Internal Barrier of Propylene Oxide from the Microwave Spectrum. IIDudley R. Herschbach and Jerome D. Swalen Citation: The Journal of Chemical Physics 29, 761 (1958); doi: 10.1063/1.1744588 View online: http://dx.doi.org/10.1063/1.1744588 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/29/4?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Microwave Spectrum of Propylene. II. Potential Function for the Internal Rotation of the Methyl Group J. Chem. Phys. 45, 1984 (1966); 10.1063/1.1727882 Microwave Spectrum and Internal Barrier of Methylphosphine J. Chem. Phys. 35, 2139 (1961); 10.1063/1.1732221 Microwave Spectrum and Barrier to Internal Rotation for TransFluoroPropylene J. Chem. Phys. 27, 989 (1957); 10.1063/1.1743920 Internal Barrier of Propylene Oxide from the Microwave Spectrum. I J. Chem. Phys. 27, 100 (1957); 10.1063/1.1743645 Microwave Spectrum and Internal Barrier of Acetaldehyde J. Chem. Phys. 24, 631 (1956); 10.1063/1.1742578

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THE JOURNAL OF CHEMICAL PHYSICS VOLUME 29, NUMBER 4 OCTOBER, 1958

Internal Barrier of Propylene Oxide from the Microwave Spectrum. 11*

DUDLEY R. HERSCHBACH, t Mallinckrodt Chemical Laboratory, Harvard University, Cambridge 38, Massachusetts

AND

JEROME D. SWALEN,t Division of Pure Physics, National Research Council, Ottawa 2, Ontario, Canada

(Received April 7, 1958)

Several long progressions of perpendicular transitions have been measured i,n the microwave spectrum of propylene oxide. These have provided an opportunity to test the theory of internal rotation over a very large range of rotational quantum numbers, J = 1 to 48 and K = 1 to 25, in the ground torsional state. For low K, the K doubling which arises from the asymmetry of the molecule is much larger than the splittings due to tun.neling through the potential barrier. The spectrum resembles that of an ordinary asymmetric rotor except that each line is split into a close doublet by the tunneling. At high K, the K doubling becomes negligible. A triplet fine structure is observed in which the spacings show a periodic dependence on K, essen­tially the same as that calculated for the case of two coaxial symmetric tops. (For a true symmetric rotor, the selection rules would prevent the observation of this pattern.) At intermediate K, the K doubling and the tunneling perturbations become comparable. The mixing of the rotational wave-functions which results permits "forbidden" lines to appear. Very good agreement with theory is obtained over the whole range of K, with the barrier height Va=895 cm-1 (2560 cal/M).

Rotational transitions have also been assigned for molecules in the first and second excited torsional states, in which the tunnel effect splittings are greatly magnified. For both the excited states, the barrier height is found to be identical with the ground state result. This supports the usual assumption that perturbations from vibratiQnal interactions with internal rotation and from the sixfold V. term in the po­tential barrier are small enough to be safely neglected.

INTRODUCTION

I N a previous paper! the treatment of internal rota­tion developed by Wilson, Lin, and Lide,2 and by

Kilb, Lin, and Wilson3 was extended to molecules such as propylene oxide,

which may have no symmetry elements other than the local threefold symmetry of the methyl group. The model assumed comprises a rigid symmetric top (CRa) attached to a rigid asymmetric frame, with internal rotation or torsion hindered by a relatively high po­tential barrier of sinusoidal shape. Tunneling between the three equivalent potential minima splits each torsional level v into two distinct sublevels, a non­degenerate level, VA, and a doubly degenerate level, VE. The effect of coupling between internal and over-all rotation differs for the A and E torsional levels. Consequently the over-all rotational transitions asso­ciated with each torsional state V show splittings which

* National Research Council Contribution. The research reported in this paper was supported in part by a

grant extended Harvard University by the Office of Naval Re­search under Contract N50ri 1866, Task Order XIV, and by a grant from the California Research Corporation.

t Charles A. Coffin, Fellow, 1956-1957. Now Junior Fellow, Society of Fellows, Harvard University.

t National Research Laboratories Postdoctorate Fellow, 1956-1957. Present address: Shell Development Company, Emeryville, California.

1 J. D. Swalen and D. R. Herschbach, J. Chern. Phys. 27, 100 (1957), hereafter referred to as I.

2 Wilson, Lin, and Lide, J. Chern. Phys. 23, 136 (1955). a Kilb, Lin, and Wilson, J. Chern. Phys. 26, 1695 (1957). For

other treatments see references 19-22.

are extremely sensitive to the barrier height. The proceduresI--3 described in I for analysis of the splittings have been applied to a number of molecules.'

Propylene oxide is particularly suitable as a test molecule for the theory because in its microwave spectrum there is accessible an unusual variety of transitions, including many with very large rotational quantum numbers and many from excited torsional states. In I, for most of the ground state transitions studied the internal rotation splittings were too small to be observed at the available resolution. Much larger splittings were observed for several transitions from molecules in the first excited torsional state, and these yielded a barrier height of 895 em-I. In an attempt to check this result, it was found that the splittings of ground state lines of high J and K could be partially resolved, but appeared to require a barrier about 6% higher. Further investigation of the theory has now shown that this discrepancy arose because the second order treatment of the coupling perturbation employed in I is not sufficiently accurate for high K levels. In the present study the ground state splittings have been observed under high resolution over a very large range of J and K. The assignment of the first excited state spectrum has also been extended, and several transi­tions in the second excited torsional state have been identified. The treatment of high K transitions is amended, and methods are described for evaluating the energy levels when the internal rotation splittings are of the same or larger order of magnitude than the K doubling of the rigid rotor levels, as is the case for many of the transitions studied here.

4 For a summary of barrier determinations from microwave spectra, see E. B. Wilson, Jr., Proc. Nat!. Acad. Sci. U. S. 43, 816 (1957).

761

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762 D. R. HERSCHBACH AND J. D. SWALEN

Experimental

The experimental work was carried out at the Spectroscopy Laboratory, National Research Council, Ottawa, Canada. The spectrum was studied from 8 kMc to 42 kMc with a square wave Stark modulated spectrograph, details of which are described elsewhere.5

Including the results of I, about 400 lines have been measured, of which 300 have been assigned.s Many other lines are present in the spectrum.

LOW K TRANSITIONS

For low K transitions7 the effects of the coupling between internal and over-all rotation may be eval­uated very simply in the case of a high barrier, since it is sufficient to treat the coupling by second order perturbation theory. The treatment given in I yields an effective rotational Hamiltonian for each torsional state. For a level of A symmetry,

HVA = AVA Pz2+ BvAPl+ CvAPx2. (la)

The rotational constants A> B> C are the inverse principal moments of inertia plus a second order cor­rection for the coupling which depends strongly on the barrier height. Since HVA has the same form as the rigid rotor Hamiltonian, the energy levels can be obtained from standard tables.8- 1o The Hamiltonian for an E torsional state is

The second order contributions which enter the pseudo­rigid rotor terms of HvE differ in sign and magnitude from those in HvA . In addition, HvE contains first order terms, linear in the angular momentum compo­nents. As discussed in I, the first order terms have little effect when they are small compared with the asymmetry splittings of the rigid rotor levels, and may be neglected or added as perturbations after evaluating the pseudo-rigid rotor energy from the tables. When this is not the case, HvE requires a more complicated treatment, described in the appendix to this paper.

The barrier dependent factors which enter the cor­rections for the coupling in HVA and HvE have been

6 C. C. Costain and B. P. Stoicheff, J. Chern. Phys. (to be pub­lished).

6 A complete list of the measured lines is available from the authors. This list, Microwave Spectrum of Propylene Oxide, has also been deposited as Document No. 5727 with the ADI Auxil­iary Publications Project, Photoduplication Service, Library of Congress, Washington 25, D. C. A copy may be obtained by citing the Document number and by remitting $2.50 for photo­prints, or $1.75 for 35-mm microfilm. Advance payment is re­quired. Make checks or money orders payable to: Chief, Photo­duplication Service, Library of Congress.

7 Throughout the paper, we use K to abbreviate the asym-metric rotor index K_1•

8 See references given in I. 9 R. H. Schwendeman, J. Chern. Phys. 27,986 (1957). 10 s. R. Polo, Can. J. Phys. 35, 880 (1957).

calculated for a barrier of the form

V(a) = !Va(1- cos3a) +! Vs(1- cos6a) + . . . (2)

and are now available in tables. ll Only the Va term will be considered here; data to be discussed later lead to the estimate: Vs/Va<O.Ol.

The structural coefficients involved in Eqs. (1) are defined in I. The principal moments of inertia of the rigid molecule, la, h, Ie, are required. These can be determined from the spectrum; for a high barrier case, the mean rotational constants defined by

(3)

etc., closely approximate12 the rigid rotor constants.

TABLE I. Additional ground state transitions.

Assign.- Obs. freq.b Calc. freq.

