Walter M. Macintyre University
of Colorado Boulder
X-ray Crystallography as a Tool for Structural Chemists
Recently Dame Kathleen Lonsdale published an account of the undergraduate program in crystallography a t the University of London (I). This article represents an attempt to supplement her paper by giving an account of some of the areas of organic chemistry and biochemistry in which X-ray crystallography h d s applications. Crystal structure analysis by X-ray diffraction is the only convenient physical procedure available to the chemist for the complete determination of molecular structure. The term "complete determination of molecular structure" means more than the assignment of a structural formula. In addition to that, X-ray analysis provides all bond lengths and angles in the molecule. Such information can then determine whether a particular bond is a single, multiple or one of an intermediate nature. The X-ray analysis of an optically active molecule can be carried out in such a way that the structural formula obtained gives the absolute configuration of the molecule. In carrying out a crystal structure analysis one is examining a molecule, not in isolation, but in a crystal where it is closely surrounded by other molecules or ions. Thus the experiment provides information about the environment of the molecule as well as information about the molecule itself. From a study of this molecular environment one can identify various types of intermolecular interactions. For example one can decide whether hydrogen bonds are formed, or whether the molecules in the crystal form charge transfer complexes, or whether the crystal is held together simply by van der Wads forces. X-ray analysis can provide a detailed picture of the thermal vibrations of each atom in the crystal. I n principle this information could be of considerable value in the interpretation of the infrared spectrum of the crystal. However this expectation has not yet been realized in practice. There is one more piece of information provided which, in the future, will be of immense importance to the chemist. Crystal structure analysis by X-ray diffraction provides the complete electron distribution in the molecule of interest, and in its environment. Since much of the theory of chemical reactivity rests on the fine details of the electron distribution in reacting molecules, an experimentally determined electron distribution is of great importance. It is unfortunate that electron distributions provided by current X-ray analyses are not sufficiently accurate to display those finer details. However, it is possible that a great deal of progress may be made soon on the experimental side of X-ray analysis. Given sufficiently numerous and 526
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precise experimental data, it is conceivable that X-ray analysis will provide electron distributions of interest to theoretical chemists. Few physical methods are capable of providing the organic chemist, or the biochemist, withsucha wealthof information about the molecules with which he works. Yet few physical methods have enjoyed so little direct use by such chemists. It is hoped that this article will show that the reasons for this paradox are no longer valid and that X-ray analysis has been developed to a point where it may be viewed as a routine procedure in structural investigations. We shall consider an X-ray analysis to be divided into three distinct stages: collection of the experimental data, determination of an approximate solution to the structure, and refinement of the approxin~atesolution. The Experimental Data of X-Ray Crystallography
A crystal may be regarded as a body produced by the regular repetition in space of identical units. The repeated unit is the unit cell. The shape of the unit cell is defined by three axes (a, b, and c) and the three angles between these axes (a,P, y ) . Each unit cell contains the same number of molecules, arranged in exactly the same way. The dimensions of the unit cells of most crystals are, very roughly, of the same order of magnitude as the wavelength of X-rays. Thus the crystal can act as a three-dimensional diraction grating for X-rays, a fact demonstrated first by Max von Lane and his associates in 1912. Diraction is a scattering phenomenon. The X-ray photons are scattered by all of the atoms in the crystal. In most directions in space the scattered photons interfere with one another. However in certain well defined directions they interfere only partially. In these directions weak X-ray beams of widely varying intensity may be observed. These are the diffraction maxima, and collectively comprise the diffraction pattern. Two properties of these diffraction maxima are important. One is the orientation with respect to the crystal and the primary X-ray beam. Knowing the distribution of the maxima about the crystal enables one to compute the shape, size and symmetry of the unit cell. The second property of interest is the intensity, which is a function of the unit cell dimensions and symmetry, and of the coordinates of all of the atoms in the unit cell with respect to the cell axes (a, b, c). I n discussing the directions along which the diiraction maxima are observed it is convenient to employ an analogy first suggested by W. L. Bragg in 1913. He
pointed out that the directions taken by the diffraction maxima were those which would be taken by the X-rays if they were reflected by rational planes in the crystal. (htional planes are those which intersect the unit cell axes in rational fractions, i.e., fractions which can be expressed as the ratio of two integers.) To each rational plane in one unit cell of the crystal there correspond other identical planes, parallel to the first, in all the other unit cells. Each of these planes may he regarded as reflecting X-rays. The condition that the reflections from successive members of this set of planes reinforce one another is given by the familiar Bragg equation, nX = 2d sin 0, where X is the wave length of the X-rays, d is the distance between successive members of the set of planes, 8 is the angle the primary X-ray beam makes with the planes and n is an integer. From a knowledge of a sufficient number of properly chosen d's one can deduce the cell dimensions. It has become customary to refer to the diffraction maxima as X-ray reflections. This is a less precise term; hut it is more descriptive and we shall use it hereafter. The relationship between the intensities of the reflections and the arrangement of the atoms in the crystal was elucidated by C. G. Darwin in 1914. Darwin's relation may he stated as follows:
electron density. These regions of high electron deusity correspond to atoms. Thus the positions of the atoms withm the cell, i.e., the atomic coordinates, may be inferred from the locations of the electron density maxima. The relation between the structure factors and the electron density distribution in the unit cell is
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4
r is a vector from the origin to some arbitrary point
-
within the unit cell, and r therefore defines that point; p(r) is the electron density a t the point within the cell +
specified by the vector r; V is the volume of the unit
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4
cell. The summation over the vector H is a triple
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summation over the three components of H. The p(r) is evaluated a t a large number of points within the cell (i.e., for a large number of different +
values of r) to give the desired electron density distribution.
