International
Tables for Crystallography Volume B Reciprocal space Edited by U. Shmueli © International Union of Crystallography 2006 
International Tables for Crystallography (2006). Vol. B, ch. 2.3, pp. 238242
Section 2.3.2. Interpretation of Patterson maps ^{a}Department of Biological Sciences, Purdue University, West Lafayette, Indiana 47907, USA, and ^{b}CABM & Rutgers University, 679 Hoes Lane, Piscataway, New Jersey 088545638, USA 
A hypothetical onedimensional 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 symmetryrelated 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 threedimensional 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 crossvectors between atoms.

Origin selection in the interpretation of a Patterson of a onedimensional centrosymmetric structure. 
Examine, for example, the caxis Patterson projection of a cuprous chloride azomethane complex (C_{2}H_{6}Cl_{2}Cu_{2}N_{2}) 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 doubleweight Cu–Cl vectors at and . Now suppose that peak I is at position , then and . Peak II should now check out as the remaining doubleweight 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 carbonatom 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.

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 symmetryequivalent atoms emerge from this milieu of Patterson vectors and such `Harker vectors' constitute the subject of the next section.

Soon after Patterson introduced the series, Harker (1936) recognized that many types of crystallographic symmetry result in a concentration of vectors at characteristic locations in the Patterson. Specifically, he showed that atoms related by rotation axes produce vectors in characteristic planes of the Patterson, and that atoms related by mirror planes or reflection glide planes produce vectors on characteristic lines. Similarly, noncrystallographic symmetry operators produce analogous concentrations of vectors. Harker showed how special sections through a threedimensional function could be computed using one or twodimensional summations. With the advent of powerful computers, it is usual to calculate a full threedimensional Patterson synthesis. Nevertheless, `Harker' planes or lines are often the starting point for a structure determination. It should, however, be noted that nonHarker vectors (those not due to interactions between symmetryrelated atoms) can appear by coincidence in a Harker section. Table 2.3.2.3 shows the position in a Patterson of Harker planes and lines produced by all types of crystallographic symmetry operators.

