Resource Papers-IX Prepared under the sponsorship of The Advisory Council on College Chemistry
Milton Orchin and H. H. J a f f i University of Cincinnati Cincinnati, Ohio 45221
Symmetry, Point Groups, and Character Tables I, Symmetry operations a n d their importance for chemical problems
Ihe concept of symmetry is important to almost every aspect of life in our universe. The replication of life at the molecular level, the motion of electrons, the relationship of form to function, as well as the creative efforts of man in art, architecture, literature, and music all involve application of symmetry principles. The existence of life remote from our planet will be probed by testing the symmetry properties (optical activity) of the molecules found there. All molecules known and yet to be discovered may be classified with respect to their symmetry properties. In the universe we know, the wave function of all electrons (i.e., their motion) is antisymmetric with respect to exchange of any two of them. The range of chemical problems whose understanding is enhanced and simplified by symmetry-derived considerations is very large and growing, and hence it is incumbent on every chemist to understand the elements of symmetry not only as an aid in understanding his discipline but as an aid in understanding himself and his ~ ~ o r l dIn . what follows we hope to provide an elementary treatment of the subject sufficient to enhance the understanding of some simple chemical problems. We also hope to provide the essential background for application to the whole range of chemical problems that are encountered in chemistry courses beyond the second-year level. The treatment is at a level intended to be comprehensible to students who have had the introductory organic chemistry course. As a matter of fact, this paper was written for the express purpose of delineating material that could hopefully find its way into such a course if it is not already there. The material which appears in fine print adds rigor to some of the arguments but This paper represents the last in a series of "Resource Papers" prepared under the sponsorship of the Advisory Council on College Chemistry (ACI)which is supported by the National Science Foundation. Professor L. Carroll King of Northwestern University is the chairman. Single copy reprints of this paper are being sent to ohemistry department chairmen of every U.S. Institution offering college chemistry courses and to others on the ACs mailing list. This is Serial Publication No. 49 of the Advisory Council.
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is not considered essential for the introductory course in organic chemistry as it is now generally structured. A Symmetry Operation.
Point Symmetry
Although what follom is applicable to all objects, we shall generally confine our discussion to molecules and their properties. If we partially rotate (rotation less than 360') any molecule around an axis which passes through its center and if, after such rotation, the molecule appears to be exactly as it was orginally, we have performed a symmetry operation on the molecule. Of course, if rve rotate the molecule 360' it will appear unchanged, but this is a trivial operation, i.e., it is applicable to every molecule. Accordingly, here we are interested in rotations of less than 360'. Thus, rotation of 180' around the z-axis in trans-1,2-dichloroethylene,I , produces a new orientation superimposable on the original. In
this molecule there are three sets of like atoms; the pair of H, the pair of Cl, and the pair of C atoms. The individual atoms of each set of two cannot be distinguished and hence the 180' rotation transforms each like at,om into its indistinguishable partner. Similarly, rotations around the z-axis of 120 or 240' in 11; 90,180, or 270' in 111; 72,144,216, or 288" in IV; 60, 120, 180, 240, or 300' in V all give new orientations which are indistinguishable from (superimposable on) the original; and hence in all cases v e have performed This paper is the first of three parts. The succeeding parts will cover classification of molecules into point groups, and character tables and their significance. The figures, tables, and footnotes rill he numbered consecutively thl.oughout t,he series. The sections in fine type indicated by a § add rigor to some of the arguments but are not considered essential to the introductory organic course a now taught.
a symmetry operation. If a molecular model of I were held in front of you and you were asked to close your eyes while someone rotated the model 180" around the z-axis, on opening your eyes, you would be unable to tell that anything had been done; when this is the case a symmetry operation has been performed on the molecule. During all symmetry operations, which can he applied to a molecule, at least one point in the molecule, the center of gravity, remains unchanged. Symmetry of this kind is therefore called point symmetry; it is thereby distinguished from other symmetry like translational symmetry, which characterizes, for example, a picket fence. If we place I in a coordinate system and assume that the center of gravity is a t the origin (i.e., the point defined as 0, 0, 0) and now perform the 180" rotation around the z-axis, we find that the coordinates which describe like atoms before the operation will also describe the like atoms after the operation. Thus, if we consider atoms as points and if C1 atoms are at the points 2, 2, 0 and -2, -2, 0 before the operation, CI atoms will be found at 2, 2, 0 and -2, -2, 0 after the operation. In the first portion of the following discussion we shall consider noint svmmetrv and its designations and later discuss aiew comparablk operations a i d notations in space symmetry. ~
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Symmetry Operations and Symmetry Elements Rotation About an Axis; Rotational Axis, C.
