Reference distance

Usually, simplified representations of the data are used to obtain preliminary structures. Thus, lower and upper bounds on the interproton distances are estimated from the NOE intensity [10], using appropriate reference distances for calibration. The bounds should include the estimates of the cumulative error due to all sources such as peak integration errors, spin diffusion, and internal dynamics. 

Although the partial differential equation Eq. 25-10 is linear and looks rather simple, explicit analytical solutions can be derived only for special cases. They are characterized by the size of certain nondimensional numbers that completely determine the shape of the solutions in space and time. A reference distance x0 and a reference time f0 are chosen that are linked by ... 

In order to discuss the properties of this solution, again it is helpful to transform Eq. 25-28 into a nondimensional form by choosing a reference distance x0 and applying Eq. 25-16 ... 

The latter technique is particularly appreciated by coordination chemists because of its simple mathematical formulation and the limited amount of experimental data to be collected (longitudinal relaxation times at a single magnetic field obtained by the inversion-recovery method (Void et al., 1968)). However, the a priori choice of a reference distance (rref) requires a partial modeling of the complex and the extracted distances strongly depend on the accuracy of this original choice. 

The quantities B (Rq) and are treated as adjustable parameters and Rq is a reference distance that can be chosen arbitrarily. In principle, the number of ligand shells considered for the calculation of intrinsic parameters is not limited, however, it is usually assumed that only the nearest neighbors of the rare-earth ion contribute significantly to the crystal-field potential. Thus, especially long-range interactions like electrostatic interactions are not accounted for explicitly. Because these interactions are most important for k = 2 parameters, in many cases only the k = 4, 6 intrinsic parameters have been considered. 

Because the NOE is due to dipolar interactions, its intensity, a, depends on the inverse sixth power of the distance r between the protons. As any macromolecule will contain protons at fixed distances, one should be able to calculate an unknown distance by comparison of its corresponding NOE with the NOE measured between protons at some reference distance rref from the simple relation... 

This treatment ignores the fact that the reference distance and unknown distance may not be subject to the same motions2 and assumes that only pairwise interactions contribute to the measured intensities.41 Now, given that one can estimate distances in a molecule, there are several ways this can be turned into a penalty function. 

It has been known for a long time that the kind of simplistic distance calibration suggested by Eq. [8] may be subject to systematic errors. First, the intensity of an NOE depends on the spectral density function for the reorientation of the vector between relaxing nuclei. This means that Eq. [8] is valid only if the reference distance and unknown distance are undergoing the same motions. As this is not likely to be the case, distance calibrations have attempted to allow for the possibility of systematic errors.23 Equation [8] also assumes that the dipolar relaxation can be considered in terms of isolated spins relaxing each other. In the presence of spin diffusion, this will lead to a systematic underestimation of distances.41 57 58... 

Shape factors—The normalized distance measurements in a body to one arbitrarily chosen reference distance in the body. 

The reference air kerma rate of a source is the kerma rate to air, in air, at a reference distance of one metre, corrected for air attenuation and scattering. This quantity is expressed in pGyh at 1 m. 

It is similarly problematic to define bonding on the basis of measured or calculated atomic distances. In order to identify a certain interatomic distance as typical of covalent bonding, suitable reference distances have to be found. Normally, this is not very difficult in cases where there is no doubt about the nature of bonding. However, for nonclassical structures or in the case of exceptional long bonds or bonds between unusual atoms the problem of the reference bond cannot be solved in an unique way. The same difficulties arise when other molecular properties are used to define and to describe chemical bonding [17]. 

The reference distance between the coordinate origins of the two charge distributions is R. The interaction is therefore conveniently expressed as the inverse Fourier transform of the convolution of the charges and the Coulomb interaction... 

The most used approach to distance measurements stems from equations 8.13 and 8.14 above and relies on a known reference distance, rxv. from which others may be calculated [26], and the assumption of uniform isotropic molecular tumbling. Recall the basic equation was ... 

All variables are functions of both time and distance. Hence at any time, t, the values at the reference distance x are ... 

To derive the equation for the conservation of eneigy, we proceed in the same way as we derived the conservation of mass equation, regarding all variables as functions of both time and distance. Hence at any time, r, the values at the reference distance x are ... 

The model potential is a function of the deviation (x-Aq) from the reference distance Xq, which is the equilibrium value when a is zero. The quadratic and cubic force constants /2 (positive) and (negative) are taken as the same for all molecules in a related series, whereas a is a perturbation expressing steric and electronic differences in ligand-metal interactions between related molecules (dotted line in Figure 5.16). The perturbation a affects AE 2md shifts the equilibrium distance to Xe = Xo + Aaq. In linear approximation the relationship is... 

Fig. 5.19. Acetal C-O distances perturbed by the stereoelectronic effects of various leaving groups. Empty circles observed filled circles calculated from two-dimensional model discussed in text. The solid line is a curve of constant bond order. The diagonal is at Ar, = - Ar2. The Ar, s are deviations from reference distances [49]...

Fewer data are available for germatranes (Table 7.4). The general trends are similar to those observed for the silatranes (compare Figure 7.16 to Figure 7.15) The correlation between the Ge-N distances and XGeO-angles is smooth. The constants used to obtain the curve in Figure 7.16 are Omax = 109.47°, c (Ge-N, ref) = 1.69 A, a = 1.01 A. The Ge-X distances follow the expected trend, but their scatter does not warrant a correlation curve to be drawn. The reference distances used to compare Ge-C with Ge-0 and Ge-Br distances are 1.89(1) [24], 1.75 [26], and 2.276(2) A [28], respectively. 

The widths of the assemblies were used as the reference distances for the calculation of their lengths from the hydrodynamic radii, when corrected for the ellipsoidal shapes of the structures. 

All summations are over nearest neighboring carbon atoms, here denoted by (mm ), t is the hopping integral between nearest neighbor carbon atom n and m, and to is the hopping at a reference distance Rq. The actual interatomic distances are denoted R n = Rn — Rrf. ... 

When low dispersible material is subjected to the high velocity impact test, particulate matter can be generated, but of aU airborne particulates up to 100 pm only a small (less than 10%) fraction will be expected to be in the respirable size range below 10 pm if the 100 limit is met. In other words, an equivalent quantity of low dispersible material less than 10 A2 could be released airborne in a respirable size range. It has been shown that for a reference distance of around 100 m and for a large fraction of atmospheric dispersion conditions this would lead to an effective dose below 50 mSv. 

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