Section 2.3.2.1. Simple solutions in the triclinic cell. Selection of the origin

A hypothetical one-dimensional centrosymmetric crystal structure containing an atom at x and at −x and the corresponding Patterson is illustrated in Fig. 2.3.2.1. There are two different centres of symmetry which may be chosen as convenient origins. If the atoms are of equal weight, we expect Patterson vectors at positions with weights equal to half the origin peak. There are two symmetry-related peaks, and (Fig. 2.3.2.1) in the Patterson. It is an arbitrary choice whether or . This choice is equivalent to selecting the origin at the centre of symmetry I or II in the real structure (Fig. 2.3.2.1). Similarly in a three-dimensional cell, the Patterson will contain peaks at which can be used to solve for the atom coordinates . Solving for the same coordinates by starting from symmetric representations of the same vector will lead to alternate origin choices. For example, use of will lead to translating the origin by relative to the solution based on . There are eight distinct inversion centres in , each one of which represents a valid origin choice. Although any choice of origin would be allowable, an inversion centre is convenient because then the structure factors are all real. Typically, one of the vector peaks closest to the Patterson origin is selected to start the solution, usually in the calculated asymmetric unit of the Patterson. Care must be exercised in selecting the same origin for all atomic positions by considering cross-vectors between atoms.

Figure 2.3.2.1 | top | pdf | Origin selection in the interpretation of a Patterson of a one-dimensional centrosymmetric structure.

Examine, for example, the c-axis Patterson projection of a cuprous chloride azomethane complex (C₂H₆Cl₂Cu₂N₂) in as shown in Fig. 2.3.2.2. The largest Patterson peaks should correspond to vectors arising from Cu and Cl atoms. There will be copper atoms at and as well as chlorine atoms at analogous positions. The interaction matrix is

which shows that the Patterson should contain the following types of vectors: The coordinates of the largest Patterson peaks are given in Table 2.3.2.1 for an asymmetric half of the cell chosen to span in u and in v. Since the three largest peaks are in the same ratio (7:7:6) as the three largest expected vector types (986:986:841), it is reasonable to assume that peak III corresponds to the copper–copper interaction at . Hence, and . Peaks I and II should be due to the double-weight Cu–Cl vectors at and . Now suppose that peak I is at position , then and . Peak II should now check out as the remaining double-weight Cu–Cl interaction at . Indeed, which agrees tolerably well with the position of peak II. The chlorine position also predicts the position of a peak at with weight 289; peak IV confirms the chlorine assignment. In fact, this Patterson can be solved also for the lighter nitrogen- and carbon-atom positions which account for the remainder of the vectors listed in Table 2.3.2.1. However, the simplest way to complete the structure determination is probably to compute a Fourier synthesis using phases calculated from the heavier copper and chlorine positions.

Table 2.3.2.1 | top | pdf | Coordinates of Patterson peaks for C2H6Cl2Cu2N2 projection Height u v Number in diagram (Fig. 2.3.2.2) 7 0.33 0.34 I 7 0.18 0.97 II 6 0.16 0.40 III 3 0.49 0.29 IV 3 0.02 0.59 V 2 0.30 0.75 VI 2 0.12 0.79 VII

Figure 2.3.2.2 | top | pdf | The c-axis projection of cuprous chloride azomethane complex (C2H6Cl2Cu2N2). The space group is with one molecule per unit cell. [Adapted from and reprinted with permission from Woolfson (1970, p. 321).]

Consider now a real cell with M crystallographic asymmetric units, each of which contains N atoms. Let us define , the position of the nth atom in the mth crystallographic unit, by where and are the rotation matrix and translation vector, respectively, for the mth crystallographic symmetry operator. The Patterson of this crystal will contain vector peaks which arise from atoms interacting with other atoms both in the same and in different crystallographic asymmetric units. The set of Patterson vector interactions for this crystal is represented in a matrix in Table 2.3.2.2. Upon dissection of this diagram we see that there are MN origin vectors, vectors from atom interactions with other atoms in the same crystallographic asymmetric unit and vectors involving atoms in separate asymmetric units. Often a number of vectors of special significance relating symmetry-equivalent atoms emerge from this milieu of Patterson vectors and such `Harker vectors' constitute the subject of the next section.

Table 2.3.2.2 | top | pdf | Square matrix representation of vector interactions in a Patterson of a crystal with M crystallographic asymmetric units each containing N atoms Peak positions correspond to vectors between the atoms and where is the nth atom in the mth crystallographic asymmetric unit. The corresponding weights are . The outlined blocks I1 and IM represent vector interactions between atoms in the same crystallographic asymmetric units (there are M such blocks). The off-diagonal blocks IIM1 and II1M represent vector interactions between atoms in crystal asymmetric units 1 and M; there are blocks of this type. The significance of diagonal elements of block IIM1 is that they represent Harker-type interactions between symmetry-equivalent atoms (see Section 2.3.2.2).   … … … 0, …             0, …               …               0,             Block I1         Block II1M                                                   …                           Block IIM1         Block IM