During a 360" rotation around the z-axis of the molecules I-V, I repeats itself twice, 11, three times, 111, four times, IV, five times, and V, six times; the z-axes in these molecules are thus called a two-fold, three-fold, four-fold, five-fold, and six-fold rotation axes, respectively. If the angle through which the molecules must be rotated in order to secure the superimposable image is designated 8, the molecules bave 360/8-fold rotational axes. The rotational axis is usually denoted as C,, where the C is for cyclic and where p = 360°/0. The subscript p is thus the number of times the superimposable orientation appears during a 360' rotation. It should be understood that the symmetry operat.ion we bave been discussing is rotation around an axis, while the symmetry element is the rotational axis C,. The symmetry operation C, frequently also is called a proper rotation, and the symmetry element C, a proper rotational axis. As we shall see, this notation distinguishes this symmetry operation and element from the ones designated as an improper rotation and an improper rotational axis. Molecules I-V all are planar and the z rotational
axis in each case is normal to the molecular plane. The rotational axes of some non-planar molecules are shown in structures VI-XI. In trans-l,2-dichlorocyclohexane (for our purposes here let us assume that the ring is planar), VI, there is a Cz axis in the plane of the molecule bisecting the bond between the two chlorine-bearing carbon atoms and the bond opposite; in ammonia, VII, there is a C3 axis passing through the N atom and the center of the triangular base formed by the three H atoms; iron pentacarbonyl, VIII, and chloroform, IX, have C3; the all cis-1,2,3,4-tetrachlorocycIohutane, X, and tungsten hexacarbonyl, XI, have C4. All linear molecules such as HC=CH and CO* (and hence also all diatomic molecules) possess C,, since rotation about the internuclear axis by any angle gives an orientation identical with the original. Although i t may appear to be trivial, it is nevertheless important to recognize that all molecules have an infinite number of Cl axes, since a 360' rotation around any or all axes passing through the center of gravity of the molecule returns it to its original position. The Identity Operation
The operation which leaves a molecule unchanged and hence in an orientation identical with the original is called the identity operation. It is desirable to distinguish between identical and equivalent orientation~. Thus, in performing the C2 operation on trans-dichloroethylene, I, we exchange like atoms and the new orientation is not identical with the original; however it is equivalent to the original because like atoms cannot be distinguished. Only a second C* operation results in the orientation identical to the original. Chloroform, IX, has a Cz axis which includes the H-C bond and passes through the center of the triangular base formed by the three Cl atoms. A 120" clockwise rotation takes C1, into CI,, CI, into Clz, and CL into CI1 and the H and C atoms into themselves. Again, the new orientation is indistinguishable from the original because one cannot distinguish chlorine atoms. However, if the first Cawere followed by a second Ca and then by a third, we would return the molecule to its original position. The two orientations obtained by C3and by C3 X C8 are orientations equivalent to the original but the Ca X C3 X CQor Ca3brings the molecule back to its original and identical orientation. Even if the C1 atoms could be distinguished, it would appear that nothing at all bad been done to the molecule by Caa. An operation which leaves the molecule identical to the original is called the identity operation, denoted by I (or by E from the German Einheit meaning unit or loosely identical) and thus Caa = I; C22 = I: C1 = I, and since all these operations by definition leave the molecule unchanged, doing nothing to the molecule is also an identity operation. The necessity of including this operation will become apparent when we consider the properties of a point group.
5 Operotions of the Same Class It should he noted that in the chloroform example performing Ca X CI or Csl gives rtn orientation that is identical with that
obtained by a Ca operation in the counterclockwise direction, i.e., Cs X Cs = C:, where the prime refera to wunterclookwise rotation. Furthermore, Ca and C's give equivalent but not identical orientations; these two operations are said to belong to the same class and we say that chloroform has 2Ca axes where 2 Volume 47, Number 4, April 1970
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H3 Figure 1 .
