5634
G. V. RXTHJENS, JR., N. K. FREEMAN, IT. D. GWINXAND KENNETH S. PITZER
[CONTRIBUTION FROM
THE
Vol. 75
DEPARTMENT O F CHEMISTRY, COLUMBIA UNIVERSITY AXD THE DEPARTMENT O F CHENISTRY AND CHEMICAL ENGINEERING, UNIVFRSITY OF CALIPORSIA]
Infrared Absorption Spectra, Structure and Thermodynamic Properties of Cyclobutane B Y G.
IA'. RATIIJENS, JR., N. K. FREEMAN, ~VILLIAM D. GWINN.4ND KENNETII s. PITZER RECEIVED Aucrrsr 3 , 1953
Infrared and Raman measurements on cyclobutane are reported. The spectroscopic data, when used in conjunction with electron diffraction measurements and the measured entropy, suggest that the equilibrium configuration for cyclobutane is one with D2d symmetry, but that the barrier hindering inversion of the molecule is sufficiently low so that even a t ordinary temperatures an appreciable number of the molecules obey D d h selection rules. A normal coordinate analysis for cyclobutane has been performed, and force constants compared with those of related molecules A possible potential function for the out-of-plane bending motion of the ring has been suggested, and some of the thermodynamic properties of cyclobutane calculated.
Introduction Cyclobutane and its derivatives have been the subjects of a number of recent investigations in molecular Strained CCC bonds in the ring suggest that the equilibrium configuration of the ring should be planar; on the other hand, torsional forces about the C-C bonds tend to cause a non-planar configuration. Previously, spectroscopic results have been interpreted as most consistent with a planar configuration for the skeleton (D4b symmetry).2 However, the entropy determination reported in the accompanying paper is in disagreement with the previous analyses. The investigations here reported were undertaken in an effort to supply further information about cyclobutane. Apparatus and Sample.-Some of the spectroscopic observations were made by (X.K.F.) at the University of California using a Perkin-Elmer spectrometer that was converted from Model 12A t o 12C during the course of the work. Potassium bromide, sodium chloride and lithium fluoride optics were used. Most of these measurements were repeated a t Columbia University (by G. W. R.) using a Savitsky-Halford modified Perkin-Elmer 12 C spectrometer.6 In addition, the spectrum of the liquid and the temperature dependence of certain vapor phase intensities were obtained. Sodium chloride and calcium fluoride optics were used. The published data represent what we regard as the best combination of the two sets of results. The Raman measurements were made by Dr. J. T. Neu t o whom we are indebted for his unpublished results. His measurements were made on a liquid sample using a Steinheil type GH three prism spectrograph. The sample used was the same as reported in the accompanying a r t i ~ l e . ~
Infrared Spectrum.-The infrared spectrum a t room temperature is shown in Fig. 1. The optics used were as indicated in the figure, except for the liquid spectrum. In that case, sodium chloride optics were used below and calcium fluoride optics above 1400 cm.-l. With increasing temperature certain of the bands diminished in both peak and integrated intensity. The most noticeable case is the 750 cm.-' band. We have estimated J ln(I/&)dv a t 459'K. to be 73 f 5% of the value a t 299'K. Otherwise, with the exception of the 1889 crn.-' combination band, (1) This work w a s assisted b y the American Petroleum Institute through Research Project 50. (2) T . P. Wilson, J . Chcm. Phys., 11, 369 (1943). (3) H.Pajenkamp, 2. Elekfrochem., 63, 104 (1948). (4) J. D . Dunitz and V. Schomaker, J. Chcm. Phys., 20, 1703 (1952). (5) G. W. Rathjens and W. D . Gwinn, THISJOURNAL, 75, 5629 (1953). Other references are also given in this article. (6) A. Savitsky and R . S. Halford, Rev. Sci. Instruments, 21, 203 (1950).