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The expression for p(r) is a Fourier series. Strictly speakmg the range of summation in a Fourier series is from minus infinity to plus infinity. Such a summation would require an infinite number of Fourier co-
-
-
H is a three component vector whose components are related to the intercepts made on the unit cell axes by the rational planes. It is clear from the definition of the rational plane that no two sets of planes may have the same intercepts with the cell axes. Thus no two d
&
sets of planes may have the same vector H, and so H may be used to identify individual sets of rational 4
-
planes. I(H) is the intensity of the radiation reflected
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by the set of planes H; f(8) is a function of 8, the Bragg angle for the set of planes H, and some universal constants. 4
F(H) is the structure factor. It is a somewhat complicated function involving, among other things, the coordinates with respect to the cell axes of all the atoms in the cell, the symmetry and dimensions of the
-
unit cell, and the vector H. I n addition the structure factor may be a complex quantity, i.e., involving 47;
efficients (i.e., F(H)'s). One cannot measure an infinite number of intensities and so, in practice, one compromises by using as many structure factors as possible. I n the case of a moderately complex organic molecule, e.g., one containing twenty atoms exclusive of hydrogen atoms, one would hope to be able to measure inteusities for between 1500 and 2000 reflections, and hence include that number of structure factors in the Fourier series. Using traditional methods it might take several months to measure the intensity of all these reflections. Thus, in the past, the time required to collect the experimental data has taken up a not insignificant fraction of the total time required for the structure analysis. The Phase Pmblem-Determination of an Approximate Structure
I n discussing the Fourier series it was assumed that the structure factors could be obtained from the intensities of the X-ray reflections. This is only partly true. It is readily seen that
+
however, we will consider only the case where F(H) is real. Since it is the structure factor that contains the structural information, the object of measuring the intensities of the X-ray reflections is to determine the values of the corresponding structure factors. Mathematically, it is difficult to extract the structural information from the structure factors. Fortunately it is possible to extract this information in a n indirect manner. A relation exists between the structure factors and the electron density distribution in the crystal cell, and it is possible to calculate the electron density a t all points within the cell. It is a simple matter to find from this distribution the regions of high
Every r a l positive number has two real square roots. These roots are equal in magnitude, but one is positive and the other is negative. Experimental measure-
-
-
ments do not reveal whether a particular F(H) should be the positive or the negative square root of the corresponding I(H)/f(B). I n other words the experimental data provide only the magnitudes of the Fourier coefficients, not their signs. Until the signs of the Fourier coefficientsare known it is not possible to sum the series. Volume 41, Number 10, October 1964
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This is the phase problem in X-ray analysis, and it must be solved anew in each crystal structure determination. There is no generd solution to the diffculty and this constitutes the central theoretical problem of X-ray analysis. The magnitude of the problem is readily appreciated. Suppose we wish to include 2000 structure factors in a Fourier series. Since each structure factor may be positive or negative we have 2%"different sign combinations possible, of which only one is correct. Systematic examination of all of these possibilities is completely impractical. Fortunately a solution to the problem usually exists in the special case where a small number of atoms in the unit cell have atomic numbers very much higher than the other atoms. This is the "heavy atom" method of solving the phase problem, discovered by J. Monteath Robertson in the period 1933-35. It is not necessary that the heavy atom be substituted directly on the molecule of interest, although this is a common procedure with certain advantages. It is sufficient that the heavy atom he sonlewherc in the unit cell. For example the structures of alkaloids are conveniently studied using crystals of the hydrobromide or the methiodide. An organic acid might be transformed to the potassium or rubidium salt. (Salt formation is useful also as a way of makmg crystalline derivatives of compounds which are gases or liquids a t room temperature, e.g., trimethylamine or acetic acid.) The presence of the heavy atom enables one to arrive a t an approximate solution to the structure very rapidly. Although there are other methods which can be used when no heavy atom is present, they are invariably slower and less certain. Thus, whenever possible, a heavy atom should be included in a crystal destined for X-ray analysis. Reflnement of the Approximate Structure