Methane: a Atrahedrd molecule.
is called the order of the class. On the other hand. in a molecule with CI as, e.g., transdichloroethylene, performing either s. C* or a C'r results in the identical orientation and hence we have only
(D is unlabelled and is the face apposite the H,:H8 edge and cannot be seen) is an equilateral triangle. The most obvious symmetry feature of such a trianale is its CI symmetry. I t is clear then-that a rotational axis wl&h passes through the center of a triangular fctee and the oorner opposite it is a C8 axis, since i t transforms corners into each other. Thus, the axis passing through the center of face A and H4is a 3-fold axis, clockwise rotation around which interchanges the atoms on face A, carrying Hainto HI, HI into HI, and HI into HZ. This axis is also a C'Baxis, and the counterclockwise rotation or C', operation carries HI into Ha, Hsinto Hz, and H* into HI. Since the tetrahedron has four faces, there are a total of 8Cs. All of the SCI are said to belong to the same "class" (or in group theory parlance, "the trsces or characters of the corresponding trrtnsformation matrices are all equal"). For the time being we can say that all 8Cs transform the four hydrogens in an equivalent way. The Inverse Operotion
We saw above that if two or more operatiom performed in sequence return the molecule to an orientation identical to the original, the product of the operations is equal to the identity. Another very simple viay to return a molecule to its original position after a. symmetry operation consists of simply reversing the operation, Thus, the Cs or clockwise rotation of the molecule CHiin Figure 1, 120' around the axis passing through HI and the base of the triangle takes H? into H8,Ha into HA, and H, into H,. Now if this were followed by C's or C a P , the inverse of CI, the molecule would be returned to its original orientation. The two operations can be considered andogous to the military exercise of right face followed by left face, since the second operation in each case cancels the effect of the first operation. For this reason, the second operation is called the "inverse" (or less desirably the "reciprocal") of the first and the multiplication may be written C3-1C3 = I, or more generally, A-'A = I , where A is any symmetry operation. I t would not make any difference as to the order in which we performed these two operations, i.e., the two operations commute. Thus, we can rotate the molecule in Figure 1, 120' around the Cs axis first counterclockwise, followed by a. 120' clockwise rotation and thus return the molecule to its origins1 position. An operation such as CS is its own inverse in either direction gives the identical orientation. since s.second (2% Reflection of o Plone:
The Mirror Plane, a (sigma)
If a molecule is bisected by a plane, and each atom in one half of the bisected molecule is reflected (the operation) through the plane and encounters a similar atom in the other half, the molecule is said to possess a mirror plane (the symmetry element). The operation and the element are denoted by a (probably derived from the first letter in the German word Spiegel = mirror). 248
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All planar molecules of course have a t least one plane of symmetry, the molecular plane. Thus, the xy plane is a horizontal mirror plane in all of the molecules I-V. BF,, 11, has in addition three vertical mirror planes, each including one BF bond and bisecting the angle between the other two B-F bonds. PtC14,111,has four vertical planes, the planes along the x- and y-axes and diagonal planes which are perpendicular to each other and which bisect the angles between the x- and y-axes. Benzene, V, has six vertical planes, a set of three which pass through opposite atoms and which may be designated 3a,, and a set of three bisecting opposite C-C bonds, which may be designated 3ad. A symmetric linear molecule such as H C k C H has an infinite number of mirror planes, all of which include the internuclear axis which here is the z-axis; in addition to m a . there is a horizontal plane aA. If we take as our object a Cartesian coordinate system, reflection in a mirror plane always results in a change in sign of the coordinate normal to this plane, and leaves the coordinates parallel to the plane unchanged. Thus, a,, changes every point x, y, z into x, y, -z and vice versa, a,, changes x to -x and vice versa, and a,, changes every y into -y and vice versa. Inversion at o Center o f Symmefry; Center o f Symmefry, i
