J. Phys. Chem. 1983, 87, 5508-5516
5508
Molecular Structure of Cyclobutanone As Determlned by Comblned Analysls of Electron Diffraction and Spectroscopic Data K. Tamagawa and R. L. Hilderbrandt' Department of Chemistry. North Dakota State University, Fargo, North Dakota 58105 (Received: March 2, 1983)
The molecular structure of cyclobutanone has been investigated by combined analysis of electron diffraction and spectroscopic data. The molecule was found to have a large-amplitude ring puckering motion with two potential minima at ring puckering angles of 10.4 f 2.7O. The potential barrier at the planar ring configuration, 1.2 f 1.5 cm-', determined from the combined least-squaresanalysis, is somewhat lower than previously reported values derived from spectroscopic measurements; however, the present result is in good agreement with the value of 2.3 f 2.3 cm-' determined from a reanalysis of the far-infrared data in this study. The results for some of the more important bond lengths (rJ and bond angles (0,) are r,(C1-C2) = 1.567 (5) A, rg(C2-C3)= 1.534 (3) A, rg(C-H)av= 1.100 (4)A, rg(C=O) = 1.202 (2) A, LC2C1C4= 90.3 ( 4 ) O , and LHCH, = 105.9 (1.6)'. A new method for the combined analysis of molecules with large-amplitudering puckering motion is also discussed.
Introduction During the past two decades there has been considerable interest in the four-membered ring molecules because of their low-frequencyout-of-plane ring puckering vibrations. In 1945, Bell1 predicted that the ring puckering vibration in four-membered ring molecules should have a large quartic term in its potential function. On the basis of their spectroscopic analysis Rathjens and co-workers2suggested a double minimum potential for cyclobutane. Several years later Gwinn and co-workers,3-6in a classic series of papers, demonstrated that the far-infrared and microwave spectroscopic data for trimethylene oxide could be interpreted with a potential function of the form
V ( x ) = ax4 + bx2
(1)
where x is a one-dimensional ring puckering coordinate, and a and b are adjustable potential function parameters. Since then, a simple one-dimensional potential function such as eq 1 has been successfully used by many investigators to interpret the spectroscopic data associated with ring puckering for a number of four-membered ring molecules containing a plane of ~ y m m e t r y . ~ ? ~ Several years ago Morino and co-workers+ll (and independently Laurie and Herschbach12J3) developed a theory which may be used to relate the vibrationally averaged molecular structure obtained from electron dif(1)R. P.Bell, R o c . R. SOC. London, Ser. A, 183,328(1945). (2)G. W. Rathjens, N. K. Freeman, W. D. Gwinn, and K. S. Pitzer, J.Am. Chem. SOC.,75,5634 (1953). (3)S. I. Chan, J. Zinn, and W. D. Gwinn, J. Chem. Phys., 33, 295 (1960). (4)S. I. Chan, J. Zinn, J. Fernandez, and W. D. Gwinn, J. Chem. Phys., 33, 1643 (1960). (5)S. I. Chan, J. Zinn, and W. D. Gwinn, J. Chem. Phys., 34, 1319 (1961). (6)S.I. Chan, T. R. Borgers, J. W. Russell, H. L. Strauss, and W. D. Gwinn, J. Chem. Phys., 44,1103 (1966). (7)T.B. Malloy, Jr., L. G. Bauman, and L. A. Carreira, "Topics in Stereochemistry", Vol. 11,N. L. Allinger and E. L. Eliel, Ed., Wiley, New York, 1979,p 97. (8)L. A. Carreira, R. C. Lord, and T. B. Malloy, Jr., Top. Current Chem., 82, 1 (1979). (9)T. Oka and Y. Morino, J.Mol. Spectrosc., 6,472 (1961). (10)T. Oka and Y . Morino, J. Phys. SOC.Jpn., 16, 1235 (1961). (11)K.Kuchitsu, J. Chem. Phys., 44,906 (1966). (12)D.R. Herschbach and V. W. Laurie, J. Chem. Phys., 37,1668, 1687 ~ .(1962). ,~ . ~~
(13)D. R. Herschbach and V. W. Laurie, J. Chem. Phys., 40,3142 (1964). 0022-365418312087-5508$0 1.5010
