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Radii pseudopotential

This rule centers on the much-used concept of ionic size, a concept that has been refined, or rather been better defined in recent years by the use of pseudopotential radii (Zunger and Cohen, 1978). Thus, the use of pseudopotential radii (r,) to define atomic size more closely, and the iden-... [Pg.335]

Burdett, J. K., G. D. Price, and S. L. Price (1981). The factors influencing solid state structure. An interpretation using pseudopotential radii structure maps. Phys. Rev. B24, 2903-12. [Pg.464]

Bundschuh T, Knopp R, Kim JI (2001) Laser-induced breakdown detection (LIBD) of aquatic colloids with different laser systems. Colloids Surfaces A 177 47-55 Burdett JK, McLaman TJ (1984) An orbital interpretation of Pauhng s rales. Am Mineral 69 601-621 Burdett JK, Price GD, Price SL (1981) The factors influencing sohd state structure. An interpretation using pseudopotential radii maps. Phys Rev B 24 2903-2912 Calas G, Brown GE Jr, Waychunas GA, Petiau J (1987) X-ray absorption spectroscopic studies of silicate glasses and minerals. Phys Chem Minerals 15 19-29... [Pg.162]

The concept of atomic or ionic size is one that has been debated for many years. The structure map of Figure 1 used the crystal radii of Shannon and Prewitt and these are generally used today in place of Pauling s radii. Shannon and Prewitt s values come from examination of a large database of interatomic distances, assuming that intemuclear separations are given simply by the sum of anion and cation radii. Whereas this is reasonably frue for oxides and fluorides, it is much more difficult to generate a self-consistent set of radii for sulfides, for example. A set of radii independent of experimental input would be better. The pseudopotential radius is one such estimate of atomic or orbital size. [Pg.4592]

Table 10.2 Average Electron Density in Typical sp Metals at 25°C Together with the Number of Valence Electrons Wy, the Metallic Density p j, and the Pseudopotential Radius... Table 10.2 Average Electron Density in Typical sp Metals at 25°C Together with the Number of Valence Electrons Wy, the Metallic Density p j, and the Pseudopotential Radius...
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. [Pg.31]

As is seen from the behaviour of the more sophisticated Heine-Abarenkov pseudopotential in Fig. 5.12, the first node q0 in aluminium lies just to the left of (2 / ) / and g = (2n/a)2, the magnitude of the reciprocal lattice vectors that determine the band gaps at L and X respectively. This explains both the positive value and the smallness of the Fourier component of the potential, which we deduced from the observed band gap in eqn (5.45). Taking the equilibrium lattice constant of aluminium to be a = 7.7 au and reading off from Fig. 5.12 that q0 at 0.8(4 / ), we find from eqn (5.57) that the Ashcroft empty core radius for aluminium is Re = 1.2 au. Thus, the ion core occupies only 6% of the bulk atomic volume. Nevertheless, we will find that its strong repulsive influence has a marked effect not only on the equilibrium bond length but also on the crystal structure adopted. [Pg.125]

Explain the concept of a pseudopotential. Aluminium is fee with a lattice constant of a = 7.7 au. It is well described by an Ashcroft empty core pseudopotential of core radius 1.1 au. Show that the lattice must be expanded by 14% for the 2n/a(200) Fourier component of the pseudopotential to vanish. [Pg.246]

Fig. 35. Dashed line Potential energy function for a valence electron in the impenetrable core model. Rc = core radius. Solid line Pseudopotential, after Austin and Heine 14 )... Fig. 35. Dashed line Potential energy function for a valence electron in the impenetrable core model. Rc = core radius. Solid line Pseudopotential, after Austin and Heine 14 )...
Fig. 1 Asymptotic structure coefficients as(j8), c ks,xG ) crw03), a CS), and aKS gOS) as fimction of barrier height parameter )S =VW/eF, where W is the barrier height and eF the Fermi energy. Corresponding values of the Wigner-Seitz radius rs for jellium and structureless-pseudopotential models over the metallic range of densities are also given. The relationship between rs and ff is via self-consistent calculations in the local density approximation for exchange-correlation. Fig. 1 Asymptotic structure coefficients as(j8), c ks,xG ) crw03), a CS), and aKS gOS) as fimction of barrier height parameter )S =VW/eF, where W is the barrier height and eF the Fermi energy. Corresponding values of the Wigner-Seitz radius rs for jellium and structureless-pseudopotential models over the metallic range of densities are also given. The relationship between rs and ff is via self-consistent calculations in the local density approximation for exchange-correlation.
Beyond a chosen cutoff radius, the all-electron and pseudofunctions (potential and wavefunction) are identical, while inside the core region both the pseudopotential and pseudowavefunction are smoothly varying. After the construction of these pseudofunctions for a single atom and ensuring that their scattering properties are almost identical to those of the all-electron functions, they can be used in any chemical environment. [Pg.17]

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. ... [Pg.476]

The connection with pseudopotential theory will also enable us to obtain other parameters of the energy bands in terms of the d-state radius r,. We now have an electronic structure based upon d states, which are coupled to each other according to Eq. (20-45). They are also coupled to free-electron bands, as seen in... [Pg.517]

Finally, lei us use the transition-metal pseudopotential theory to estimate matrix elements between d states and s and p states. These are not so useful in the transition metals themselves since the description of the electronic structure is better made in terms of d bands coupled to free-electron bands, ti k /(2m) d-<01 IT 10>, rather than in terms of d bands coupled to s and p bands. However, the matrix elements and so forth, directly enter the electronic structure of the transition-metal compounds, and it is desirable to obtain these matrix elements in terms of the d-state radius r. We do this by writing expressions for the bands in terms of pseudopotentials and equating them to the LCAO expressions obtained in Section 20-A. [Pg.519]

For analysis of the transition metals themselves, the use of free-electron bands and LCAO d states is preferable. The analysis based upon transition-metal pseudopotential theory has shown that the interatomic matrix elements between d states, the hybridization between the free-electron and d bands, and the resulting effective mass for the free-electron bands can all be written in terms of the d-state radius r, and values for have been listed in the Solid State Table. [Pg.520]

Each pseudopotential is defined within a cut-off radius from the atom center. At the cut-off, the potential and wavefunctions of the core region must join smoothly to the all-electron-like valence states. Early functional forms for pseudopotentials also enforced the norm-conserving condition so that the integral of the charge density below the cut-off equals that of the aU-electron calculation [42, 43]. However, smoother, and so computationally cheaper, functions can be defined if this condition is relaxed. This idea leads to the so called soft and ultra-soft pseudopotentials defined by Vanderbilt [44] and others. The Unk between the pseudo and real potentials was formaUzed more clearly by Blochl [45] and the resulting... [Pg.340]

In early implementations of CP, norm-conserving pseudopotentials have been used. [70] In such a pseudopotential, pseudowavefunctions match the all-electron wavefunctions beyond a specified matching radius (core-radius) rc. Inside the r. ... [Pg.113]

Much effort has been put to improve the pseudopotentials. [71,72] The most successful pseudopotential model is the so called ultra-soft pseudopotentials, proposed by Vanderbilt. [73] The model allows one to work with optimally smooth pseudopotentials. Thus the number of plane waves needed to express the pseudo-wavefunctions can be greatly reduced. In the model, pseudo-wavefunctions ipiif match the true orbitals outside a given core radius re, within Tc, are al-... [Pg.114]


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