it has generally not been possible to estimate quantitatively changes in intensity, and in some cases there is a question as to whether there actually was a change. Bands whose intensities were observed to decrease on heating are marked in Table 11. In addition to the 750 cm.-l band, two other bands assigned as fundamentals, 1453 and 1261 cm. -I, exhibited changes in structure on heating. These cases are discussed in the section on interpretation. Comparison of the liquid spectrum with that of the gas has been of aid in making certain assignments; the single peak in the liquid a t 750 cm.-l seems to indicate that the two observed peaks in the gas are part of the same band, and that the expected third branch does not appear; the pronounced shoulder on the 1261 em.-' band seems to indicate definitely two assignments in that region; and the marked change in structure in the 2100 cm.-' region indicates that the very complex gas phase spectrum is in part probably due to rotational structure of the various combinations that occur in the region. In the assignment of combinations (Table 11) this information has been used. With minor exceptions, the spectra obtained by us are in agreement with that reported by Wilson.2 Because of the accessibility to us of lithium fluoride optics, our spectrum shows better resolution in the higher frequency region, and it is in this region where our spectrum differs most significantly from that published by Wilson. We have also observed two weak bands in the gas phase, but not in the liquid, a t 955 and 860 cm.-' which were not observed by Wilson. (These may be due to impurity.) Raman Spectra.-Table I1 gives the Raman lines as observed by Neu, their estimated intensities, states of polarization and assignments. A comparison with Wilson's2 data and that of Pajenkamp3 shows certain differences. Wilson observed doubling of the lines a t 1444, 1220 and 928 cm.-l while neither Pajenkamp's nor Neu's data show such doubling. The resolution available in the spectrum reported here seems to indicate that the doubling is not genuine. Three lines, 1148, 1515 and 634 cm. --I not reported by Wilson were observed by Xeu
.
Symmetry Consideration and Selection Rules.As previously indicated, the planarity or nonplanarity of the carbon skeleton in cyclobutane is dependent on the balance of the forces tending to cause the two configurations. The most likely
Nov. 20, 1953
SPECTRA, STRUCTURE AND
THERMODYNAMIC PROPERTIES OF CYCLOBUTANE
symmetries in the two cases are, respectively, Dlh and D2d. Table I is a summary of the vibrations belonging to each class in the two symmetries, the selection rules, and the behavior of each class as the molecule goes from one symmetry to the other. From the table, it can be seen that Blu, Alu, Azu and E, vibrations become Raman active as the symmetry goes from Dlh to D2d, and Bzg and E, vibrations become infrared active as the symmetry goes to D z d . It also can be seen that there are 23 fundamental vibrations and that for D4h symmetry eleven fundamentals should be Raman active, and six infrared active with no coincidences. There are also six inactive fundamentals for this model. For D 2 d symmetry nine vibrations are Raman active only, and twelve vibrations are active in both Raman and infrared. Two fundamentals remain inactive. Since the number of strong infrared bands and Raman bands actually observed is nearer the number predicted for D4h, this is a factor in leading one tentatively to select D4h symmetry. Wilson2 interpreted his data on this model. As a result of the discrepancy between the entropy measurements and the value calculated from spectroscopic data with the assumption of a planar carbon skeleton, it has become necessary to consider more carefully the possibility of a non-planar skeleton, and to try to interpret the spectra in
5635
terms of D2d symmetry. If this model is assumed, not all of the allowed bands or coincidences between Raman and infrared are observed. This may be due to intrinsic weakness of the bands, especially inasmuch as the deviation from skeletal planarity is expected to be small. There are, however, three observed coincidences (630, 750 and 1220 cm.-l) in the low frequency range, where accidental coincidences are less likely. Since these are observed in the liquid state, and might otherwise be interpreted as violations of (Ddh) selection rules, conclusions therefrom regarding symmetry are equivocal. Nevertheless, it seems unlikely that all three coincidences would arise from a planar molecule. The following section will clarify the point of view from which neither symmetry is strictly correct, and from which both must be considered. Interpretation From a thermodynamic point of view the most serious problem in the interpretation of the cyclobutane spectra has been in the assignment of the B1, out-of-plane bending vibration (assuming for the time being D a symmetry) since this vibration is expected to be of very low frequency and extremely anharmonic. Wilson assigned a value of 145 cm.-' to it, and CottreW has used this assign(7)T.L. Cottrell, Trans. Faraday SOC.,44, 716 (19481.