fraction to the structure obtained from spectroscopic experiments. The most significant result of this theory was the development of a method for the simultaneous analysis of spectroscopic and electron diffraction data.14 A number of molecules have thus far been studied with this method. Unfortunately one of the shortcomings of the original theory for combined analysis was that it could only be applied to molecules for which the small amplitude harmonic vibration theory was valid. In particular it could not be applied to molecules with large amplitude vibrational motions such as ring puckering and pseudorotation. Recently Kuchitsu and co-workers15have shown that it is possible to apply a variation of the original theory to such molecules provided that spectroscopic information on the potential function for the large-amplitude motion is available. This recent development has opened up a new range of possibilities for performing precise structural studies on gas-phase molecules which contain large-amplitude motions. Whereas it has always been possible to determine the coefficients in the potential function expansion from far-infrared and microwave spectroscopic data, most of the potential functions thus far determined were obtained as a function of a dimensionless coordinate. Such functions give the magnitudes of the barriers, and the overall relative shape of the potentid, but they do not provide any detailed information on how the atoms move. Electron diffraction, on the other hand, can do very little to improve upon the accuracy of the potential associated with the large amplitude motion, but it is capable of further characterizing the actual atomic trajectories associated with the largeamplitude motion (in effect, adding dimension to the dimensionless coordinate). The present study was initiated in order to determine the precise molecular structure of cyclobutanone by using all of the available experimental information in a combined diffraction-spectroscopic analysis. The molecule has previously been studied by both far-infrared and microwave spectroscopy, and it is therefore a good candidate upon which to test the newly developed methods for analysis. Borgers and StrausslGhave reported a 5-cm-l (14)K. Kutchitsu and S. J. Cyvin, "Molecular Structure and Vibrations", S. J. Cyvin, Ed., Elsevier, Amsterdam, 1972,Chapter 12. (15)K. Karakida and K. Kuchitsu, Bull. Chem. SOC.Jpn., 48,1691 (1975). (16)T.R. Borgers and H. L. Strauss, J.Chem. Phys., 45,947(1966).
0 1983 American Chemical Soclety
Molecular Structure of Cyclobutanone
planar barrier for cyclobutanone from an analysis of their observed far-infrared spectrum of the molecule. Laurie and c o - ~ o r k e r s ~have ~ J * determined rotational constants in the ground vibrational state and in the first ten excited states for the ring puckering vibration of cyclobutanone. They have accounted for most of the features of the observed variation of rotational constants with ring puckering state and of the far-infrared transition frequencies using a quartic-quadratic potential with a Gaussian barrier. A barrier of 7.6 f 2 cm-l was obtained from this analysis. They have also determined the substitution structure for this molecule using the rotational constants for six isotopic species; however, no estimate was given for the ring puckering angle a t the potential minima. We have recently written two computer programs which have enabled us to apply combined analysis of electron diffraction and spectroscopic data to molecules having large amplitude degrees of freedom. The first program performs the calculation of the reduced mass function which may be used to obtain information about the dimensioned coordinate associated with the large-amplitude motion. The second program, which was based upon the classical paper by Ueda and Shimano~chi,'~ performs a variational calculation to obtain the vibrational energy levels and wave functions for the large-amplitude motions. In addition to the calculation of the energy levels, this program also calculates the vibrational distribution function for each level, the overall thermal average distribution function, and the vibrational contribution to the rotational constants for each vibrational level. One of the more significant results which we are able to obtain from the program is the true thermal-averaged quantum mechanical probability distribution function. It is quite revealing to compare this function with its classical analogue. The present study is the f i i t application of these new programs to the study of a molecule having a large-amplitude motion.