"a-type"

101-000 12633.52 Mc 12633.60 318-212 36783.14 36783.96 312-2u 38974.35 38974.89 322-221 37900.09 37900.81 321-220 38036.22 38037.01

"b-type"

Ito-101 12072.14 12072.24 2u-202 12837.32 12837.06 918-909 34611.20 34610.22

101.9-100.10 40469.72 40468.93

220-2n 34058.49 34058.97 321-312 33120.72 33121.10

Ih.9-1It.l0 32883.48 32884.70 122.10-121,u 35690.23 35692 .16 132.U-131.12 39388.43 39391.38

22.-212 36216.40 36216.72 32r313 37333.28 37333.56 ~8-414 38834.10 38833.70

413-322 18972.89 18973.54 4-..-321 11502.18 11503.73 ~-533 19105.71 19107.43 ~6-582 16786.89 16789.40

726-634 33018.36 33019.91 726-633 28938.18 28940.23 936-846 33248.35 33251.69 937-8 •• 32347.54 32351.24

"e_type"

1.0-000 24705.86 24705.84 2u-101 38070.06 38070.09 221-2n 34024.55 34024.79 322-312 32950.24 32950.71

A 18023.72 B 6682.12 C 5951.48

- The upper level of the transition is written first. b Estimated accuracy of measurements is ±0.1O Mc. For the

lines which have been resolved into doublets (see Table II), the listed frequency is the average of the A and E components.

u D. R. Herschbach, J. Chern. Phys. 27, 975 (1957). 12 See the introduction to the tables described in reference 11.

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IN T ERN ALB A R R I E R 0 F C 3 H 60 FRO M M I C ROW A V ESP E C T RUM 763

TABLE II. Analysis of ground state splittings.

Assign. Obs. (VE-VA) Calc. (VE-VA)" V3b

413-322 0.28±0.04 0.28 Me 895 em-I 6,..-533 0.62±0.04 0.57 885 6.,-532 0.34±0.06 0.36 900 725-634 0.52±0.05 0.51 892 936-8" 0.98±0.02 0.93 889 937-844 0.35±0.04 0.36 897

523-514 -0.27±0.04 -0.28 898 6..-61, -0.27±0.08 -0.27 895 725-716 -0.25±0.02 -0.25 895 826-817 -0.21±0.07 -0.23 906 927-916 -0.22±0.05 -0.20 880

894±25

"Calculated for V3=895 em-I. b Barrier required for exact fit to observed splitting.

The moment of inertia of the internal top (CH3)

along the symmetry axis, la, and the direction cosines between the top axis and the principal axes, Aa , Ab, Ac, are also required. Only two of the direction cosines are independent. Moreover, for propylene oxide Ab is negligibly small. Hence it is sufficient to specify only one direction cosine, say Aa.

In fitting the observed E- A splittings, the single adjustable parameter is the barrier height Va, once Aa and la are fixed. The quantity and variety of data obtained in the present work made it feasible to include

Rp PR

30 /' f ... s

/ K_I

20

-I

-2

10

o 50

TranSition Frequency. Kilomegacycles/Second

FIG. 1. Pattern of r-family transitions in propylene oxide. Solid circles represent the reported spectrum.

Aa and l", as adjustable parameters in some of the calculations. The results agreed quite well with the values which had been calculated in I with the aid of some structural assumptions, and these values (given in Table X) have therefore been adopted for the final analysis.

Table I lists the additional low K transitions which have been identified for the ground torsional state; this supplements Table I of Part 1. The observed fre­quencies are compared with predictions calculated from the rigid rotor constants determined in I, which are the average constants of Eq. (3). Since the barrier is high enough to make the rotational constants for the A and E levels nearly identical (see Table IX), only very small splittings will appear.

Several of the lines of Table I, and also some of those which were reported in Part I, have been resolved into symmetrical doublets, and the observed splittings are given in Table II. The estimated uncertainty of measurement is based on the reproducibility obtained for six or more recorder tracings. The resolution obtained in a slow sweep over the line was often limited by klystron instability.

The second order contributions to the calculated splittings in Table II were obtained by the difference method given in Eq. (14) of 1. For the K-:( 2 levels, the first order terms in HvE were entirely negligible; for the K = 3, 4 levels, the first order contribution was added as a perturbation, evaluated from Eq. (22).

A good ftt is obtained with Va=895 cm-I • Because of the smallness of the splittings, experimental error contributes more than usual; an error of ±0.05 Me in a splitting of 0.30 Mc corresponds to an uncertainty of about ±25 cm-I in Va. If only the splittings in Table II were available, the uncertainty would have to be increased to about ±75 cm-I, because the values adopted for Aa and lex could not be satisfactorily tested.

HIGH K TRANSITIONS: T FAMILIES

A striking feature of the propylene oxide spectrum is several regular progressions of perpendicular type transitions which extend to very large values of J and K. These form both PR(~J=+1, ~K=-1) and RP(~J = -1, ~K = + 1) branches. The large cancella­tion which occurs because J and K change in opposite directions permits transitions between levels of very high rotational energy to fall in the microwave spec­trum. For example, the 4&4~4726 transition at 35401 Mc or 1.2 cm-I is the difference between levels having about 720 cm-I of rotational energy.

As illustrated in Fig. 1, the P Rand R P transitions are most naturally grouped according to T, the average value of T= K_I- K+I for the initial and final levels. These "T families" show regularities which make their identification and analysis quite simple even for high J and K, unlike the usual situation in asymmetric rotor spectra. Some formulas are given below which may

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764 D. R. HERSCHBACH AND J. D. SWALEN

prove useful for the analysis of other molecules having T-family transitions.

In an asymmetric rotor, the K-degeneracy char­acteristic of a symmetric rotor is removed. At low K, this K doubling of the levels is large, and consequently each T family consists of two separate branches, as indicated in Fig. 1. As K increases, the asymmetry splittings decrease rapidly and for sufficiently large K the pairs of rigid rotor levels having the same value of K become nearly degenerate. When this occurs, the

TABLE III. High K transitions, ground torsional state.

Assign.

157-148 17s-16. 19.-1810 2ito-20u 23u -2212 2512-2413 2713-2614 2914-2815 3h,-3016 3316-3217

3517-3418 3718-3619

3919-3&0 4ho-4O:!1

167-158

188-17. 20.-1910 221O-21u 24u-2312 2612-2513 2813-2714 3014-2915

2611.2514 2814-271, 3015-2916 3216-3117

3417-3318 3618-3519 3819-3720 4{)zo-3921 4211-4122

~2-4323 ~-4524 4&..-472,

7,-85

97-10, 11s-127 13.-148 1510-16. 1711-1810 1912-20u 2113-2212 2314-2413

Obs.

(v,-vo)

Obs.

7=0; T= - h- 1 and -2<--+2

14760.68 16721.31 18682.74 20644.92 22607.49 24570.46 26533.12 28496.19 30459.18 32422.83 34386.62 36349.51 38313.23 40276.05

7=-1;

27575.98 29530.32 31486.76 33444.73 35402.99 37361.64 39320.40 41279.30

7=1;

13759.98 15725.98 17692.55 19659.50 21626.39 23594.05 25561.43 27529.07 29497.04 31464.96 33433.28 35401.45

7=3;

27429.04 25480.04 23527.00 21569.20 19609.28 17646.86 15682.39 13716.40 11478.57

0.57 0.56 0.54 0.48 0.40 0.31 0.21 0.12 0.01

-0.08 -0.17 -0.24 -0.30 -0.35

3.16 3.56 4.04 4.39 4.78 5.09 5.96 6.59 7.46 7.79 8.06 9.29 9.76

11.10

T= -2<--0 and -3<--+1

0.57 9.61 0.56 12.69 0.54 15.55 0.48 18.14 0.40 21.29 0.31 24.67 0.21 28.42 0.12 32.36

T=0<--2 and -1<--3

0.21 0.12 0.01

-0.08 -0.17 -0.24 -0.30 -0.35 -0.38 -0.38 -0.37 -0.34

-15.02 -17.57 -20.51 -23.71 -26.72 -30.46 -33.83 -37.40 -41.25 -45.77 -49.85 -53.69

0.52 5.78 0.56 9.43 0.57 12.89 0.56 18.13 0.54 23.32 0.48 29.33 0.40 36.12 0.31 43.41 0.21 51.72

(v,-vo)

Calc.

2.97 3.41 3.87 4.35 4.84 5.36 5.90 6.46 7.06 7.68 8.34 9.03 9.75

10.52

10.33 12.72 15.35 18.24 2l.38 24.78 28.44 32.36

-14.94 -17.60 -20.47 -23.54 -26.81 -30.30 -33.97 -37.85 -41.91 -46.00 -50.45 -54.92

6.00 9.25

13.19 17.82 23.14 29.15 35.85 43.23 51.28

spectrum will in many respects resemble that of a symmetric rotor, even though the molecular asym­metry may still give important contributions to the energy levels. An example is the nearly equal spacing of the high K members of a T family, which is evident in Fig. 1.