5636
G. W. RATHJENS, JR., N. K. FREEMAN, W. D. GWINNAND KENNETH S. PITZER
Vol. 75
assumed shapes for potential functions are grossly in error, or both. Our temperature dependence measurements were initiated with the hope that increase in intensities Symmetry class and selection rules on heating would locate definitely the difference Vibration type Cm.-1 Dab D2d bands involving the B1, vibration. No such inC H stretching 2870 creases have been observed, and we have been C H I deformation 1444* Ai, R.(P.) forced t o conclude that no such differences are C C stretching 1003* sufficiently intense to appear in our spectrum. (PosAI R.(p.) C H stretching 2921 sible exceptions are the 650 and 1355 ern.-' bands). CHI rocking (878) BI, inactive However, a number of diminutions in intensity have Ring bend. (out-ofb been observed. Such behavior is consistent with a plane for D h h ) model in which the potential energy function for CH? wagging (1299) Asr inactive the puckering of the ring is of the form illustrated AZ inactive CHI twisting 1104 (1155) BzU inactive in Fig. 2. If the potential is of the illustrated shape and the barrier hindering inversion is sufCHI wagging (1288) ficiently low, an appreciable fraction of the moleC C stretching 928* R.(dp.) B1 R.(dp.) cules will be in states whose energies lie above the 1148 (1155) AI, inactive CHz twisting barrier peak. These molecules and those slightly C H stretching 2981 below the peak have effectively D d h symmetry CHz deformation 1450* Bzg R.(dp.) while molecules whose energies lie well below the Ring bend. (in-plane for barrier peak will have D 2 d symmetry. On heating, Dad 750* BI IR.. R.(dp.) the former population will be increased at the exC H stretching 2896 pense of the latter, and consequently there will be CHz rocking 901* A b IR. a decrease in intensity of all infrared bands that C H stretching 2960 are allowed in D 2 d symmetry but are forbidden in CHz twisting 1220* E, R.(dp.) D a symmetry. This model is quite consistent CHz rocking 890C' with the observations, and also explains the weak C H stretching 2974 E IR., R.(dp.) Raman lines observed a t 634 and 1148 cm.-'. CHz deformation 1453 (1420) E, I R The strength of the 750 ern.-' band, and the 1361* CHt wagging 1220 cm.-' shoulder in the liquid spectrum rather Ring deformation strongly suggest that an appreciable fraction of (stretch -bend.) 626* the molecules in the liquid obeys D 2 d selection rules. Frequencies used to evaluate force constants marked In the discussion that follows we shall generally with an asterisk; calculated values in parentheses. See prefer to use the D,h nomenclature while keeping in text. e Assigned on the basis of combination bands. mind the effects of D 2 d symmetry considerations. From a consideration of the paraffins and particment to calculate thermodynamic properties for both harmonic and quartic potential functions. ularly the other saturated rings, the C-H stretchThe measured entropy reported in the accompany- ing frequencies for cyclobutane may be expected to ing article is considerably greater than either of the fall in the range from 2900 to 3100 ern.-'. We values calculated. This suggests either that the have assigned the strongly polarized Raman band frequency is even lower than 145 cm.-', or that the a t 2870 to the AI, vibration. No other polarized fundamentals should appear in this part of the spectrum for D 4 h symmetry, but for Dza symmetry the B1, vibrations become active; Also there are overtones of vibrations in the region of 1450 cm.-' which are totally symmetric and could yield polarized bands in the vicinity of 2900 cm.-'. We have rather arbitrarily chosen to assign the 2921 cm.-' band to the B1, fundamental. All actual states in this region are presumably the resultants of the interaction of overtones and fundamentals, but we have no means to sort out the respective contributions. Strong bands appear in the infrared a t 2974 and 2896 ern.-'. These have been assigned to vibrations of symmetry E, and Azu, respectively, since both are allowed even in D4h. CHP deformation vibrations are expected to lie a t approximately 1450 cm.-'. I n the infrared spectrum there is a strong band in this region. The band appears to be of somewhat complex struc0.4 0.2 0 0.2 0.4 ture, and changed qualitatively with increasing z (A,). temperature. We have interpreted this band as Fig. 2.-Potential function and vibrational levels for out-of- being the superposition of an E,, band at 1453 plane bending motion. cm.-' and a Bfg band a t 1450 ern.-'. The latter TABLE I FUNDAMENTAL VIBRATIONS,a SYMMETRY CLASSES AND SELECTION RULES
NOV.
