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An Excel Spreadsheet for a One-Dimensional Fourier Map in X-Ray Crystallography
W
William Clegg Chemical Crystallography Group, School of Natural Sciences (Chemistry), University of Newcastle, Newcastle upon Tyne NE1 7RU, United Kingdom;
[email protected] The teaching of crystal structure determination with single-crystal X-ray diffraction at undergraduate level faces numerous challenges. Where it occurs at all, it is usually in a chemistry major program, often only as an option at a relatively late stage in the program. This is the most powerful technique available for the detailed geometrical structural characterization of both molecular and nonmolecular materials in the solid state. Single-crystal X-ray diffraction is used in a vast range of chemical research projects and forms the basis for a high proportion of structural results that are presented to high-school, undergraduate, and graduate chemistry students in their courses. Yet, it suffers the consequences of outdated views and misconceptions on the part of students, teachers, and those responsible for course contents and structure. These include the notions that crystallography is intrinsically complex and difficult, that it is a slow or an expensive technique, that it is irrelevant at the undergraduate level, that it is exclusively the realm of specialists, and that its teaching requires a sophisticated level of mathematics. In such respects it contrasts, for example, with the similarly important structural investigation technique of NMR spectroscopy, which typically enjoys a high profile in undergraduate chemistry courses, frequently with an empirical approach of simple “rules of thumb” for the assignment and analysis of spectra in terms of chemical shifts, coupling patterns, and signal intensities. In fact, students of chemistry can gain much from a short course highlighting the modern context, importance, and principles of X-ray crystallography, in which they are shown something of its capabilities and limitations, the main experimental procedures involved, and the meaning and significance of the results it delivers. For such a course, it is not necessary to go into the details of such classical (but, frankly, uninspiring) topics as the derivation of Bravais lattices, reciprocal space, twinning, and Lorenz-polarization corrections. Such details are more appropriate in advanced courses for practicing specialists, such as graduate students who are to be directly involved in research projects with some hands-on crystallographic experience of their own. The author teaches a second-year course of 10 lectures, together with some practical exercises and small-group tutorial sessions, which is taken by all chemistry major students at this institution. The core material of this course has been published (1) and has been adopted by many other university chemistry departments for courses at a similar level. It is considerably more compact and less detailed than other modern classical texts on the subject (2). This approach has proved to be an effective and efficient use of the limited time available. A Problem in Teaching X-ray Crystallography: Fourier Transformation There are some concepts that need to be explained at this level, which are unlikely to be featured elsewhere in an 908
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undergraduate chemistry course. Among these is the meaning and the use of Fourier transformation as the basis of the diffraction experiment, relating the single-crystal sample to its measured diffraction pattern, and the reconstruction of the image of electron density from the amplitudes and (initially unknown) relative phases of the diffracted X-ray beams. Although Fourier transformation also features in modern NMR, IR, and other spectroscopic techniques, its application there is essentially an automatic procedure, the exact meaning and practical details of which are unimportant at this level of teaching. By contrast, the place of Fourier transformation in crystallography is central to an understanding of the technique as anything beyond a “mystery black box”, being intimately tied up with the “phase problem” that lies at the heart of practical structure solution. This, like some other central crystallographic concepts, calls for the use of graphical and other visual tools to overcome the reluctance or inability of many students to grapple with mathematical symbols alone. The main problems associated with explaining the meaning and application of the forward and reverse Fourier transforms in crystallography are their (literally) complex mathematical notation, complete with multiple summations or integrals, and the fact that even the simplest crystal structures involve large numbers of calculations, obscuring the principles by floods of numbers. Introducing a standard research crystallography computer package to illustrate Fourier transformation tends to cloud the issue with its user interface and black-box approach, if it is the only visual aid. I have found the use of optical transforms invaluable in illustrating the fundamental properties of crystallographic Fourier transforms and usually take examples from the excellent compilation in the Atlas of Optical Transforms (3). However, there is still a need for something as close as possible to a real crystal structure application, while minimizing the complexity of the example. Solution to the Problem: A One-Dimensional Example The solution is a molecular structure as nearly linear as possible, extended approximately along one unit cell axis, with minimum overlap of symmetry-related molecules within the unit cell. A suitable example will give effectively atomic resolution in one dimension. If it contains one relatively heavy atom per molecule, it can be used to illustrate the Patterson synthesis as well as full and difference density maps, and such structures are usually also amenable to solution by direct methods, provided the atoms do not lie in special positions. All these requirements are fulfilled by the crystal structure of racemic 3-bromooctadecanoic acid, which has two _ molecules related by inversion symmetry in space group P1 (4). The unit cell, like the molecules themselves, is extended in one direction, and the molecules lie approximately paral-
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a c
0
b
Figure 1. The contents of one unit cell of racemic 3-bromooctadecanoic acid.