When a straight line is drawn from any atom in a molecule through the center of the molecule and, if continued in the same direction, encounters an equivalent atom equidistant from t,he center (the operation), and if the same operation can be performed on all atoms, then the molecule possesses a center of symmetry, designated i for inversion. Since each atom is thus reflected through the cent,er into an equivalent atom, atoms must occur in pairs (with the exception of any atom which may lie on the center itself) with the members of the pair equidistant but in opposite directions from the center. Thus, molecules I, 111, V, and X I have i, but 11,IV, and VI-X do not. The operation of inversion changes the sign of t,he three coordinat,es which define the posit,ion of an atom. Actually an atom should be considered a point but for our purposes we may consider the atom as having some volume. Thus, a point on the upper part of the upper right C1 atom in trans-dichloroethylene, I, may be described as being at x, y, z in the coordinate system shown. On inversion, this, point becomes - x , -y, -2, and is found on t,he lower part. of the equivalent lower left Cl atom. Since such an equivalent point was found on inversion, and every other point on every atom can be similarly transformed, t,he molecule has a center of symmetry. Rotation obouf on Axis, Followed by Reflection of o Plane Norm01 fo this Axis: Rotofion-Reflection Axis, S,
If a molecule is rotated around an axis and the resulting orientation is reflected in a plane perpendicular to this axis (the operat,ion) and if the resulting orientation is superimposable on the original, the molecule is said to possess a rotation-reflection axis (the element). The axis around which the rotation was performed is the rotation-reflection axis, and it is designated as S,, where p, as usual, is the order. This axis is also called
an alternating axis or improper axis, and is thus distinguished from a rotational or proper axis, C,. I n frans-dichloroethylene, I, the x-axis is an 5%axis, since, e.g., the lower right H atom on a 180' rotation around the x-axis followed by reflection in the yz plane (which is normal t o the x-axis) places the H atom on top of the upper left one in the original. The other atoms are similarly transposed. I t should be noted that in this example the S , axis which coincides with the x-axis is not a C2 axis, which here is the z-axis. It should also be noted that the S2 operation achieves exactly the same result as i. Thus, if we focus on a point in front of the molecule and on the right-hand chlorine atom in I and rotate the molecule 180' around the x-axis, the initial point x, y, z goes to x, -y, -'z, and reflection across rv, gives -x, -y, -2, so that Sz reverses the signs of all .three coordinates of each point. But this is precisely what i does, and so S2 is equivalent to i. Thus, if a molecule possesses i, the 8%need not be separately specified because it is equivalent and implied. Actually, any axis through a molecule having a center of symmetry is an Sz axis. Some molecules have alternating axes of order twice that of the highest rotational axis, and C,. Thus, in the chair form of cyclohexane, XII, the vertical axis which passes
alent bo S2 but a is equivalent to S I . Usnallg it is easier to find the plane of symmetry than the 8, axis, but the & axis will he any axis perpendicular t,o the plane of symmet,ry. Thus, in chloroethylene, XIV, the x-axis is an & axis; rotation of 360' around x followed by reflection in the ?/z plane gives a molecule identical with the original. Traditionally, a student is taught to look for a plane or center of symmetry in a molecule because these are more easily recognized, but it should now be clear that these element,^ are equivalent to S1 and S2,respectively. Summary of Point Symmetry Operations
If a molecule can be rotated or reflected around its center in such a wag that after t,he operation the molecule is identical with, or superimposable upon, the orginal, the molecule possesses point syn~met,rg of some kind. There are only two fundamental point symmetry operations: simple rotation around an axis, C,; and rotation-reflection about an axis, S,. C1 is equivalent to doing nothing, the identity operation, I; reflection in a mirror plane, a,is equivalent. to S1; and inversion, i, is equivalent to 8%. Some Applications of Symmetry Operations Optical Activity
through the center of the molecule is a Ca axis (e.g., clockwise 120' rotation transforms hydrogen 1 into 5, 5 into 10, and 10 into 1, as well as 2 into 6 , 6 into 9, and 9 into 2), but this axis is also an Sflaxis (e.g., clockwise 60" rotation followed by reflection in the normal plane takes 1 into 4, 4 into 5, 5 into 8, etc.). A vertical axis through the carbon of methane, XIII, is a Sa axis, taking 1 into 2,2 into 3 , 3 into 4, and 4 into 1. Inscribing the tetrahedron of CHain the cube helps to illustrate other symmetry properties of the tetrahedral structure. Thus, the S I axis described above passes through the cent,er of opposite faces of t,he cube and since there are 6 faces there are three S n axes. And because for each S 4 (cloclcwise) there is a (counterclockwise), there are thus a total of 6S* operations in a