David Y. Curtin University of Illinois Urbano, 61801
Stereo Pair Drawings of Crystal Structures Prepared by a Desk Ca~cuhor-computer
That there exists an explosion of interest in the X-ray crystallography of organic compounds is shown by the fact that more than 3900 structures had been published a t the end of 1968 and that the number of structure-determinations appears to be doubling approximately every three years (I). In spite of this, little if any attention is paid to this suhject in most college and university chemistry courses. Although not all students will develop an interest in carrying out crystal structure determinations, i t seems that the time has come when an understanding of the use of the crystal structure data accumulating in the literature should be a part of the chemical education of all students of organic chemistry. With the greatly increased accessibility of desk-top calculator-computers and other small computers, there is availahle a means of introducing the basic principles of crystallography in a way which is much more readily understood than has been possible in the past. This paper describes a convenient method of using such a calculatorcomputer equipped with a plotter or oscilloscope display to produce from data reported by the X-ray crystallographer stereo pair pictures of individual molecules or, if desired, drawings of many molecules showing the mode of crystal packing. If a plotter is not availahle such drawings can he plotted by hand on ordinary graph paper hut only with a considerably greater expenditure of effort. Crystallographic data suitable for plotting are published regularly not only in the specialty journals such as Acta Cystallogmphica but also in the Journal of the American Chemical Society, Journal of Organic Chemistry, and most other journals where organic chemical work is regularly published. Valuable compilations of crystal structure references (1) and data ( 2 )are also availahle. To produce a stereo pair drawing, the computer draws two images side by side, one to be seen by the left eye and the other simultaneously by the right. Together they give the observer the illusion of seeing a truly three-dimensional object. The pictures can be viewed by many people (with practice) without special equipment although viewing is greatly facilitated by the use of special glasses which are available commercially (3); a possible substitute can be constructed from a pair of magnifying glasses and some Scotch tape. An example of such a stereo pair drawing is the three-dimensional structure of a molecule of b e n d (I) shown in Figure 1. The system described here was employed with a Hewlett-Packard 9100A Calculator with the 9101A Extended Memory and 9125A Calculator Plotter hut a number of other suitable calculations are available.
phistication from a level suitable for a high school chemistry student to that of a college student beginning the study of organic or physical chemistry. Although these programs were designed primarily for the plotting of crystal structures they can of course he used for drawing other types of three-dimensional objects. In this paper the operation of the program will be described first, then the description of the programming method will be discussed, and finally the mathematical basis will be indicated. Certain elementary principles of crystallography will be introduced as needed. The limitation of space prohibits the presentation here of the mathematical details of the calculations referred to. These are availahle in a mathematical appendix available from the author on request. Operation of the Plot Program The coordinates of each point to he plotted together with information as to whether the pen should lift before going to the point are entered into the memory with the use of a special program. Crystallographers typically employ a coordinate system with axes not necessarily a t right angles to one another. In addition units of length for measuring distances along each of the three axes may differ. The advantage of this unorthodox approach to analytical geometry will become abundantly clear as we proceed. In any case, it is often the coordinates in such a reference frame which are entered a t this stage. After information about the coordinates for all points and how they are to he connected is stored, the master plotting program is entered in the memory and executed. The first request from the computer is for the unit cell data-that is the units of measurement of length along each of the three axes (dimensions of the unit cell) a, b, and c and the angles between axes a , 8,and y. ( a is the angle between the positive ends of the b and c axes, 8 the ac angle, and y the ab angle.) These parameters are now used by the computer to calculate and store the elements of a transformation matrix which will he used later to transform the coordinates to others referred to conventional rectangular axes. The number of points to he plotted is entered. The program adjusts the scale of the plot automatically but if a
n
I n The method of plotting described here can be made the basis of a variety of projects varying in difficulty and so-
Figure 1. A stereo pair picture of a molecule of benzil (I) as it exists in the crystalline state. (4). The upper drawing is made looking down the c-crystallographic axis. The lower drawing shows how the rotation pmgram can be used to bring out a different perspective. All paints where straight lines join are positions of carbon atoms. The oxygen atoms are indicated by the "0's.'' Hydrogen atom positions are not shown. (They Were not determined in this investigation.)
