1040
J. Phys. Chem. 1988, 92, 1040-1051
Pseudorotation of Cycbpentane and Its Deuterlated Derivatives L. E. Baumant and J. Laane* Department of Chemistry, Texas A&M University, College Station, Texas 77843 (Received: June 3, 1987; I n Final Form: August 20, 1987)
The gas-phase mid-infrared Raman spectra of cyclopentane, cyclopentane-d, cyclopentane-1,1-d2,cyclopentane-1,1,2,2,3,3-d,, and cyclopentane-dlohave been recorded and analyzed in order to investigate the vibrational potential energy surface associated with the pseudorotationaland radial motions. For cyclopentane two separate combination band series involving the pseudorotation and a CH2 deformation or a CH2 wag were observed. The spacings in those series were about 5.4 and 4.1 cm-I, respectively. Similar series with spacings of about 5.0 and 3.7 cm-' were observed for the d, species. Cyclopentane-dloshows one combination band series near 1000 cm-' with a spacing of 4.0 cm-I. In addition to these combination bands, pseudorotational fine structure was observed in the overtone region of the radial band (420-560 cm-' for the different isotopic species). Previously, the absorption in this region had been mistakenly assigned to a vibrational fundamental. The radial band itself was observed for each species in its Raman spectrum. Computational methods for calculating the energy levels for this type of two-dimensional problem have been developed. Reduced mass calculations for each isotopic species have also been carried out. The observed spectra have been analyzed both by using a one-dimensional approximation model and by calculating a two-dimensional potential energy surface. The results demonstrate that the pseudorotation in cyclopentane is relatively unhindered and that the barrier to planarity is about 5.5 kcal/mol (values ranging from 1808 to 2090 cm-' were calculated by using different models for the various isotopic species).
Introduction The two out-of-plane ring vibrations of saturated five-membered rings, such as cyclopentane, which have a high barrier to planarity are best described as a pseudorotation and a radial vibration. The pseudorotation resembles a wave moving around the ring. The maximum amplitude of puckering moves around the ring so that the molecule rapidly interconverts between different bent and twisted forms without passing through the planar configuration. The radial vibration is a nearly harmonic vibration in which the amount of puckering oscillates about an equilibrium value. The concept of pseudorotation was first proposed by Kilpatrick, Pitzer, and Spitzer in 1947 in order to explain the unusual thermodynamic properties of cyclopentane.' Their calculations showed that the bent and twisted structures very nearly had the same conformational energy and the pseudorotation should be essentially unhindered. If the assumptions of unhindered pseudorotation and an infinite barrier to planarity are made, then the two-dimensional problem separates into two one-dimensional problems. The energy levels then resemble those of a planar rigid rotor for the pseudorotation (hence the name "pseudorotation") and those of a harmonic oscillator for the radial mode. On the basis of their thermodynamic calculations, Kilpatrick, Pitzer, and Spitzer calculated a value of 0.472 A for the equilibrium outof-plane displacement for cyclopentane.2 In 1959, Pitzer and Donath performed more detailed calculations on the conformations and strain energy in cyclopentane and substituted cyclopentane~~ and predicted barriers to pseudorotation for many of the cyclopentane derivatives. The first spectroscopic evidence of the pseudorotation vibration was found for tetrahydrofuran in 1965.4 In the original work the observed far-infrared transitions were analyzed in terms of free pseudorotation. Later work showed that there were small twofold and fourfold barriers to pseudorotati~n.~-~ The pseudorotation vibration has also been observed and analyzed for 1,3-dio~olane,~~~ which has nearly free pseudorotation. Some other molecules which have been analyzed and which have barriers to pseudorotation are thiacy~lopentane,~ silacyclopentane,'O silacyclopentane- 1,l -d2,1' selenacyclopentane,I2 1,l -difluorocyclopentane,I3 cy~lopentanone,'~ and germacyclopentane.15 Reference 16 summarizes the results for these molecules. Two-dimensional analyses have been performed on germa~yclopentane,'~ cyclopentarlone," and ~ilacyclopentane.'~The pure pseudorotational spectrum has not been observed for cyclopentane, but, in 1969 Present address: Department of Physics, Mississippi State University, MS 39162.