Experimental Section A sample of cyclobutanone was obtained from Aldrich Chemical Co. The purity of the sample was checked by gas chromatography, and was found to have less than 1% of a less volatile impurity so it was used without further purification. Electron diffraction data were collected a t room temperature (298 K) on the North Dakota State University electron diffraction instrument by using nozzle-to-plate distances of 243 and 92 mm. The accelerating voltage used was 40 keV, and the background pressure was maintained at 1.4 X torr during exposure. Exposure times for the 0.6-kA beam current were 45 s for the long camera distance and 140 s for the short camera length. The Kodak 4 X 5 in. electron image photographic plates were developed a t room temperature with nitrogen burst agitation. Approximate voltage/distance calibrations were obtained from digital voltmeter readings and cathetometer measurements; however, all final calibrations were made on the basis of benzene calibration plates which were obtained under operating conditions identical with those used for the sample. Four photographic plates for each camera distance were traced on the NDSU microcomputer-controlled microdensitometer. The optical densities were obtained at intervals of 0.150 mm. The data were corrected in the usual manner for emulsion saturation, plate flatness, and sector imperfections after which they were interpolated a t inte(17)L.H.Scharpen and V. W . Laurie, J. Chem. Phys., 49,221(1968). (18)W. M. Stigliani and V. W. Laurie, J. Mol. Spectrosc., 62, 85 (1976). (19)T.Ueda and T. Shimanouchi, J. Chem. Phys., 47,4042 (1967).
The Journal of Physical Chemistry. Vol. 87, No. 26, 1983 5509
TABLE I: Observed and Calculated Ring Puckering Vibrational Frequencies for Cyclobutanone (cm-I ) transition
obsda
calcd
0-1
35.85 56.4 64.6 71.8 77.0 81.1 84.8 89.9 93.1 95.9 99.0
37.36 54.78 63.38 70.45 76.27 81.29 85.7 2 89.71 93.33 96.70 99.81
1-2 2-3 3-4 4-5 5-6 6-7 7-8 8-9 9-1 0
10-11 a
obsd - calcd 1.51 1.62 1.22 1.35 0.73 -0.19 - 0.92 0.19 -0.23 --
-0.80 - 0.81
Observed frequencies taken from ref 16.
gral q = [40/X) sin (0/2)] for analysis. The data were analyzed by a least-squares procedure similar to the one employed by Gundersen and Hedberg20with elastic scattering factors and phase shifts calculated by Schlifer, Yates, and Bonham.21
Analysis Far-Infrared and Microwave Spectroscopic Data Analysis. The far-infrared vibrational transition frequencies for the ring puckering vibration reported by Borgers and StrausslGwere analyzed with a one-dimensional Hamiltonian of the form
H = 1/22(g44(0)P2
+ g44(2)PX2P + g44(4)PX4P) + a x 4 + bX2 (2)
where X is the one-dimensional ring puckering coordinate which is defined as half of the perpendicular distance between the ring diagonals; P = (h/i)(d/dX) is the momentum operator; a and b are adjustable potential function constants; and gu(i)are coefficients in the expansion of the reduced mass, g,, which is the Wilson G matrix element corresponding to the ring puckering vibration. The reduced mass function was calculated for a semirigid model (described below) to be g44
=
0.61125
X
- 0.1346
X
+
10-lX2 0.8881
X
10-2X4 (3)
where g4, is expressed in amu and X in A. The coefficientsa and b for the potential energy function of eq 2 were then adjusted by a least-squares routine in the computer program to obtain a fit between the observed and calculated far-infrared frequencies. The Hamiltonian was set up by using 100 harmonic oscillator functions as a basis set. For computational convenience, the calculations were handled internally in a dimensionless coordinate; however, the results were transformed back to a dimensioned coordinate system once the results were obtained. The final form of the potential function obtained from the analysis is as follows:, V(X) = (0.4042 f 0.0252) X 106X4- (0.1945 f 0.0959) X 104X2 (4)
where V is expressed in cm-l and X in A, and where the uncertainties represent 3 standard deviations. This function has a small barrier of 2.3 f 2.3 cm-l at the planar ring configuration, and a double minimum a t X = h0.049 f 0.013 A. The location of the potential minima corre(20) G. Gundersen and K. Hedberg, J. Chem. Phys., 51, 2500 (1969). (21)L. Schifer, A. C. Yates, and R. A. Bonham, J. Chem. Phys., 55, 3055 (1971).
