Observed and calculated structure

Table 1. Observed and calculated structure factors far Ag3Co(CN)e.

The results of the X-ray study are contained in Tables II-IV. Anisotropic B s and a table of observed and calculated structure factors are available (H)). 

The difference Ap(r) between the total electron density p(r) and a reference density pref(r), is a measure for the adequacy of the reference density in representing the system. Difference densities Ap(r) are obtained by Fourier summation in which the coefficients AF are equal to the difference between the observed and calculated structure factors. If k is the scale factor, as defined in chapter 4, the difference structure factor AF is given by... 

Figure 3 Comparison of observed and calculated structures of 1,2,3,4-thiatriazoles (a) observed structure of 5-amino-1,2,3,4-thiatriazole (10) (b) calculated CNDO model of 1,2,3,4-thiatriazole.

The residual index, or R factor, gives a measure of the difference between the observed and calculated structure factors and therefore of how well the structure has refined. It is defined as... 

Table 1. Observed and Calculated Structures for Some Cyclopropanes and Cyclobutanes...

Another useful equation due to Cruickshank (1950) is the relation of the standard deviation of electron density a(p) and the discrepancies between observed and calculated structure factors ... 

Normally, if the assumed model for a crystal structure has an R value of 0.5 and resists attempts to refine to a lower residual, then the model structure is rejected as false, and a new model is tried until a fit between the observed and calculated structure factors yields an acceptable residual (R < 0.25). (Other models were tried for this complex, but they either gave Fourier maps which were uninterpretable or they converged to the present model). However, the normal crystal structure is solved with data obtained from crystals which have dimensions of the order of 0.1 mm. In the crystals available for this experiment, two of the dimensions were of the order of 0.01 mm. Thus, long exposures were required to give a small number of relatively weak diffraction spots. (Each Weissenberg photograph was exposed for five days with Cuka radiation 50 kv., 20 ma. loading, in a helium atmosphere). 

The results of this crystal structure analysis should then be viewed as having a reliability intermediate between standard single crystal structures where R is often 0.1, and crystalline polymer structures where only qualitative agreement is expected between observed and calculated structure factors. 

Table II. Atomic Coordinates of Model Structure and the Observed and Calculated Structure Factors...

Structure refinements of the AH 200 and AH 300 samples were conducted in the same way. Unit cell constants, final atomic parameters, and R indexes are given in Table I. (Observed and calculated structure factors are available from the authors.) Interatomic distances and angles are given in Table II. Estimated errors on the population and position of the cations may in some cases be greatly underestimated especially for atoms with low occupancy factors. 

A listing of observed and calculated structure factors is available for all 4 crystals by interlibrary loan from Hamilton Library at this university see your librarian. 

As far as the structure of 6GT is concerned, a fully extended planar chain fits well in the simple triclinic unit cell this is confirmed by the good agreement between observed and calculated structure amplitudes for electron diffraction (10). 

Two approaches have been applied to estimate the net charges of atoms from the observed electron-density distribution in crystals. The first method is a direct integration of observed density in an appropriate region around an atom (hereafter abbreviated as DI method) (64). The second is the so-called extended L-shell method (ELS method) (19, 81) in which a valence electron population of an atom is calculated by a least-squares method on the observed and calculated structure amplitudes. 

To fit the 6-fold symmetry, the tripod is disordered between two orientations. Addition of fixed atomic positions for the TPA(OH) complex reduced the weighted R from 0.14 to 0.06, mainly because of changes in calculated intensities for low-angle intensities. Tables II and III give the atomic coordinates and interatomic distances. A listing of the observed and calculated structure factors is available as supplementary material. 

Some refinement procedures include, at certain stages, criteria for stereochemical regularity in addition to the agreement between observed and calculated structure factor magnitudes. 

Check agreement of observed and calculated structure factors. 

At this point it must be clear to the reader that the electron density maps or other Fourier maps obtained are only as good as the relative phases used in their calculation. The structure amplitudes are important in this calculation, but if the relative phases are not approximately correct, the crystallographer may be in the situation of refining an incorrect structure, often to a quite reasonable result, but to a local minimum in deviations of observed and calculated structure amplitudes. Calculated relative phases... 

Maximum likelihood method, similar to maximum entropy, works in reciprocal space and results in finding a raw model that has the best chance to be improved by applying small steps to achieve full agreement between the observed and calculated structure amplitudes, rather than the final structure. Maximum likelihood and maximum entropy techniques are often combined together. - ... 

A serious complication arises, however, in the application of the least squares approach to structure refinement using X-ray diffraction data. (One seems to appear in almost every aspect of X-ray crystallography, doesn t it ) Here we are trying to minimize the difference between the observed and calculated structure amplitudes Fobs and Fcaic that is, Fobs — Fcalc j or more properly, the sum of their squares taken over the entire data set. 

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