The Static Contribution

For complete generality we may allow the static potential in cell, to consist of both a local and a non-local part. 

A convenient way to construct the cellular potentials is by projection. We define a cellular projection operator 0(r) such that in co-ordinates 

The cellular decomposition of the local part of V is trivially exact and allows a complete correspondence between the geometric arrangement of the ligands in space, and their additive contributions to V. The decomposition of the non-local part is however not so obvious and is worth looking at in more detail. To do this it is convenient to recall Eq. (6-13 b) for the screened exchange operator A typical matrix element of can be put in the form, using (6-27) 

Since is a rotation matrix, the rf-orbital di) can be represented in polar coordinates, with Z-axis along the metal-ligand bond, as 

If we construct the B-tensors with spherical harmonics taken with Of as polar axis, and use their orthogonahty, the angular integrations can be easily done, and then (6-35) reduces to, 

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Since F1(0)/2m = — e/2m is the Dirac moment of the negaton, we deduce that F2 0) is the static anomalous moment of the electron, t.e., F2(0) is the static contribution to its magnetic moment over and above its normal Dirac value. 

Besides a number of advantages demonstrated above, IET was also shown to have a few disadvantages that revealed themselves at high concentrations. This is the false asymptote f 3/ 2 intimated in Sections VIII.E (Fig. 3.53) and IX.A (Fig. 3.56) and the absence of the static contribution to the normalized initial distribution of transfer products proportional to the quencher concentration c (Fig. 3.33). In a similar way, IET looses the concentration dependence of the... 

The static contribution l. , incorporates heat transfer by conduction and radiation in the fluid present in the pores, conduction through particles, at the particle contact points and through stagnant fluid zones in the particles, and radiation from particle to particle. Figure 19-20 compares various literature correlations for the effective thermal conductivity and wall heat-transfer coefficient in fixed beds [Yagi and Kunii, AlC hE J. 3 373(1957)]. 

Calculations of the vibrational contributions to the static polarizability and hyperpolarizability have also been attempted. As far as the EFISH experiment is concerned, which depends on the square of an optical frequency field, it is assumed that there will be no direct contribution to (—2static electronic contribution to /1(0 0,0). An indirect vibrational effect through the linear polarizability of the solvent molecules is more important. Calculations of the vibrational effects in pNA cannot be carried out reliably even for the static case since the second term in the perturbation theory is much larger then the first and there is no evidence of convergence. 

The static contribution to the electron exchange mediated triplet energy transfer was determined by measuring the triplet carotenoid rise time for each compound in a polystyrene glass at ambient temperatures, where large-scale intramolecular motions are precluded. As expected for the similar covalent... 

The total laser-induced intermolecular energy shift is given by the sum of the two contributions (31) and (34) and applies for radiation propagating in a fixed direction relative to the two oriented molecules. A molecular average can be carried out for all possible orientations of A and B relative to each other. In this case, the static contribution (34) vanishes while the dynamic term becomes... 

A. Salam, Two Alternative Derivations of the Static Contribution to the Radiation-Induced Intermolecular Energy Shift. Phys. Rev. A 76 (2007) 063402. 

K.A. Muller on phase transitions or the Encyclopedia of Magnetic Resonance edited by Grant and Harris with, for instance, the contributions of F. Borsa, A. Rigamonti, R. Blinc and others to the high relevancy of NMR and NQR techniques to the studies of phase transitions in solid-state physics are very important sources of knowledge.18-22 The end of the section is particularly devoted to the static contribution of the average structure to the NMR lineposition and lineshape, and methods to gain resolution and sensitivity. The dynamics will be treated in the next section. 

X%r is the static contribution of the effective conductivity of the catalyst bed and has a value equal to ... 

A, is the static contribution to the effective thermal conductivity of the bed and will be discussed in detail in Sec. 11.10. The friction factor/ now remains to be specified in the pressure drop equation. 

Zehner and SchlUnder [74, 75] arrived at the following formula for the static contribution ... 

The increasing residual scattering at higher temperatures (see Fig. 9) is straightforward explained by the temperature dependent fluctuations of the interfacial surfactant layer covering the dispersed droplets (26). The fluctuations can be observed since the static contributions of the refractive index increments of the dispersed particles and the solvent (oil) are optically matched. Hence, except for the fluctuations of the surfactant molecules in the monolayer the microemulsions discussed in the present paper show a remarkable monodispersity. 

The static contribution (of permanent dipoles) to the van-der-Waals interaction 132,s is reduced when the separating medium 3 contains mobile ions. In contrast, the dispersion contribution Ai32,d = Ai32,o Ai32,s remains unaffected because the fluctuations of electrons are much faster than the Brownian motion of ions. Mahanty and Ninham (1976, as cited in Israelachvili 1992) describe the screening for Kh 1 as follows ... 

Figure 4.10.70 Effective radial conductivity of a packed bed of spherical particles for different aspect ratios c/r/c/p, gases (Pr = 0.7), and high Rgp values (Rgp = Usdp/v) lines correlation given in VDI (2002), see Eq. (4.10.194), for a value of the static contribution A,bed/ fiuid 4 area range of literature data (Yagi and Kunii, 1960 Yagi and Wakao, 1959 Calderbank and Pogorski, 1957 Demirel, Sharma, and Al-Ali, 2000 Dixon, 1985 Li and Finlayson,...

The effective dispersion coefficients of heat and mass (Xax, Dax) te calculated by the Pedet numbers [Pem,ax= Wsdp/(eDax), P h.ax = WsCpPmoidp/kax]- Both numbers are approximately 2 (Section 4.10.6.4). Correlations that also consider the static contribution are (VDI, 2002) ... 

To avoid having to treat unnecessary mathematical complications, we will adopt the simplest type of model that allows us to relate residence time and velocity distributions. It consists in an array of independent parallel channels of equal lengths L as schematized in figure 15- Each channel is representative of a particular flow path between two successive intersections. The description of the static contribution to the dispersion phenomenon is obviously quite oversimplified. 

The static contribution can be measured experimentally (11) or estimated from the model of Kunii and Smith (12). The dynamic term is obtained by first integrating the heat flux over the hemispherical surface between 6=0 and 0=90° in Fig. (6). After some algebra, the total heat flow is given by, 2... 

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