Pseudopotentials definition

Atomic spin-orbit pseudopotential definition and its relation to the different relativistic approximations. /. Chem. Phys., 123 (2005) 034106. 

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. 

However, this is definitely the technique for future calculations involving a large number of metal atoms. Furthermore, the idea behind the pseudopotential method is also applied in other types of Hamiltonians described below, e.g., valence effective Hamiltonian and semi-empirical methods. 

The electronic structure is reformulated in terms of free electrons and a d resonance in order to relate the band width W, to the resonance width T, and is then reformulated again in terms of iransilion-metal pseudopotential theory, in which the hybridization between the frce-electron states and the d state is treated in perturbation theory, The pseudopotential theory provides both a definition of the d-state radius and a derivation of all interatomic matrix elements and the frce-electron effective mass in terms of it. Thus it provides all of the parameters for the L.CAO theory, as well as a means of direct calcidation of many properties, as was possible in the simple metals. ... 

Fig.4.1. Estimates of the extent (bottom B, top A) and position (centre C) of the s, p, and d bands of non-magnetic chromium obtained by the third-order expression (3.50), definitions (2.22), and the potential parameters in Table 4.1. Expression (3.50) spans only the square-well pseudopotentials V for the s and p bands, i.e. Vd is outside the energy window...

Finally, some spectroscopic applications for pseudopotentials within SOCI methods are presented in section 3. We focus our attention on applications related to relativistic averaged and spin-orbit pseudopotentials (other effective core potentials applications are presented in chapters 6 and 7 in this book). Due to the large number of theoretical studies carried out so far, we have chosen to illustrate the different SOCI methods and discuss a few results, rather than to present an extensive review of the whole set of pseudopotential spectroscopic applications which would be less informative. Concerning the works not reported here, we refer to the exhaustive and up-to-date bibliography on relativistic molecular studies by Pyykko [21-24]. The choice of an application is made on the basis of its ability to illustrate the performances on both the pseudopotential and the SOCI methods. One has to keep in mind that it is not easy to compare objectively different pseudopotentials in use since this would require the same conditions in calculations (core definition, atomic basis set, SOCI method). The applications are separated into gas phase (section 3.1) and embedded (section 3.2) molecular applications. Even if the main purpose of this chapter is to deal with applications to molecular spectroscopy, it is of great interest to underline the importance of the spin-orbit coupling on the ground state reactivity of open-shell systems. A case study is presented in section 3.1.4. 

Let us first discuss the usual spin-orbit pseudopotentials, which can be defined in a general way via relativistic two-component pseudopotentials Uf r). They originate fi om the definition of a Schrodinger-like valence model Hamiltonian in a two-component form shown here for an atom... 

It is also clear that having the inner orbitals present in the definition of the pseudopotential and in the definition of the orbital x sis arbitrary linear combinations is probably redundant. We have introduced the outer orbital x a nodeless function on two grounds ... 

The presence and form of this last term has some important consequences. The first is that the definition of the pseudopotential is determined to some extent by the definition of the pseudospinors. This means that the pseudospinors and the pseudopotentials are not independent. The only dependence on the valence spinors in yOPK comes from the spinor eigenvalue, but the remainder terms and the term that came from the renormalization contain the valence pseudospinors, and therefore the detailed shape of the potential is dependent on the shape of the valence pseudospinors. 

This effect presents some serious problems for the development of pseudopotentials. A pseudopotential that depends critically on the shape of the pseudospinor and for which the results are sensitive to the valence occupation is of no value. The problem was overcome (Christiansen et al. 1979) by the definition of the so-called shape-consistent pseudospinors and the corresponding pseudopotentials. 

---