The high K levels are of special interest here because the near degeneracy in K which would be present for a rigid rotor is partially removed by the internal angular momentum associated with hindered rotation. The resulting fine structure is very similar to that which would be obtained if perpendicular transitions were possible in a symmetric hindered rotor. For the ground torsional state, however, the internal rotation perturba­tion is small enough that it can be added at the end of the analysis, after computing the rigid rotor and cen­trifugal distortion contributions as for an ordinary asymmetric rotor.

In Table III are listed the transitions for which the rigid rotor asymmetry splittings are less than 0.1 Mc, and hence negligible at the present resolution. The lower members of the 7= 0, -1, 3 families were stud­ied in I.

Rigid Rotor Analysis of T Families

We shall write p( J, K) for the frequency of the transition

(4)

As in the foregoing, K denotes the prolate index K_1 ;

the oblate index K+l will be omitted whenever the asymmetry splittings are negligible, as in Table III. From (4) there follows the convenient relation:

J+7=2K-1. (5)

The formulas given below refer to apR transition if p( J, K) is positive; if p( J, K) turns out to be negative, the actual absorption, -p (J, K), is a R P transition, and corresponds to the reverse of (4). To apply the formulas to a near-oblate rotor, interchange K_l and K+1, and the A and C rotational constants.

As an initial estimate,

p(J, K) = [3 (B+C) - 2AJ(K -J) - (B+C)7. (6)

This expression neglects terms involving Wang's asymmetry parameter, b= (C-B)j(2A-B-C). These terms are important for propylene oxide (b= -.0.0312056), and contribute several thousand Me. Recently there have been published extensive tables9 •10

of the coefficients which appear in an expansion of the rigid rotor levels in powers of b; Schwendeman9 has tabulated numerical values for J:( 40. This expansion converges well for the high K members of T families, even for rather large values of b.

The spacing between successive members of a T

family,

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IN T ERN ALB A R R I E R 0 F C 3 H 60 FRO M M I C ROW A V ESP E C T RUM 765

varies quite slowly; furthermore its value is relatively insensitive to the asymmetry parameter. Once the lower I members of a r family are located with the aid of calculations from the tables, the series can easily be extrapolated to higher I. For K>3, ~ is given by

~=[3(B+C)-2A]

+[A-HB+C)](b2/16)[156+g(1', K)+g( -.1', -K)]

(7) where

g(T, K) = {[T2(T+l)2]/[K(K+1)]1

- {[(T+3)2(T+4)2]/[(K+1) (K+2)]1

and terms of order b4 and higher are neglected. As K increases, g(T, K)-->O, and ~ tends to a constant. This limiting interval is the same for all r families, and is approached quite rapidly if l' is small in absolute magnitude. For propylene oxide, the limit of Eq. (7) is 1964.51 Mc, only 5% of which is contributed by the term in b2• For K> 10, the ~'s calculated from the tables are within 2 Mc of this limit. The observed ~'s include a centrifugal distortion contribution which does not tend to a limiting value, but this is small, and may be approximated by Eq. (10). At high K, the observed ~'s for the propylene oxide r families become constant within a few tenths of a megacycle, and are all in the range 1958-1968 Me.

It may be noted from Eq. (6) that if C=B=tA, all members of a r family would coincide at

11= -(B+C)1'

in the rigid rotor approximation; the two families cor­responding to ±1' would then represent coincident RP and pR branches. If the deviation from C=B=tA, as measured by ~, is only a few thousand megacycles or less, many 1'-family transitions will appear in the microwave region. More generally, whenever the rotational constants of a molecule approximately satisfy the relation C=B=A/n, where n>2 is an integer, the P Rand RP transitions will form regular monotonic series similar to those in Fig. 1. These series proceed as follows: .. 'II( I, K), II( I +n-1, K + 1), "', with an increase of n-1 in I for each unit increase in K.13

Centrifugal Distortion

The centrifugal distortion shifts, IIr igid- 110, given in Table III, are surprisingly small, considering the large values of I and K involved. An approximate treatment described below shows that the distortion parameters are not usually small, but their contributions largely cancel in r-family transitions. For example, in the 4~4f-4726 transition, centrifugal distortion displaces

13 An example of the n = 5 case is the series of very high J transitions in methylene fluoride studied by D. R. Lide, Jr., J. Am. Chern. Soc. 74, 3548 (1952).

2.5,----,-----,----,--------,----,

2.0

15

S'v Me T ... -I

T-' 10

/ , 0.5

, ,

, / r-o r--

0

-0.50:-------c":----::'::-------=-':------,l:-...J

J

FIG. 2. Centrifugal distortion shifts (divided by J) for T

families. Points are experimental, solid lines are calculated from the constants of Table IV. To conserve space, the negative of the actual shift has been plotted for the T= -1 family.

the energy levels by about 1 cm-\ but shifts the transition frequency only 50 Mc or 1.7XlO-3 cm-l. As is evident in Table III, the extent of this cancellation is characteristically different for each r family; it becomes less complete as I increases and as the abso­lute magnitude of l' increases. Figure 2 shows that the major contribution to the distortion shifts is nearly proportional to 1'12.

Pololo has derived a convenient expansion of the Kivelson-Wilsonl4 distortion formula in powers of I(I+1). The diagonal contribution to a rotational level W(JK) is, for K>3,

W ri• iC W=Dd2(I+1)2+DJKI(I+1)K2

+DKK4+[4I(J+1)-10K2]Rs, (8)

if terms involving the asymmetry parameter are neg­lected. Note that even in this approximation the Rs terms enter in addition to those which would be present for a symmetric rotor molecule. On introducing Eqs. (4) and (5) into (8), the centrifugal distortion correction for r-family transitions is obtained as a cubic polynomial in I,

IIr - 110= ao+alI +~12+a3J3. (9)

The correction to the ~ of Eq. (7) is

-2(al+2~+3a3) -4(~+3a3) I-6a312. (10)

14 D. Kivelson and E. B. Wilson, Jr., J. Chern. Phys. 20, 1575 (1952).

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766 D. R. HERSCHBACH AND J. D. SWALEN

Here

112= -3Dxi/2

al= 18Rs+!DJK (T2+1) -!DK(3r2+1) (11)

ao= 10Rsr- !DKr(T2+ 1).

Table IV gives the distortion constants, which were evaluated by a graphical procedure. For the r= 0 family, the coefficients ao and 112 vanish; therefore a straight line results if the quantity 071= (7Ir-7Io)/ J is plotted versus J2; the slope of this line is aa. Since the term aaJ2 is independent of r, and is small, it may be conveniently subtracted from the observed distortion shifts of each T family. In Fig. 2 the resulting values of 0'7I=07l-aaJ2 are plotted versus J; the slopes give the 112 coefficients, and the extrapolated intercepts at J = 0 give the al coefficients; the ao's are too small to determine. The observed values for the various

TABLE IV. Centrifugal distortion constants.

DK=19.S0±O.S kc DJK=4.20±2 DJ=2.97±O.S R6=10.94±O.S a3=4DJ-~DJK-~DK

=O.040±O.OO3

T families agree satisfactorily with the relations re­quired by Eqs. (11). It is not possible to obtain a consistent fit without including the Rs terms, however. Seventy-five percent or more of the distortion shifts for the r>O families is contributed by the il2J2 term; this is simply proportional to r and involves only one distortion constant, DK •

Although the distortion shifts calculated from Eq. (9) and the constants of Table IV are in good agree­ment with the observed values in Table III and Fig. 2, the predicted shifts are found to be 5-10% too small for a few members of the r=4, 5 families which were assigned.s The over-all fit was not improved by a least squares analysis including these data and adding terms omitted in Eq. (8), which contain additional distortion constants multiplied by powers of the asym­metry parameter. This calculation indicated that DK is well determined, but the other constants given in Table IV should be regarded as coefficients in a semi­empirical formula rather than as the actual distortion constants.

Internal Rotation

As mentioned in the foregoing, in a rigid rotor the asymmetry splitting or K doubling of the levels becomes negligible at high K. This also applies to the A levels of a hindered rotor, regardless of the barrier height (since the effective rotational Hamiltonian for an A torsional

FIG. 3. Recorder tracings of high K transitions, showing fine structure due to hindered internal rotation.

state contains only even powers of the angular mo­mentum operators). For the E levels, however, the K degeneracy will be removed by the odd order terms which are introduced into the Hamiltonian by the coupling between internal and over-all rotation. This splitting, like that between the A and E levels, is strongly barrier dependent, and vanishes for a suffi­ciently high barrier.

In the transitions of Table III, such splittings of the E lines are small but resolvable, whereas the rigid rotor K doubling is negligible and consequently the A lines are K degenerate at the present resolution.

25

20

K_I

15

10 T

3 II

I 0

0 0

-I 0

5 -1.0

1/". - 1/. Me

FIG. 4. Observed and calculated splittings of the high K transi­tions in Table III (compare Fig. 3). If the molecule consisted of two coaxial symmetric tops, the splittings would be a periodic function of K; it is seen that (aside from a change of scale) the presence of asymmetry merely introduces a gentle damping of the K dependence.

