J. D. Dunifs
Swiss Federal Institute of Technology 8006 Zurich, Switzerland
1
I
The TWO Forms
M o s t students of chemistry know that there are two forms of the cyclohexane molecule: a "chair" form, which is rigid, and a "boat" form, which is flexible and hence really corresponds to one of a continuous manifold of "twist-boat" or "skewboat" forms (Fig. 1). The rigidity of the chair form and the flexibility of the boat form are both immediately apparent from a study of models, e.g., Dreiding models, in which bond angles are held more or less constant but rotation about bonds is not restricted. It is easy to be
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Figure 1. Tap: Rigid choir form of cyclohexane, Dwl3ml symmetry Bottom: Three of the flexible forms 1, boat, Gvlmm21 symmetry; 2, generol twirl, Cn121 symmetry; 3, rymmetricd or fully extended twist, D l 2 2 2 ) symmetry.
persuaded from a study of such models that any deformation of the chair form is associated with a deviation of the bond angles from their standard value (usually, although not necessarily correctly, taken as 109" 28', the tetrahedral angle), whereas bond-angle distortion is apparently not required in order to pass from one of the flexible forms to another. It is difficult, however, to be quite sure that some bond-angle distortion has not taken place during the latter process, for the bond angles incorporated in the models are not infinitely stiff. Some bond-angle deformation must occur, for example, in a 5-membered ring built from tetrahedral units, since the sum of the internal angles is only 540" (5 X' 108") for a planar ring and even less for a nonplanar one. Nevertheless, a 5-membered ring can be constructed from the usual units without difficulty, and the small angle deformations that must occur may easily be overlooked. Once the dangers of drawing conclusions from mechanical models become apparent, the student may try to examine the properties of the 6-membered ring from a purely mathematical standpoint. The mathematical problem can be expressed in terms of the number of torsional degrees of freedom of an equilateral, equiangular-&membered ring. If there are no torsional degrees of freedom, then the ring is rigid; it can be 'SACHSE, H., Ber., 23, 1363 (1890); Z. physik. Chen., 10, 203 moer -eives the detailed mathematical .. analysis. P., A N D OOSTERHOFF, L. J., D k . Farad. Sac., HAZFIBROEK, 10,87 (1951).
01 Cvrlohexone
deformed only by changing the bond lengths or bond angles. If the ring possesses a torsional degree of freedom, it is flexible and can be deformed without alteration of the bond lengths and angles. The object of the present paper is to provide a simple mathematical derivation of the flexibility of the boat form and of the rigidity of the chair form. The result is, of course, not new. It was already given by Sachsel in his very detailed, long-neglected analysis of the 6membered ring. More recently, it has been derived in a more elegant way by Hazebroek and Oo~terhoff.~ However, many students will have neither the patience to work through Sachse's somewhat tedious analysis, nor the mathematical ability to follow the arguments of Hazebroek and Oosterhoff. The derivation presented here may,, therefore, be of interest, and the methods can be applied to numerous other problems involving geometric constraints in cyclic and polycyclic systems. Geometrical Preliminaries
The relative positions of n points (n atoms) are specified by 3n - 6 independent parameters, which may be chosen, for example, as 3n - 6 Cartesian coordinates or as 3% - 6 internal parameters, such as distances, angles, and torsion angles, or in any other way. The torsion angle w(1234) = w a between atoms 1 and 4 about the bond 23 (Newman projection angle) is defined as the angle between the directions 21 and 34 in projection down the bond 23 (Fig. 2). It is taken as positive if the sense of rotation from 21 to 34 in projection down 23 is clockwise, and negative if anticlockwise. Given the bond distances dl2, &a, d31 and bond angles 02,O3 the torsion angle wZ3is related to the nonbonded distance d ~ by r dl,' = dll'
+ dm1 + dwa - ~ ~ ~ ~ Z C O-S 2d>ad~4~08& O X +
2 d l 2 d a l ( ~ o ~ 0 ~c osin02sinO3co~ou) ~O~
A chain of n atoms contains exactly n - 1 bond lengths, n - 2 bond angles and n - 3 torsion angles; these add up to 3n - 6. Thus the bond lengths, bond angles and torsion angles in a chain molecule are in-
(1R92). The -~~second \----,
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Figure 2. The tonion angle I12341 is positive labout +45'1 this drawing.
oin
Figure 3. Numbering of atoms in 6-membered ring.
dependent; if they are specified then the relative positions of the atoms are completely defined. In a ring of n atoms the n bond lengths, n bond angles, and n torsion angles are not independent and they must be related by 3 equations of constraint (ring) closure conditions). These constraints lead to important restrictions on the topographies of cyclic molecules especially when, as is often the case, additional constraints arising from ring fusion or from symmetry relationships are present. Some of these restrictions are obvious; some are well-known to anyone experienced in handling molecular models; but some only become evident through a detailed analysis of the system. If the system of n atoms is taken to possess elements of symmetry, the number of degrees of freedom (freely assignable parameters) is correspondingly reduced. Thus a regular tetrahedral arrangement o f 4 atoms about a central atom (n = 5,3n - 6 = 9) is defined by only 1 parameter which is essentially a scale parameter (for example, the bond length). The bond angles in a regular tetrahedral arrangement of atoms are not adjustable; they are fixed and equal to 109' 28'. Thus the symmetry conditions lead here to the loss of 8 of the 9 independently variable parameters of the 5-atom system. Constraints in 6-Membered Ring
A 6-membered ring contains 3 distinct 1,4-distances and each of them can be expressed in terms of the bond distances, bond angles, torsion angles in t u ~ oways, corresponding to the two ways (clockwise and counterclockwise) in which we can run round the ring. For example, dla does not depend on whether we take the 4atom chain as 1-2-3-4 or 1-6-5-4 (Fig. 3). For an equilateral, equiangular ring we have
other special forms with higher symmetry belonging to this class of solutions (Fig. 4). II. The solution was = -wsa corresponds to a ring with at least s. mirror-plane relating the segments 1-2-3-4 and 1-6-54 mnd passing through the atoma 1 and 4. It also requires that
III. By the same arguments we must have either a twofold rotation axis or a mirror plane relating the segments
The solutions with rotation axes do not add anything newthey are included in the various forms of I. The solution with mirror planes through 1 and 4, and through 2 and 5 implies 1,4: 2,.5:
-la = -us wr4 = - w a
wrr = - w e wcs = -wsa
wzz = -oat w n = -wrs
and hence Thus the presence of the two mirror planes through 1,4 and 2,5 implies the existence of a thirdmirror-plane through 3,6.