5510
The Journal of Physical Chemistry, Vol. 87, No. 26, 1983
Tamagawa and Hiiderbrandt
TABLE 11: Comparison of the Rotational Constantsa for the Ring Puckering Vibrational States of Cyclobutanone with Those Calculated from the Empirically Determined Rotational Constant Expansionsb (MHz)
B
A obsd
u
calcd obsd 0.58 -0.29 -1.73 -0.54 0.70 1.62 2.83 2.41 -9.31 2.11 1.64
0 1 2 3 4 5 6 7
10785.2 10737.7 10706.4 10671.0 10636.6 10602.9 10569.1 10537.0 8 10516.2 9 10472.2 10 10440.0
C
obsd
calcd obsd
obsd
calcd obsd
4806.69 4817.68 4824.18 4831.62 4838.73 4845.78 4852.70 4859.36 4863.60 4873.15 4880.10
0.19 -0.30 0.17 -0.01 -0.13 -0.29 -0.39 -0.27 2.27 -0.53 -0.71
3558.47 3578.40 3589.14 3601.30 3612.57 3623.43 3633.97 3643.82 3649.77 3663.13 3673.56
0.27 -0.52 0.35 0.03 -0.19 -0.43 -0.67 -0.46 3.45 -0.80 -1.04
Observed rotational constants taken from ref 17. A , = 10818.91 - 10.7813 X 103(X*),,- 2.1168 X 105(X4),,. B , = 4799.63 + 2.3933 X 103(X2),,+ 0.4135 X 105tX4),,. C, = 3545.03 t 4.8204 X 103tx2),, + 0.3645 x 105(x4~,,. a
sponds to the ring puckering angle of 10.4 f 2.7', where the ring puckering angle is defined as the dihedral angle between the L C ~ C plane ~ C ~ and the K2C3C4plane. The observed and calculated transition frequencies are given in Table I. The calculated transition frequencies fit moderately well with the observed frequencies, and the agreement is much better than that obtained by Borgers and Strauss.lG If the ring puckering motions are not interacting with any other modes of vibration, and the planar ring has a plane of symmetry, then the variation of the rotational constants with ring puckering vibrational states can be expressed as a power series expansion in the expectation values of the puckering coordinate, X, as follow^:^^^^^^
p, = $0)
+ p'2'(x2),, + p(4'(x4),, + ...
(5)
where p, is the A , B, or C rotational constant in the vth excited vibrational state of the puckering mode. The coefficient p(n)in eq 5 can be either treated as an empirical parameter or calculated from a dynamic model of the ring puckering motion. Observed rotational constants as a function of puckering vibrational quantum number were least-squares fit by eq 5 with expectation values of X2 and X4 calculated from the puckering potential function of eq 4. The following expressions were obtained A, = 10818.91 - 10.7813 X 103(X2),, - 2.1168 X 105(X4),, (64 B, = 4799.63 + 2.3933 X 103(X22),, + 0.4135 X 106(X4),,, (6b) c, = 3545.03 4.8204 X 103(X2),, + 0.3645 X 105(X4),,
+
(6c)
where A,,,B,, and C,are in MHz units. Table I1 shows that the calculated rotational constants obtained from these expressions agree well with the experimental values except for the rotational constants for u = 8 which are perturbed (22) D. 0. Harris, H. W. Harrington, A. C. Luntz, and W. D. Gwinn,
J. Chem. Phys., 44, 3467 (1966).
Figure 1. The numbering of the atoms used in defining structural parameters for cyciobutanone.