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I N T ERN ALB A R R I E R 0 F C 3 H 6 0 FRO M M I C ROW A V ESP E C T RUM 767

These transitions therefore show a triplet fine structure consisting of two E lines and a single A line of twice the intensity. Several examples are shown in Fig. 3. It is seen that the relative positions of the three com­ponents vary rapidly as K changes. At K = 8, the two E lines are nearly coincident, and the transition ap­pears as a doublet with the E member slightly broad­ened. At K = 10, the E lines are slightly split. At K = 12, one of the E lines has shifted to a position almost under the A line, and at K = 16 it appears on the opposite side of the A line. By K = 20, both E lines have shifted below the A line.

Again we may expect to obtain a good first approxi­mation by considering the limit of zero asymmetry, in which the molecule consists of two coaxial symmetric tops. In this case,15 the splitting of the energy levels produced by internal rotation is independent of J and is simply a periodic function of K. For a true symmetric rotor, therefore, internal rotation does not give rise to splittings in the rotational spectrum,16 since only parallel transitions are allowed in which K does not change. However, the propylene oxide T

families are perpendicular transitions, and the sym­metric rotor terms which enter in zeroth order give the major contribution to the K dependence of the ob­served splittings. The transitions studied ascend in integer steps to K = 25, making it possible to map the K dependence over almost a complete cycle (see Fig. 4).

The problem of the coupling between internal and over-all rotation can be solved exactly in the case of two coaxial symmetric tops.I5 A separation of variables is effected by transforming to an axis system which is not fixed in either top, but is defined to move in such a way that the relative angular momentum of the two groups vanishes when viewed from this internal axis system. The energy can then be expressed simply as the rigid rotor terms plus an internal energy W vu'

The latter is obtained from a Mathieu equation, but the relevant eigenvalues differ from the tabulated8

solutions employed in I because the boundary condi­tions have been altered by the transformation to internal axes. The periodic K dependence which is thereby introduced into W vu may be regarded as a modulation of the internal motion by over-all rotation.

It is convenient to expand W vu as a Fourier seriesP

15 H. H. Nielsen, Phys. Rev. 40, 445 (1932); J. S. Koehler and D. M. Dennison, Phys. Rev. 57, 1006 (1940).

16 The small splittings which have been observed for some symmetric rotor molecules are due to the interaction between hindered rotation and vibrations. See D. Kivelson, J. Chern. Phys.22, 1733 (1954); 27, 980 (1957).

17 The form of Eq. (12) may be established from the boundary conditions formulated by Dennison and co-workers, although it is not identical with the expansion used by them. Equation (12) has the advantage that it refers to a level of given symmetry, as specified by the index u, where u=O for A levels, u= ±1 for E levels, and the selection rule is du=O. Their formulas refer to a different index r (not to be confused with the rigid rotor notation used above) whose value depends also o.n the value of K. The relation is K +r+u= 1 modulo 3. Compare reference 20.

For an A level

W VA =WO+WI COSO+W2 cos20+ . . . (12)

where

(13)

The two E levels are obtained from the A by h O± (2'11/3). The w's are the same for the A and E levels of a given torsional state v, and depend only on the reduced barrier height s= 4 Vs/9F defined in Eq. (8) of I. As s decreases or v increases, WI and the higher coefficients increase rapidly; but for the ground torsional state (v= 0) it is sufficient to retain only WI when s> 25 (for propylene oxide s= 68.0). At K = 0, the eigenvalues W vu are the same as the tabulated Mathieu eigenvalues Evu employed in I, and substitu­tion into Eq. (12) yields the relations

Wo= +HEvA+2EvE)

WI= -j(EvE-EvA ) = -i~o. (14)

Thus in the high barrier case the Wvu for K> 0 are easily obtained from the tabulated K = 0 solutions by use of Eqs. (14). Similar relationsI2 .I8 which make use of other tabulated material have been developed to provide expressions for W2, Wa, etc., and to extend Eqs. (14) so that lower barrier cases or excited tor­sional states may be treated.

The Wo term in Eq. (12) may be absorbed into the vibrational energy. The contributions to the rotational energy from the WI term, to be added to those from rigid rotation and centrifugal distortion, are

WvA = - i~o cosO= - i~K

WVEo,= -i~o cos (0±27r/3) =t~K±OK (15)

where ~K= ~o cosO and OK= 3-!~o sinO. In rotational transitions v does not change and the A and E levels do not mix. As mentioned above, Eq. (15) gives no contribution for parallel transitions, but for the perpendicular transitions of Eq. (4) splittings appear:

VE±-VA= (~K-I-~K)±(OK_I-OK)' (16)

Relative to the average position of the three com­ponents,

VO=t(VA+VE+ +VEJ,

the A line is displaced by

VA-VO= -j(~K-I-~K)'

(17)

(18)

Before comparing Eqs. (16) and (18) with the ob­served fine structure of the transitions in Table III, two adjustments must be introduced to account for the asymmetry of the propylene oxide molecule.

When one of the tops is asymmetric, there are several ways in which a transformation to an internal

18 D. R. Herschbach, J. Chern. Phys. 27, 1420 (1957).

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768 D. R. HERSCHBACH AND J. D. SWALEN

axis system may be defined,19.20 but it is no longer possible to remove entirely the explicit coupling terms from the Hamiltonian. The general formulation used by Hecht and Dennison21 and by Lide and Mann22 is particularly suitable for our purpose, although most of the formulas developed by these authors refer to the more typical case in which the asymmetry splittings are large rather than negligible.

The complications are of three kinds. (i) The bound­ary conditions associated with the internal axis system are complicated. However, Eqs. (12)-(18) still hold in zeroth order, after making a change of scale which is required because the z axis of the internal axis system does not in general coincide with the axis of the symmetric group (CH3). If the internal axes defined by Hecht and Dennison are employed, it is only necessary to replace the factor I al1z in the defini­tion of fJ given in Eq. (13) by the quantity p, where

p2= LoCXolallo)2. (19)

The relations (14) are unaffected. (ii) The Hamiltonian remains diagonal in the symmetry index (J', but not in the torsional quantum number v. Hecht and Dennison have shown that the nondiagonal terms can be neg­lected for sufficiently high barriers (s>25, if v=O). (iii) The process of diagonalizing the Hamiltonian in K is more complicated than for an ordinary asym­metric rotor. In the high barrier case, the main difficulty comes from the additional terms nondiagonal in K which arise because the asymmetric rotor energy is not expressed in a principal axis system. If the contribution from asymmetry is small, as in the present application, the splitting of the energy levels may be conveniently developed as a perturbation series in powers of the parameter

(20)

(The analogous parameter involving Xy which would enter in the general case is small enough to omit here.) To second order in j3, the perturbation calculation23 only mixes states differing by unity in K, but this introduces a J dependence into W vu• The results can be reduced to the same form as Eqs. (15) if the trigo­nometric factors in ~K and OK are replaced as follows:

cosfJ----7cosfJ-tj32(A cosfJ-B sinfJ) (21a)

sinfJ----7sinfJ-t/32(A sinfJ+B cosfJ) (21b) ----

19 D. G. Burkhard and D. M. Dennison, Phys. Rev. 84, 408 (1951); D. G. Burkhard, Trans. Faraday Soc. 52, 1 (1956).

20 T. Itoh, J. Phys. Soc. Japan 11, 264 (1956). 21 K. T. Hecht and D. M. Dennison, J. Chern. Phys. 26 31·

(1957). ' 22 D. R. Lide, Jr., and D. E. Mann, J. Chern. Phys. 27, 868

(1957). . 2' This calculation is the same as that given by Lide and Mann III reference 22 except that in the present case it is convenient to use the ~ymmetric r?tor representation, .rat~er than the Wang representatIOn. Accordmgly, Eq. (21a), which IS an even function of ~, can be shown to be identical with the diagonal terms of their HI; :-vhereas Eq. (21b), which is odd in K, is now diagonal, although It corresponds to the off-diagonal terms of their HI.

where

A = [J(J+1) -K2J(1- cosfJl )

B=K sinfJl

with fJ l = (271/3)p. As Fig. 4 shows, the observed and calculated split­

tings for the transitions of Table III agree very closely over the entire range of K studied. As expected, the splittings of the i= 3 family, which comprises RP transitions, are opposite in sign to those of the other families, which consist of p R transitions; for simplicity in Fig. 4 the negative of the i= 3 splittings has been plotted.

The periodic K dependence of the splittings which would appear if propylene oxide were actually to consist of two coaxial symmetric tops is merely damped mildly by the terms involving j32 in Eqs. (21) .24 Al­though the j32 terms are in fact relatively large for the levels of high J and K, most of their contribution cancels out the transition frequencies, making the f32 terms less effective than the purely periodic functions of K, which enter in zeroth order. The vertical scale in Fig. 4 or the K periodicity is determined by the parameter p of Eq. (19). Both p and j3 depend solely on the molecular structure; the values used (given in Table X) were calculated from the structure given in I and no adjustment was necessary. The scale factor for the amplitude of the splittings is the quantity ~o of Eq. (14), which depends very strongly on the barrier height. The only structural parameter which enters into ~o is the reduced moment of inertia of the methyl group, defined in Eq. (3) of I; its value is determined when p and j3 are chosen. The curves in Fig. 4 were calculated for V3= 895 cm-l

; a change of more than 5 cm-l in the barrier height gives curves which differ significantly from those observed.