20, 1953
SPECTRA, STRUCTURE AND
THERMODYNAMIC PROPERTIES OF
5637
CYCLOBUTANE
TABLE I1 SPECTRA AND ASSIGNMENTS Infrared
Gfd CUI.-] I
611 625 640 650
740 758 850 885 901 922
Liquid
Cm-1
I
Cm.-*
5
630
5
0
648
0
2"
750
3
901
10
Raman
1210 1228 1246 1261 1277 1355
Pol.
DzdClassD 4 h
Selection rules DZd D4h
E
E,
IR, R
IR
Assignment
634 f 5
0.5
Ring deformation
750 f 5
1
Ring bending (in-plane for
D4h)
Bz
1 10
928 i 1 955
1
8
CHZrocking
Bz
Azu
IR,R
IR
dp.
CC stretching
Bi
Blg
R
R
p.
Ai
dp.
CC stretching CHg twisting CHt twisting
Az Bi
Au Bzu Am
R ia. R
R ia. ia.
dp.
CHz twisting
E
Eg
IR, R
R
CHZwagging
E
E,
IR, R
IR
E
E,
IR, R
IR
CHZdeformation
Bz
Bo
IR,R
R
CHz deformation
E
Eu
IR,R
IR
Ai
Ai9 Eu
Eu
R IR,R IR, R IR, R
R IR R IR
Eg
IR, R
R
1R.R IR,R IR, R IR,R IR,R IR,R
R R R IR IR IR
IR, R IR,R IR,R IR, R IR, R IR,R IR, R IR,R IR,R IR,R IR,R IR,R IR,R IR, R
IR IR IR R IR R IR IR IR IR R IR IR R
IR,R IR,R 1R.R 1R.R R
IR IR IR IR R
1
2' 2
1147 1220
10
1253
1
1338
1
1002.6 f 0 . 5 1104 f 5 1148 f 5
10 0.5 2
3
1220
*1
5
10
625
O?
1444 f 1
+ 750 = 1375
10
1530
0
1550 1605 1630 1650 1683
1
1817 1879 1898 2035 2047 2098 2113 2135 2150 2176 2191 2202 2264 2279 2357
0 1' O? O?
3'
1549 1605
0 0
1650
0
1817
0
1886 2039
2
1 2 6 2 1 2' 0
2098
2
2123
7
2"
2258
2
3
2358
3
2404
0
2397
1
2437 2455 2603 2675 2694
1 2
2441 2467 2577
3 0 1
2680 2715 2760
3 2 1
2755 2775
1"
1
O? 4
1 1
2183
7
1444 f 1 1515 i 5
1
p.?.