Durig and Wertz reported infrared combination bands (obtained under low resolution) of the pseudorotation vibration with a methylene deformation.20 The combination bands were separated by approximately 5 cm-I. These bands were analyzed assuming unhindered pseudorotation with a pseudorotational constant of 2.54 cm-I, which gives an equilibrium out-of-plane displacement of 0.479 A. The radial band is inactive in the vapor-phase infrared spectrum' but has been observed at low temperatures for the solid and liquid phases.21 It also gives rise to a very weak, broad Raman band in the liquid phase at 288 cm-I. In 1972, Carreira, Jiang, Person, and Willis observed three Q branches for the radial vibration in the gas phase.22 Using this data for a two-dimensional analysis they calculated a barrier to planarity of 1824 cm-'. This was the first direct spectroscopic measurement of the barrier to planarity of cyclopentane. Their calculations showed a negative deviation of the pseudorotational levels from those predicted one-dimensionally because of the finite barrier to planarity. The
(1) Kilpatrick, E.; Pitzer, K. S.; Spitzer, R. J. Am. Chem. SOC.1947, 69, 2483. (2) The calculated value of 40 was reported as 0.236, but an error was made in the symmetry number for pseudorotation and their calculations were off by a factor of two. This was pointed out in ref 3. (3) Pitzer, K. S.; Donath, W. E. J . Am. Chem. SOC.1959, 81, 3213. (4) Lafferty, W. J.; Robinson, D. W.; St. Louis, R. V.; Russell, J. W.; Strauss, H. L. J. Chem. Phys. 1965, 42, 2915. (5) Engerholm, B. G.; Luntz, A. C.; Gwinn, W. D.; Harris, D. 0.J . Chem. Phys. 1969, 50, 2446. (6) Laane, J. In Vibrational Spectra and Structure; Durig, J. R., Ed.; Dekker: New York, 1972; Vol. I, pp 25-50. (7) Greenhouse, J. A.; Strauss, H. L. J. Chem. Phys. 1969, 50, 124. (8) Durig, J. R.; Wertz, D. W. J. Chem. Phys. 1968, 49, 679. (9) Wertz, D. W. J . Chem. Phys. 1969, 51, 2133. (10) Laane, J. J. Chem. Phys. 1969, 50, 1946. (11) Durig, J. R.; Willis, J. N. J . Mol. Spectrosc., 1969, 32, 320. (12) Green, W. H.; Harvey, A. B.; Greenhouse, J. A. J . Chem. Phys. 1971, 54, 850. (13) Millikan, R.; Gwinn, W. D.; private communication. (14) Carreira, L.; Lord, R. C. J. Chem. Phys. 1969, 51, 3225. (15) Durig, J. R.; Willis, J. N. J . Chem. Phys. 1970, 52, 6108. (16) Laane, J.; In Vibrational Spectra and Strucfure;Durig, J. R., Dekker: New York, 1972; Vol. I, pp 25-50. (17) Durig, J. R.; Le, Y.S.; Carreira, L. A. J. Chem. Phys. 1973,58,2392. (18) Ikeda, T. K.; Lord, R. C. J . Chem. Phys. 1972, 56, 4450. (19) Durig, J. R.; Natter, W. J.; Kalasinsky, V. F. J . Chem. Phys. 1977, 67, 4756. (20) Durig, J. R.; Wertz, D. W. J. Chem. Phys. 1968, 49, 2118. (21) Ewool, K. M.; Strauss, H . L. J. Chem. Phys. 1976, 64, 2697. (22) Carreira, L. A,; Jiang, G. J.; Person, W. B.; Willis, J. N. J . Chem. Phys. 1972, 56, 1440.
0022-3654/88/2092-1040$01.50/0 0 1988 American Chemical Society
Pseudorotation of Cyclopentane
Figure 1. Reaction scheme for the preparation of deuteriated cyclopentane.
