Difference Fourier synthesis

Ba(2), Ba(3), and Ba(4) refined to 1.9(1), 1.2(1), 2.0(5) and 0.8(5), respectively. Because the Ba(3) and Ba(4) positions had nearly become coincident, they were represented as 3 Ba + ions at an intermediate position, designated Ba(3). The occupancies at Ba(l) and Ba(2) were fixed at 2 and 1 in further refinement. A subsequent difference Fourier synthesis served to locate the water molecules at 0(5). Numerous attempts to locate additional water molecules failed. 

This function is known as the difference Fourier synthesis. Calculated structure amplitudes, iF c(H), can be derived by the Fourier transform of the second term in Eq. (5) as... 

Figure 2a. Difference Fourier synthesis of oxymyoglobin, showing the electron density for the bound oxygen. The Fourier synthesis was computed with Fobserved — Fcalculated] as coefficients the calculated structure amplitudes were derived from the positions of all the atoms except the two oxygens.

The difference Fourier synthesis, phased by the P, Cl, N, C and O atoms, revealed the hydrogen atoms with their expected locations. Thus, the final refinement could be performed on the entire set of atoms including hydrogens with fixed isotropic thermal parameter factor, BH = 4 A2. Final R and S values are 0.022 and 0.948, respectively. The last difference Fourier map showed no values to be greater than 0.3 eA 3 (Table 15). Atomic scattering factors were corrected for anomalous dispersion from Cromer and WaberS8). 

Unless direct methods are used to locate heavy atom positions, an understanding of the Patterson function is usually essential to a full three-dimensional structure analysis. Interpretation of a Patterson map has been one of two points in a structure determination where the investigator must intervene with skill and experience, judge, and interpret the results. The other has been the interpretation of the electron density map in terms of the molecule. Interpretation of a Patterson function, which is a kind of three-dimensional puzzle, has in most instances been the crucial make or break step in a structure determination. Although it need not be performed for every isomorphous or anomalous derivative used (a difference Fourier synthesis using approximate phases will later substitute see Chapter 10), a successful application is demanded for at least the first one or two heavy atom derivatives. 

One of the most commonly employed types of difference Fourier synthesis uses as coefficients 2/< / - Fcaic, and phases calculated from the current model. This shows the investigator positive density superimposed on the model where the model is correct, with no density appearing where the model contains incorrectly placed residues or atoms and positive density appears where the model should be. There are also other higher order difference syntheses, but they are less frequently used. 

Idealized OH group with tetrahedral X—O—H angle. As with HFIX 137 the irutial torsion angle is derived from a difference Fourier synthesis and a rigid group refinement is performed. 

Excepf for the H-atoms involved in the disorder, all hydrogen positions can be seen in the difference Fourier synthesis (see the Q-peaks in the file ga-04.res). To validafe fhe restrainfs, examine fhe file ga-04.1sf (especially the lines 137-175). The density peaks Q(12), Q(23), and Q(24) (0.65, 0.62, and 0.61 electrons) correspond to the hydrogen atoms bonded to N(2), N(3), and N(4), respectively. The following H FIX instructions cause SHELXL to geometrically calculate the hydrogen... 

This time all atoms refine well, and we find residual electton density maxima representing the three independent hydrogen atoms on the nitrogen atoms N(l) (Q(15) and Q(17)) and N(2) (Q(3)) in the difference Fourier synthesis. These three hydrogen atoms have been included into the file zr5-07.ins (don t forget the DFIX commands for the N—H distances). We should also include the appropriate HFIX commands to include the hydrogen atoms that bind to carbon. 

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