Volume 50. Number 7 1 , November 1973 / 775
larger or smaller scale is desired the altered scale factor is entered and stored. On instruction to continue, the computer then draws first a picture to he viewed by the left eye and then a picture for the right eye. If it is desired to rotate in space the object being plotted, a special program is called, given information as to which axis of rotation to use and how many degrees of rotation are desired. When executed the program operates on the elements of the stored transformation matrix leaving a new matrix which during execution of the plot program both rotates and transforms to rectangular coordinates. As many successive rotation operations as desired can he made before plotting (hut only after the original transformation matrix is stored). To show the structure of a solid i t is, of course, not enough to plot the structure of a single molecule. Enough molecules must he pictured to allow the viewer to visualize the arrangement of large numbers of molecules filling three-dimensional space. The beauty of the crystallographic method is that once the crystal symmetry ("space group") and coordinates of a single molecule are known the coordinates of all other molecules in the crystal can (given sufficient time) be calculated from simple arithmetic (5, 6). The space group is always indicated in the report of a crystal structure and to plot more of the molecules in a crystal we need to go very briefly into the significance of this symbol. The simplest space group from our present point of view is PI. (The letter P designates the kind of lattice, in this case primative, and the number 1 indicates that the only symmetry element present is a l-fold rotation axis.) A crystal which belongs to this space group is made up simply of rows of many molecules all with the same orientation in space, the rows proceeding in each of the three axial directions with the separation between molecules along each row equal to the unit cell axial length in that direction. One of the few organic compounds (7) which crystallizes in such a simple structure is shown in Figure 2. To obtain the coordinates of the first molecule along the a direction, it is only necessary to add 1 to the xt for each atom in the structure of the parent molecule. Addition of 1 to each of the yi's of the parent molecule gives coordinates of the atoms of the first molecule in the b direction and so on. T o facilitate this proeess a translation operator is stored in the calculator memory. When it is called, given any values Ax, Ay, Az, and executed Ax is added to each rt, Ay is added to each yi and Az and to each zi, leaving the sums in the original locations in the memory. (Bear in mind that addition of l to xi shifts the molecule a distance a, in the direction of the a axis.) Crystals may have more than one molecule "in the asymmetric unit." This means that there are two or
Figure 2. Illustration of space group P1 with one molecule per unit cell (2 = 1). The crystal structure of iadodeaxyuridine (11) (7). The important point in the present discussion is fhat the structure consists of row upon row of moiecules completely identical in their internal structure and orientation. The rows are arranged in the directions of the three crystallographic axes. The molecular structure chosen is unfortunately camplex but the choice was extremely limited because of the rarity of this kind of crystal packing. Incidentally, this compound often known as IDU has been widely used in ophthalmology in the treafment of herpes virus keratitis, "the first clear-cut demonstration that true viral disease can be effectiveiy treated without obvious harm to the host" (7).A larger scale drawing of the structure with a somewhat different selection of molecules shows beautifully the hydrogen bonding which together with a weak 1. . . .O intermolecular band holds the crystal together in a three-dimensional network.
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more molecules not related by translation or another symmetry operation and each of which has its own independent set of coordinates. This increases the amount of work required to enter coordinates hut otherwise there is no chanee - in the wav the structure is ~ l o t t e d :all of the molecules in the asymm'etric unit are treated in the same way as if there were only one. Most crystal structures contain, in addition to the parent molecule and those molecules related to i t by translation, one or more sets of molecules related to the parent by symmetry. Since a report of the crystal structure in such a case gives only coordinates of the parent molecule it is necessary to know how to derive the coordinates of these additional molecules from the description of the crystal symmetry (the space group). There are needed certain symmetry operator programs in addition to the translation program. These carry out the appropriate operations on the stored coordinates xi, yi, zi, and then leave the results stored in the original locations. Although there are 230 possible kinds of symmetry (230 space groups) only four additional operators are needed to calculate the coordinates of any symmetry-related molecule in any of the space groups. These operator programs are stored in the computer together with the plot and rotation proarams and called when needed. They will be presented briefly and then their use will he illustrated. Inversion Ouerator. Reflects the molecule throuah a center of symmetry by changing the signs of all xi, Yi,-zito -xi, -yi, -zi (generally and more conveniently written as % gi, ti). Reflection Operator. Changes in signs of one of the three coordinates of each atom chosen by the chemist. For example xi yl, zi changed to xi, g , zi amounts to a reflection of the parent molecule in the ac plane. Three-fold-Axis Operator. This operator changes xi, y,, zi to Ti, XI - yi, zi which rotates the molecule around a three-fold axis (parallel . ):ot for examples, the point 2,3,4 would he transformed to 3,1,4. Interchange Operator. This operator interchanges any two coordinates specified. For example xi, yi, zi might go to vr. x,. zi. In a rectaneular coordinate svstem this am&ntsto reflection of the molecule in the plane = x. The "International Tables for X-ray Crystallography"
Figure 3. A P1 structure (8) with "2 = 2," i.e.. there are two molecules of malonic acid (I1I) (labeled A and B) i n the unit cell. These have different orientations but careful examination will show that one is derived from the other by reflection through a center of symmetry at the center of the cell. Mblecule B was, in fact, plotted from coordinates derived from Molecule A by means of the reflection operator (reflection through the Point, 0. 0, 0) and then translating that molecule (not shown) by adding 1 to ail xi's, y,'s. and 2,'s. This is equivalent to reflection through a point at the center of the cell. The labeling of the atoms other than carbon is indicated for an A-type molecule in the lower part of the picture. Note that there are infinite chains of acid molecules heid together by hydrogen bonds between the carboxyl groups
0-H---0
-C
/ \
\cI
O---H-0
The hydrogen atoms were not found in the X-ray analysis so their spproximate positions have been calculated (9) in order to complete the drawing.