CHI deformation 625 890 = 1515 625 901 = 1526 625 928 = 1553
'+ + +
4
-
+ 890 = 1640 890 + 928 = 1818 625 + 1261 = 1886 890 f 1003 = 1893 928 + 1104 = 2032 890 + 1148 = 2038 878 + 1220 = 2098 901 + 1220 = 2121 890 + 1261 = 2151 928 + 1261 = 2189 750 + 1444 = 2194 750 + 1453 = 2203 1104 + 1148 = 2252 1003 + 1261 = 2264 901 + 1444 = 2345 1148 + 1220 = 2368 1104 + 1286 = 2390 1148 + 1261 = 2409 1148 + 1299 = 2447 1003 + 1453 = 2456 1148 + 1453 = 2601 1220 + 1453 = 2673 1261 + 1444 = 2705 2 X 928 + 901 = 2757
750
4
2" 1'
1444 z!= 1
.
+
2870 f 1
8
p.
+
625 928 1220 = 2773 CH stretching
E E E
E
Eg
E
Eg
E
Eg
E
EO
B2
Azu E, Eu
E E
E E E Bz
E Bz E Bz E B2 E
Bz E
Eu Eu
Eu Bzg Eu
Bzg Eu A% E, Azu E, Azu
E
Eu El
E
Eu
E
Eu Am Eu Au
Br E Ai
G. W. RATHJENS,
5638
JR.,
N. K. FREEMAN, W. D. GWINNAND KENNETH S. PITZER
Vol. 75
TABLE I1 (Continued) Cm.-1
Gas
Infrared
I
Liquid
Cm.-'
I
Cm.-1
Raman
DzdClassD4h
I
Selection rules Dad D4h
Assignment
Pol.
Azu IR, R CH stretching Bz B1u R CH stretching Ai E Eg IR, R CH stretching E IR, R Eu CH stretching 2974 20 Ba lR,R 2981 =t3 2 p.? CH stretching B2 E Eu IR,R 901 928 1220 = 3049 3058 3 E 2 Eg IR, R 1003 1220 = 3151 928 3145 4" 3330 E IR,R Eu 750 1220 1261 = 3231 3238 0 E Eu IR,R 2 X 1220 901 = 3341 3341 0 E Eu IR,R 928 1220 -I- 1261 = 3409 3401 0 E E, IR,R 625 2960 = 3585 E IR, R 625 2974 = 3599 Eg 3585 3"? E IR,R Eu 625 2981 = 3606 A*, IR, R 2921 = 3671 750 Bz 3690 4 E E, IR, R 890 2921 = 3811 3832 4"? E Eu IR,R 878 2960 = 3838 E E, IR, R 1148 2960 = 4108 4100 5 E 1220 2981 = 4201 Eg IR, R 4190 3"? E IR, R 1220 2974 = 4194 Eu E 1299 2974 = 4273 E, IR,R 4290 3 IR, R 1450 2921 = 4371 BP Alu 4390 6 E IR,R 1453 2960 = 4413 E, * See text. The CH stretching region was not investigated in the liquid. a Diminishes in intensity on heating. calibration was made on the liquid spectrum above 3300 cm.-l. The 4190 crn.-l band appears slightly stronger in the than in the gas, and the 4290 cm.-l band perhaps slightly weaker; otherwise, there is no apparent difference.
2896
20
2921 =k 3 2960 zk 3
7 7
p. dp.