tetrahedral molecule, and all of these S 4 transform in a similar manner and hence belong to the same class. Note again that these S4 axes are not C4 axes; they coincide with C2axes. The highest fold axis implies a great deal about the symmetry properties of a molecule. Thus, the most outstanding symmetry property of the chair form of cyclohexane is its six-fold alternating axis, S6,which transforms all axial hydrogens into each other, as well as all equatorial hydrogens into each other, and thus the twelve hydrogens in CBH12belong to two sets of six chemically equivalent hydrogen atoms. The Sn operation on CHI transforms all H atoms into each other and hence the four H atoms in CHI constitute one set of chemically-equivalent atoms. The S axis is called an alternating axis because the equivalent atoms transformed by the operation lie alternately above and below the plane of reflection. It should also be pointed out that not only is i equiv-
The criterion for optical activity of a molecule involves the test of superimposability of its mirror image on the original; if the mirror image can be superimposed, the original and the mirror image are identical and the molecule is inactive (unresolvable). If the original cannot be superimposed on its mirror image, the compound is active. Thus, in the 1,2,-dichlorocyclopropanes XV-XVII we see that the cis-isomer, XVa, and , its mirror image; X V ~are directly superimposable, and hence identical. On the I contrary, the trans-isomer, XVI, and its mirror image, d" W O li '. XVII, are not superimposable; accordingly, trans1,2-dichlorocyclopropane is resolvable into the optically active enantiomorphs, XVI and XVII. d X", 1, c i x v , , It is always relatively easy to draw the mirror image of a structuire, but to test whether or not the mirror image is superi~ n~osableon the orieinal can be a trickv exercise. The problem of comparison is made &ore complicated by the practice of using projection formulas. If a molecule is represented by a three-dimensional drawing, there are no restrictions on the manipulation of the drawing; it can be rotated out of the plane of the paper or in the plane of the paper in any manner desired for comparison because the spatial relationship of each atom to every other atom is specified in the three-dimensional drawing. Hovever, there are restrictions in the way that plane projection formulas can be manipulated because there is no way of knowing (except by convention) which atoms were projected forward and which atoms projected back. Let us examine the optically-active compound
cA" "AC, XVb
I"
A
-
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shown in XVIII. I n drawing XVIIIa, the Br and F atoms are shown projecting in front of the paper, the C1 behind the paper, and the H and C atoms in the plane of the paper. The mirror image of XVIIIa is shown as XIXa and the two structures cannot be superimposed. The plane projection formula of XVIIIa is shown as XVIIIb, and the plane projection of the enantiomorph XIXa is shown as XIXb. Now if we wish to test for the superimposability of XVIIIb aud XIXh, we might be tempted to rotate XIXb 180" out of the plane of the paper around a vertical axis in the plane of the paper. After such a rotation, XVIIIb and XIXb would be superimposable; but in fact they are not. The error arises in the rotation of the plane projection of XIXb out of the plane of the paper, because in so doing all the atoms which were behind the plane of the paper are now in front, and those in front are now behind. Accordingly, in the comparison of these plane projection formulas for superimposability, one is not permitted to rotate the molecule out of the plane of the paper. Furthermore, it should also be apparent that such a plane projection formula cannot be rotated 90' in the plane of the paper because such a rotation would again correspond to interchanging atoms from behind the plane to in front of the plane of the paper. However, rotation of 180' in the plane of the paper is permitted because such a rotation does not exchange front and back groups. It should also be noted that when projection formulas like XVIIIb and XIXb are used, the exchange of one pair of atoms around an asymmetric carbon constitutes the transformation of the original to its enantiomorph. Thus, it is seen that exchange of the Br and F atoms in XVIIIh leads to the enantiomorphic molecule XIXb. Two exchanges, however, result in retention of configuration. Transformation of XVIIIb to XVIIIc represents two exchanges of atoms: first, Br and F were exchanged to obtain XIXb (the enantiomorph of XVIIIb) and t,hen Br and C1 were exchanged to obtain XVIIIc, identical with the original XVIIIb. That XVIIIc and XVIIIb are identical may perhaps be more readily seen if XVIIIc is drawn as XVIIId. Rotation of XVIIId, 120" counterclockwise around a vertical axis passing through t h e H-C bond gives the structure XVIIIa, whose plane projection is XVIIIb. The test of superimposability is readily performed by visualization %hen simple molecules are involved, but when complicated molecules are examined such visualization is difficult. Of course, a molecular model and its mirror image may be constructed to determine whether the pair are identical or enantiomorphic, but this is cumbersome and time-Consuming. Accordingly, it is desirable to determine from the symmetry properties of the molecule whether it is o~ticallvactive or nut. All molecules which lack a rotation-reflection axis of any order (fold) (S,) are said to be dissymmetric and are optically active, i.e., the molecule and its mirror 250