lel to the long cell c axis (Figure 1); in fact, the angle of inclination is such that pairs of carbon atoms almost completely eclipse each other in projection on this axis, so the one-dimensional electron density consists of well-resolved maxima, most of which correspond each to two superimposed C atoms, together with two very large peaks for the Br atoms (Figure 2). The two halves of the one-dimensional unit cell are related by inversion symmetry at the point z = 1兾2; there are also inversion centers at z = 0 and z = 1. Reducing from three dimensions to one simplifies the Fourier transform equations considerably. In the case of a centrosymmetric structure, there is further simplification in the reverse Fourier transform equation (the recombination of the diffracted beams to form an image of the electron density), since each reflection phase must be either 0 or 180⬚ (0 or π radians). Exponential functions of a complex variable become simple cosines, and the problem of determining the unknown phase of each reflection reduces to a problem of deciding whether each reflection contributes to the sum positively (cos 0 = 1) or negatively (cos π = ᎑1). Thus the full three-dimensional Fourier synthesis 1 V
ρ ( x, y, z ) = =
1 V
∑ ∑ ∑ F ( h, k, l ) exp[ −2πi ( hx h
k
l
+ ky + lz )]
∑ ∑ ∑ F ( h, k, l ) exp[ i ϕ(h, k, l )] exp[−2 π i ( hx h
k
l
+ ky + lz )]
simplifies drastically to ρ( z ) =
1 ∑ F ( l ) sign[F ( l )] cos(2π lz ) c l
where sign[F(l )] is either +1 or ᎑1 for each reflection.
the unit cell (all other terms are placed on a common scale relative to this). In the spreadsheet, each of columns A to V represents one term of the Fourier summation, and contains, in rows 20–220, values of |F(l )|sign[F(l )]cos(2πl z) at intervals of 0.005 (1兾200) in z, between z = 0 and z = 1. The first few rows contain correct values of the amplitudes |F(l )| and various other terms derived from them as described below, the correct values of sign[F(l )], and some alternative sets of signs to illustrate various effects; in each case, these signs may be +1 or ᎑1, or 0 in order to omit a term completely from the summation. A set of simple button-operated macros copies various combinations of amplitudes and signs into the working cells from which the rest of the column contents are calculated. The right-most column X is the sum of all the others, and represents the one-dimensional Fourier synthesis equation given above. This sum is shown in a graphical chart, plotted against z, which is superimposed on part of the spreadsheet and is automatically updated whenever any of the contributing terms is changed. The effect of individual terms in the Fourier sum is best illustrated first, by setting all amplitudes correctly and all signs equal to zero (using two of the macro buttons), and then manually setting a number of signs to +1, one at a time. Thus it can be shown that F(0), which always has the largest value of all amplitudes for a data set, just spreads the total available electron density uniformly across the unit cell, while all other terms shift electron density from some positions and concentrate it in others, while making no net contribution. Terms with small values of l provide broad electron density features, and increasing values of l lead to finer contributions and hence a greater resolution of structural features; these correspond to X-ray diffraction measurements at higher Bragg angles (Figure 3). Obviously (though the obvious does often need to be
Excel Spreadsheet: Implementation of the One-Dimensional Fourier Synthesis The meaning and practical use of the Fourier synthesis can be easily demonstrated in a lecture, or in a self-teaching exercise for students, by the implementation of this simplified equation in a Microsoft Excel spreadsheet. The number of terms in the summation (X-ray reflections) is only around 20. Reflections with l = 3 to 21 were measured as part of a full three-dimensional X-ray diffraction data set by Abrahamsson and Harding in 1966 (4). For teaching purposes, it is useful to include also the terms with l = 1 and l = 2, which can be calculated from the known crystal structure, and the term with l = 0, which is the number of electrons in www.JCE.DivCHED.org
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Figure 2. One-dimensional electron density for 3-bromooctadecanoic acid, calculated from 22 terms with their correct relative signs.