negative deviations had been predicted for cyclopentane as well as for tetrahydrofuran and 1 , 3 - d i o ~ o l a n e . ~ ~ * ~ ~ Due to the fivefold symmetry of cyclopentane, only ten- and twentyfold barriers to pseudorotation are allowed. In an effort to observe evidence for a barrier to pseudorotation, Chao and Laane examined the combination bands of cyclopentane at a higher resolution25 and found the combination bands to be split into doublets. These doublets were believed to be due to Coriolis coupling of the molecular rotation with the pseudorotation vibration. N o evidence for a barrier to pseudorotation was found, but a small barrier (
+ Cp' cos 28 + A: + Bq2+ C:
cos 28
(36) (37)
+ q3)e-q2/2ezB
(38)
The Hamiltonian matrix elements are integrals of the basis functions over the Hamiltonian operators (39) Finding the matrix elements by integration would be a lengthy process, so instead the following shift operators were used
P+= (ql + iq,) - i(pl+ ip2) P-= ( q l - iq,) - i(pl - ip2) F+= (ql + iq,) F -= (ql - iq,)
(40) (41)
+ i(pl + ip2) + i(pl - ip2)
(42) (43)
where
+
+ 1 + 2))'/ZJV+1,1+1) P - l u , l ) = (2(u - I + 2))I++l,l-l)
P+IU,l) = -(2(u
Dl cos 28 + E l cos 48 (23)
where
(25)
A = (1/8)(3Al/m1zy12 + 3A2/mz27z2 + C12/mlmzyly2) (26) (30) Carreira, L. A,; Mills, I. M.; Pearson, W. B. J . Chem. Phys. 1972, -~56. -, -1444. . . .. (31) Wang, S. C. Phys. Rev. 1929, 34, 243. (32) Harris, D. 0.; Engerholm, G. G.; Tolman, C. A,; Luntz, A. C.; Keller, R. A.; Kim, H.; Gwinn,W. D.J . Chem. Phys. 1969, 50, 2438.
(44) (45)
F+lU,l)= (2(u - l))lqv-l,l+l)
F = (h2N, 1016/hc)(1/4)(y1 + 7,) = (N, = Avogadro's no.) (24) 33.715(1/4)(y1 + y,) G = 33.715(1/4)(~1 + 7 2 )
+ 2q3
[(2( 1!))/2!](2~)-'/~= ( 2 ~ ) - ' / '
13,l) = (2/7r)'/,(-2q
(22)
The Hamiltonian is now
(34)
As an example, the basis function for u = 3 and 1 = 1, 13,1), is found as follows: w=(u-111)/2=1, S = I ~ I = I r, = w + 1 = 2 (35)
where the kinetic energy operators are abbreviated as P? = (-h2/2)(d2/dq,2)
(32) (33)
L:(p) = (ds/dps) [(ePd'/dp')(p'e-P)]
(15)
cos 28
w = ( 1 / 2 ) ( ~- 111)
F-lu,l) = -(2(u
(46)
+ 1))1~2~V-l,l-l)
(47)
Each term in the Hamiltonian can be expressed as a combination of shift operators and the matrix elements found by their application p2 = p 1 2 +p22 = - ( 1 / 4 ) ( F + - P + ) ( F - - P -
-1 (48)
+
p 2 COS 28 = pI2 - p2' = (-1 / 1 6 ) ( ( F + - P++F -- P-)2 (F+ - P+- F -+ P-)2)(49) q4 = (41'
+ 422)2= (1/16)(F"+ + F+)'(P-+ F-)'
(50)
1044 The Journal of Physical Chemistry, Vol. 92, No. 5, 1988
+
+ F+)(F’-+ F-) (51) = ( 1 / 8 ) ( ( F + F+)’ + (P+ F-)2)
q2 = q I 2 qZ2= ( 1 / 4 ) ( P +
q2 COS 20 = 41’ - 92’
(52)
q4 cos 26 = (912 + 4Z2)(4I2- 4z2)
+ F + ) ( P +- F-)X ((P+ + F+)2+ (P+ F-)2) (53) q4 COS 40 = 2(q12 - 42’)’ + (41’ + 92’) = ( 1 / 3 2 ) ( ( P + + F+)’ + (P+ F - ) 2 )-2 = (1/32)(P+
(1/16)(p+
+ F + ) ( P+- F-)lu,l) = -y4(2(u - 1 + 2))1/2(2(u+ 1 + 2))1/21u+2,1)+ y4(2(u + I))1/2(2(u+ I))1/21u,I)+ f/4(2(u - I + 2))’l2(u - I + 2))1/21u,I) - Y4(2(u + I))1/2(2(u- I))1/21u-2,1) ( 5 5 ) !/4(Ft+
( u + 2 , l ( q z J ~ , l=) - ( 1 / 2 ) ( ~ - 1