It should be pointed out that the Hamiltonians (1) used for the low K transitions are not accurate for high K. The treatment given in I employs a principal axis system fixed in the asymmetric part of the mole­cule, in order to retain simple boundary conditions and to simplify the K diagonalization. In such a fixed axis system, the coupling between internal and over-all rotation appears explicitly in the Hamiltonian, and the coupling terms become too large to be treated ade­quately by second order perturbation theory when the product pK is greater than about 0.25. It is then usually preferable to transform to an internal axis representation. The connection between the two treat­ments has been discussed elsewhere,1us and it has been shown that they will agree closely for low K transitions which are encountered in most applications. (Se~ Table VII.) In general, the internal axis representation

24 The {32 terms also cause the splittings to differ slightly for each T family because of the relation (4) between J K and T' however, this difference is too small to observe in the r~gion i~ which the families overlap.

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INTERNAL BARRIER OF C 3 R 6 0 FROM MICROWAVE SPECTRUM 769

is the simpler to use when the contributions from asymmetry are small, whereas the fixed axis representa­tion is often the more convenient when asymmetry splittings are large, and may be more easily extended to intermediate barrier cases.25

INTERMEDIATE K TRANSITIONS: "FORBIDDEN LINES"

We now consider transitions for which the asym­metry splittings and the torsional splittings are of the same order of magnitude. Table V lists several members of the r= -1 and r= - 2 families which represent this case.

TABLE V. Intermediate K transitions, ground torsional state.

VA (Mc) VB-VA YE*-JlA

Assign. Obs. Obs. Calc. Obs. Calc.

r=-1

83.-7 •• 20053.31 1.27 1.25 -4.05 -4.12 836-7'3 19598.51 -5.21 -5.24 a 0.13 10'6-9 •• 21774.11 2.63 2.56 -1.72 -1.66 10'7-954 21701.65 -3.20 -3.00 1.12 1.22 12.r 11 66 23680.51 3.45 3.43 -1.22 -1.15 12.8-116• 23670.17 -2.77 -2.67 1.95 1.91 1468-1377 25623.66 b 4.90 a 0.12 1469-1376 25622.24 b -3.28 1.42 1.49

r=-2

11'7-10 •• 34724.83 1.99 2.13 -2.48 -2.36 11'8-10 •• 34571.04 -0.61 -0.77 3.88 3.72 13.8-12.7 36565.29 3.14 3.15 -1.39 -1.45 13.9-1266 36542.27 -1.35 -1.65 3.16 2.96

• Not resolved. bE lines too weak to observe; see text.

For a rigid rotor, each T family separates into two branches in the region of intermediate K (see Fig. 1), with corresponding transitions 1I( I, K) separated by the sum ~ W K-l + ~ W K of the asymmetry spli ttings of the initial and final levels. The asymmetry doubling can be approximated by well-known formulas and is another characteristic property of a T family which may aid the assignment of high I lines.13 At a given K, ~ W K increases rapidly as I increases and as r decreases.

In the transitions of Table V the introduction of asymmetry splitting does not simply break up the triplet pattern observed in the high K limit into two A, E doublets of the kind observed for low K transi­tions. The odd order term OK present for the E torsional levels is responsible for a complete change of selection rules in passing from the limit of low K, in which ~W K»OK, to that of high K, where ~W K«OK. As illustrated in Fig. 5, lines which are forbidden in each of these limits appear in the intermediate K region.

2. W. J. Tabor, J. Chern. Phys. 27, 974 (1957).

25

I I I I I I I I I I I I

-I----l-----'.-L I I

: 116 I

5 10 15 FREQUENCY, Me

E A E" E"A E E A E"

1 I I I

T

I I I I 1

I I I I I I I I I

E'A E

E

A

E A

E

A A E

FIG. 5. The 12.--116 transitions in the ground torsional state for which the asymmetry splittings are comparable to the split~ tings due to internal rotation ("intermediate K" case). The E* transitions indicated by dashed lines would be "forbidden" in a rigid rotor, but gain intensity at the expense of the "allowed" E transitions, as the rotational wave functions become mixed by the coupling of internal and over-all rotation.

The odd order term OK does not transform according to the four-group, as the even order perturbations and the rigid rotor terms do. Let the pair of rigid rotor levels I K,J-K and I K,J-K+l be denoted by W + and W_, respectively, so that ~WK=W+-W_>O. If we choose the matrix representation which diagonalizes the asymmetric rotor energy, OK will appear on the antidiagonal and connect the levels W + and W _; if instead we use the symmetric rotor representation, OK is diagonal, but the asymmetry terms would be off-diagonal. When OK is comparable to ~W K neither of these representations is suitable for a perturbation treatment, and it is necessary to solve a quadratic secular equation.26 Thus the E levels corrected for the odd order perturbation are given by

W±'=!(W++W_)±K(~WK)2+4oK2J!. (22)

Equation (21a) remains a sufficient approximation for the even order part of the internal rotation perturba­tion. Hence the A levels are given by W ± - i~K, as before, and the E levels by W ±' +t~K. The asymmetry splitting of the pair of A levels is the same as for a

26 In the ground torsional state, whenever they are comparable both OK and ilWK are small enough to permit the neglect of higher order contributions, which arise from propagation of the perturbation via its influence on the neighboring rigid rotor levels.

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770 D. R. HERSCHBACH AND J. D. SWALEN

rigid rotor, while the spacing AW K' = W +' - W _' be­tween the E levels, as given by the radical in Eq. (22), is larger.

The normalized rotational wave functions corre­sponding to Eq. (22) are linear combinations of the original pair of asymmetric rotor functions,

1/!+' = 1/!+ cosw-1/!_ sinw

1/!-'=1/!+ sinw+1/!_ cosw (23)

where the extent of mixing is specified by the angle w, defined such that

tanw= 2oK /(AWK +AWK ') (24)

and

O;(W;(t1l'.

By use of Eqs. (23), the matrix elements of the dipole moment operator are easily obtained as linear com­binations of those for the rigid rotor. The selection rules are now, so to speak, the corresponding mixture of the rigid rotor rules.

In the case of perpendicular transitions, the mixing tends to interchange the role of the two dipole moment components which are perpendicular to the axis of quantization. (In the present notation, the "allowed" perpendicular transitions are ±f-t=F, and the "for­bidden" transitions are ±f-t±.) Thus, for the transi­tions of Table V, which arise from f-Ib, the "forbidden" transitions observed correspond to transitions of the type which would be made active by f-Ic in a rigid rotor.27 .28 To a good approximation,29 the distribution of the rigid rotor line strength Sr between the "for­bidden" lines (denoted by E*) and the "allowed" lines (denoted by E) can be given very simply in terms of the mixing angles for the initial and final levels:

SE.'= Sr[sin(wi+wI) J2

(25)

Equation (25) of course shows that the "allowed" transitions have almost all the intensity when both Wi and WI are near zero; this is the low K case repre­sented by Table II. The interesting point is that the "allowed" transitions disappear when both Wi and WI

are near 11'/4. This is the high K limit: all the E transi­tions of Table III are thus "forbidden" lines. Here both the initial and final states approach equal mix-

27 Forbidden lines of the type discussed here were first observed for methylamine by Nishikawa, Itoh, and Shimoda, J. Chern. Phys. 23, 1735 (1955).

28 Since propylene oxide has a small c axis dipole moment com­ponent, transitions such as those shown by the dashed lines in Fig. 5 are not strictly forbidden even in the rigid rotor limit. In the discussion above JJ., has been neglected because (JJ.eI JJ.b)2 is only about 0.05.

29 These formulas assume that the line strength is the same for two members of a K doublet, a good approximation when the asymmetry splitting is small, and neglect (see reference 28) the contribution from p.,.

tures, 1/!±'= 2-t(1/!+=F1/!_)' of the rigid rotor wave functions, and the derivation of Eqs. (25) shows that the intensity of the "allowed" transitions literally cancels out. Equal mixing exactly undoes the Wang transformation which is required to make the wave functions remain invariant under the four-group operations appropriate for a rigid asymmetric rotor, and effectively imposes instead the C3 symmetry of a hindered symmetric rotor.

We may remark that the situation is quite different for parallel transitions. The forbidden lines which are induced by the parallel component of the dipole moment do not correspond to any of the types per­mitted for a rigid rotor. In Eqs. (25) the sum of the mixing angles is replaced by their difference, (Wi-WI).

Consequently, the allowed transitions will appear regardless of the extent of mixing, whereas the for­bidden lines will vanish at both the limits Wi= w,= 0 and Wi=WI=1I'/4, and can never become more intense than the allowed ones. Moreover, further examination shows that in most parallel transitions the difference in the mixing of the initial and final states is not large enough to give appreciable intensity to the forbidden lines. This explains why such lines have not been observed in a number of cases for which the mixing itself is large.