+ + + + + + + + + + + + + + + + + + +
+
vibration should be infrared active in D 2 d symmetry, but not for Dh; therefore, a change in observed band structure on heating is to be expected though the intensity change might not be marked if the E,, absorption is much stronger than that of the BPgvibration. The B2gvibration should be Raman active for both symmetries, and we believe the 1444 cm.-' line observed in the Raman must be assigned BZg. I t is also necessary to assign 1444 cm.-' to the expected AI, vibration since no other line appearing in the Raman spectra can reasonably be assigned to it. The 1515cm.-'band is at a higher frequency than would be expected. The questionable state of polarization of the 1444 cm. -l line is consistent with the possibility that it is actually two lines. CH2 twisting and wagging frequencies may be expected to lie in the region 1100 to 1400 cm.-'. The E, twisting vibration should appear in both Raman and infrared if the symmetry is D 2 d ; for D l h symmetry it is Raman active only. A Raman line is observed at 1220 cm.-' and a pronounced shoulder appears in the infrared spectrum of the liquid at the same value. In the gas phase two shoulders appear a t 1228 and 1210 cm.-l; because of the changing rotational structure of the 1261 cm.-l band, it is not possible to decide whether there is any variation in intensity of the 1228 cm.-l shoulder; however, the shoulder a t 1210 ern.-' does appear to diminish in intensity on heating. We believe these two to be branches of the E, twisting vibration. In view of the uncertainty in locating the peaks and the absence or obscuration of the third branch we have used the 1220 cm.-' observed in the liquid phase in our assignment. The 1261 cm.-' infrared band can only be the E, wagging vibration (allowed in infrared, for Dq symmetry). The R1 wagging vibration should be Raman active
IR ia. R IR R IR R IR IR IR IR R IR IR IR IR IR R IR IR IR IR No liquid
only, regardless of symmetry. The observed 1148 cm.-' Raman band may be that of this vibration; however, correlation with the 1261 cm.-' E, wagging vibration is not very satisfactory. A large interaction constant is required in the normal coordinate analysis, and the resulting calculated value for the A2, wagging vibration is well above the expected value. We, therefore, have assumed that the B1, wagging vibration did not appear in Neu's spectrum, and have assigned the observed 1148 cm-1 line to the A1 twisting vibration (allowed in the Raman for D 2 d symmetry but not for D4h.) The B2Utwisting vibration is forbidden in Raman and infrared for both symmetries. However, we have tentatively attributed the very weak Raman band a t 1104 em.-' to it assuming a selection rule violation in the liquid. CH2rocking frequencies may be expected a t from 900 to 1000 cm. -I. The 901 cm. infrared band is almost certainly the A2, vibration; it is quite clearly a parallel type band, and the P-R splitting is as predicted using the formula of Gerhard and Dennison.* Both the Blg stretching and E, rocking vibrations should appear in the same region of the Raman spectra as depolarized lines. Since only one line was observed, 928 cm.-', we have chosen to assign it to the stretching vibration, and to believe the rocking vibration was not observed. We have assigned a value of 890 cm.-' to the latter. Such an assignment is certainly a weak point in our analysis, but it is the only assignment we can advance consistent with the observed diminution in intensity of the 1630 and 1650 cm.-l combinations on heating. The polarized Raman line a t 1003 cm.-l is clearly the A1, C-C stretching vibration. The E, stretching-bending vibration should be active in infrared (8) S. 1, Gerhard and D. M. Dennison, Phys. Rew , 43, 197 (1911)
Nov. 20, 1953
SPECTRA, STRUCTURE AND THERMODYNAMIC PROPERTIES OF CYCLOBUTANE 5639
