To judge on the influence on the model for all error model options the refinement with the in Chapter 3 determined strategy have been carried out. No weighting scheme was applied. To get a first impression the residuals of the least squares refinement are reviewed (Table 18). The lowest R-value is achieved by using the error model option 0. A general trend visible from Table 18 is that the options which use a g parameter per experimental run seem to exhibit a higher R-value than the error model options utilizing one overall parameter g. The R-values of the error model options 1, 3 and 5 differ from their corresponding options 2, 4, and 6 by about 0.05%. Furthermore, it can be seen that the introduction of the refinable parameter K introduces an increase in R-value (error model option 0 R(F2) = 2.43%; error model option 3 R(F2) = 2.63%). However, it needs to be noted that the R-value is influenced by a lot of factors such as the number of reflections and the data to parameter ratio. While the number of parameters is the same for all of the compared models the number of reflections is not. Therefore, also the data to parameter ratio has been compared (Table 18). It may be seen from Table 18 that the data to parameter ratio does only deviate for the error model options 5 and 6.
While for error model option 5 (48.8099) the ratio is slighlty higher than for options 1 – 4 (48.8079) it is slightly lower for error model option 6 (48.7980).
Table 18: Final R-values, Goodness-of-Fit, data to parameter ratio and highest peak and deepest hole for compound 1 using error model options 0 to 6.
Error model R(F2) GoF Ndata/Npar highest peak / eÅ-3 deepest hole / eÅ-3
0 2.43% 1.7991 48.8079 0.158(34) -0.174(34)
1 2.43% 1.7991 48.8079 0.158(34) -0.174(34)
2 2.49% 1.6980 48.8079 0.159(34) -0.178(34)
3 2.63% 2.1082 48.8079 0.161(34) -0.179(34)
4 2.68% 2.3039 48.8079 0.170(34) -0.186(34)
5 2.60% 2.2491 48.8099 0.159(34) -0.178(34)
6 2.65% 2.3165 48.7980 0.168(34) -0.189(34)
A generally accepted indicator of model quality is the residual density. From the values for the highest peak and deepest hole in Table 18 it may be seen that also these also show a narrow distribution. The lowest value of the positive residual density is given as 0.158 eÅ-3 when using error model option 0 or 1. The smallest value for the deepest hole is also achieved by using error model options 0 or 1 (-0.174 eÅ-3). However, taking the deviation into account the values do not differ significantly. Since the values for the highest peak and the deepest hole do not give any information about the spatial distribution of the residual density the fractal dimension of the residual density is examined. Figure 61 shows an overlay of the plots of the fractal dimension versus the residual electron density for all refined models. It needs to be noted that this these plots are displayed at a different scale (X-axis range -0.2 to 0.2 eÅ-3) than usual (X-axis range from -1 to 1 eÅ-3). The highest maximum of the fractal dimension is achieved by using error model option 0 (Df(max) = 2.764). As discussed in Chapter 3 the plot of the fractal dimension of the residual density for error model option 5 exhibits a small shoulder in the negative region. It can be said that the main differences can be spotted in the regions
Figure 61: Plot of the fractal dimension of the residual density for all error model options. Error model 0 Df(max) = 2.764; error model 1 Df(max) = 2.764; error model 2 Df(max) = 2.761; error model 3 Df(max) = 2.758; error model 4 Df(max) = 2.756; error model 5 Df(max) = 2.759; error model 6 Df(max) = 2.757.
of high positive and negative residual density. The error model options 4 (cyan) and 6 (black) seem to perform slightly worse than the other options. These options show a less featureless distribution. The error model options 0 (green) and 1 (red) are exactly the same which is why the red curve can not be seen in Figure 61. However, the shape is almost paraboloid and sufficiently flat and featureless for all models. Also the number of gross residual electrons within the unit cell egross
shows that all models perform good. The lowest value for the egross is achieved by using the error model option 0 or 1. Furthermore the same trend as from the fractal dimension plots is visible. Error model options 5 and 3 are almost even. Error model option 2 performs slighty better than error model options 5 and 3. The error model options 4 and 6 perform worst. The overall variance of the egross equals to 1.5 %
which can be considered a low value.
In addition the normal propability plots for all error model options were analysed. It can be seen from Figure 62 that the distribution of the squared differences between expected and experimental structure factors is not normal for
Table 19: egross for
compound 1.
Coefficient of variation:
1.5%.
Error model egross
0 30.818 1 30.818 2 31.137 3 31.679 4 31.961 5 31.595 6 31.823
Figure 62: Overlay of the normal propability plots for error model options 0 to 6 for compound1.
the error model options 3 to 6 (Figure 62). For the error model options 0 to 2 the distribution does also deviate from normal distribution but not as pronounced as for the other options.
The analysis of the resolution dependence of the ratio between mean observed and calculated structure factors reveals that there is an influence on it. For error model option 0, 1 and 2 no value does exceed a variation of 5% from unity. The other error model options however do only slightly exceed this value (maximum deviation of∑
∑ for error model option 3 0.9482, error model option 4 0.9456, error model option 5 0.9485 and 0.9456 for error model option 2). When comparing the course of the plots in Figure 63 it may be seen that the main effects of the error model can be seen in the low resolution range from 0.0 to 0.2 sin(θ)/λ.
Besides the model quality indicators also the refined parameters need to be closely examined. The properties calculated after the multipole model refinement are solely based on these parameters. Thus it needs to be revealed if deviations of these parameters are introduced by the error model. To test for deviations bigger than 3σ between the different models an automated program was written. The program first plots the parameter values for all error model options and then tests if the prameters deviate more than 3σ from each other. For the present compound the
a) b)
c) d)
e) f)
g)
Figure 63: Resolution dependence of the ratio between mean observed and calculated structure factors. a) error model option 0; b) error model option 1; c) error model option 2; d) error model option 3; e) error model option 4; f) error model option 5; g) error model option 6.
monopole population and the population of the octupole O3+ for nitrogen atom N1 and thus also the thereupon constraint mono- and octupole population N4 deviate more than 3σ. Furthermore, the values of the expansion-contraction parameter κ for the nitrogen atoms N1 and N4 deviate (Figure 64). The monopole parameter M1 for nitrogen atom N1 shows a significantly higher value when using error model option 0, 1 and 2 whereas the linked expansion-contraction parameter κ shows a
Figure 64: Plots of the values of the parameter a) monopole M1 for N1, b) κ1 for nitrogen atoms N1 and N4 and c) octupole O3+ for the nitrogen atoms N1 and N4 d) scale factor number 1 e) scale factor number 1 f) scale factor number 5 in dependence of the error model option used. Error bars indicate 3σ range.
a) b)
c) d)
e) f)
significantly lower value. The graphs in Figure 64 a) and b) show an inverse relationship. This is of course a consequence of the correlation between these two parameters. The trend observed from the model quality indicators can also be found in these plots. The error model options 0 to 2 form a group while a second group is formed by the error model options 3 to 6. Within the second group the absolute values of the error model options 3 and 5 are almost even. The same is true for options 4 and 6.
In summary it can be stated that for compound1 only slight differences in the refined model are detectable. However, it needs to be examined if this does have an impact on the thereupon derived properties.