Even in the case of perpendicular transitions, the changeover from negligible to equal mixing occurs very abruptly in the region of intermediate K, so that the "allowed" and "forbidden" lines will be of comparable intensity for only a few members of a T family.

The pattern shown in Fig. 5 is typical of those observed for the transitions of Table V. If propylene oxide were a rigid rotor there would appear two transi­tions (1257+--1166 and 1268+--1165) separated by AW6+ AW6= 10.19+0.07= 10.26 Mc. The two observed A lines are displaced from the rigid rotor predictions by centrifugal distortion, and by internal rotation, but in the present approximation their separation is not affected, and is observed to be 10.34 Me. For the E levels, the upper pair (125) is primarily split by asym­metry (W5= 11 °12'), while the lower pair (116) is primarily split by internal rotation (W6= 44 °33'). The forbidden lines (E*) are observed to be more intense than the allowed ones (E) in about the ratio 2.5 given by Eqs. (25). The calculated separation of the two allowed transitions is AW6'+AW/= 11.02+4.58= 15.60 Mc, in good agreement with the observed value 15.74 Mc. For the forbidden lines the calculated separa­tion is AW/-AW6'=6.46 Mc, and 6.35 Mc is ob­served.

In Table V, the observed E- A splittings agree well with those calculated from Eqs. (21) and (22) with the same values of Ao, p, and {3 which were used for the high K transitions. These parameters were also con­firmed independently by fitting them to the four differ­ent splittings provided by each group of six lines; the

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IN T ERN ALB A R R I E R 0 F C 3 H 60 FRO M M I C ROW A V ESP E C T RUM 771

corresponding values obtained for the barrier height (b) are all in the range Va= 89S±S em-I.

EXCITED TORSIONAL STATES

0

2 I I , ," -50 0 +50

°E"OA (0)

2E IA IE

2A OBSERVED

-350 :-300 -50 :0 + 50 PREDICTED

FREQUENCY, MEGACYCLES/SECOND

A rotational transition in an excited vibrational state is decreased in intensity by the Boltzmann factor of the vibrational energy and is displaced from the ground state line because the effective moments of inertia are somewhat different in the excited state [see Fig. 6(b)]. For the torsional vibration the satellite pattern is complicated by the splitting into A and E lines. As illustrated in Fig. 6(a), the splitting is greatly magni­fied in an excited torsional state, since the area lying under the potential barrier and above the vth excited torsional level is considerably less than that for the ground state. The tunneling in propylene oxide is increased by factors of about fifty and twelve hundred

FIG. 6. Torsional satellites of the 2,_1 0, transition: (a) ob­served and predicted positions of the satellite doublets; (b) dis­placement of the "center of gravity" of the doublets.

Assign

101-0 00

202-1 0, 300 -2 02 3,3 -2,2 3[2- 211

322 -22, 32,-2,0

2'2- 101

202 -1"

211 -20, 817- 808 9'8- 90'

22,-2,2 322-3'3

220-211

321 -312

826 - 817 11,.9-11'.'0 12'.10-12'.11

2,,-1 01 22,-211

322-3[2

Obs.

12614.92 Mc 25196.48 37710.31 36736.13 38910.96 37844.07 37979.00

35845.46 13855.30

12825.07 29220.78 34408.24

36198.17 37306.66

34055.65 33124.24 29289.95 32777 .28 35537.88

38021.11 34022.78 32956.15

A B C

• Calculated for V3= 895 cm-'. b Not resolved.

TABLE VI. Additional first excited state transitions.

Calc. Obs.

"a-type"

12614.75 18.60 25195.83 36.42 37710.04 54.24 36736.01 46.53 38910.37 63.39 37844.25 56.02 37978.47 57.22

"b-type"

35845.28 28.99 13854.94 39.11

12824.79 9.08 21219.65 159.57 34408.25 203.77

36198.00 9.52 37306.25 17.18

34056.34 -8.41 33124.44 -15.24 29290.22 3.29 32779.50 102.54 35538.13 152.35

"e-type"

38020.61 45.58 34022.67 -8.78 32956.55 -17.09

A level E level

18010.82 18016.15 6669.93 6669.96 5944.82 5945.05

b b b 0.90 b

25.28 -24.15

5.21 -3.83

5.15 b

-2.21

-25.14 -1.25

55.86 31.21 12.51 5.49 b

5.59 -25.15

b

0.25 0.55 0.89 1.27 0.32

25.10 -23.62

5.67 -3.68

5.27 -0.99 -3.99

-25.03 -1.21

56.21 32.26 12.68 4.49 0.88

5.82 -25.12 -0.41

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772 D. R. HERSCHBACH AND J. D. SWALEN

TABLE VII. First excited state splittings, J-:::<'3. in the first and second excited states, respectively. Molecules in the excited states are roughly equivalent to ground state molecules having barrier heights of about 480 cm-1 for v= 1 (still a "high barrier" case) and 225 cm-1 for v= 2 (an "intermediate barrier" problem).

Assign.

111-000

202-111

211-1 01 212-1 01

211-202

30a-212

312-30a

221-212

221-211 220-211

322-313

321-312

4.65 Me -3.83

5.59 5.21 5.15

-4.31 4.82

0.90

-25.14 -25.15

55.86 -1.25 31.21

25.28 -24.15

4.17 -3.74

6.11 4.94 5.82

-4.21 5.80

0.45

-24.65 -25.82

55.60 -0.39 31.13

24.71 -24.05

• Calculated for Va=895 em-I.

4.48 -3.68

5.82 5.67 5.27

-4.23 4.70

1.27

-25.03 -25.12

56.21 -1.21 32.26

25.10 -23.62

Calc. I

890 em-1

890 900 905 899 893 892

938

895 895 895 895 897

895 894 895±5

b Barrier required for exact fit to obs erved splitting.

The analysis of the splittings observed in the excited torsional states will be based on the model used for the ground state, in which the framework and the internal top are regarded as separately rigid. It is convenient to use the Hamiltonians (1) since low K transitions are to be considered; the matrix elements needed to apply Eqs. (1) to excited torsional states are now available in tablesY It should be emphasized that the semirigid model is used only to compare the rotational spectra of molecules in the A and E sublevels of a given torsional level v. For this purpose, the E- A splittings in Fig. 6(a) are regarded as simply superposed on the displacements of Fig. 6 (b). The displacements are of course not calculable from this model (which would require them to vanish), but are accounted for em­pirically by the use of different moments of inertia for each torsional state (see Table IX).

Assign

101-000

2 0;-1 01 3 03-2 02

31a-212

31,-211

322-2'1

32c 220

111-000 212-1 01 303-212

211-2 02

31,,3 03 41a-404

514-5 0,

61,-6 06

716-7 07

817-8 08

918-909

Obs.

12598.99 25165.43 37666.36 36700.24 38852.54 37796.16 37926.73

24032.19 35913.04 26918.07

12901.27 14087.32 15777.87 18053.85 20995.83 24663.52 29075.76 34201.56

Calc.

12598.83 25164.89 37665.85 36699.85 38851.84 37795.90 37926.54

24032.20 35913.29 26917.43

12901.29 14087.30 15777.72 18053.92 20996.10 24663.72 29076.39 34201.97

A B C

• Calculated for V,=895 em-I.

TABLE VIII. Second excited state transitions.

Calc.

o o o 0.07 0.21 0.59 0.59

0.30 0.35

-0.34

0.53 0.73 0.93 1.17 1.37 1.51 1.48 1.14

Obs.

36.50 71.90

106.41 92.74

126.20 109.02 114.36

58.24 119.36

39.12 71.75

112.43 167.67 237.72 318.19 405.58

A level

18091.93 Me 6658.26 5940.57

Obs.

"a type"

12595.01 25156.18 37651.04 36774.08 38757.75 37853.03 37855.93

"b type"

35566.15 27240.85

14084.31 15741.10 18015.71 20963.10 24653.60 29082.36 34254.08

Calc.

12595.03 25156.15 37653.28 36775.85 38759.20 37852.95 37857.84

Calc.

-1.02 -2.60 -1.92 88.63

-88.10 64.80

-63.20

Obs.

3.98 9.25

15.32 -73.84

94.81 -56.87

70.80

Calc.

3.80 8.74

12.57 -76.00

92.64 -57.05

68.70

Va

Calc.