by a resonance interaction with the carbonyl bending m0de.l' The values of B(O)obtained from these equations provide effective rotational constants for the planar configuration of the molecule in which X = 0. It is possible to deduce something about the dynamics of the ring puckering vibration from model calculations which attempt to reproduce the variation in the rotational constants with puckering vibrational quantum number. In these calculations the molecule was treated as a semirigid framework which undergoes puckering by folding along the tie line connecting atoms C2 and C4.21The assumptions made were as follows: (1)The molecule has C, symmetry and maintains this symmetry throughout the motion. (2) All of the bond lengths remain constant throughout the puckering motion. (3) All C-H bond lengths are equal. (4) All LHCH valence angles are equal and remain constant. (5) The LCCC planes are the perpendicular bisectors of the respective LHCH planes. (6) The C=O bond remains on the bisector of the L C ~ C valence ~ C ~ angle. (7) The L C ~ C ~and C ~,!C2C3C4 ring angles maintain constant values during puckering, and only the L C ~ C ~ C ~ and ,!C1C4C3 angles decrease with the puckering motion. The atomic numbering used in defining the structural parameters for cyclobutanone is illustrated in Figure 1. Using this molecular model, we calculated instantaneous rotational constants for a number of values of the puckering coordinate, X, and the results were fit to a power series expansion in X similar to that of eq 5. Substitution of the expectation values for X 2 and X4 for a given vibrational state into these expressions should then reproduce the observed variation in the rotational constants. The calculated variations in the rotational constants are compared with the experimentally observed variations in Figures 2-4. The solid line designated as e@)= 0.0 represents the results of the semirigid model calculation and the experimental results are indicated by solid circles. From these figures it is obvious that the proposed semirigid model does not reproduce the observed variations. The motion is evidently more complex than that which is represented by this highly constrained model. Extensive model calculations were therefore performed in which assumptions 5-7 were relaxed. Various models which correlated the rocking motion of the CH2 groups with the puckering were tried as well as models which coupled the out-of-plane bending of the C=O bond with the puckering motion. Most of these models either had little effect on the variation or produced an undesirable slope for one or more of the three curves. Finally it was found that the observed variations could be well reproduced by a model which coupled the variation in the
The Journal of Physical Chemistry, Vol. 87, No. 26, 1983 5511
Molecular Structure of Cyclobutanone
400
150
’
I
300 I h
4
v
2
&200
a
100
0
1
2
3
4
5
6
7
8
9
10
V
Flgure 4. Comparison between the experimental variation in the C 0
1
2
3
4
6
5
7
8
9
10
V
Figure 2. Comparison between the experimental variation in the A
rotational constants and that calculated from the model for the ring puckering motion. The black circles represent the experimentally determined variation, while the lines are calculated variations (seetext).
0
1
2
3
4
5
6
7
8
9
10
V
Figure 3. Comparison between the experimental Variation in the B
rotational constants and that calculated from the model for the ring puckering motion. The black circles represent the experimentally determined variation, while the lines are calculated variations (seetext).
LC2C1C4,0, with the puckering angle 4. More specifically, the angle 0 was expressed in terms of the angle @I by the function 0 = ($0) - p 4 2 (7) where is the effective value for the L C ~ C angle ~ C ~when @J = 0, and is a quadratic coupling constant. Figures 2-4 show the comparison between the experimental and calculated variations in the rotational constants for several values of the parameter O(2). As can be seen from the plots, both the A and B rotational constants are quite sensitive to the value of e(”,. The best fit with the experimental data was obtained for a coupling parameter O@) = 0.0013. Since the lC1C2C3and K2C3C4angles are dependent functions of 0 and @I, their values can also be expressed in the form of eq 7 . The observed values of the coupling amplitudes, 0(2),for these two angles are 0.0029 and 0.0014 for LC,C2C, and LC2C3C4,respectively. In retrospect, this model is
rotational constants and that calculated from the model for the ring puckering motion. The black circles represent the experimentally determined variation, while the lines are calculated variations (see text).
