Auxiliary coefficient

The GEM force field follows exactly the SIBFA energy scheme. However, once computed, the auxiliary coefficients can be directly used to compute integrals. That way, the evaluation of the electrostatic interaction can virtually be exact for an perfect fit of the density as the three terms of the coulomb energy, namely the nucleus-nucleus repulsion, electron-nucleus attraction and electron-electron repulsion, through the use of p [2, 14-16, 58],... 

For molecules containing light atoms, we accordingly neglect this effect of finite nuclear volume or field shift, but other effects prevent exact application of isotopic ratios that one might expect on the basis of a proportionality with in formula 13 instead of total F. For this reason we supplement term coefficients in formula 8 for a particular isotopic species i with auxiliary coefficients [54],... 

We examine a few particular radial coefficients in detail. Coefficients Mq pertaining to adiabatic corrections in Y R) appear in only an expression for auxiliary coefficient Zq q, explicitly in this form [55] ... 

According to apphcation of Dunham s formalism to analysis of molecular spectra, as for GaH and H2, these radial coefficients of seven types represent many Dunham coefficients Ym and their auxiliary coefficients Zki of various types that collectively allow wave numbers of observed transitions to be reproduced almost within their uncertainty of measurement through formula 54. Mostly because of inconsistency between reported values of frequencies of pure rotational transitions [118,119], the reduced standard deviation of the fit reported in table 3 is 1.25, slightly greater than unity that would be applicable with consistent data for which uncertainty of each measurement were carefully assigned. 

This condition is similar to that used by Onsager in his own unpublished derivation of the result, Eq. (34), for the spontaneous magnetization of the Ising model. It is to be noted, however, that Devinatz retains conditions (d), (e), and (/). In our theorems, on the contrary, we retain condition (c) but replace (/) by another (slightly stronger) condition on the auxiliary coefficients p and qn defined in Eqs. (29) and (30). To state this we define the monotonically decreasing coefficients... 

Auxiliary coefficient in TDMA outline Partial pressure of species i... 

Auxiliary Data ivlass Absqrption Coefficients Sodium... 

The easiest way to obtain the coefficient of uni l 1 vn l 1 is to differentiate rq — l — 1 times with respect to u, and n2 — l — 1 times with respect to v. This causes all terms of lower degree in each auxiliary variable to vanish. Then by placing u and v equal to zero all terms of higher degree vanish, and the desired coefficient remains, multiplied by ( 4-Z- 1) (n2-l- 1) . 

The mass transfer equations, Equations (ll.l)-(ll.lO), remain valid when A, replaces A,. Equations (11.27) and (11.28) contain one independent variable, 2, and two dependent variables, ai and Ug. There are also two auxiliary variables, the interfacial compositions a and a. They can be determined using Equations (11.5) and (11.6) (with A, replacing A). The general case regards K/f in Equation (11.4) as a function of composition. When Henry s law applies throughout the composition range, overall coefficients can be used instead of the individual film coefficients. This allows immediate elimination of the interface compositions ... 

Because of the irrelevant condition sy/b s ), the reverse model, viz. the estimation of x from /, according to Eq. (6.7) and therefore the coefficients by and ay> are not of direct relevance in analytical calibration. Notwithstanding, their estimates will be given here for completion and as auxiliary quantities for further calculations ... 

As the reaction rate may not be assessed directly, the definition invokes an auxiliary parameter pk (e.g., an enzyme concentration) that is assumed to act only on the rate v. Note that X may stand for an arbitrary steady-state property with the coefficients for concentrations Cs and flux CJ as the most important examples. 

Finally, and more profoundly, not all properties require explicit knowledge of the functional form of the rate equations. In particular, many network properties, such as control coefficients or the Jacobian matrix, only depend on the elasticities. As all rate equations discussed above yield, by definition, the assigned elasticities, a discussion which functional form is a better approximation is not necessary. In Section VIII we propose to use (variants of) the elasticities as bona fide parameters, without going the loop way via explicit auxiliary functions. 

Equations (9) and (11) indicate how the auxiliary radial function for the rotational factor becomes separable into contributions from atomic centres of types A and B. An analogous separation is practicable for both the vibrational g factor and the total adiabatic corrections for the latter quantity this separation is effected in the original quantum-chemical calculations. Accordingly we express these calculated values of rotational and vibrational g factors, presented in Table 1, and adiabatic corrections, presented in Table 3, of He H" to generate coefficients of radial functions for atomic centres of either type. He or H. The most useful variable for these functions is z, defined in terms of instantaneous R and equihbrium R internuclear distances as... 

The new indices i, i are auxiliary in the sense that they do not appear in the final coefficient tensor VF and must be contracted over in some fashion. The simplest arrangement is to contract the indices sequentially from one ij/n tensor to the next. 

Because of the choice of enumeration, the vectors of logarithms of reaction rate constants form a convex cone in which is described by the system of inequalities lnfc2i> lnfc,y, (/,/)t (2,1). For each of the possible auxiliary systems (Figure 4) additional inequalities between constants should be valid, and we get four correspondent cones in These cones form a partitions of the initial one (we neglect intersections of faces which have zero measure). Let us discuss the typical behavior of systems from these cones separately. (Let us remind that if in a cone for some values of coefficients dp then,... 

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