The approximate solution to the structure, obtained in solving the phase problem, is approximate only in the sense that the bond lengths and angles given by the atomic coordinates still have relatively large errors. The atoms will be in their correct relative positions. If the X-ray analysis is being undertaken only to establish the structural formula then the analysis can usually be terminated a t this point. Thereafter, refinement of the structure to determine exact bond lengths and angles is largely a matter of arithmetic. The amount of arithmetic involved is vast, but modern, large, high-speed electronic computers will normally do this arithmetic in a few hours. Examples of Structures Solved
It will be possible to mention only a very few of the major contributions of X-ray analysis to organic and biochemistry. One crystallographer whose work is of profound importance to organic chemistry is J. Monteath Robertson. The reader is referred to some of his publications for a more comprehensive account of work on organic structures (2, PI). First Organic Structures. The earliest organic structures to be solved were hexamethylbenzene by K. Lonsdale, and the pair naphthalene and anthracene by J. M. Robertson. These analyses were carried out in 528
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the early 1930's and the bond lengths and angles found provided vital confirmation of the values predicted by the new quantum mechanical theories of valence which were being developed a t about the same time. X-ray crystal structure analysis has continued to be a major source of bond data on which further theoretical developments depend. Penicillin and Vitamin BIZ. There are numerous examples of the determination of structures for which no structural formula was available a t the outset of the analysis. Two outstanding analyses were those of penicillin and vitamin BIZby D. C. Hodgki and her associates. These analyses arc noteworthy in the degree of cooperation between the crystallographers and the organic chemists who were doing degradative studies a t the same time. The information provided by the organic chemists contained clues without which the strnctures might not have been solved. Alkaloids. On the other hand the brilliant work on alkaloid structures by J. Monteath Robertson and his associates in the past six years has required no assistance from degradative organic chemistry (4). In fact, in some cases the organic chemists' findings were contra&ctory, and in others the structures suggested by the organic chemists were wrong. In each case the X-ray analysis proceeded swiftly towards a unique structure which invariably was the correct one. In this work the power of X-ray analysis in molecular structure determination is beautifully demonstrated. Proteins. Two extremely important structure analyses are those of the proteins myoglobin and hemoglobin by J. C. Kendrew and M. Perutz respectively, and their associates. Kcndrew and Perutz shared the 1962 Nobel prize in Chemistry for this work. Probably the most dficult problem in proteim crystal structure analyses is the preparation and crystalliiation of the appropriate heavy atom derivatives of the proteins. Thus the work of Kendrew and Perutz includes sonlc rather brilliant biochemical manipulations. Hydrogen Bonding. X-ray crystallographic studies of hydrogen bonding have been just as important as the purely structural studies. The X-ray analysis of crystals of the amino acids by L. Pauling and R. B. Corey provided the molecular dimensions of the amino acids and also the lengths and angles of the hydrogen bonds formed. This information was essential to the formulation of Pauling's or-helix theory of protein structure. Similarly the information on molecular dimensions and hydrogen bonding characteristics of purines and pyrimidines, obtained by W. Cochran and his associates, was of great importance in the development of the theory of the structure of DNA by F. H. C. Crick and J. D. Watson. Absolute Crmfisuratias. Conformational questioils frequently may be answered by X-ray analysis. The most remarkable of such results is the determination of the absolute configuration of tartaric acid by J. M. Bijvoet and his associates in 1951. They showed that the Fischer convention for the configuration of tartaric acid was correct. This single result placed a large part of organic stereochemistry on an absolute basis. Furthermore Bijvoet's method of determining absolute configuration is of general application. It may be used to determine the absolute configuration of molecules
whose configuration cannot be related chemically to that of tartaric acid. Nucleic Acids. Another important conformational question is that of the planarity of the pyrimidines and purines in nucleic acids. I n the original theory of DNA structure, Crick and Watson assumed that these molecules were planar and that the atoms directly substituted on the ring were in that plane also. However, recent very precise work in three diierent laboratories, including this laboratory, has shown that in some nucleosides and nucleotides the heterocyclic rings depart considerably from planarity. The significance of these observations has not been fully worked out yet. A related result is one from this laboratory in which it is shown that two different conformations of the deoxyribose ring in nucleosides are equally probable. These two conformations can lead to two different nucleic acid structures in which the distances between neighboring base pairs may differ by as much as 1 A. Recent Technological Developments