quite plausible since it distributes the valence angle strain associated with the puckering motion over the entire ring rather than localizing it in the LC1C2C3and L C ~ C angles ~C~ exclusively. In fact, the relative magnitudes of the coupling parameters are in accordance with what one might predict on the basis of relative magnitudes of bending force constants. Attempts to couple the rocking motions of the CH2and C=O groups to the puckering using the method of Malwere inconclusive. The variations in the rotational constants produced by such couplings were all much smaller in magnitude than those produced by the coupling of the skeletal valence angle bending to the puckering motion. In summary, the model which couples the skeletal angle bending to the puckering motion appears to be a reasonably good approximation to the true dynamic behavior of the ring as evidenced by the variation in rotational constants with puckering vibrational quantum number. Normal Coordinate Analysis of Small-Amplitude Motions. Although the infrared and Raman spectra of cyclobutanone have been studied by several investigat o r ~ , no ~ complete ~ - ~ ~ force field has previously been reported for this molecule. A Urey-Bradley force field for cyclobutanone was therefore determined by fitting the observed vibrational frequencies assigned by Cataliotti and ~o-workers.A ~ ~least-squares ,~~ program written in our laboratory was used for the force field refinement. As a starting point in the analysis the Urey-Bradley force constants for alkanes determined by Schachtschneider and Snyder2’ were used together with Shimanouchi’s UreyBradley force constants28for the carbonyl portion of the molecule. In the course of the refinement, the HCC-HCC’ bending interaction was introduced in order to obtain better agreement between the observed and calculated vibrational frequencies of the methylene rocking and twisting motions. Furthermore, a few minor modifications (23) T. B. Malloy, Jr., J. Mol. Spectrosc., 44, 504 (1972). (24) K. Frei and H. H. Gunthard, J . Mol. Spectrosc., 5 , 218 (1960). (25) R. Cataliotti, M. G. Giorgini, G. Paliani, and A. Poletti, Spectrosc. Lett., 7, 563 (1974). (26) R. Cataliotti, M. G. Giorgini, G. Paliani, and A. Poletti, Spectrochim. Acta, Part A , 31, 1879 (1975). (27) J. H. Schachtschneider and R. G. Snyder, Spectrochim. Acta, 19, 117 (1963). (28) T . Shimanouchi, Pure Appl. Chem., 7, 131 (1963).
5512
The Journal of Physical Chemlstty, Vol. 87, No. 26, 1983
TABLE 111: Vibrational Force Field for Cyclobutanone
force constanta KC-C KC-H
Kc=o Hccc "CC
HHCH Hcco F c . . .c
value 1.963 4.123 11.49 1.028 0.256 0.542 0.432 0.4
force constant
T A B L E V : Calculated Mean Amplitudes and Shrinkage Corrections for Cyclobutanonea
value
-0.09 0.646 0.52 Y,=, 0.236 K(CH,) 0.03 H7(C-C) 0.007 H ~ c c - H ~ c c ' -0.123 FH. ..H
FH.. .c F c . . .o
a Bonded stretching force constants and nonbonded Urey-Bradley force constants have units of mdyn/A. All other force constants have units of mdyn A . Y,,o is the C=O out-of-plane bending force constant, and K(CH,) is t h e intermolecular tension.
T A B L E IV : Calculated and Observed Frequencies
for Cyclobutanonea assignment
Tamagawa and Hilderbrandt
obsd
calcd
A , species 2934 a-CH, stretch p-CH, stretch 2892 C=O stretch 1814 a-CH, deform 1470 p-CH, deform 1401 a-CH, wag 1291 ring deform 1017 ring deform 848 ring deform 667
2946 2902 1814 1431 1417 1297 1048 878 649
a-CH, stretch p-CH, twist or-CH, twist a-CH, rock
A, species 2979 1196 1163 909
2984 1178 1137 879
a-CH, stretch a-CH, deform p-CH, wag a-CH, wag ring deform ring deform /IC= 0
B, species 2934 1401 1330 1243 1125 957 457
2924 1425 1373 1248 1102 940 464
a-CH, stretch p C H , stretch or-CH, twist a-CH, rock p-CH, rock l C = O bend ring pucker
B, species 3005 2979 1209 1073 734 395 63
2996 2971 1201 1032 809 395 67
a Observed frequency assignments were taken from ref 23 and 24 with minor modifications were necessary. The quoted assignments are based o n t h e magnitudes of t h e potential energy distribution values obtained from the normal coordinate calculation.