Buerger (1946) noted that Harker sections can be helpful in spacegroup determination. Concentrations of vectors in appropriate regions of the Patterson should be diagnostic for the presence of some symmetry elements. This is particularly useful where these elements (such as mirror planes) are not directly detected by systematic absences.
Buerger also developed a systematic method of interpreting Harker peaks which he called implication theory [Buerger (1959, Chapter 7)].
The previous two sections have developed some of the useful mechanics for interpreting Pattersons. In this section, we will consider finding heavyatom positions, in the presence of numerous light atoms, from Patterson maps. The feasibility of structure solution by the heavyatom method depends on a number of factors which include the relative size of the heavy atom and the extent and quality of the data. A useful rule of thumb is that the ratio should be near unity if the heavy atom is to provide useful starting phase information (Z is the atomic number of an atom). The condition that normally guarantees interpretability of the Patterson function in terms of the heavyatom positions. This `rule', arising from the work of Luzzati (1953), Woolfson (1956), Sim (1961) and others, is not inviolable; many ambitious determinations have been accomplished via the heavyatom method for which r was well below 1.0. An outstanding example is vitamin B_{12} with formula C_{62}H_{88}CoO_{14}P, which gave an for the cobalt atom alone (Hodgkin et al., 1957). One factor contributing to the success of such a determination is that the relative scattering power of Co is enhanced for higher scattering angles. Thus, the ratio, r, provides a conservative estimate. If the value of r is well above 1.0, the initial easier interpretation of the Patterson will come at the expense of poorly defined parameters of the lighter atoms.
A general strategy for determining heavy atoms from the Patterson usually involves the following steps.
Detailed and instructive examples of using Pattersons to find heavyatom positions are found in almost every textbook on crystal structure analysis [see, for example, Buerger (1959), Lipson & Cochran (1966) and Stout & Jensen (1968)].
The determination of the crystal structure of cholesteryl iodide by Carlisle & Crowfoot (1945) provides an example of using the Patterson function to locate heavy atoms. There were two molecules, each of formula C_{27}H_{45}I, in the unit cell. The ratio is clearly well over the optimal value of unity. The P(x, z) Patterson projection showed one dominant peak at in the asymmetric unit. The equivalent positions for require that an iodine atom at , , generates another at and thus produces a Patterson peak at . The iodine position was therefore determined as 0.217, 0.042. The y coordinate of the iodine is arbitrary for yet the value of is convenient, since an inversion centre in the twoatom iodine structure is then exactly at the origin, making all calculated phases 0 or π. Although the presence of this extra symmetry caused some initial difficulties in the interpretation of the steroid backbone, Carlisle and Crowfoot successfully separated the enantiomorphic images. Owing to the presence of the perhaps too heavy iodine atom, however, the structure of the carbon skeleton could not be defined very precisely. Nevertheless, all critical stereochemical details were adequately illuminated by this determination. In the cholesteryl iodide example, a number of different yet equivalent origins could have been selected. Alternative origin choices include all combinations of and .
A further example of using the Patterson to find heavy atoms will be provided in Section 2.3.5.2 on solving for heavy atoms in the presence of noncrystallographic symmetry.
As early as 1939, Wrinch (1939) showed that it was possible, in principle, to recover a fundamental set of points from the vector map of that set. Unlike the Harker–Buerger implication theory (Buerger, 1946), the method that Wrinch suggested was capable of using all the vectors in a threedimensional set, not those restricted to special lines or sections. Wrinch's ideas were developed for vector sets of points, however, and could not be directly applied to real, heavily overlapped Pattersons of a complex structure. These ideas seem to have lain dormant until the early 1950s when a number of independent investigators developed superposition methods (Beevers & Robertson, 1950; Clastre & Gay, 1950; Garrido, 1950a; Buerger, 1950a).
A Patterson can be considered as a sum of images of a molecule as seen, in turn, for each atom placed on the origin (Fig. 2.3.2.3). Thus, the deconvolution of a Patterson could proceed by superimposing each image of the molecule obtained onto the others by translating the Patterson origin to each imaging atom. For instance, let us take a molecule consisting of four atoms ABCD arranged in the form of a quadrilateral (Fig. 2.3.2.3). Then the Patterson consists of the images of four identical quadrilaterals with atoms A, B, C and D placed on the origin in turn. The Pattersons can then be deconvoluted by superimposing two of these Pattersons after translating these (without rotation) by, for instance, the vector AB. A further improvement could be obtained by superimposing a third Patterson translated by AC. This would have the additional advantage in that ABC is a noncentrosymmetric arrangement and, therefore, selects the enantiomorph corresponding to the hand of the atomic arrangement ABC [cf. Buerger (1951, 1959)].