4 = change in angle between 2 and Y (see Fig. 3) where for D4h symmetry, and both infrared and Raman Z is fixed in the plane of the CHg group and Y in active for D2d symmetry. We believe that the inthe plane of the carbon skeleton. This is the cofrared band a t 625 cm.-' and the Raman line a t 634 ordinate for CH2 twisting motion. cm.-' correspond to it. The expected value of the C$ = change in angle between Y and X (see Fig. 3), where frequency is much higher, but we can offer no other Y is fixed in the plane of the carbon skeleton and X is the bisector of the CH2 angle. This is the coexplanation for the observed infrared band and ordinate for the CHs wagging motion. Raman line. Our interpretation requires a rather e = change in angle between X and 2 where 2 is fixed large interaction constant in the normal coordinate perpendicular to the skeletal plane and X is the analysis. In hope of support for our assignment we bisector of the CH2 angle. The coordinate correhave studied the published spectra of cyclobutane sponds to the CH2 rocking motion. derivatives. Because of the removal of degeneracy, these should exhibit pairs of corresponding frequencies. Unfortunately, infrared investigations on derivatives seem to have been confined to the region above 700 cm.-'. Kohlrausch and coworkersg have, however, obtained the Raman spectra of a large number of derivatives, and that of methylenecyclobutane is also available.10 The \ corresponding lines are not found in cyclobutanone or methylenecyclobutane but, in general, lines that may correspond to this vibration are found for the Fig. 3.-Vectors for describing CH2 motions. derivatives of lower symmetry. We have assigned the B2g ring bending vibraA redundancy relationship about each C atom tion as 750 cm.-l. This vibration should be only exists between a, p and x Raman active for D4h symmetry, but also infrared active for Dzd. This assignment (compared with (sin cos a + (cos sin p (sin X 0 ) x o = o Wilson's value of 595 cm.-1)2 is supported by the observance of the corresponding lines in the Raman where the zero subscripts refer to equilibrium anspectra of methylenecyclobutane (657 cm. - 9 9 and gles, and x,, is the normalized sum of the four HCC cyclobutanone (674 c ~ n . - l ) ,and ~ also in other deriv- bond angles, xc = '/z(xi x j 4- xlr XI). This a t i v e ~ . The ~ observed diminution in intensity on redundancy relationship has been used to reduce heating of the infrared band a t 750 cm.-l is perhaps the three coordinates a, and x to a new mutually the strongest single point supporting a low-barrkr orthogonal pair a' and p' where double minimum potential function for the out-of-plane bending. sin* xo cos* 0 sin* 5 a - sin - cos cos 9 sin - p (sin x0)x 2 ao 2 2 2 2
/
?
2)
+
+
The Assignment of Inactive
CY'
=
(
+
Vibrations.rIn order to calculate most of the frequencies of the inactive vibrations and also the force constants, we have resorted to a normal coordinate analysis, using Wilson's F and G matrices and the CH stretching frequencies have been factored out by his method." The analysis has been made on the basis of Dq Symmetry. 'Since the deviation from planarity in going to Dzsymmetry is small, this is permissible. The fact that frequency shifts are not observed on heating may be taken as support for this procedure. The BlUout-of-plane bending vibration has such a low frequency that interaction between this and other B1, vibrations can be neglected. This has been done in our calculations, and the out-of-plane bending treated separately. The symmetry coordinates have been obtained in terms of the internal coordinates defined below 5
= change in C-C bond length a = change in CCC bond angle @ = change in HCH bond angle
x
= change in HCC bond angle
7 = p =
change in C-H bond length (above plane) change in C-H bond length (below plane)
(9) K. W. F. Kohlrausch and R. Skrabel, Z . Ekktrochcm., 48, 282 (1937); Monatsh., TO, 44 (1937); A. W. Reitz and R. Skrabel, ibid., 70, 398 (1937); R. Skrabel, ibid., 70, 420 (1937). (10) F. F. Cleveland, M . J. Murray and W. S. Gallaway, J . C h c n . Phys., 15, 742 (1947). (11) . . E. B. Wilson, Jr., ibid., 9, 76 (1941). We acknowledge also the aid of some preliminary calculation of C. S. Lu on cyclic molecules.
+
+
[(
sin2 xo
+ cos2 02 sing -2 + sin2 0 cos2 2 2
1
PO
p'
=
sin
[
xo (sin
XO)~
sin2 xo
x (cos -;sin -31
+ cos2 32 sin2 le,2
The coordinates q and p have been transformed to a mutually orthogonal pair
'I' = 1/2/12 ('I
+
P)
and
P' =
I/v'%'I
- P)
s
The definition of the vectors is straightforward except in the case of that associated with the carbon atom for a 0'vibration. In this case there is an CY component as well as a p one; thus sin sin2 xo
CYO
sin PO
+ cos2 92 sin2 2
Symmetry coordinates are defined in Table 111. The choice for the E, skeletal deformation was particularly made so that a stretching-bending interaction term would occur in the potential energy function. In evaluating the elements of the G matrix, we have used the following parameters: C-C bond length 1.568 A., C-H bond length 1.096 A., HCH bond