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Figure 3. Individual Fourier summation terms for l = 1 and for l = 10.
Figure 4. The Fourier sum with random signs for the contributions.
Figure 5. The one-dimensional Patterson synthesis, obtained with |F|2 values and all signs positive.
Figure 6. The Fourier sum with correct amplitudes and with signs calculated from the Br atoms alone.
Figure 7. A difference Fourier synthesis based on the Br atoms alone.
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demonstrated), smaller amplitudes make smaller contributions to the sum; this does not show with individual contributions, as the chart is always normalized to the same maximum, but can be demonstrated easily by displaying one large contribution (e.g., l = 6), and then adding a small one to it (e.g., l = 8), whereupon the chart does not change very much. Setting all amplitudes and signs to their correct values (known only after the crystal structure has been determined!), gives the result shown in Figure 2. The maxima correspond to atoms, with Br having a much larger electron density than the others; these Br atoms are found to have z close to 0.1 and 0.9. Using random signs gives an unrecognizable structure, usually with some deep minima; an example is in Figure 4, and is produced by one of the macro-generated sets of signs provided. With 21 contributing terms in the summation in addition to F(0), each having two possible signs, the number of possible permutations is 221, which is in the millions. This illustrates the futility of any pure trial-and-error procedure for solving the crystallographic “phase problem”, the fact that the correct signs are not obtainable from the diffraction experiment. Excel Spreadsheet: Heavy-Atom Method A number of commonly used methods in crystallography can be illustrated by other sets of “amplitudes” or signs. Thus, replacement of the amplitudes |F | by |F |2, with all signs set to +1, gives a one-dimensional Patterson synthesis (Figure 5). The two main features are the large maximum at z = 0 and at z = 1, which is common to all Patterson syntheses, and the next largest maxima at z = 0.2 and 0.8, approximately. These latter correspond to vectors between pairs of heavy atoms and lead very simply to the location of the Br atoms in the structure without prior knowledge of any reflection phases: 0.2 = 0.1 − (᎑0.1) and 0.8 = 0.9 − 0.1. Similar procedures in three dimensions are commonly used for the location of heavy atoms in crystal structures. Once the heavy atoms have been found, it is possible to calculate what the diffraction pattern would be for a structure containing only these and no other atoms; this calculation is not carried out in the spreadsheet, but its results are available as a set of calculated amplitudes and corresponding calculated phases. Combination of these amplitudes and phases merely reproduces the model structure with Br atoms only (together with some “series termination” ripples owing to the inclusion of only a finite number of terms in the summation). However, combining the correct, observed amplitudes (experiment) with the phases calculated from Br atoms alone (the nearest approximation we have at this stage to the unknown correct phases) generates a new one-dimensional electron density (Figure 6), in which the lighter atoms now appear (compare with Figure 2; there are small differences, especially in the variation of peak heights, because not all the calculated phases are correct at this stage). Alternatively, the light atoms can be revealed more clearly by using the difference amplitudes |Fo − Fc| instead of |Fo| (o stands for observed and c for calculated), together with the calculated phases (Figure 7); now the dominance by the Br atoms is removed. These two syntheses, a normal electron density and a difference electron density, together with other related variations, are routinely used in three-dimensional structure determinations, to develop a complete structure from an initial partial model.