Figure 4. An example of space graup P1 with Z = 1. Crystal structure of the triclinic form of p-dichlorobenrene (iV) (10). This is the crystalline form of this substance stable above 42%. In this case a and y a r e 93" although they may appear to be right angles. One at the challenges in making good stereo pair drawings is to show the third dimension clearly. A rule to remember is that a goad deal of overlap of lines is tolerable as long as they cross each other at a rather steep angle; they should not overlap when they run nearly parallel or a sense of confusion will be obtained by the viewer.
(6) lists all of the possible space groups, shows the symmetry relations in each, and gives the "coordinates of equivalent positions," that is the algebraic change in coordinates required to convert the parent molecule to each of the symmetry related ones. A simple example is the structure (8)of malonic acid (Illshown ) in Figure 3. The space group in this crystal is P I (primative cell with a one-fold inversion axis). The "International Tables (6) give for coordinates of equivalent positions "x,y,z; ?,?,T." In this case there are two kinds of molecules in the crystal-the coordinates of the second molecule being derived from those of the first by changing all of the signs. The reflection operator does this and leaves the new coordinates in the registers formerly occupied by the coordinates of the parent molecule. Operation of the plot program draws the second kind of molecule and both molecules are translated with the translation operator to show adjacent cells as desirsd. Another example from space group P1 illustrates two frequent complications. p-Dichlorohenzene (IV) has the structure shown in Figure 4.
m
lv
V
It will be seen that there is only one kind of molecule (in a single orientation). The first complication is that although there are two chlorine atoms and six carbon atoms coordinates are presented in the literature for one chlorine and three carbon atoms (the unprimed atoms). The explanation is that in this case the center of the molecule is at a crystallographic center of symmetry. As a result the symmetry operation xyz FyZ does not produce a new molecule hut converts one half-molecule to the other half. As a consequence, there is a single molecule per unit cell and it is only necessary to give coordinates of one-half of the atoms. In the present illustration, for example, the coordinates of C1 are those of C1' with all of the signs changed. The second complication illustrated by the p-dichlorobenzene is that authors do not always give coordinates of atoms all of which are in the same molecule; in this example, as can he seen in Figure 4, C1 and C1 project into the next unit cell where the x coordinates would be negative. The authors have reported the coordinates of these two atoms as positive by adding 1 to their xi's. In order to get a sensible plot the 1 has to he subtracted. This practice of reporting part of the coordinates for atoms in an adjacent molecule is confusing to the novice hut causes no difficulty once it is known that it can happen. As an illustration of the use of the reflection operator, o-nitrohemaldehyde (V) .crystallizing in space group P21 (primative cell with a 2-fold screw axis), is shown in Figure 5. In this case there are two equivalent positions with co-
-
Figure 5. An example ot space group P2r. o-Nitrobenzaldehyde (11) (V) iS made up entirely of molecules with the shape of the one labeled A. The nitro and aldehyde groups are each twisted about 30" out of the plane of the ring. Since they are held rigidly in this position the molecule is chiral In the crystal. Molecule 8 is simply an A molecule which has been rotated around an imaginary twa-fold axjs and advanced, in the b direction a distance of b / 2 . The B molecule thus has the same chirality as A. as do all other molecules in the crystai. This is a not uncommon exampie of a substance which is achiral in solution or in the vapor phase but Which form9 chiral crystals-all of the molecules in any given single crystal being either D or L. How would you obtain a plat of the mirror image of the crystal above?