in the assignments of Cataliotti et al. were made. The refined force field and calculated vibrational frequencies are shown in Table I11 and Table IV, respectively. The vibrational amplitudes, shrinkage corrections, and the corrections required to extrapolate the average structure to 0 K are shown in Table V. In these calculations, the contribution of the large amplitude ring puckering motion to the calculated amplitude matrix was removed by setting the normal mode amplitude for this low-frequency ring puckering motion to zero. Electron Diffraction Analysis. In the initial stages of structural refinement, only the electron diffraction data were analyzed. The molecule was treated as a semirigid framework where the structure was changed only by the ring puckering motion. In addition to the assumptions 1-6 employed in the earlier section of this study, it was also
514 5 20 374 790 548 559 601 563 1019
G-C, C,-C3
c=0 C-H c,. 4, c,. . .c, c,. -0 c,. . .o C. . .Hgem *
16 29 43 148 22 15 12 11 64
0.0000 0.0002 0.0008 0.0004
a Atomic numbering employed is illustrated in Figure 1. lij parameters are t h e parallel mean amplitudes in A . rg - r, values are t h e shrinkage corrections in A , and r,(O) - r , ( T ) values are t h e temperature corrections in A . Only calculated values for prominent distances are shown.
assumed that the L C ~ C ~ valence C, angle, 8, was related to the puckering angle 4 by the equation
o
=
e(0)
+ 0.001342
(8)
where O(O) = LC2C1CJ0)is an independent structural parameter. In the electron diffraction analysis, the following structural parameters were used to define the molecular model: (1)the C1-C2 bond length (2) the C2-C3 bond length (3) an average C-H bond length (4) the C=O bond length (5) the LC~C~C~(O) valence angle (6) an average LHCH valence angle (7) the wagging angle of the H3C2H4and H5C4H6groups, Pweg,which is defined as 1/2~C1C2C3 - v where p is the angle between the H3C2H4plane and the C1-C2 bond. In addition, the following nine amplitude parameters were also varied in the analysis: C1-C2, C2-C3, C-H, C=O, Cl--C,, C2*..C4, C-H em, Cz-.O, and C1-0. The C2.*.C4 and the C1.-0 amplitudes were introduced as the following functions of 4: zc*c4 = l(O)C*C4 ZClO
=
l(0)ClO
+ (0.0010~)2
(94
- (O.O011q5)2
(9b)
These were used to take into account the rather large variations in these parameters with ring puckering angle. The asymmetry parameters, k , for bonded distances were estimated by using a diatomic a p p r o ~ i m a t i o nwith ~~ Morse anharmonicity parameters, a, of 2.0 A-1 for the C-C bonds, 2.5 A-1 for the C-H bonds, and 2.3 A-1 for the C=O bonds. The k parameters for the nonbonded distances were all assumed to be negligible in the analysis. A dynamic model involving a double minimum potential function was investigated throughout the analysis. For computational convenience, the potential function was expressed in the form
V ( d = Vd1 + (4/40)4 - 2(4/40)21
(10)
where Vois the potential barrier at the planar ring configuration and rb0 represents the location of the two minima in the potential function. Both Vo and 4o were chosen as adjustable independent parameters. We found, however, that the correlation between these two potential function parameters was very strong (-99%), and that it was dif(29) K. Kuchitsu, Bull. Chem. SOC.Jpn., 40, 505 (1967).