TABLE I: Matrix Elements operator v+2 u v-2 pz cos 28 u 2
+
v v-2 q4
u+4
+ 2 ) ’ / ’ ( ~+ I + 2)ll2
(u,llq21u,l) = ( u
+ 1)
( ~ - 2 , 1 1 q ~ 1 ~=, 1-)( 1 / 2 ) ( ~ + I)’/’(u - I)’/*
(56) (57) (58)
The matrix elements are listed in Table I. The q“ and q2 elements were checked against those found by Shaffer;34however, a few sign errors in the q4 elements have been pointed out by Bell.33 The application of the Wang transformation takes the basis functions from e‘@to cos 10 and sin 10 functions. To begin with, the Hamiltonian matrix is split into even an odd blocks due to the fact that only even powers of xl and x2 are included in the Hamiltonian. The Wang transformation splits the Hamiltonian matrix into four blocks-even cosine, even sine, odd cosine, and odd sine blocks. The new Hamiltonian matrix can be found by matrix multiplication, H’ = FEET, where T is the transformation matrix, t denotes transpose and H’ is the new Hamiltonian matrix. Figure 3 shows the transformation matrix for the even block and maximum u, vmax, of 6. The matrix would be the same for the odd block, excluding the I = 0 elements. When setting up the Hamiltonian matrix in the four symmetry blocks, the Wang transformation has the following effects on the matrix elements as listed in Table I. 1 . p 2 , q2, and q4 terms are diagonal in 1 and are unchanged. 2. cos 20 terms a. Multiply matrix terms by 2 1 / 2for I = 0. b. In the odd blocks, the 1 = f l terms become diagonal in I for I = + 1 and for 1 = -1 the signs are reversed. 3. q4 cos 40 terms a. Multiply matrix terms by 2]12 for 1 = 0. b. In the even blocks, I = f 2 terms become diagonal in I for I = 2 and for I = -2 the signs are reversed. c. In the odd blocks, I = f l and I f = r3 terms become I’ = 1 f 2 terms for I = *l and 1 = ~3 (multiplied by -1 for the -1 terms). Figure 4 illustrates the effect of the Wang transformation on the even block. In the computer program the first matrix blocks are set up, changed to tridiagonal form, and then solved for the eigenvalues and eigenvectors, using the Givens-Householder matrix diagonalization procedure. The first 15 eigenvalues and eigenvectors were calculated for each of the four blocks and then ordered. The least-squares adjustment was included based on that written by Ueda and S h i m a n o ~ c h i . ~The ~ change in each force constant, AF, is given by A F = (J’WJ)-’(J‘WAv) (33) Bell, S. J . Phys. B 1970, 3, 745. (34) Shaffer, W. H. Reu. Mod. Phys. 1944, 16, 245. (35) Ueda, T.; Shimanouchi, T. J . Chem. Phys. 1967, 47, 4042.
(59)
q2
I‘
u‘
(OD)
P2
+ F+)2(P”+ F - ) 2(54)
As an example, the matrix elements for the q2 operator were found as follows q21u,l) =
Bauman and Laane
1 l 1
(0
I f2 1 f 2 I f 2 I
v+2
1
V
I
u-2 v-4 u+2
I
V
I
1 1
I If 2 u I f 2 v-2 lf2 44 cos 28 v + 4 1 f 2 u-2
q2 cos 29 u
+2
2 2 u-2 2 u-4 2 q4 cos 40 u 4 I f 4 u+2 u
+
1 I 1 I