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image cannot be made to coincide in space by any kind of rotational or translational motion of the whole molecule. It will be recalled from our earlier discussion that a mirror plane is equivalent to S1and a center of inveraion is equivalent to Sz and hence molecules possessin~these symmetry elements have mirror images that are identical mith the original and hence are inactive. If we examine trans-1,2-dichlorocyclopropane, XVI, we see that there is a C2 axis in the plane of the three-memhred ring passing through the bond between the chlorine-bearing carbon atoms and the methylene carbon atom. Since the molecule does not possess an S, axis, however, it is active. On the other hand, because XVI and XVII have a Cz axis, they do not lack symmetry, i.e., they are dissymmetric but are not asymmetric. The only symmetry that dissymmetric (optically active) compounds possess are one or more C, axes; although many common dissymmetric compounds have C2, most optically-active compounds are asymmetric as well as dissymmetric and hence have only C1.' The spiro compound shown in Figure 2 has an Spaxis bisecting both rings and the nitrogen atom, but no planes of symmetry (81 axes). I t has been syn-
Figure 2. A rpim compound with an the nitrogen atom.
SI
axis bisecting both rings
and
tbesized and found to be optically inactive. The S4 axis is also necessarily a C2 axis, but as pointed out above, if C2 were the only symmetry element present the molecule would be active. The condition for optical activity, asymmetry, and dissymmetry can be concisely summarized as follows: If a molecule possesses only C,, it is dissymmetric and optically active; if p = 1, the molecule is asymmetric as well as dissymmetric; and if p > 1, the molecule is dissymmetric; if a molecule possesses S, with any p, it cannot be optically active. It is important to recognize that practically all molecules exist in some conformation that possesses only C, symmetry, and hence if all the molecules of that compound could be frozen in that conformation, the
' One useful classification of stereoisomers (isomers that have the same number and kinds of atoms attached to each other hut which differbecause of different fixed geometry) is a division into two classesstereoisomers that are enantiomeric and those that are not; the latter class are called diastereomers. Thus cis- and trans-Qhutene are diastereomers by this definition. The enantiomeric relationship is essentially the relationship between right or left hand helices or the handedness in the screw sense of being right or left handed. The handedness of s given enantiomer is called its chirality (pronounced kirality, from Greek cheir = hand).
compound would be optically active. Take as an example, 1,2-dichloroethane,XXa, in the saw-horse representation of the staggered form (or conformer) with chlorine atoms trans. This form is also drawn by means of the Newman projection formula in XXb. The two other staggered forms (gauche) are shown in XXI and XXII. Conformation XXa (and its equivalent XXb) have i = Sz as well as u = 81and C1; XXI and XXII have C2only, the broken line which bisects the C-C bond. Conformations XXIII-XXV are the three eclipsed conformations (drawn slightly askew for clarity); XXIII and XXIV have C2 only and XXV also has 2u = 2S1. There are an infinite number of conformations resembling XXVI which are neither completely eclipsed nor completely staggered; all of these have Cz only. Thus, practically all conformations other then XXb and XXV have only Cz symmetry. However, the barrier to rotation around the C-C bond is very small, and at all reasonably accessible temperatures, the compound exists as a mixture of all conformations. Thus, statistically there are as many molecules in conformation XXI as in XXII, and hence any optical activity due to XXI is exactly cancelled by its mirror image XXII. The statistical probability of finding equal populations of enantiomorph~at all times is true of all conformations, and hence the molecule is inactive because the aggregate of molecules is at all times a mixture of racemic mixtures; or more simply stated, on the average, for every optically-active molecule with a left-handed configuration there is one with a right-handed configuration. As a practical matter, it is sufficient to determine whether a molecule in any readily-accessible conformation has an S, axis. Chemical Equivalence and Isomers