Molecular Structure of Cyclobutanone
The Journal of Physical Chemistry, Vol. 87, No. 26, 1983 5513
TABLE VI: Structural Parameters for Cyclobutanone Obtained from Least-Squares Analysis" ED
ED t MW
1.569 ( 5 ) 1.533 ( 5 ) 1.101 ( 5 ) 1.202 ( 2 ) 89.4 ( 8 ) 104.4 (2.5) 6.1 (3.3) 0.8 (1.1) 10.4d 0.050 ( 4 ) 0.047 ( 4 ) 0.080 ( 4 ) 0.032 ( 2 ) 0.056 ( 1 0 ) 0.049 ( 9 ) 0.080 ( 1 2 ) 0.062 ( 3 ) 0.053 (11)
1.567 ( 5 ) 1.534 ( 3 ) 1.100 ( 4 ) 1.202 ( 2 ) 90.3 ( 4 ) 105.9 (1.6) 7.3 ( 3 . 7 ) 1.2 (1.5) 10.4d 0.050 ( 3 ) 0.049 ( 4 ) 0.081 ( 4 ) 0.033 ( 2 ) 0.049 ( 9 ) 0.049 ( 8 ) 0.082 ( 1 4 ) 0.063 ( 3 ) 0.061 (11)
parameters C-C, 2' -3'
C-Hav c=0 LC,ClC, f HCH, Pwag b
e
VQC $0
lOOK
k,-c, k,-c , IC-Hav
I
-0.2
-0.1
0.0
, -50
-40
-30 -20 -10
0.2
0.1 X(+
10
0
,
,
,
,
20
30
40
50
k=o IC,.
0P)
lc.
Figure 5. Comparison of the classical and quantum mechanical probability distribution functions for cyclobutanone. Enclosed circles represent the classical distribution values, while the lines are the calculated quantum mechanical values.
ficult to refine them simultaneously. We were therefore obliged to constrain the parameter 4, to the value of 10.4O, determined from the analysis of the far-infrared data, and refine only the potential barrier, V,, as an adjustable parameter. The series of puckering pseudoconformers was generated by varying the ring puckering angle, 4, between 4 = Oo and 4 = 6O0. Various tests of the least-squares analysis for 6, 8, and 10 pseudoconformers confirmed that the results obtained were independent of the number of puckering models used in the analysis. The composite molecular intensity curve for the mixture of conformers was calculated in the usual manner as
n
dependente fClC,C,
89.1 ( 6 ) 88.3 ( 2 ) 92.8 ( 3 ) 92.2 ( 5 ) a Distances ( A ) are reported as rg parameters and angles (in degrees) are reported as r, parameters. Quoted errors are 3 u values obtained from least-squares analysis plus systematic error caused by t h e uncertainty in the assumed @Qvalue. ED refers t o analysis based o n electron diffraction data alone, while ED t MW refers t o data analysis of t h e combined electron diffraction and microwave data sets. pwag = I/2fC,C,C, - M where fi is the angle between the H3-C,-H, plane and the C,-C, bond. Units of V,are cm-'. Assumed value. e Errors for t h e dependent parameters were calculated from errors in t h e independent parameters by transforming the elements of t h e error matrix. Lc2c3c4
1
I
h
K
v
Y
(12)
where Y,, is the eigenfunction and E,, is the eigenvalue for the nth level. The classical distribution function, on the other hand, is given by the equation
pCd4J= exp(-V(rbJ/RT)/C exp(-V(@,)/RT)
. "gem
1c2.* .o IC,. . .o
c
sMb) = W,(4,).sM(s,4,) (11) where sM(s,4,) is the molecular intensity for the ith pseudoconformer, and W(@Jis the weight assigned to the particular conformer. One of the novel aspects of the newly developed modeling programs is the ability to calculate the exact thermally averaged quantum mechanical distribution function for the puckering coordinate. It provides an excellent test for the validity of the classical distribution function which may only be expected to be valid in the limit of high temperatures. The exact distribution function was calculated according to the equation
pq,(4,) = CIYn(@,)12 exp(-E,,/RT)
. 'C,
IC,. . 'C4
(13)
1
where V(4J is the potential function for the puckering motion of the molecule. Figure 5 shows a comparison of the classical and quantum mechanical probability distribution functions for cyclobutanone a t several different temperatures. It is obvious from the plots that the classical distribution function is a very good approximation at room temperature and that we are justified in using it in the analysis of the data. It is important to note, however, that the validity of the classical distribution formulation for a double minimum quartic oscillator is limited to temperatures for which V,/RT