Two technological developments in the last few years have transformed X-ray crystallography from a tool accessible only to specialists into one that any welleducated chemist can use effectively. One is the development of large electronic computers. The other is the development of automatic equipment for the measurement of the intensities. These two events turn out to be very closely related. Solving the phaae problem and the subsequent structure refinement involves performing an enormous amount of arithmetic. Until quite recently analyses of all but the very simplest structures required months, and even years, of calculation. Few chemists who had a structural problem felt they had the time necessary to become involved in such a project. Those who did tended to become specialists in X-ray analysis and to give up their work in the chenlical laboratories. In addition to limiting the number of structures worked on, the complexity of the calculations affected the precision of the results in those analyses which were carried out. The crystallographers used the intensities of only a small number of the available reflections (usually about 10%) in order to keep the time required for the calculations to within reasonable limits. Furthermore, once the phase problem had been solved in any structure analysis, the crystallographer had to make rather brutal approximations in the refinement calculations for the same reason. About eight years ago there became available, for the first time, electronic computers large enough and fast enough to perform rigorous crystallographic calculations. These machines became available commercially in large numbers and very soon most crystallographers in Europe and North America had access to them. I n a few hours the machimes were able to carry out calculations requiring tens of years of hand calculation. Thus the arithmetic involved in X-ray analysis suddenly became quite trivial. Of course once the arithmetic ceased to be the bottleneck in X-ray analysis crystallographers began to use all of the experimental data they could collect. This created a new bottleneck.
When crystallographers used only 10% of the available data, they could measure all the intensities they required in two or three weeks. Furthermore, since necessarily the strncture was refined only partially, data of moderate precision sufficed. (In fact the author recalls a statement by an older crystallographer that he could solve his structures knowing only whether the intensities were zero or nonzero.) It soon became clear that many months were required to measure the intensities of all the reflections from a typical organic crystal. This was true particularly when attempts were made to use traditional methods to collect intensity data with the higher precision justified by the more rigorous calculations. Full use of the new computers gave much more precise results, but the total time required for these results to appear remained about the same as before. The development of new methods of collecting the intensity data has been stimulated as a result (5). One such development has taken place in the author's laboratory (6, 7). A machine has been built, called CASCADE I, which is con~pletelyautomatic in its operation. It works twenty-fours a day for several days with trivial supervkion. CASCADE I is controlled by a paper tape produced by an IBM 1620 computer and it in turn produces another paper tape, containing the intensities, which is processed by the computer. I n three weeks CASCADE I can collect a set of intensities which graduate students previously required a full year to collect; also the precision of the data collected by CASCADE I is higher than that conveniently obtainable by traditional methods. Further developments in this laboratory are leading to CASCADE 11,an entirely different type of machine. It should be capable of collecting intensities of all reflections from a crystal, even a protein crystal, in a matter of two or three hours. Thus given the availability of a machine like CASCADE I, one or more heavy atoms in the unit cell, and access to a large, fast computer, it is clear that the X-ray analysis of the crystal could be carried out in a very few weeks. Presumably most chemists with a structural problen~would be prepared to set aside a few weeks in which to carry out a structure analysis. Should CASCADE I1 perform as expected it is conceivable that this time could be reduced to a very few days. X-ray crystallography is therefore on the verge of becoming a physical tool to be used by the chemist directly. This event will mark the beginning of a new era in organic and biochemical analysis. Literature Cited (1) LONSDALE, K., J. &EM. EDTIC.,41,240 (1964). (2) EWALD,P. P., Editor, "Fifty Years of X-ray Diffraction," Oostoek, Utrecht, Netherlands, 1962. (Especially part IV, chap. 10 and 13.) (3) ROBERTSON, J. MONTEATH, "Organic Crystals and Mole cules," Cornell University Presa, Ithaca, New York, 1953. (41 ROBERTSON. J. MONTEATE. Prm. C h a . Soc. (London). , 1963,229: ( 5 ) ABR*HUS,S. C., Chem. Eng. News, 41, June 3,108 (1963). (6) "Research Topic of the Week," Chem. Eng. Nms, 40, Aug. 27,36 (1962). (7) cowm, J. P., MILCINTYRE, W. M., AND wEEmMA,G. J., AclaCqst., 16,221 (1963).
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