Figure 6. The crystai structure of benzil ( I ) , space group P3,21. This crystal is in the trigonai system so that it is understood (and not stated in the publication of this structure) that a = b, a = 0 = 90' and y = 120' The space group is extremely rare. (Note that this crystal like the one in Fig. 5 is chiral.)
Figure 7. The structure of a.a'-dipyrldil (VI), an example of a variant of the mast common space group of P2,/c. This structure is platted as P2,ln (see the table). Note that as in the structures of o-nitrobenzaldehyde (V) end benzii (I) the molecuies in the crystal are locked in a chiral conformation. However, in this case (which is certainly more typlcal) there are two D-molecules and two L-molecules per unit cell. (Which molecules have the same chirality?)
+
ordinates x, y, z and f , y 112, i . The coordinates of the second molecule are generated from those of the first by using (1) the inversion operator to give a, g, t, (2) the reflection operator to give f , Y, t and, (3) the translation operator to add 0.5 to the yi's. The use of the 3-fold axis and the coordinate interchange operators is illustrated by the structure of henzil (I) (Fig. 6) space group P3121 (No. 152) (primative, with a 3-fold screw axis and perpendicular 2-fold axes) for which the structure of a single molecule was shown earlier (Fig. 1). The coordinates for equivalent positions are x , y, z; 9, x-y,z+1/3;y-x,?,z+2/3;y,x,i;E,yy-x,1/32; x - Y, 7 , 213 - 2. (However, with benzil the last three sets of coordinates produce the second half of the parent molecule.) The three-fold axis operator converts x, y, z to 9, x - y, z (that is the xi's are replaced by -yi's and the yl's by xl - yr) and then the translation operator used to add 0.333. . . to each of the zi's. (The net effect is to rotate and translate the molecule along the 31 axis one-third of a cell length.) Note that the application of these two operations to the second molecule produces the third and that the application to the third produces the first again hut translated one unit along c. The fourth set of coordinates Volume 50, Number 71, November 1973
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START CALL PROGRAM Explanatory Notes
.
Calc. I
(1) Calculates and stores the elements
Enter a, b,c,n,P,y
:
~--.
ttj of transformation
matrix T
(2) Calculates and stores d = ( 0 2 (3) StoresS
S (scale factor) and N (number of paints to be plotted)
(2) and (3) used
+ b2 +
for scaling. ca)1/2
(4) Calculates and stores the address Aof the last register to be used for storage of coordinates A.. = Ao + 4N-I where A. is address of first register (whereXI is to bestored)
7
(Al
Start new drawing without necessity of changing unit cell parameters, scale, and no. of points
To draw right picture
Resets A =Ao
(B)
L-i Sets (RL) = 0
E&l Extracts A from memory
"-+
Lift
IExtracts rr from resister with address v
A, incrementsA Extracts y, from address A, increments A Extracts r , from address A, increments A Extracts (PC), from A, incrementsA
(lift pen?)
I Cale. III (see appendix)
(RL)is indicator to determine whether the right or left picture is being drawn
Resets the address indicatorA toAo. Resets the (RL)indicator to 1 (="draw left picture")
(PC),is pen lift control indicator far point i.
(PC), for each paint has been stored at the same time as coordinates as x , y , and a.
CALCULATE X, Y, Z from xi,yi,ri and MatrixT
YRLand XL for left picture
g+Q
Calc. I1
Calculates for right picture
Calc. I11
set?
Pen to XL, YRLand down (if not
Return to Paint B
set?
Pen to XR,YRLand down (if not
-L
Return to Point B
Figure 8. Schematic of stereo plot program.
is produced by use of t h e interchange operator to interchange xi and yt followed by the reflection operator t o change t h e sign of the zi's. (This rotates the molecule around the two-fold axis.) T h e fifth and sixth sets of coordinates can he produced by successive rotation and translation a s was used t o produce the second and third sets from the first. Finally, t o maintain perspective, it should b e noted t h a t although a s has been mentioned, there are 230 space 778
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Journal of Chemical Education
groups well over half of the organic compounds whose crystallographic symmetry is known belong in only four space groups. These are listed in the table together with their equivalent positions and their number assigned in the International Tahles.6 Note t h a t numbers 14 and 15 occur in several variations due t o alternative possible choices of axes. Each of the common variants is eiven. An example from the most common space group is the structure of a,a'-dipyridil (VI), a close relative of benzil
Coordinates of Equivalent Positions of the Most Common Space Groups6 Space Group
I.T.C.DNo.