I n molecules such as methane, ethane, benzene, and ethylene, all the hydrogen atoms are chemically equiual a t and in each of these molecules all the hydrogen atoms belong to one set. We now ask how exactly we can demonstrate that the hydrogens discussed above are members of a set and chemically equivalent. If hydrogen atoms are chemically equivalent and members of a set, replacement of any of the members of the set by another atom must result in an identical molecule. This criterion of chemical equivalence may he called the atom replacemat test. Thus, in the above examples, replacement of any of the four hydrogens in methane by a chlorine atom gives the identical monochloromethane; similarly, there is only one monosubstitution product of ethane, benzene, and ethylene. With respect to symmetry requirements, groups
(methyl group, e.g.) or atoms are equivalent if they can be exchanged by rotation about a symmetry axis, C, (a > p > I), of the molecule or group. Thus, if we refer hack and examine methane, XIII, shown inscribed in a cube we see that the C2axis, passing through the center of the top and bottom faces, carries HI into H8 and vice versa and Hz into Ha and vice versa. The Cz axis passing through the center of the side faces carries HI into Hz and vice versa and Ha into Ha and vice versa, and since HI can thus be carried into all the other H atoms by Cz operations, the hydrogens are all equivalent. We could alternately show the equivalence by -meansof the C3axes. Thus, rotation around the C3 axis passing through the corner occupied by HI, the center C atom and the corner opposite, takes HI into HI, Hginto H4,and Hb into Hz. Rotation around the C3 axis passing through Ha, C and the corner opposite exchanges atoms 1 , 2 , and 4. All four hydrogens are equivalent and can be transformed into each other by appropriate C3 operations, and hence there is only one chloromethane.
Examination of CH3C1,XXVII, shom that there is a C3 axis passing through the CI and C atoms and the center of the triangular base formed by the three H atoms. Rotation around this axis transforms the hydrogens into each other and hence the three hydrogens are equivalent; substitution of any hydrogen atom, e.g., by chlorine, gives only one product. This demonstration of the equivalence of the three hydrogen atoms in chloroform is the most satisfying demonstration that there is only one possible dichloromethane, CHzCIs. Analogous arguments for the numbers of isomers formed by substitution of hydrogens by other atoms or groups can be used for all molecules and prove equally satisfying. Consider ethane, XXVIII, in the staggered conformation. Rotation around the Cs axis perpendicular to the paper and passing through the center of the two carbon atoms carries the hydrogens of each individual methyl group into each other, and the Cz operation around the axis (broken line) in the plane of the paper and perpendicular to the C3 axis carries the top front H atom into the bottom rear H atom and vice versa, etc. Hence, all H atoms are equivalent and only one monosubstitution product is possible. The hydrogens are equivalent not only in this conformation but in all the infinite number of possible conformations, since they all have the C3 and C2described above. The chemical equivalence of hydrogen atoms in planar compounds can be ascertained by exchange of Volume 47, Number 4, April 1970
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hydrogens on reflection in mirror planes. Such mirror planes will always include a C2 axis. Thus, if we examine naphthalene, XXIX, we see that HI and Ha, Hg and H,, HZ and Ha, Hs and H7 are exchanged by reflection through the xy plane, while reflection through the xz vertical plane exchanges HI and Ha, Hq and HI, HZ and H7, H3 and HE. Thus there are two sets of hydrogens: HI, Ha, Ha, HI; and Hz,Hz, Hs, Hi and thus only two possible monosubstituted naphthalenes. Enontiotopk Hydrogens and NMR Spectroscopy
