Observed and calculated structure factors

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... 

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... 

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. 

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. 

X,R represent the set of atomic positions and vibrational parameters, respectively. Fobs and ivaic are respectively the observed and calculated structure factor amplitudes. R is the crystallographic residual between Fobs and Fcak-... 

In a crystallographic problem the parameters to be determined are the positional and thermal parameters for each atom (x,y,z,B). The observables are the structure factor amplitudes F bs(fj), where h represents the reflection hkl. Following the method of Lagrange, the best parameters are those for which the sum of the squares of the differences between the observed and calculated structure factors is a minimum, i.e.,... 

For a random structure, / = 0.83 for a centric distribution and R = 0.59 for an acentric distribution (which is always the case with proteins in three dimensions) [112,113]. In a small molecule structure R values of <0.10 are routine and many have R <0.05. For proteins an R 0.30 at 2.5 A resolution usually indicates that most of the structure is correct but several errors may remain. An R < 0.2 is usually satisfactory. Luzzati [114] has shown that if the errors in position are normally distributed and that if these errors are the sole cause of differences between observed and calculated structure factors, then at 2 A resolution a mean error in atomic position of 0.2 A gives rise to an R = 0.23, and an error of 0.1 A gives rise to an R of 0.12. The Luzzati estimate of errors, which is frequently used in protein crystallography, is usually an overestimate because other sources of error also contribute to the residual R. 

Refinement of the model then consists of minimizing the difference between the observed and calculated structure factors. The refinement is usually monitored by calculating the overall agreement (R factor) between the observed and calculated structure factor amplitudes, as follows ... 

There are several, say a dozen, individual major discrepancies between observed and calculated structure factors. 

The observed and calculated structure factors, if not published, are not made available in accord with instructions in a footnote. 

When the application of Eq. (11) to a least squares analysis of x-ray structure factors has been completed, it is usual to calculate a Fourier synthesis of the difference between observed and calculated structure factors. The map is constructed by computation of Eq. (9), but now IFhid I is replaced by Fhki - F/f /, where the phase of the calculated structure factor is assumed in the observed structure factor. In this case the series termination error is virtually too small to be observed. If the experimental errors are small and atomic parameters are accurate, the residual density map is a molecular bond density convoluted onto the motion of the nuclear frame. A molecular bond density is the difference between the true electron density and that of the isolated Hartree-Fock atoms placed at the mean nuclear positions. An extensive study of such residual density maps was reported in 1966.7 From published crystallographic data of that period, the authors showed that peaking of electron density in the aromatic C-C bonds of five organic molecular crystals was systematic. The random error in the electron density maps was reduced by averaging over chemically equivalent bonds. The atomic parameters from the model Eq. (11), however, will refine by least squares to minimize residual densities in the unit cell. 

It has also been demonstrated that molecular dynamics can play a useful role in the refinement of protein structures against X-ray data.420a By adding an effective potential that represents the difference between the observed and calculated structure factors (Eq. 99) to the standard empirical potential function (Eq. 6), simulated annealing4206 can be used to automatically refine a crude X-ray structure. In this way much of the manual rebuilding of the model structure, that is, the most time-consuming part of standard structure refinement,421 can be avoided. 

Observed and calculated structure factors for fully dehydrated, fully Cs+-exchanged zeolite A, CSj2 A 3/4Cs. In the table, the... 

---