NO.of Equivalent Pmitions
(I), whose structure is shown in Figure 7. Careful study of the structures used for illustration here and other such structures in the literature provides, probably, the easiest method of learning the basic crystallographic principles as well as a method of developing a sound grasp of the concept of symmetry and its importance.
Acknowledgment
The author gratefully acknowledges support of the Natioral Science Foundation (Grant NSF GP 34545 X) and the Advanced Research Projects Agency of the Department of Defense (Contract ARPA HC-15-670221). Literature Cited (11 Kennard. 0..and Wstson. D. G.. "Malecular Structures and Dimonamm,"A. Omthmk. Domatraat 11-13, Utreeht, Netherlands Vol. 1 and 2. 1970; Val. 3, 1971. These and athor works on crystallography rehned to a n available from the Polvcrvstsi Book Service. P. 0. Box 11567. Piftsbureh. " .Pennwlvania. 15238. (2) ~ k k o f i R. . W. G . . '"~6st.i ~tructurep;"2nd Ed., Wiloy~lnterse~enee. N w York, Vol 5. l%966: Vol. 6, part I. 1969: Vol 6. part 2. 1971. (31 Hubbard Scientific ~ o m p & y , Northbmak, 1 1 l ~ ~The ~ i reader ~. interested in other applications of stereo psi* vieanng should see Teehter. D.. Ok", E., and McLesn. A.. "Steremmm Bmk of Rock. Minerals. and Gems." Hubbsrd Press.
.
VI For the serious X-ray crystallographer there are much more sophisticated plotting programs of which the one in most general use is that called ORTEP (13). Reference (13) gives a description of the program and also references to earlier work on stereoscopic drawing. It should be mentioned also that an ingenious stereo plot program using only the limited memory of the Hewlett-Packard 9100A without the memory extender was earlier written by Dr. D. B. Pendergrass, Jr. In this case the coordinates of each point must he entered as the point is plotted. Description of the Program
The heart of the program is the stereo plot program which is most easily described with the aid of Figure 8. Since computing systems vary, no description of the auxiliary programs will he given. Details of all programs will be provided by the author on request.
(4) Brown. C.J., sndSsdansga,R.,Actn Crysfollogr., 18,158(19651. (5) stom, G. H.. and Jemen, L. H., "X-RayStructure Determination..) The Mac.mil. lar Company. New York. 1968. C h s p 3. Buergec, M. d.. "Eiementary Cryrtsllogrsphy," John Wiley 8: Sons, New York, 1963. See Kapecki. J . A . 3. CHEM. EDUC., 49,231 (19721 for a brief intmductian t o X-ray structure determination. (6) "International Tables far X-Ray Crystallography," Vol. 1, lnfcrnational Union of Crystallography, Kynoeh Press, Birmingham. Eniland. 1969. (71 Camermsn, N., andTmtfor. J . A m Cryaldlogr. 18,203 (1965). IS) Goedkmp, J. A,, and MeeMllavry, C. H.,Acto Cry~follogr,10,125 (1957). (91 This cdcuiation war carried out bL a pmgrsm adapted for the HealettPackard 91WA calcuiafor fmm tho Fortran pmgiam of Wiberg. K. B.. "Computer Pm~amrningforChemists,"W.A. Benjamin, lnc. NewYork, 1965, p. 53. (10) Houaty. J., andClaatre,J..Aeto Cvstollogr.. 10.695(1957). (111 Coppena. P..Acro Crysfallogr., 17.573(19M). (12) Hirokawa, S., and Ashids. T.,Acto Clylnllogr.. 11,774 (1961). (13) Johnson, C. K., "Drawing Crystal Structures by Computer..) io "Crystallogrsphk Computer..) (Editors: Ahmed, F. R., Hall, S. R., end Hubcr. C. P I . Munks. gaard. Copenhagen. 1970. p. 227.
Volume 50, Number 11, November 1973
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