If a molecule has a Cn axis, the atoms exchanged by the C, operation are equivalent and substitution of any of the equivalent atoms leads to only one compound. Now let us examine the situation where similar atoms are exchanged, only by reflection through a mirror plane, or more generally, by an S, operation. In a general case, RCH2R1,XXX, the molecule does not possess C,, p > l , but does possess a mirror plane bisecting the HICHz angle. Reflection through this plane exchanges HI and H2. Now if we use the atom substitution test for equivalence, we see that suhstitution of HI by another atom leads to the enantiomorph of the compound obtained by substitution of Hz by the same atom. If substitution is by a C1 atom, the enantiomorphs are XXXI and XXXII. I n this situation the hydrogen atoms of XXX hear a special r e l a tionship to each other and they may be said to he pseudo-equiualat or symmetry equivalent. In recently approved nomenclature, such atoms are called enantiotopic atoms. Now let us examine a molecule such as RCHIR where the R groups are equivalent; the simplest example in the hydrocarbon series is propane where R = CH,. Let us assume that propane has the staggered conformation shown in XXXIII. The C2 axis which bisects the HCH angle of the methylene group takes H7 into H,, H, into H,, HI into HI, and H2 into H4. But nmr evidence as well as the fact that only two different monochloropropanes are known tells us that there are only two sets of H atoms, one with two and the other with six atoms. Logically the first set is assigned to H7 and Hz, the secondary hydrogen atoms, and the other six must belona.-to one set. How can we justify this? The answer lies in the following two facts: (1) If R in CH2R is an atom, e.g., CI, the molecule has a three-fold axis. But we cannot consider CH;CH?-.the actual R group in propane, as an atom. However, ( 2 ) rotation around the C-R axis is unhindered, so that this axis is a pseudo-symmetry axis, or an a x i s of local symmetry. The free rotation about this axis makes H atoms 1, 2, and 3 .equivalent as well. as Ho, Hz, and H, and since rotation around the real C2axis takes HI into HI, all six hydrogens 1-6 are equivalent and members of a set. Chemically equivalent protons give only one signal in the nmr. Thus, we would expect one proton signal in methane, ethane, ethylene, benzene, chloroform, and methylene chloride and two different signals in propane. Although the symmetry equivalent hydrogen atoms (atoms that can he exchanged mly by reflection through a mirror plane) are not chemically equivalent, such protons still give only one proton signal. The nmr ~
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cannot distinguish enantiotopic hydrogen atoms except perhaps in an optically active solvent. (If either R or R' is optically active, then the methylene hydrogens no longer give the same signal). A mixture of the (+) and (-) forms of the enantiomorphs of say lactic acid, CH3CHOHCOZH,in an optically-inactive solvent gives a single signal for the proton on the asymmetric carbon atoms, because the nmr is a probe which cannot distinguish enantiomorphs. Since there are thousands of compounds of the general formula RCH2R1, this problem is of general importance. It is quite likely that the optical activity of biological compounds is induced in non-active compounds by the interaction of the latter with optically active enzymes. Thus it is conceivable that enzymatic oxidation of one of the enantiotopic methylene hydrogens in propionic acid, CH,CHzCO2H, could lead to optically-active lactic acid.
5 I n the compannd RCHZR', the carbon atom is said to be
a prochiral atom because exchange of one of the hydrogen atoms, as pointed out above, by a group or atom other than H, R, or R' leads to an optically-active compound and a c h i d carbon atom. Replacement of one of the hydrogens would lead to the enantiomorph with the S configuration, while replacement of the other H atom would lead t o the R configuration. The hydrogen atoms can therefore be designated ss pro-S and pro-R hydrogen atoms. For example, if we consider ethylbenzene and intend to replace a. hydrogen by a chlorine atom, the hydrogens are designated as shown in XXXIV
Dipole Moments
A dipole moment is a vector property, i.e., it has both magnitude and direction, and it results from the unequal sharing of electrons between atoms. Although a vect,or property, the dipole moment is a stationary and not a dynamic property of a molecule. Stationary properties must remain unchanged by every symmetry operation of a molecule, and in order to remain unchanged the dipole moment vector must lie on each of the symmetry elements. Molecules with a center of symmetry cannot have a dipole moment because a vector cannot lie on a point. Also, molecules with more than one C, axis (p>l) cannot have a dipole moment because a vector cannot be coincident with two different axes. Thus, molecules with one C, (p>l) only or with only a and no C, (p>l) may have dipole moments. In addition, molecules that have C,(p>l) and planes of symmetry that include the C, also may have dipole moments. Such molecules are H20, XXXV, and NH,, XXXVI; the former has C2 and two a which intersect at the C2, while pyramidal NHI has 3a which intersect at the C3 axis. In these and in all cases the dipole moment vector must lie in all symmetry elements of the molecule. The direction of the dipole moment is thus determined; however, the magnitude and positive and negative ends of the dipole cannot be determined from symmetry arguments only.