With its equal bond lengths and bond angles, and with the derived pattern of torsion angles, this form is seen to be the familiar chair form of D 3 symmetry. ~ The projection of the 6-membered ring with Dad symmetry is a regular hexagon, specified by a single parameter (the projected bond length). To specify the actual ring requires a further parameter, i.e., the amount by which successive atoms are alternately above and below the plane of projection. Thus two independent parameters are required to specify the ring. In terms of internal parameters, one of these must be a scale factor (bond length) and the second can be taken as the bond angle or the torsion angle. If one of these is specified, the other must be fixed. Now we can choose Cartesian coordinates as indicated in Figure 5 and we have
+
d d = r P 42%= 1 (say) dl,' = 32% = 2 - 2eos8 = 4za 4r2 = 3 - 2cose = 3 - 4cosB ZcosPB- 2sinPBcosw (from the previously derived formula)
Hence
+
+
Hence wna = fwss
cosw =
c0sPB- COSB sin9@
The sense of a torsion angle is not changed by a rotation or translation of the system; but it is changed by a reflection or inversion of the system-the mirrorimage of a right-handed corkscrew is a left-handed corkscrew!
or, in a more symmetrical form
I . The solution w~ = wss corresponds to a ring with at least a 2-fold rotation axis of symmetry relating the segments 14-34 and 1-6-54 Thus, it also requires that
and hence o cannot be changed without changing 0 a t the same time. Note that for 0 = 109' 28' (tetrahe-
cosw
+ coswcoss + cos8 = 0
In addition to the general form with C2symmetry, there are two
-$.-$.& z
Figure 4. Torsion angle patterns in throe forms of cyclohexane: twist;' "symmetrical b o a r and "symmetrical twist."
"general
1'
7 --+T,-z
Figure 5. Cartesian coordinates for 6-membered ring with Dad symmetry.
I
Figure 6. Assignment of Cartesian coordinates to atoms of 6-membered ring with a twofold mtdion axis (parollel to 1) not pmrring thmugh ony @oms or any bonds.
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dral angle, cos 0 = -'/a) w = 60' (synclinal partial conformation), and that as 0 increases towards 120°, its maximum possible value, w decreases rapidly towards 0°, the value corresponding to a planar ring. The bond angle in the actual cyclohexane moleculea is about l l Z O , appreciably larger than the tetrahedral value, so that the molecule is distinctly flattened compared with a ring built from tetrahedral carbon atoms. Thus all equilateral, equiangular 6-membered rings can be divided into two classes (1) Rings that contain a.twofold rotation axis not passing through
any rttomsor through any bonds. (2) Chzir-form rings of DMsymmetry in which the torsion angle is a function of the bond angle. Such rings are rigid.
It remains to be shown that rings of class (1) are not rigid, i.e., that they possess one or more torsional degrees of freedom. We consider any 6-membered ring with a twofold rotation axis not passing through an atom or a bond. To define the positions of the 6 atoms in terms of Cartesian coordinates it is necessary to specify the values of at least seven parameters (Fig. 6; .one of the atoms can be assigned coordinates 0, yr, 0, the other two symmetry-independent atoms are in general positions xz,y2, z2 :53, y3,21). The ring may also be defined in terms of internal parameters-bond lengths, bond angles, torsion angles, etc.-and for each such symmetry-independent parameter that is assigned a fixed value, one degree of freedom
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is removed. Since there are seven Cartesian parameters, there must also be seven independent internal parameters. Clearly there are three symmetry-independent bond lengths and three symmetry-independent bond angles. If these are assigned fixed values there is still an additional internal parameter that can be varied. This free parameter may be taken as a torsion angle. Of course, it is possible that this free parameter "disappears" in certain special cases. For example, if the three symmetry-independent bond angles are chosen so that their sum is exactly 360' then the ring degenerates into a planar hexagon in which all the torsion angles are zero. The sum of the symmetry independent bond angles must be less than 360" for a puckered ring. If a torsional degree of freedom is a property of all ~uckered6-membered rinns with a twofold rotation axis not passing through any atoms or any bonds, then it should also be a property of all puckered, equilateral, equiangular &membered rings with this symmetry property-which is what we wanted to show. I should like to thank Professors E. Eliel (Notre Dame), E. Heilbronner (Bade), L. J. Oosterhoff (Leiden), J. Waser (Pasadena) and B. L. van der Waerden (Zurich) for helnful criticism.
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8 D ~ v 1 sM., , (1963).
AND
HASSEL,O., Acla Chon. Scand., 17, 1181
