Structural parameters, standard deviation

Within this group of structures, none of the different combinations of factors gave an R-value significantly worse - at the 5% level of probability - than the best. Thus there is no justification for choosing any one in preference to any other, and so the mean was taken as the best structure. The standard deviations of the parameters of this model were calculated from the overall variance within a structure factor set. That is, the contribution to uncertainty due to errors in the structure factor set have been omitted. Thus these standard deviations are almost certainly underestimates, for whilst the structure factors of Yokouchi et al might be significantly better than those of the other authors, they are certain to contain some error. 

The atomic parameters from the last refinem t are given in Table 1. The observed structure factors, standard deviations, and differences are given in Table 11. ... 

Experience has shown that correlations of good precision are those for which SD/RMS. 1, where SD is the root mean square of the deviations and RMS is the root mean square of the data Pfs. SD is a measure equal to, or approaching in the limit, the standard deviation in parameter predetermined statistics, where a large number of data points determine a small number of parameters. In a few series, RMS is so small that even though SD appears acceptable, / values do exceed. 1. Such sets are of little significance pro or con. Evidence has been presented (2p) that this simple / measure of statistical precision is more trustworthy in measuring the precision of structure-reactivity correlations than is the more conventional correlation coefficient. 

Two models of practical interest using quantum chemical parameters were developed by Clark et al. [26, 27]. Both studies were based on 1085 molecules and 36 descriptors calculated with the AMI method following structure optimization and electron density calculation. An initial set of descriptors was selected with a multiple linear regression model and further optimized by trial-and-error variation. The second study calculated a standard error of 0.56 for 1085 compounds and it also estimated the reliability of neural network prediction by analysis of the standard deviation error for an ensemble of 11 networks trained on different randomly selected subsets of the initial training set [27]. 

Figure 8.38. Structural parameters of an ensemble of needle-shaped soft domains in a poly(ether ester) as a function of elongation . D (open circles) is the average needle diameter, a/D (filled circles) is the relative standard deviation of the needle-diameter distribution. Square symbols demonstrate the lateral compressibility of the soft needles during elongation... 

It is important to be able to assess the accuracy of a new or published structure from the relevant standard parameters (R value, estimated standard deviations). The review by Jones (1984) is a useful antidote to uncritical acceptance of published figures. Finally, all, or almost all, the bond lengths, angles and torsional angles in a structure will be normal. If the object of the exercise was to determine whether a particular structural feature is or is not normal, a comparison with appropriate controls will be necessary. For bond lengths a compendium of standard values is available (F. H. Allen et al., 1987). 

The molecular structure of bis(trifluoromethyl) peroxide can be determined by electron diffraction spectrometry. The following molecular parameters and their estimated standard deviations (3a) are obtained by this method Bond distances in pm units are 0-0 141.9 2.0, C-O 139.9 0.9 and C-F 132.0 0.2 bond angles are ZC-O-O 107.2 1.2°, ZF-C-F 109.0 0.5° and dihedral angle C-O-O-C 123.3 14.0°. A staggered conformation with a small tilt for the group axis is observed for the CF3 groups. A C2 molecular structure may be assumed . ... 

Pei et al. looked at the structure of the orthorhombic phase of Ba1 xKxBiOs powders by time of flight powder neutron diffraction (59). Neutron diffraction intensities should be considerably more sensitive to oxygen atom positions than are X-ray diffraction intensities for compounds such as this. The structural parameters at 300K for x=0.1 and x=0.2 are presented in Table 5. The crystal structure is the same as that of BaPbOs (Table 2) with small differences in cell parameters and atomic coordinates. The refinements show that the atoms are displaced from the ideal positions by many standard deviations, and the thermal parameters for the oxygen atoms make physical sense. The crystal structure is one in which relatively regular Bi-O octahedra (Bi-O distances... 

Serious correlations can sometimes be reduced by a careful investigation of the corresponding dF/dpitj and their dependence on different classes of hkly regions in reciprocal space, etc. and a critical elimination of those reflections for which both derivatives are nearly proportional to one another. This method was successful for a highly pseudo-symmetric structure of type la with several r ranging between 0.80 and 0.98 (19). Reducing the numbers of observations increases the estimated standard deviations of the parameters. Values of r < 0.4 or 0.5 are not serious. 

In the various tables of crystallographic data presented throughout this chapter, the mean values of the obviously chemically equivalent structural parameters arc reported. The larger of the standard deviations from the mean ([2, (x,- - x)2/(N — 1)],/2) or the reported standard deviations (as derived from the inverse matrix) are given. 

A typical CC-measurement procedure (for a pH-sensitive LAPS structure) is depicted in Fig. 6.3. From the raw data material of the CC-mode measurement of all measurement spots under the pH-sensitive layer, a calibration plot can be derived. For example, for the above example, an average pH sensitivity of 54.2 mV/pH with a standard deviation of 0.5 mV/pH between the different measurement spots can be calculated. This initial calibration measurement allows furthermore the determination of different measurement parameters, e.g., the hysterisis, overall drift, stability, selectivity and the influence of external disturbances such as light and temperature. These parameters are important to evaluate the performance of the complete LAPS-based measurement system. 

The number of parameters adjusted was generally in the range 10-30 (7 to 51 in the x-ray cases reviewed by Young (6.)). The standard deviations in atom position parameters are typically 1-4 parts in 1,000 in the x-ray cases reviewed. Fig. 3 shows the x-ray patterns for a fluorapatite, CasfPOOsF, case in which 28 parameters were varied (20 of them structural), Rp= 12.1%, and RWn= 13.9% (4.). In this Fig., the observed pattern is indicated by the data points with vertical error bars, the calculated pattern is the continuous curve overlying them, the difference between observed and calculated patterns is shown by the lower curve, and the positions of Bragg peaks are shown by the vertical bars between the upper and lower curves. In this case the fit both looks to be very good and is so indicated by the R value. 

Standard Deviation of Structural Parameters Caused by Main Factors in Structure Analysis... 

The structural parameters of this model, together with two standard deviations, are given in Table Vll. If this is compared with Table 1, and the reasonable assumption made that similar uncertainties apply there, then it is seen that the differences between the original models are not significant, and that they do not differ significantly from this new one. 

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