 98, 6540 (1993) Erratum 99, 10089 (1993)) or the so-called spherically averaged pseudopotential (SAPS) model (M. D. Glossman, J. A. Alonso and M. P. Iniguez, Phys. Rev. B 47, 4747 (1993)). This is a simplified atomistic scheme, in which the external potential (written as the sum of the atomic pseudopotentials) acting on the electrons is developed in spherical harmonics around the cluster center of mass, and only the spherical component is retained in the solution of the KS equations. 

Using a linear combination of plane waves is a very effective approach to modehng the behavior of valence electrons, but not so for the core. Here, the electron density varies rapidly, and so many plane waves are required to describe the core wavefunction that the calculation quickly becomes unfeasible. Once again, we see that there is a problem with modeling the core electrons in an explicit way, and the solution used is the same as discussed above in Section 3.2.2, namely to use a pseudopotential. Thus, to summarize, in solid-state simulations we construct delocalized basis sets using plane waves to model the kinetic energies of the valence electrons and atomic pseudopotential functions to mimic the effects of the core electrons. 

In molecular DFT calculations, it is natural to include all electrons in the calculations and hence no further subtleties than the ones described arise in the calculation of the isomer shift. However, there are situations where other approaches are advantageous. The most prominent situation is met in the case of solids. Here, it is difficult to capture the effects of an infinite system with a finite size cluster model and one should resort to dedicated solid state techniques. It appears that very efficient solid state DFT implementations are possible on the basis of plane wave basis sets. However, it is difficult to describe the core region with plane wave basis sets. Hence, the core electrons need to be replaced by pseudopotentials, which precludes a direct calculation of the electron density at the Mossbauer absorber atom. However, there are workarounds and the subtleties involved in this subject are discussed in a complementary chapter by Blaha (see CD-ROM, Part HI). 

Although the pseudopotential is, from its definition, a nonlocal operator, it is often represented approximately as a multiplicative potential. Parameters in some chosen functional form for this potential are chosen so that calculations of some physical properties, using this potential, give results agreeing with experiment. It is often the case that many properties can be calculated correctly with the same potential.43 One of the simplest forms for an atomic model effective potential is that of Ashcroft44 r l0(r — Rc), where the parameter is the core radius Rc and 6 is a step-function. 

The dynamics of carbon-halogen bond reductive cleavage in alkyl halides was studied by MP3 ab initio calculations, using pseudopotentials for the halogens and semidiffuse functions for the heavy atoms [104], The effect of solvent was treated by means of the ellipsoidal cavity dielectric continuum model. Both a concerted (i.e., a one-step) and a stepwise mechanism (in which an anion radical is formed at first) were... 

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]. 

One interesting scheme based on density functional theory (DFT) is particularly appealing, because with the current power of the available computational facilities it enables the study of reasonably extended systems. DFT has been applied with a variety of basis sets (atomic orbitals or plane-waves) and potential formulations (all-electron or pseudopotentials) to complex nu-cleobase assemblies, including model systems [90-92] and realistic structures [58, 93-95]. DFT [96-98] is in principle an ab initio approach, as well as MP2//HF. However, its implementation in manageable software requires some approximations. The most drastic of all the approximations concerns the exchange-correlation (xc) contribution to the total DFT functional. 

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. 

Analytic derivatives have been reported for both the LSCF and GHO models, making them attractive options for MD simulations (Amara et al. 2000). Their generalization to ab initio levels of theory through the use of core pseudopotentials (along the lines of the pseudohalogen capping atoms described above) ensures that they will see continued development. 

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