+ 2 ) ( u + I + 2))1’2 I / 2 ( ( u - I)(u + l ) ) l / 2 - l / 4 ( ( u f I + 2)(u f 1 + 4))’/2 --I14((u 0 l)(u f 1 + 2))”2 - l / 4 ( ( u + l ) ( u F 1 - 2))1/2 l / 4 ( ( u + 1 + 2)(u + I + 4)(u - 1 + 2 ) ( u - I + 4))1/2 -(u + 2 ) ( ( u + I + 2 ) ( u - 1 + 2))1/2 ’ / 2 ( 3 V 2 - 1‘ + 6~ + 4 ) -(u)((u + l ) ( u - 1 ) ) ’ / 2 + l ) ( u + 1 - 2 ) ( u - l)(u - I - 2))1/2 --1/2((u + 1 + 2 ) ( u - 1 + 2))1/2 ( u + 1) --1/2((u + I)(u - 1 ) ) 1 ’ 2 I14((u f 1 + 2)(u f I + 4))”2 - l / 2 ( ( u f I)(u f 1 - 2))1/2 + l ) ( u + I - 2))”2 -‘/g((U f 1 + 2)(V f 1 + 4)(U f I + 6 ) ( ~ I + 2))1/2 1 / 4 ( 2 up I + 3 ) ( ( u f 1 + 2)(u f I + 4))1/2 -3/4(u + l ) ( ( u f I)(u f 1 + 2))’/2 1/4(2u f 1 + l ) ( ( u + I - 2 ) ( u + 1))1/2 - l / g ( ( u + l ) ( u + I - 2)(u + 1 - 4)(u f ‘/g((U f 1 + 2)(U f I + 4)(0 f 1 + 6 ) ( f ~ + 8))1/2 + l ) ( u f 1 + 2 ) ( u f I + 4)(u f 1 + 6))’12 3 / 4 ( ( u + I)(u + 1 - 2 ) ( u f I + 2 ) ( u f 1 + 4))1/2 + I)(u + I - 2 ) ( u + 1 - 4)(u f 1 + 2))“2 I / g ( ( u =F l ) ( u + I - 2 ) ( u 7 1 - 4 ) ( u + I -I + 1)
l/z((u
f f f f
u+2
I f 4
u
l f 4
u-2
I f 4
u-4
1 f 4
i
I))l/*
6))’12 0
-2
-4-6
x 2-*
= T
-i -i
Figure 3. Wang transformation matrix for the even block of the Hamiltonian matrix with u,, of 6 . All elements not shown are zero.
WANG TRANS.
blank
t,$,Jare
\\
cos28
UH
q4cos4e terms. x21’2for I=O,
unchan rd
terms, x 2 ’11
for 1.0
diagonal I terms for 1.2 and x - l for I = - 2 Figure 4. The effect of the Wang transformation on the even block of the Hamiltonian matrix with umax = 6 .
where Av is a diagonal matrix of the differences between observed and calculated frequencies, W is a diagonal matrix of weighting factors which are input to the program, J is the Jacobian matrix, and t denotes transpose. The elements of the Jacobian matrix are found as follows
The Journal of Physical Chemistry, Vol. 92, No. 5, 1988 1045
Pseudorotation of Cyclopentane
= dvi/dFj = dVn,/dFj
Jij
100
(60)
7
W
where Y, is the n-to-m calculated transition, F, is the force constant being adjusted, dVn,/dF, is the difference between expectation values of the operator corresponding to the Fj potential constants for the n and m eigenvectors. This is given by
0 2
75
i-
where L, is the mth eigenvector and VU) is the matrix representation of the j t h operator. In calculating the J matrix, the eigenvectors and V G ) matrices must come from the appropriate symmetry blocks. The program was written so that up to five isotopes could be used in the least-squares adjustment by averaging the U s for each molecule. Damping factors for each isotope were included so that different weights could be given to each isotope. A subroutine was also included to calculate intensities for transitions for which the dipole operator is ql = q cos 0 or q2 = q sin 0. The matrix elements were found by using
e = ( i / 4 ) ( ~ ++ F+ + P-+ F-) (q sin e) = ( 1 / 4 ) ( P + + F +- P-F-) COS
(62)
50
i-
2 W
0 E W
a
25
n "
I
I200
I
1
1400
1600
CM-1
Figure 5. Infrared pseudorotational combination bands of cyclopentane. Resolution = 0.25 cm-I; path length = 10 cm; vapor pressure = 52 Torr; no. of scans = 1024.
Results A . Pseudorotational Combination Bands. The infrared spectrum of cyclopentane was reexamined at a resolution of.0.l cm-I. In addition to the main combination band series centered a t 1460 cm-I, a less intense series was observed near 1250 cm-I.
