Crossing vector

In Eqs. 4.25-4.27, the numerators contain cross (vector) products and the denominators are scalar triple products equal to the volume of the real-space unit cell. It may be computed from its determinant (see practice problem 7). 

Note that vertices of degenerate cases can be distinguished by an appropriate label. Also note that in earlier works [112,197,198] a somewhat different convention was used, where the endpoints were omitted. If the graph gi has n vertices, then these labels can be stored in an n-dimensional crossing vector C, ... 

The crossing vector is well defined as long as the projections of all the crossings are regular. 

The Patterson function of a crystal consists of two parts self vectors, which are a set of vectors within the particular molecule, and cross vectors, which are a set of vectors from atoms in one molecule in the crystal to atoms in another molecule. Suppose that a vector map is calculated for a molecule or part of a molecule whose structure is known. Then the set of self vectors in this vector map will be identical to those in the Patterson map computed for the crystal, but will probably be rotated by some set of angles. If the two Patterson functions are superimposed, there would be no particular agreement except when the two sets of selfvectors have the same orientation. If one of these Patterson maps is rotated on the second, keeping the locations of the origins of the two... 

FIGURE 8.22. A set of translation vectors between two benzene molecules, (a) The two molecules with atomic numbering used, and (b) their cross vectors with the atoms involved in each vector so marked. Compare these with the self vectors in Figure 8.19. The origin of the vector (Patterson) map is labelled as such. The highest peak in the... 

Another problem that frequently arises with multiple isomorphous derivatives is that of handedness. In space group P2i2i2i, Patterson maps for two independent derivatives may be interpreted to yield a set of symmetry related sites for one derivative and, independently, a second set for the other. Because handedness is completely absent in a Patterson map (because it contains a center of symmetry), there is an equal chance that the heavy atom constellation for the first will be right handed, and the constellation for the other will be left handed, and vice versa. This won t do. The two heavy atom sets will not cooperate when used to obtain phase information. There are ways of unraveling this problem too, and once again, it involves difference Patterson maps between the two derivative data sets and cross vectors. This case can also be resolved by calculating phases based on only one derivative and then computing a difference Fourier map (see Chapter 10) for the other. 

Advancement of knowledge in the kaleidoscopic environment of esoteric philosophy and mysteries is perilous. The cardinal rule is that evidence must always exist for any hypothesis. The main purpose of this book is to assemble and cross-vector arcane knowledge in a new way, to underpin a new and better understanding of the subject. As in a court of law, some evidence is more reliable than other evidence. The author will attempt to warn in advance of the more contentious areas. 

---