Pseudo-potential Transferability

A useful pseudo-potential needs to be transferable, i.e., it needs to describe accurately the behavior of the valence electrons in several different chemical environments. The logarithmic derivative of the pseudo wave-function determines the scattering properties of the pseudo-potential. Norm-conservation forces these logarithmic derivatives to coincide with those of the true wave-functions for r r . In order for the pseudo-potential to be transferable, this equality should hold at all relevant energies, and not only at the energy, j, for which the pseudo-potential was adjusted. Norm-conservation assures that this is fulfilled for the nearby energies, as [49,72] 

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The assignment of the intense band in the beam to a LMCT transfer is supported by molecular orbital calculations [31]. Pseudo-potential ab initio... 

FIGURE 6.15 Fermi surfaces of LiC6 calculated using empirical pseudo-potentials and a self-consistent determination of the charge transfer the Fermi surfaces for the lower (a) and upper (b) bands. (From Ohno, T., J. Phys. Soc. Jpn. 49(Suppl. A), 899, 1980. With permission.)... 

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... 

For a wide range of chemically interesting events, such as bond breaking and formation, an accurate description is required only for the valence electrons. Such an accurate description can be obtained using a pseudo potential description of the nuclei. This technique is well established in the plane wave community. We take advantage of the experience with this scheme using the pseudo potentials of Goedecker et al. (GTH) [12,13]. These accurate and transferable pseudo potentials have an analytic form that allows for an efficient treatment of all terms within the GPW method. 

AEband can also be calculated from perturbation theory via the pseudo-potential matrix elements " ). The pseudo-potential approach, however, is only justified if the parent metals have the same valency and the same Fermi vector, and if furthermore no charge transfer takes place and the AB compounds have the same atomic volumes as A and B ). 

From a conceptual point of view, this technical problem is related to a qualitative question would it be possible to define a universal monoelectronic potential characteristic of an atom, say the C atom Any chemist would consider that atoms in molecules keep some of the characteristics of the isolated atoms. Is it possible to give a quantitative justification of this idea It would be interesting, for instance, to estimate whether singly, doubly and triply bonded carbon atoms can be considered as fixed carbon atoms in the series of saturated and conjugated hydrocarbons. It will be shown below that quantum chemistry is able to determine purely monoelectronic (pseudo)potentials of atoms in molecules and to check their transferability. 

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. 

In some cases, like the alkali metals, the use of a nonlinear core-valence xc scheme may be necessary to obtain a transferable pseudo-potential. In these... 

Pseudo-Kolbe electrolysis is the name given to anodic decarboxylations where the electron transfer does not occur from the carboxylate but from a group attached to it [31]. These oxidations are characterized by potentials that are much lower than the critical potential for the Kolbe electrolysis. The salt of p-methoxyphenylacetic acid can be oxidized in methanol to afford the corresponding methyl ether as the sole product. The low oxidation potential of 1.4 V (see) suggests, that the electron is being transferred from the aromatic nucleus (Eq. 39) [31]. 

Early studies of ET dynamics at externally biased interfaces were based on conventional cyclic voltammetry employing four-electrode potentiostats [62,67 70,79]. The formal pseudo-first-order electron-transfer rate constants [ket(cms )] were measured on the basis of the Nicholson method [99] and convolution potential sweep voltammetry [79,100] in the presence of an excess of one of the reactant species. The constant composition approximation allows expression of the ET rate constant with the same units as in heterogeneous reaction on solid electrodes. However, any comparison with the expression described in Section II.B requires the transformation to bimolecular units, i.e., M cms . Values of of the order of 1-2 x lO cms (0.05 to O.IM cms ) were reported for Fe(CN)g in the aqueous phase and the redox species Lu(PC)2, Sn(PC)2, TCNQ, and RuTPP(Py)2 in DCE [62,70]. Despite the fact that large potential perturbations across the interface introduce interferences in kinetic analysis [101], these early estimations allowed some preliminary comparisons to established ET models in heterogeneous media. 

The same pseudo-ensemble concept has been used by Camp and Allen [44] to obtain a pseudo-Gibbs method in which particle transfers are substituted by volume fluctuations of the two phases. The volume fluctuations are unrelated to the ones required for pressure equality (10.7) but are instead designed to correct imbalances in the chemical potentials of some of the components detected, for example, by test particle insertions. 

In contrast to the pseudo 3-D models, tmly multi-dimensional models use, in general, finite element or finite volume CFD (Computational Fluid Dynamics) techniques to solve full 3-D Navier-Stokes equations with appropriate modifications to account for electrochemistry and current distribution. The details of electrochemistry may vary from code to code, but the current density is calculated almost exclusively from Laplace equation for the electric potential (see Equation (5.24)). Inside the electrolyte, the same equation represents the migration of ions (e g. 0= in SOFC), elsewhere it represents the electron/charge transfer. In what follows, we briefly summarize a commonly used multi-dimensional model for PEM fuel cells because of its completeness and of the fact that it also addresses most essential features of SOFC modeling. 

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