Figure 5 shows both of these series. The second of these has an average spacing of 4.1 cm-I. The bands were observed at 1251.6, 1255.3, 1259.8, 1263.9, 1268.1, and 1272.1 cm-I. Figure 6 shows the complicated series of pseudorotational combination bands with a methylene deformation. All of the bands are split into doublets, many into triplets. The doublet splittings are about 1 cm-l, ranging from 0.5 to 1.5 cm-I. Two combination bands series were also observed for cyclopentane-d, near 1400 cm-' and near 1300 cm-', both very weak. Figure 7 illustrates these series of bands. Only difference bands were observed in combination with the methylene deformation with an average spacing of 5.0 cm-I. Figure 8 shows these bands. The second series has an average spacing of 3.7 cm-I, with bands at 1315.8, 1313.0, 1307.5, 1303.3, and 1301.0 cm-I. Zhizhin, Durig, Kasper, and Vasinas7 have previously reported pseudorotational combination bands with a methylene deformation for the d , . However, their sample had an impurity of 20% cyclopentane and their combination bands appear to be due to the cyclopentane impurity which has a more intense spectrum. Pseudorotational combination bands for cyclopentane-dlo were observed at approximately 1000 cm-' (Figure 9). A difference band series was observed and there also appear to be sum bands, but the series is obscured by other peaks. Figure 10 shows the difference bands that were observed. The average spacing of the bands is 4.0 cm-I. At longer path lengths, the band at 546 cm-I of cyclopentane shows pseudorotational fine structure (Figure 1 1). The analogous band for the d l (Figure 12) and d2 (Figure 13) molecules also show considerable structure. Not enough sample was available to obtain a good spectrum of the d6 molecule in this region. However, a broad band was observed at 455 cm-'. The infrared solid-phase spectrum of the d6 shows sharp bands at 493, 462, 468, and 445 cm-I. The analogous band in the d l o molecule is at 427 cm-' and is shown in Figure 14. The broad band at -490 cm-' shows fine structure at long path length. In crystal phase I11 of cyclopentane the pseudorotation is frozen out3*and in the solid-phase infrared spectra bands were observed at 547 and 428 cm-l for the do and d l omolecules, and doublets were centered at 529, 522, and 464 cm-I for the d , , d,, and d , molecules. B. Radial Bands. The infrared spectrum was searched in the region of the radial band for all five samples and nothing was observed. However, the radial band was observed in the Raman spectrum of each compound. These were very weak bands and the spectra were taken under optimum conditions at the instrumental limit of detection. The spectrum of the do was very similar to that observed by Carreira and co-workers.22 The Q branches were observed at Raman shifts of 272.5, 264.5, and 257.0 cm-'.
(36) A much higher u,, can, of course, be attained on a larger computer such as our Amdahl 470 V/6 but there are major cost and convenience advantages to using a small computer.
(37) Zhizhin, G. N.;Durig, J. R.; Kasper, J. M.; Vasina, T. V. Zh. Srrukr. Khim. 1975, 16, 56. (38) Mills, 1. M. Mol. Phys. 1970, 20, 127.
and the elements are
+ 2))1/2 ( u i l , I + l ( q sin Blu,l) = -(1/4)(2(u + I + 2))'12 (uii,i-ilq COS elu,i) = +(i/4)(2(u + i))l/2 (uf1,l-l(q sin Blu,l) = -(1/4)(2(u + I))'/'
(uii,i+ilq
COS
elu,i) = -(i/4)(2(u i I
(63)
(64)
The calculated intensities are then found by
I
a
Io:
vnme-(udlkT(nlq cos elm)'
(65)
vnme+~)IkT(n(q sin
where v,,,,, is the calculated frequency of the n-to-m transition, the exponential term is due to the Boltzmann factors, and the integral is the transition moment integral found by matrix multiplication. The intensities are normalized so that the most intense transition has a value of 1.0. The calculated eigenvalues were checked by comparing to previous calculations on 2,5-dihydrofurans0 and germacyclopentane." The intensities were checked by comparison to those calculated for 2,5-dihydrofuran. The computer calculations were done on a CDC-1604A computer with a memory of only 32K words. The words are 48 bits, so single precision arithmetic was adequate. The use of the Wang transformation reduces the memory required, so the u,,, of 26 could be used.36 This corresponds to matrices of 105 X 105, 91 X 91, 91 X 91, and 91 X 91. Without the transformation, the largest u,,, would be 18. The program as written by Carreira, Mills, and Personso had a u,,, of 20. For the same umax,this program is much faster due to the smaller matrix sizes in the diagonalization routine. The larger allowable quantum number is very important in calculating higher energy levels accurately. This was especially important for the calculations on cyclopentane. More recently, we have checked these calculations on a VAX 11/780 computer.