889 em-1

889 877 900 892 895 891

35579.48 -194.72 346.89 333.80 890 27228.56 188.75 -322.78 -311.13 892

14080.14 15743.81 18015.56 20967.98 24653.50 29086.57 34245.55

E level

17960.65 6658.20 5937.85

111. 56 74.97 55.78 45.98 39.05 31.94 38.53

3.01 36.77 38.16 32.73

9.92 -6.60

-52.52

7.16 891 33.91 904 38.36 895 28.12 915 10.22 895

-10.18 915 -43.58 922

897±1O

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IN T ERN ALB A R R I E R 0 F C 3 H 60 FRO M M Ie ROW A V ESP E C T RUM 773

First Excited State

Table VI gives additional data for the first excited torsional state. This supplements Table VI of Part I. No lines due to the smaller components of the dipole moment, /Joa and /Joe, had been identified in I, but the new data required only slight adjustment of the rota­tional constants. As before, the observed 1A lines agree satisfactorily with the pseudo-rigid rotor calculations from Eq. (1a). For some of the h transitions it is . ' necessary to mclude the first order terms in Eq. (1b). For the K = 0 levels, the first order terms were negligible. For the K = 1 levels, 2QpllPz contributed a small cor­r~c.tion of 1 Mc or less; this was added to the pseudo­ngld rotor terms as a perturbation,

± (2QPll)2/ (B- C) J(J+ 1) (26)

where the plus sign goes with the level having the larger value of T= K_1- K+1• For the K = 2 levels of low J, however, the asymmetry splitting is not large compared to the first order term 2QpllPz which con­nects them. A continued fraction procedure designed for this case is given in the Appendix; to a good ap­proximation this reduces in the present application to Eq. (22) with OK replaced by 2QpllK. The first order term caused quite large shifts from the K = 2 pseudo­rigid rotor levels, about 40 Mc for J = 2 and about 16 Mc for J = 3. The excellent agreement found for the eight h transitions involving K = 2 levels is a partic­ularly sensitive confirmation of the barrier height and structural parameters of Table X.

Table VII, which includes some data from paper I, compares the treatment used here with results obtained by the methods of Hecht and Dennison. The close agreement illustrates the general conclusion that the two approaches are practically equivalent for low K transitions. The barrier height determined from the average of the v= 1 data, a total of thirty doublet splittings, is 894±8 cm-l •

Second Excited State

The transitions assigned to the second excited torsional state are listed in Table VIII. Preliminary calculations were an important aid in the initial search since t?e v= 2 lines are rather weak and in many case~ are shifted far from the corresponding ground state lines, as in the example of Fig. 6(a). To predict a set of mean rotational constants for the second excited state, the difference between those observed for the ground and first excited state was extrapolated in the manner described under Eq. (27) below. These mean constants, as defined in Eq. (3), enabled the displace­ments illustrated in Fig. 6(b) to be estimated. The constants for the 2A and 2E states were then predicted by adding to the mean rotational constants the second order coupling computed from Eq. (lOa) of I with the parameters of Table X.

Since the analysis of the 2A lines could be based on

Eq .. (1 a) , a pseudo-rigid rotor Hamiltonian, they were studled first, and were easily identified. The initial es~in:ates of the rotational constants proved to be wlthm a few megacycles of the final values which were adopted. I~ the final analysis, two corrections to Eq. (la) were mcluded. (These were too small to mention for the ground and first excited states.) The corrections a:e a slight rotation of the principal axis system, as glven by Eq. (lOb) of I, and a fourth order contribu­~ion from the coupling perturbation (denoted by v A (4)

m Table VIII), which is analogous in form to a cen­trifugal distortion correction.12.25 These corrections were small, but improved the agreement with the observed lines, without introducing any new parameters.

The analysis of the 2E transitions is awkward because the first order coupling term 2QP22Pz in Eq. (1b) is t?~ large to be treated as a perturbation of the pseudo­ngld rotor terms. It contributes to the K = 1 levels about 400, 200, and 100 Mc for J = 1,2,3, respectively, and to the K = 2 levels about 1300 Mc for J = 2 and 3. The 2NP22Px term gives the main contribution to the K=O levels of low J, which is only 5 Mc or less· it is almost negligible for the K>O levels. As J incr~ases, both the P z and P x contributions tend to decrease except for the K = 0 levels, for which the P z contribu~ tion increases rapidly. (2M P22P y is still negligible for v= 2, as it was for v= 0 and v= 1.) The first order con­tributions (denoted by II E(l) in Table VIII) were eval­uated by numerical methods which are described in the Appendix.

Lines found in the neighborhood of the initial predictions were accepted as candidates for assignment as 2E transitions if they exhibited the correct relative intensity and qualitative Stark effects. (In most cases, the S~a.rk effect was barel:y discernable; the mixing of the ngld rotor states whlCh occurs in the 2E levels introduces large terms linear in the Stark field and these are difficult to observe for weak lines.) From each of the observed frequencies, the calculated v E(l)

contribution was subtracted. The usual rigid rotor techniques were then applied to compare the re­mainders v- v E(l) with the pseudo-rigid rotor terms of Eq. (1b). After deciding on a trial assignment and adjusting the rotational constants accordingly, the VE(l)

were recalculated, and this procedure was iterated. The corrections to Eq. (1b) which were added for the final analysis are similar to those mentioned for the 2A state, and include the rotation of axes, the fourth orde: coupling terms, with v E(4) = - tv A (4) to a good ap­proxlmatlOn, and a small third order contribution12

involving P z3. The latter is qualitatively similar to the first order P z term and has been included in v E(1).

. The parallel transitions could be readily assigned, smce for them most of the internal rotation contribu­tion cancels out, particularly for K = o. Use was made of the sum of the 303~202 and 202~lo1 parallel transitions in identifying the 303~ 212 and 212~ 101 perpendicular

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774 D. R. HERSCHBACH AND J. D. SWALEN

transitions; because of the inaccuracy in the calculated position of the 212 level, there were several possible assignments for these two transitions, but only one choice gave the correct sum.

The pair of J=3+-2 parallel transitions with K=2, which are separated by 130 Mc in the 2A state, nearly coincide in the 2E state. The mixing of the rigid rotor functions which is induced in the 2E state by the large first order coupling terms tends to remove the effect of the molecular asymmetry. This illustrates for parallel transitions the same situation which was encountered for the intermediate K and high K perpendicular transitions in the ground torsional state.

In Table VIII, the observed and calculated 2A transitions agree very well. The discrepancies found for some of the 2E transitions are not much greater than those found in the ground and first excited state data, when compared with the increased magnitude of the internal rotation contributions. An E state might be expected to be more sensitive to perturbations which were neglected in the above treatment than is an A state (for which any terms of odd order vanish by symmetry). In the Q branch series, it will be noticed that the differences between the observed and calcu­lated 2E lines show trends with J and the parity of J; this suggests that the approximations made in the functional form of the Hamiltonian are responsible. When the structural parameters were varied from the values of Table X, it was found that a somewhat im­proved fit could be obtained for either the parallel or the perpendicular 2E transitions, but not for both; and the trends in the Q branch series persisted. Therefore the structural parameters of Table X have been re­tained in the barrier analysis. The barrier height found is Va=897±lOcm-1•

DISCUSSION

The upper section of Table IX compares the para­meters of Eqs. (1) for the A and E levels. The large increase of the perturbation terms with v makes the three torsional states quite different problems from the point of view of numerical calculation. Furthermore, the torsional states differ greatly in their sensitivity to perturbations which have been omitted in the present treatment, including the details of the barrier shape and vibrational interactions with internal rotation. Our aim in studying the excited states was to test whether the approximations adopted in the analysis would cause the different torsional states to have different apparent barrier heights. The excellent agree­ment found in the barrier heights determined for the three torsional states demonstrates that this is fortu­nately not the case.

The large magnification of the splittings which occurs in excited torsional states offers a means to determine the barrier height in many molecules for which the barrier is high enough to make the splittings

TABLE IX. Comparison of parameters for the torsional states.

Quantity v=O v=1 v=2

Differences of rotational constants for A and E levels

0.10 (0.002) (0.005) 0.51

(0.08)

-5.33 -0.03 -0.23

-28.0 -4.7

131.28 Mc 0.06 2.72

670 110

Variation with v of mean rotational constants'

9.35 12.17 6.51

19.31 Mc 23.90 12.72

Torsional energy parameters (for V3=895 cm- I )

4.11 105.2

-227.9 308.4

• These are defined by A,= ~ (A ,A +2A,E), etc.

5480 Mc 495.7 cm- I

unresolvably small in the ground torsional state. As discussed in I, the natural unit for characterizing the splittings is s= 4 Va/9F, where F is the reciprocal of the reduced moment of inertia for internal rotation. The ground state splittings will be too small to be observed with a conventional spectrometer if s is larger than about 75, whereas the upper limit is increased to about s= 130 for v= 1 and to s= 180 for v= 2. (The limits in particular cases may be considerably lower, e.g., about s= 55 for v= ° if only parallel transitions are accessible.) In terms of Va, the upper limits would be about 950, 1600, and 2300 cm-1 for v= 0, 1, 2 respectively, if the value taken for F corresponds to a CHa group which rotates against a framework having heavy off-axis atoms (e.g. CHaCH2F or propylene oxide); the value of F, and therefore the limits for Va, can be much larger if the off-axis atoms are light (e.g. CHaOH), and will be much smaller if the top is heavy (e.g. CFaCH2F) .