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Excel Spreadsheet: Direct Methods The sequence of Fourier syntheses described above serves to demonstrate the heavy-atom method of solving crystal structures. The spreadsheet can also be used to demonstrate the principles on which “direct methods” are based. Such methods aim to deduce approximately correct reflection phases from the amplitudes themselves, without any other information. In fact, this common simple description is incorrect, because other information is, in fact, both available and used. The most important information concerns the nature of the desired electron density distribution itself. It can never have a negative value, and it is concentrated in discrete regions (approximately spherical in shape in three dimensions), which we recognize as atoms; low values of electron density are found between atoms. This result must be obtained by adding up the diffracted waves with their correct relative signs (phases). Apart from F(0), or F(000) in three dimensions, all the waves consist of positive crests and negative troughs, with a mean value of zero. They must therefore be added together in such a way that discrete, compact positive regions are built up, and negative regions, caused by overlap of many large troughs, are avoided. This places significant restrictions on the relationships among the phases of different reflections, especially the most intense ones, and this is the basis of direct methods. To illustrate this, the first step is to demonstrate that the largest amplitudes contribute most to the overall sum. Displaying the electron density derived from all the amplitudes and their correct signs, and then setting to zero the signs of the weakest reflections (one of the macro buttons), shows that these make little difference. Thus, direct methods concentrate on the strongest reflections, reducing the number of phases (signs) that need to be determined. (This is actually an oversimplification, but is adequate for teaching at this level.) To demonstrate the meaning of restrictions on phase relationships, we choose three strong reflections such that l1 + l2 = l3, for example 5, 6, and 11. The correct signs for reflections 5 and 6 are both ᎑1. Setting these two signs appropriately, by hand, with all others at zero, shows the contribution of these two intense reflections (large amplitudes) to the overall electron density distribution. Now add in reflection 11, with its two possible signs, one after the other. With sign +1, electron density peaks are enhanced, while the sign ᎑1 diminishes peaks and produces deeper troughs (Figure 8). Thus the sign +1 for l = 11 is more likely to be correct than sign ᎑1, on the basis of these three reflections alone. Since (᎑1) × (᎑1) = (+1), the sign relationship is given by sign [F (11)] = sign [F (5)] sign[ F ( 6 )] or φ(11) = φ( 5 ) + φ( 6 ) and the probability that this relationship is correct increases with the amplitudes of the three reflections, because their contributions become more important. For this particular one-dimensional structure, it is found that all such “triple-phase” relationships are, in fact, correct for reflections with amplitudes greater than 20, with one exception, sign[F(4)] ≠ sign[F(2)] × sign[F(2)]; this can be seen www.JCE.DivCHED.org
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Figure 8. Contributions of reflections 5, 6, and 11 with probable (left) and improbable (right) sign combinations.
by examining the list of correct signs. In general, for crystal structures, there will be some incorrect relationships and some contradictory indications, and a balance of probabilities has to be found; the ultimate test of success is the production of an electron density distribution recognizable as the true structure. The essential point illustrated by the spreadsheet is that probability relationships lead to the prediction of some phases from combinations of others, so that likely sets of approximately correct phases can be developed for the strongest reflections in a data set. Modern direct methods for crystal structure determination are highly sophisticated, but this is the fundamental basis on which they work. Acknowledgments The author is grateful to Bob Gould, Edinburgh, for showing how this structure can be used to illustrate one-dimensional Fourier syntheses as a group hand-calculation exercise, to numerous colleagues and students for their helpful comments and suggestions in developing the spreadsheet, and to Horst Puschmann, Durham, for supplying an initial set of macros. Supplemental Material The complete spreadsheet is available in this issue of JCE Online and also by request from the author. It can readily be modified to work with data for other appropriate structures, and can be used in lecture demonstrations or, with a suitable set of instructions, as a self-study exercise. For further related material, an extensive collection of links to online tutorials in crystallography, including some Fourier syntheses in one and two dimensions, primarily from a macromolecular crystallography viewpoint, may be found at the web site of Bernhard Rupp at Lawrence Livermore National Laboratory (5). W
Literature Cited 1. Clegg, W. Crystal Structure Determination (Oxford Chemistry Primer number 60); OUP: Oxford, 1998. 2. (a) Giacovazzo, C.; Monaco, H. L.; Viterbo, D.; Scordari, F.; Gilli, G.; Zanotti, G.; Catti, M. Fundamentals of Crystallography; OUP: Oxford, 1992; 2nd ed., 2002. (b) Glusker, J. P.; Lewis, M.; Rossi, M. Crystal Structure Analysis for Chemists and Biologists; Wiley-VCH: Weinheim, Germany, 1994. (c) Woolfson, M. M. An Introduction to X-ray Crystallography, 2nd ed.; CUP: Cambridge, 1997. 3. Harburn, G.; Taylor, C. A.; Welberry, T. R. Atlas of Optical Transforms; Bell: London, 1975. 4. Abrahamsson, S.; Harding, M. M. Acta Crystallogr. 1966, 20, 377. 5. Macromolecular Crystallography Web Server. http://wwwstructure.llnl.gov (accessed Mar 2004).
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