Pseudo potential approximation

Bliimel, R. (1993a). On the integrability of the two-ion Paul trap in the pseudo potential approximation, Phys. Lett. A174, 174-175. 

An adequate theoretical basis for the calculation of slow neutron scattering from chemically bound systems exists in the pseudo-potential approximation introduced by Fermi in 1937 [1]. The fundamental cross section of interest for neutron thermalization is the differential cross section g(Eo,E,6) for energy transfer Eq- E with scattering through an angle 0 in the laboratory system. The calculation of this cross section, even in the pseudo-potential approximation, depends on the detailed dynamics of the atomic motion in the moderator. The dynamics of atomic motion in crystals and liquids is complicated and not as yet known in detail. The direction of most fundamental interest, therefore, is to determine these dynamical properties from experimental measurements of slow neutron scattering. 

This condition follows from the principle of microscopic reversibility, and is explicitly contained in the Fermi pseudo-potential approximation. 

A. Theoretical basis. The theoretical basis for the calculation of slow neutron scattering from chemically bound systems is the pseudo-potential approximation. This approximation can be derived [13] by replacing the strong, short-range /S-wave interaction between neutron and nucleus by a boundary condition on the wave function at small neutron-nucleus separation. This corresponds to replacing the actual interaction by a pseudo-potential interaction... 

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. 

When using the pseudo-potential approximation, the external potential, Uext, is simply the sum of the pseudo-potentials of all the atoms in the system. If atom a. is located in the unit cell at Tq, and its pseudo-potential is Wa r, r ), the external potential is... 

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.)... 

The wave functions are expended in a plane wave basis set, and the effective potential of ions is described by ultrasoft pseudo potential. The generalized gradient approximation (GGA)-PW91, and local gradient-corrected exchange-correlation functional (LDA)-CAPZ are used for the exchange-correlation functional. 

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... 

Solution of the Kohn-Sham equations as outlined above are done within the static limit, i.e. use of the Born-Oppenheimer approximation, which implies that the motions of the nuclei and electrons are solved separately. It should however in many cases be of interest to include the dynamics of, for example, the reaction of molecules with clusters or surfaces. A combined ab initio method for solving both the geometric and electronic problem simultaneously is the Car-Parrinello method, which is a DFT dynamics method [52]. This method uses a plane wave expansion for the density, and the inner ions are replaced by pseudo-potentials [53]. Today this method has been extensively used for studies of dynamic problems in solids, clusters, fullerenes etc [54-61]. We have recently in a co-operation project with Andreoni at IBM used this technique for studying the existence of different isomers of transition metal clusters [62,63]. 

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]. 

Diatomics-in-Molecules (DIM) Method Semi-Empirical Valence-Bond Methods Approximate Pseudo-Potential Theories... 

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