A perturbation treatment of the V6 correction to the barrier shape in Eq. (2) has been carried out and the results tabulatedll ; the parameters in the Hamiltonians (1) become power series in (V6/Va) , but their form is unaffected. It is found that Va and Va cannot be separately determined from splittings within a given torsional state v, so that it is necessary to compare data for different torsional states. Inspection of the tablesll shows that for propylene oxide the Va contribution to the splittings increases by a factor of 200 over the range v=O, 1, 2; however, since the Va contribution increases by a factor of 1200, the relative importance of Va actually decreases in the excited states. It is clear that V6 must be very small in propylene oxide, since the three torsional states gave the same value for Va when V6= ° was assumed. A detailed analysis yields only a nominal upper bound, Va < 10 cm-I, because of the experimental uncertainties and the lack of precise

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INTERNAL BARRIER OF C a H 6 0 FROM MICROWAVE SPECTRUM 775

TABLE X. Parameters used in the analysis of internal rotation.-

Ia=3.194 amu A2 F=5.856 cm- l

Aa=0.8921 Ab = 0.0046 Ac=0.4518 p=0.1030 /3=0.1644

Ao=4.1l Mc (ground torsional state) Va=895±25 cm-'=2560±70 cal/mole

- In reference 21 the quantities la, p, and F are denoted by C2, (C';C) *, and (!fi2) (C/C,C2) *, respectively.

information about how vibrational interactions change with v.

Tables VI and VIII include the displacements ~vo of the centroid of the VA and VE lines from the ground state line [see Fig. 6(b)]; it is seen that these bear no simple relation to the E- A spliUings which were used to determine the barrier height. In principle, informa­tion about the barrier height could be extracted from the ~vo by use of a more general Hamiltonian,so which includes terms for the vibrational interactions with internal and over-all rotation. Some of these terms may be responsible for the discrepancies found in the 2E spectra. However, in the case of an asymmetric rotor the large number of new parameters which must be evaluated is prohibitive.

We have tried the following expression as a semi­empirical formula for the change in rotational constant with torsional state, which is one effect of the vibra­tional interactions.

(27)

where Ev~ is the torsional energy (Mathieu eigenvalueS) in the state specified by vu, and XV~= (T- V>v~ is the difference in the average torsional kinetic and potential energy (diagonal matrix elementsll). The constants B t, /317 /32 are independent of the torsional state, and represent averages over the other vibrations. In the limit of a very high barrier, where Ev~->hw(v+!), Xv~->O, as for a harmonic oscillator, Eq. (27) reduces to the well-known formula for an ordinary vibration; and for the K = 0 transitions of a symmetric top, Eq. (27) can be rearranged to give Kivelson's formula.16

Equation (27) was used with (32= 0 to obtain initial estimates for each of the three mean rotational constants of the v= 2 state from those observed for v= 0 and v= 1, and the results were within 1-3 Mc of the observed values in Table IX. With the values of (31 and (32 deter­mined from the observed mean rotational constants, the vibrational contributions to the differences BvE- BvA ,

etc., were also estimated. These contributions were only a few tenths of a megacycle for v= 2, and de­creased rapidly with V; their inclusion would not alter the derived barrier height.

30 D. Kivelson, J. Chern. Phys. 23, 2~30, 2.2~6 (1955); 27, 980 (1957); K. T. Hecht and D. M. Denmson, tbtd., 26, 48 (1957); P. R. Swan, Jr., and M. W. P. Strandberg, J. Mol. Spectroscopy 1,333 (1957).

The constants which were used in the analysis fo internal rotation are collected in Table X. The struc­tural parameters have been calculated fron: ~able III of I, and are experimentally confirmed wlthm a few percent. The estimated uncertainty in the barrier h~i!?ht has been increased somewhat beyond the preCIsIOn indicated by analysis of the observed spliuings and the structural uncertainties, in an attempt to allow for the fact that some of the idealizations of the molecular model used may not affect the nature of the fit to the splittings, but will change the derived barrier height. An accuracy of ±3% is suggested as a reasonable estimate by experience with other I?olecule.s, pa:­ticularly the comparison of results for dIfferent IsotOPIC species.

ACKNOWLEDGMENTS

We have enjoyed many discussions with Professor E. B. Wilson, Jr. We are very much indebted to D~. C. C. Costain for the use of his spectrometer and for hIS assistance and advice. We thank Dr. G. Herzberg and Dr. A. E. Douglas for helpful criticism of the manu­script. One of us (D. H.) wishes to express his ap­preciation for the hospitality extended by the National Research Council during his visit.

APPENDIX

In Eq. (lb), when the first order coupling terms,

-2(NPx+M Py+QPz)pvv,

are not small compared with the energy differences between the pseudo-rigid rotor levels which they con­nect perturbation techniques become inconvenient. Tw~ methods of more general applicability will be described.

Let the Hamiltonian matrix of Eq. (lb) be set up in the appropriate symmetric rotor representation,31 with the most nearly unique principal axis of the molecule taken as the axis of quantization (z axis). It will be advantageous, however, to omit the usual Wang transformation, since its application would remove the P terms from the diagonal. The elements of the re­d~ced energy matrix of KHC31 (prolate representation) are replaced by

EKK=K2+qK

EK,K±I=!(m±in)[1(J+1)-K(K±1)]l (Al)

EK,K±2=bP(J, K±1)

where q= (2QPvv)/[AvE-!(BvE+CVE)], and nand m are defined by replacing Q by N or M; the other nota­tion is that of KHC.31 The matrix (Al) is of order 21+1, as K takes on the values I, 1-1, ... , 0, "',-1.

The p. term is often much more important than the P" and Py terms, since the rigid rotor levels corre-

al King, Hainer, and Cross, J. Chern. Phys. 11, 27 (1943).

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776 D. R. HERSCHBACH AND J. D. SWALEN

sponding to ±K may be nearly degenerate, whereas the K, K±l levels are usually well separated; further­more, in many molecules the principal axis nearest the symmetry axis of the internal rotor group is the z axis, and then Q> N or M. If the P x and P y terms can be neglected or added later as perturbations, the p. term can be conveniently treated by a continued fraction method. Without the (K I K±l) elements, odd and even K are not connected, so the matrix (AI) factors into two blocks, of order J and J + 1. In the secular determinant for one of these blocks,

(A2)

suppose interest lies in the eigenvalue X associated with a particular diagonal element EKK. The rows and columns having I K' I> I K I may be eliminated from (A2) by the well-known procedure of "outer pivotal condensation" which is used in constructing continued fractions.32 (In the present application, this condensa­tion ordinarily must not be carried past the boundary I K' I = I K I, in order to avoid small denominators.) This treatment reduces (A2) to a determinant of order I K 1+ 1, and replaces the corner elements EKK and E_K._K by continued fractions, eK and e_K. These may be defined by a recursion relation,

eK=EKK- (EK.K+2)2(eK+2-X)-l (A3)

from which the continued fraction is generated by successive substitutions on the right-hand side; e_K is obtained by replacing all subscripts in (A3) by their negatives.

For K = 0, the secular equation reduces to

For K = ± 1, a quadratic secular equation is obtained whose solutions are

X= Hel+e-l ) ±![(e1-e_l)2+4El._12J!. (AS)

When evaluated with a trial value of X, the right-hand side of (A4) or (AS) provides an improved approxi-

32 M. W. P. Strandberg, Microwave Spectroscopy (Methuen and Company, Ltd., London, 1954), p. 13.

mation; an iteration converges very rapidly to the eigenvalue. For J = 1 or 2, Eq. (AS) becomes the formula given by Kilb and Pierce.33 For K=±2, a cubic secular determinant is obtained. However, the K = 0 row and column may be removed by an "inner condensation;" this yields a quadratic secular deter­minant in which the off-diagonal elements are also continued fractions. The solutions thus have the form of (AS), but with the e±l replaced by

8±2- Eo'} (Eoo - X)-l

and E l .-l replaced by E022(Eoo-X)-l. For larger values of K, the above treatment naturally becomes more complicated (e.g., quartic secular equations are ob­tained for K = 3 or 4) and the factorization into quadratic blocks is best carried out numerically. The method may be applied to any odd function of p., by adding the same function of K to EKK in (AI); standard continued fraction methods are availablel 4.3l

for even functions of p •. Inclusion of the Px and P y terms in (AI) spoils the

odd and even K factoring, and makes the continued fraction approach awkward to use. When the P x or P y

terms are more important than the p. term, it is con­venient to transform (At) to the asymmetric rotor representation. This situation often arises for the K = 0 levels; and for them perturbation calculations in the symmetric rotor basis tend to converge slowly.lO In the transformed Hamiltonian matrix, 1'-lHvE T, the diagonal elements represent the pseudo-rigid rotor terms of Eq. (lb) and thus may be obtained from tables. Since the only nondiagonal elements comprise transformed terms in Q, N, and M, the diagonalization process is simplified. Except for J = 1 or 2, it is neces­sary to carry out the operations numerically. The transformation matrices T have recently been tabu­lated at intervals of 0.1 in K (Ray's asymmetry param­eterl ) by Schwendeman and Laurie.34

33 R. W. Kilb and L. Pierce, J. Chern. Phys. 27, 108 (1957). 34 R. H. Schwendeman and V. W. Laurie, unpublished work,

Harvard University, 1957. Copies of a deck of IBM cards con­taining these tables may be obtained from Professor E. Bright Wilson, Jr.

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