For the multipole model refinement, the existing IAM refinement was used as a starting point. The model was modified so that the hydrogen atoms were no longer set with an HFIX command on idealised positions but the positions thereof were
Figure 22: Definition of the Li-N bond distances d1 and d2, and the distances d3and d4 from the lithium atoms to the RNR plane.
-2
identified by difference Fourier analysis. The resulting IAM was taken and the hydrogen atoms were moved to neutron distances along their bond vectors.[79]
On the basis of this modified routine structure the starting model for the multipole refinement with the XD2006 program suite[29] has been created with the XDINI program.
The resulting *.mas files need to be modified so that local coordinate system, highest order of multipoles and order of thermal parameters fit to the necessities of the current compound. For the carbon and nitrogen atoms the highest order of multipoles was set to hexadecapoles. Different expansion-contractions parameters and ′ were used for the individual atom types and atoms of the same type in different chemical environments. The local coordinate systems were set up so that the highest reasonable symmetry could be used (see Table 5). The hydrogen atoms were refined with monopole and bond directed dipole and quadrupole. The expansion-contraction parameters and ′ were fixed to the values proposed by Volkov et al ( = 1.1; ′ = 1.18).[80] As previous studies showed, refining valence
Table 5: Definition of the local coordinate systems (columns one to five) for the non-hydrogen atoms of compound 1. Maximum level of thermal parameters (TP), maximum level of multipoles (LMX), local symmetry (Site Symm) and chemical constraints (Chem Con)
Atom Atom 1 Ax1 Atom 2 Ax2 R/L TP LMX Site Symm Chem Con
density at alkaline metals proves to be challenging[81–86]. Refinement of a monopole for the lithium atom has been tried but initial steps suggested a lack of refinable valence electrons.
Hence, lithium was refined as Li+ using the entry for the cation in the database assembled bySu, Coppens and Macchi[87] included parameters are refined. At the end of the block all current parameters are refined together and the next block starts.
As discussed in Chapter 1.5 the choice of refinement strategy and the set of refined parameters is crucial to get the best results. Therefore, k-fold cross validation[24,26,88] was used to judge on the refinement strategy. Specifically, it was used to test whether refining Gram-Charlier coefficients up to third order, loosening chemical constraints and loosening the local symmetry of the atoms would overfit
Table 6: Detailed description of the refinement strategy (Abbreviations: Sca: scale factor (refined in every step, only mentioned in the first one); CC: chemical constraints; LS:
local symmetry; SIGOBS: data with / smaller than the number is excluded from the refinement; M: monopoles; D:
dipoles; Q: quadrupoles; O: octapoles; H: hexadecapoles; ,
′: expansion-contractions parameters; (H)XYZ: positional parameters; U2: displacement parameters; U3: third order Gram-Charlier coefficients).
the data. A strategy was set up that includes all the above mentioned groups of parameters using the block refinement approach (cf. Table 6).
First all monopoles and multipoles are introduced on top of the IAM geometry (steps 1 to 4). Afterwards the thermal parameters and coordinates of the non-hydrogen atoms were added in steps 5 to 10. After every refinement of the coordinates of the heavy atoms the hydrogen atoms were reset to neutron distance along their bond vectors.[79] Then the expansion-contraction parameter , coordinates of the hydrogen atoms and ′ were gradually introduced (steps 11 to 21) and refined. In the next four refinement steps the cut-off was lowered to zero so no data was excluded from the refinement. After this the local symmetry of the atoms was disregarded (steps 26 to 29) and third order Gram-Charlier coefficients were introduced (steps 30 to 35). At last the chemical constraints have been taken out of the refinement. Subsequent to the refinement the increase or decrease inRwork
andRfree is calculated for every refinement step (Figure 24).[24,26]
Figure24:CourseoftheRfreeoverthewholerefinementstrategy.
As already discussed (cf. Chapter 1.5) a decrease in Rwork in combination with an increase in Rfree is used as a sign for overfitting It can be seen that for all steps prior to the loosening of the chemical constraintsRfree andRwork both either increase or decrease. The slight increase in R-value in the steps 23, 24 and 25 is a consequence of the inclusion of (weak) data. This indicates that neither loosening the local symmetry of the atoms nor introducing ten third order Gram-Charlier parameters to the methyl carbon atoms has to be seen as overfitting the data.
However, the steps 36 and 37 show a different picture. The disregarding of the chemical constraints shows an increase in Rfree while the Rwork decreases. As can be seen from Table 5 only six heavy atoms were refined when chemical constraints were enabled. This means that the number of multipole parameters increases by a factor of more than four (from 150 to 650). To avoid this overfitting, the loosening of the chemical constraints was removed from the strategy.
In addition to the strategy explained above another strategy was tested where the local symmetry was loosened after the introduction of the third order Gram-Charlier coefficients. This was done to test if the change in order results in the conclusion that loosening the local symmetry already overfits the data. This was not the case so the final strategy consists of the steps one to 35 in Table 6.