Standard Deviations from Least-squares Refinements

Standard Deviations from Least-squares Refinements. The standard deviations for the parameters are obtained by taking the square root of the diagonal elemenents of the moment matrix for the parametors  

The correlation coefficients are obtained from the off-diagonal elements of 

The weight matrix which gives minimum variances for the parameters is proportional to (M )-  

Pearsen and H. O. Hartley, Biometric Tables for Statisticians, 3rd edn, Cambridge Univ. Press, Cambridge, 1966, Vol. 1 see also ref. 34rf. 

The electron-difiraction data have been shown to be highly correlated as one would expect. The correlation depends on the experimraital background. Assuming that depends only on z = — si, Sdp et al.  

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These standard deviations will be denoted by in the following while standard deviations from least-squares refinements will be given as cr when a distinction is necessary. Using one set of plates the standard deviations (7 are quite small, sometimes even smaller than found by least-squares refinement with a diagonal weight matrix on the average intensity curve. ... 

SHELXL (Sheldrick and Schneider, 1997) is often viewed as a refinement program for high-resolution data only. Although it undoubtedly offers features needed for that resolution regime (optimization of anisotropic temperature factors, occupancy refinement, full matrix least squares to obtain standard deviations from the inverse Hessian matrix, flexible definitions for NCS, easiness to describe partially... 

When bond lengths with different chemical sorrounding are compared to each other, the experimental error limits for the individual values are of great importance. It has become common practice in ED studies to report estimated uncertainties, which usually are 2 to 3 times the standard deviation of the least-squares refinement. It is assumed that such uncertainties account for possible systematic errors. Most MW studies also report estimated uncertainties, which may be considerably larger than the standard deviations derived from fitting the rotational constants. In X-ray crystallography it is common use to cite standard deviations from the least-squares analysis of the structure factors. 

ESD is the estimated standard deviation obtained from the least squares refinement correlation... 

One way to assess the precision of a structure determination is to use the estimated standard deviation (e.s.d.) of the geometric quantity of interest. This is obtained from the least-squares refinement (see Chapter 10). The more precisely a measurement is made, the smaller is the e.s.d. of that measurement. The equations for calculating the e.s.d. values of a bond length and of a bond angle are given in Figure 11.8. 

Bond angles (interbond angles) can be calculated by use of Equation 11.4. Estimated standard deviations for atomic coordinates are derived from the least-squares refinement, as described in Chapter 10. These can be applied to the bond lengths and interbond angles. 

Figure 7.39. Relative shifts of individual atoms (left-hand scale) and average shift (A) to standard deviation (cr) ratio during the first 12 cycles of the least squares refinement of the coordinates of all atoms in the model of the crystal structure of the anhydrous monoclinic FeP04. Both the P-0 distances and O-P-0 angles were soft restrained with the weight of 4 during the first five cycles. The weight was set to 10 beginning from cycle No. 6. The first five cycles indicate erratic shifts of P and O atoms. Beginning from the sixth cycle, the shifts of all atoms steadily decrease and approach zero after cycle No. 12.

The structural parameters (r, u, sometimes k, and possibly other parameters) are then obtained by least-squares fitting, usually on the intensity curves, but occasionally on the RD curves (c/. p. 45). If the intensity data are used, the data from all the nozzle-to-plate distances are not necessarily combined into one composite curve an individual scale factor may be refined for each set of data. Since the observed data are considerably correlated, i.e. each point cannot be regarded as an independent observation, a number of problems are encountered in the application of the least-squares method. This problem is most important for the estimation of the standard deviations, since the parameters may be obtained fairly satis-factorally by conventional least-squares refinement. (A more detailed discussion is set out on p. 45.)... 

It is possible to carry out the least-squares refinement treating some of the parameters formally as observables, i.e. the sum S [equation (77)] is increased if the parameters deviate from given values. However, in most cases the calculation is carried out simply with fixed values for some of the parameters. The least-squares criterion may still be applied, but the uncertainties in the fixed parameters should be included in the standard deviations, for example by the methods suggested by Kuchitsu or by Vilkov. A somewhat simpler approach, which probably is satisfactory in most cases, is described below. ... 

Two sets of intensity data were colleded at different neutron wavelengths. They were fully corrected for absorption and extinction errors. Least-squares refinement started from the parameters from a good X-ray analysis (1966), and converged at an K-value of 2.2%. All seven hydrogen atoms were located with standard deviations of their positions of. 002 A. All amino-hydrogen atoms take part in N-H O bonds, details of which are set out (in the usual form X - O, X-H, H O, ZX-H O) —... 

Least-squares refinement yields not only the structure parameters, but also their standard deviations. The R values defined above can serve as figures of merit if w represents the weighting function employed in the refinement. A more important point for assessing the quality of a structure analysis, however, is that the results fit with the wealth of experience of crystallographers and chemists, and deviate from prior knowledge only in justifiable cases. 

The geometrical parameters as determined in the electron diffraction analysis are shown in Table 7 The results appeared to be little sensitive to the changing conformational properties of the models. It was of great advantage that the 0...0 distance with great accuracy was available from the microwave spectra. Besides, in addition to the requirement of convergency in the least-squares refinement, an important criterion for any acceptable structure was the consistency with the rotational constants derived from the microwave spectra. It is stressed, however, that such a criterion was considered to a certain limit only. Discrepancies up to 2% were acceptable since the parameters were not corrected for the effects of the intramolecular motion, and, what may be even more important, the standard deviations were relatively large. 

The structure of the closely related molecule, 1,2-cyclopentenophen-anthrene, has been determined and refined with partial three-dimensional data by least-squares methods by Entwhistle and Iball (1961). Independent confirmation of the correctness of this structure has been provided by Basak and Basak (1959) who did not, however, carry out any refinement of the structure. Entwhistle and Iball s results show that the molecule is not planar the deviations of the carbon atoms from the mean molecular plane are shown in Fig. 9 (the standard deviations of the atomic coordinates lie between 0-009 and 0-015 A). The three aromatic rings appear to be linked in a slightly twisted arrangement. Atoms H and K, which are bonded to the overcrowded hydrogen atoms, are displaced almost the same distance on opposite sides of the mean plane. In the five-membered ring, atoms C and E are below the molecular plane by about 0-10 A while atom D lies 0-18 A... 

The standard deviations for each refined parameter according to the least squares method are calculated from... 

Typically, 10 ml of 0.5 to 10 mM solutions of the samples were preacidified to pH 1.8-2.0 with 0.5 M HCl, and were ttien titrated alkalimetrically to some appropriate high pH (maximum 12.0). The titrations were carried out at 25.0 0.1 °C, at constant ionic strength using NaCl, and under an inert gas atmosphere. The initial estimates of pKg values were obtained from Bjerrum difference plots (nH vs. pH) and then were refined by a wei ted nonlinear least-squares procedure (Avdeef, 1992,1993). For each molecule a minimum of three and occasionally five or more separate titrations were performed and the average pXa values along with the standard deviations were calculated."... 

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