Structural coefficients

Using the structural coefficients in table 3.3 implies knowledge of the conditions of internal disorder of the various cations in the structure. In most cases (especially for pure components), this is not a problem. For instance, table 3.4 shows the decomposition into structural components of the acmite molecule, in which all Na+ is in Vlll-fold coordination with oxygen, Fe + is in Vl-fold coordination, and Si is in tetrahedral coordination. In some circumstances, however, the coordination states of the various elements are not known with sufficient precision, or, even worse, vary with T. A typical example is spinel (MgAl204), which, at T < 600 °C, has inversion X = 0.07 (i.e., 7% of the tetrahe-drally coordinated sites are occupied by Al +) and, at T = 1300 °C, has inversion X = 0.21 (21% of tetrahedral sites occupied by AF ). Clearly the calculation of... 

Table 3.4 Coefficients of modified Haas-Fisher polynomial 3.79 for acmite component of pyroxene (NaFeSi20g), after application of structural coefficients in table 3.3. The resulting Cp is in J/(mole X K) (adapted from Robinson and Flaas, 1983).

In order to ensure reliable convergence, we generally employ VB optimization procedures that require first and second derivatives with respect to all of the variational parameters. Expressions for these derivatives are most easily derived by considering the first- and second-order changes in Evb with respect to the VB parameters defined by O andc . These maybe generated from combinations of cf Eqs. (3) and (4)) and the analogous operator for the structure space, E/i. For the first-order variations in orbitals and structure coefficients, we find ... 

There are even more possibilities for optimization procedures if one considers also a partitioning of the optimization problem. Work in this area is currently ongoing, and we present for the first time, in Section 5, results obtained using a Davidson-like strategy for the (linear) optimization of the VB structure coefficients [51, 52]. Further details of this new procedure will be presented in a future publication. 

Symmetry projection may be applied after an optimization procedure is completed, so as to remedy a symmetry-broken solution. Alternatively it may be applied during an optimization procedure by substituting for Pye in Eq. (6) or Eq. (12). An attractive third option is to employ the projection operator in combination with orbital and structure-coefficient constraints. For examples of these different possibilities we refer to previously published accounts [4]. 

One of the most useful types of constraint is the restriction of the spin coupling to just a single mode. Many molecular systems are described rather well by the perfect pairing mode of spin coupling, for example. A useful alternative, especially when this is not the case, is to base the structure coefficients on the CASSCF wavefunction in the VB orbital basis ... 

See text. For the OVB method die weights add up to at least 0.990 because VB structure coefficients less than 0.05 have been neglected. 

To investigate possible approximate schemes to the procedure just discussed, we first observe that, taking into account the properties of the structure coefficients one can... 

STRUCTURAL COEFFICIENTS AND WEIGHTS OF VALENCE BOND WAVE FUNCTIONS... 

By performing the above procedures, the solvent effect is taken into account at the VBSCF level, whereby the orbitals and structural coefficients are optimized till self-consistency is achieved. Like VBSCF, the VBPCM method is suitable for diabatic states, which are calculated with the same solvent field as the one for the adiabatic state. Thus, it has the ability to compute the energy profile of the full state as well as that of individual VB structures throughout the course of a reaction, and in so doing to reveal the individual effects of solvent on the different constituents of the wave function. In this spirit, it has been used to perform a quantitative VBSCD analysis of a reaction that exhibits a marked solvent effect, the Sn2 reaction Cl- + CH3CI —> CH3CI + Cl- (55). 

BOVB Breathing orbital valence bond. A VB computational method. The BOVB wave function is a linear combination of VB structures that simultaneously optimizes the structural coefficients and the orbitals of the structures and allows different orbitals for different structures. The BOVB method must be used with strictly localized active orbitals (see HAOs). When all the orbitals are localized, the method is referred to as L-BOVB. There are other BOVB levels, which use delocalized MO-type inactive orbitals, if the latter have different symmetry than the active orbitals. (See Chapters 9 and 10.)... 

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.

The Lagrange multiplier X is determined by requiring that the derivatives of the Lagrangian with respect to all optimised variables like the structure coefficients C are zero ... 

In the middle of cells and in faces that are perpendicular to the flowing direction, the borders are branched, which means that the effective number of borders, equivalent to that in a real system, is different from five. The number of independent borders with constant by height radius and length L can be determined by the electro-hydrodynamic analogy between current intensity and liquid flow rate through borders, both being directly proportional to the cross-sectional areas [6,35]. This analogy indicates that the proportionality coefficients (structural coefficients B = 3) in the dependences border hydroconductivity vs. foam expansion ratio and foam electrical conductivity vs. foam expansion ratio, are identical [10]. From the electrical conductivity data about foam expansion ratio it follows... 

In a polyhedral foam the liquid is distributed between films and borders and for that reason the structure coefficient B depends not only on foam expansion ratio but also on the liquid distribution between the elements of the liquid phase (borders and films). Manegold [5] has obtained B = 1.5 for a cubic model of foam cells, assuming that from the six films (cube faces) only four contribute to the conductivity. He has also obtained an experimental value for B close to the calculated one, studying a foam from a 2% solution of Nekal BX. Bikerman [7] has discussed another flat cell model in which a raw of cubes (bubbles) is shifted to 1/2 of the edge length and the value obtained was B = 2.25. A more detailed analysis of this model [45,46] gives value for B = 1.5, just as in Manegold s model. 

The difference in the structural coefficients B and Bf at border and film conductivities, can be most easily illustrated in a cubic bubble model... 

Analysing the vast experimental data Lemlich [56] has pointed out that the structural coefficient B monotonously changes from 1.5 to 3 with the increase in foam expansion ratio. He proposed an expression that is in good agreement with the experimental data and approximates the B(net) relation... 

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