Atoms ABCD, arranged as a quadrilateral, generate a Patterson which is the sum of the images of the quadrilateral when each atom is placed on the origin in turn. 
A basic problem is that knowledge of the vectors AB and AC also implies some knowledge of the structure at a time when the structure is not yet known. In practice `goodlooking' peaks, estimated to be single peaks by assessing the absolute scale of the structure amplitudes with Wilson statistics, can be assumed to be the result of single interatomic vectors within a molecule. Superposition can then proceed and the result can be inspected for reasonable chemical sense. As many apparently single peaks can be tried systematically using a computer, this technique is useful for selecting and testing a series of reasonable Patterson interpretations (Jacobson et al., 1961).
Three major methods have been used for the detection of molecular images of superimposed Pattersons. These are the sum, product and minimum `image seeking' functions (Raman & Lipscomb, 1961). The concept of the sum function is to add the images where they superimpose, whereas elsewhere the summed Pattersons will have a lower value owing to lack of image superposition. Therefore, the sum function determines the average Patterson density for superimposed images, and is represented analytically as where is the sum function at x given by the superposition of the ith Patterson translated by , or Setting (m and can be calculated from the translational vectors used for the superposition), Thus, the sum function is equivalent to a weighted `heavy atom' method based on the known atoms assumed by the superposition translation vectors.
The product function is somewhat more vigorous in that the images are enhanced by the product. If an image is superimposed on no image, then the product should be zero.
The product function can be expressed as When (h and p are sets of Miller indices), Successive superpositions using the product functions will quickly be dominated by a few terms with very large coefficients.
Finally, the minimum function is a clever invention of Buerger (Buerger, 1950b, 1951, 1953a,b,c; Taylor, 1953; Rogers, 1951). If a superposition is correct then each Patterson must represent an image of the structure. Whenever there are two or more images that intersect in the Patterson, the Patterson density will be greater than a single image. When two different images are superimposed, it is a reasonable hope that at least one of these is a single image. Thus by taking the value of that Patterson which is the minimum, it should be possible to select a single image and eliminate noise from interfering images as far as possible. Although the minimum function is perhaps the most powerful algorithm for image selection of well sharpened Pattersons, it is not readily amenable to Fourier representation.
The minimum function was conceived on the basis of selecting positive images on a nearzero background. If it were desired to select negative images [e.g. the correlation function discussed in Section 2.3.3.4], then it would be necessary to use a maximum function. In fact, normally, an image has finite volume with varying density. Thus, some modification of the minimum function is necessary in those cases where the image is large compared to the volume of the unit cell, as in lowresolution protein structures (Rossmann, 1961b). Nordman (1966) used the average of the Patterson values of the lowest 10 to 20 per cent of the vectors in comparing Pattersons with hypothetical point Pattersons. A similar criterion was used by High & Kraut (1966).
Imageseeking methods using Patterson superposition have been used extensively (Beevers & Robertson, 1950; Garrido, 1950b; Robertson, 1951). For a review the reader is referred to Vector Space (Buerger, 1959) and a paper by Fridrichsons & Mathieson (1962). However, with the advent of computerized direct methods (see Chapter 2.2 ), such techniques are no longer popular. Nevertheless, they provide the conceptual framework for the rotation and translation functions (see Sections 2.3.6 and 2.3.7).
The power of the modern digital computer has enabled rapid access to the large number of Patterson density values which can serve as a lookup table for systematic vectorsearch procedures. In the late 1950s, investigators began to use systematic searches for the placement of single atoms, of known chemical groups or fragments and of entire known structures. Methods for locating single atoms were developed and called variously: vector verification (Mighell & Jacobson, 1963), symmetry minimum function (Kraut, 1961; Simpson et al., 1965; Corfield & Rosenstein, 1966) and consistency functions (Hamilton, 1965). Patterson superposition techniques using stored function values were often used to image the structure from the known portion. In such singlesite search procedures, single atoms are placed at all possible positions in a crystal, using a search grid of the same fineness as for the stored Patterson function, preferably about onethird of the resolution of the Patterson map. Solutions are gauged to be acceptable if all expected vectors due to symmetryrelated atoms are observed above a specified threshold in the Patterson.
Systematic computerized Patterson search procedures for orienting and positioning known molecular fragments were also developed in the early 1960s. These hierarchical procedures rely on first using the `self'vectors which depend only on the orientation of a molecular fragment. A search for the position of the fragment relative to the crystal symmetry elements then uses the crossvectors between molecules (see Sections 2.3.6 and 2.3.7). Nordman constructed a weighted point representation of the predicted vector set for a fragment (Nordman & Nakatsu, 1963; Nordman, 1966) and successfully solved the structure of a number of complex alkaloids. Huber (1965) used the convolution molecule method of Hoppe (1957a) in three dimensions to solve a number of naturalproduct structures, including steroids. Various program systems have used Patterson search methods operating in real space to solve complex structures (Braun et al., 1969; Egert, 1983).
Others have used reciprocalspace procedures for locating known fragments. Tollin & Cochran (1964) developed a procedure for determining the orientation of planar groups by searching for origincontaining planes of high density in the Patterson. General procedures using reciprocalspace representations for determining rotation and translation parameters have been developed and will be described in Sections 2.3.6 and 2.3.7, respectively.
Although as many functions have been used to detect solutions in these Patterson search procedures as there are programs, most rely on some variation of the sum, product and minimum functions (Section 2.3.2.4). The quality of the stored Patterson density representation also varies widely, but it is now common to use 16 or more bits for single density values. Treatment of vector overlap is handled differently by different investigators and the choice will depend on the degree of overlapping (Nordman & Schilling, 1970; Nordman, 1972). General Gaussian multiplicity corrections can be employed to treat coincidental overlap of independent vectors in general positions and overlap which occurs for symmetric peaks in the vicinity of a special position or mirror plane in the Patterson (G. Kamer, S. Ramakumar & P. Argos, unpublished results; Rossmann et al., 1972).
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