1046
The Journal of Physical Chemistry, Vol. 92, No. 5, 1988
Bauman and Laane
n I-
IW
z
n 0 a W 1450
1400
1500
Figure 6. Infrared pseudorotational combination bands of cyclopentane combined with E{ methylene deformation. Top spectrum for 52 Torr; bottom spectrum for 31 1 Torr. Other parameters are the same as for Figure 5. 100
100
/
w
0
2 t z
/
/
75
z
9
/PI-
50
a LL
tk-
z
25
LL
w
a 0
0
1000
1100
1200
CM-'
CM-l
Infrared pseudorotational combination bands of cyclopentane-d. Resolution 0.25 cm-I; path length = 10 cm; vapor pressure 280 Torr: no. of scans = 512.
Figure 7.
Infrared pseudorotational combination bands of cyclopentane-d,@ Resolution = 0.25 cm-I; path length = 0.75 m; vapor pressure = 10 Torr; no. of scans = 1024. Figure 9.
50
80
i_i,
W
z a
i-
I
2
30
a K
c k-
z 0 W
"r, i
W
I
IO
I
i I
30 I375
I
1
1400
1425
C M-I
Figure 8. Infrared pseudorotational combination bands of cyclopentane-d
950
I 1051
Io00 CM-'
Infrared spectrum of the combination bands of cyclopentane-dlo(expanded spectrum of Figure 9). Figure 10.
(expanded spectrum of Figure 7).
Only one Q branch was observed for the d l at 268.0 cm-I. The recorded Raman spectrum of d2 is shown in Figure 15. Three Q branches were observed at 261 .O, 252.0, and 244.0 cm-' (broad). The d6 molecule showed two Q branches at 240.0 and 228.0 cm-l while only one Q branch, at 216 cm-I, was observed for the d,,, molecule. C. Calculations of Reduced Masses. It is necessary to calculate reduced masses for the out-of-plane ring vibrations in order to relate the calculated potential functions to actual conformations of the molecule. The model used to calculate the reduced masses is given by eq 1, where the only motion of the carbon atoms is perpendicular to the ring, i.e., in the z coordinate. This assumption
TABLE II: Calculated Reduced Masses for Cyclopentane, Cyclopentane-d, -l,l-d2, -1,1,2,2,3,3-d6, and -dlo reduced masses, amu phase angle
4, deg
do
dl
d2
d6
dl0
0
29.00 29.00 29.00 29.00 29.00 29.00
32.11 31.82 31.06 30.12 29.36 29.07
34.91 34.36 32.92 31.14 29.70 29.14
44.20 43.14 40.38 36.96 34.20 33.14
45.99 45.99 45.99 45.99 45.99 45.99
18
36 54 72 90
of the rectilinear motion yields exactly the same reduced mass value for the planar ring as does the use of curvilinear coordinates.
The Journal of Physical Chemistry, Val. 92, No. 5, 1988 1047
Pseudorotation of Cyclopentane
I
I, 550
5b0
600
650
I
CM-'
Figure 11. Infrared spectrum of cyclopentane in the radial overtone region. Resolution = 0.25 cm-I; path length = 5.25 m; vapor pressure = 20 Torr; no. of scans = 512.
,
250
300
1
200
I
RAMAN SHIFT (CM-II
Figure 15. Raman spectrum of the radial mode of cyclopentane-l,l-d2. Exciting line = 5145 .& with -1 W at the sample; spectral bandwidth = 4 cm-I; pen period = 100 s; sensitivity = 75 cps; scan speed = 0.01 cm-I/s; maximum vapor pressure at 25 "C.
TABLE III: Infrared Subband Structure for Cyclopentane, Derived by Mills for a Pseudorotating Model" sym subband structure E,' ugSUb= ug C - 2CCt - B 2k[C - Cct - E ] El" voSUb= u0 D - C - E - 2Cr23 + 2C{, f (2X[D - C{23] + 2k[C - cl23 + crt - 41 Efl u p b = YO + D (2XD + 2k[Ctt - CCI~]! upb= vg D + C - E Cr23 + 2Crt F (2X[D + Cf23] - 2k[C E; + cl23 + cct -
+ +
+
01
1
450
500
'
1
550
1
600
I
650
E;
u p b = YO 2cc23
CM-l
Figure 12. Infrared spectrum of cyclopentane-d in the radial overtone region. Resolution = 0.25 cm-'; path length = 3.75 m; vapor pressure = 20 Torr; no. of scans = 1024.
*
+
Pure Pseudorotation F (4X[D
+ 4 0 + C - B + 4c{23
- B1l
'/2C
