Theory of torsion-vibration-rotation interaction in ethane and analysis of the band ν11 + ν4

JOURNAL

OF MOLECULAR

SPECTROSCOPY

49, 1-17 (1974)

Interaction Theory of Torsion-Vibration-Rotation and Analysis of the Band vll + vi l

in Ethane

JOEL STJSSKIND 2 Molecdar

Spectroscopy Laboratory, Sclzool of Clremistry, liniz~ersily oj Minnesota, Minneapolis, 554.55 and Department of Physics, Florida State University, Tallahassee, Florida 32306

Minn.

The theory of torsion vibration-rotation interaction as applied to high barrier to internal rotation ethanelike molecules is discussed. It is shown that the effects of internal rotation can be described by one parameter X which introduces an oscillatory behavior in subband origins. Also, the effects of xy Coriolis interaction between A and E vibrational states on the energy levels and effective band constants of the E state are discussed. The band VII+ ~1 of ethane has been measured with high resolution and analyzed. The barrier to internal rotation in Y,I has been found to be 3330 & 23 cal/mole.

I. INTRODUCTION

The barrier to internal rotation in ethane and its vibrational dependence has been of interest for some time but has not been easily accessible experimentally. The molecule has no permanent dipole moment and hence no microwave spectrum. Moreover, the torsional vibration is forbidden in the first approximation in both infrared and Raman spectra. The torsion is made weakly infrared active via xy Coriolis interaction. Weiss and Leroi (I), using high pressure, high path length conditions, have observed the torsional fundamental and first hot band in gaseous &He, CHSCDS, and C2D6. From these, they calculated a barrier of 2928 f 2.5 Cal/mole (1024 f 9 cm-‘) to internal rotation in ethane. There is some uncertainty as to the vibrational state from which these torsional transitions arise. The consistency of the barriers obtained from each molecule makes it seem quite likely that these transitions do arise from the ground state. The barrier to internal rotation in ethane in its ground state and excited vibrational states can also be calculated from the effects of torsional splittings in the high resolution infrared spectra of ethane. These effects are most prominent in combination and difference bands involving changes in torsional quantum number. The effects of torsion on fundamentals are too small to be observed. They may become significant in certain hot bands arising from excited torsional states but not involving changes in the torsional quantum number. In general, the strongest bands involving changes in torsional quantum number will be combination and difference bands involving one quantum of an E, vibration and one quantum of the torsion. From these, the barrier to internal rotation in the vibrational state with the excited torsion can be calculated. 1This work was supported in part by Air Force Grant #AFOSR-570-67. 2 Present Address: Goddard Institute for Space Studies, 2880 Broadway, 1

Copyright 0 1974by Academic Press. Inc. All rights of reproduction in any form reserved.

Sew York, N. Y. 10025.

2

SUSSKIND Table

Species Designation

D

A A

A A E

“8 3 vlu 31 Y12

E E E E E

Species

1c’ 1% 1g 1” 2u 2” ” ” u g g i:

Approximate Frequency

3tl

A

A

I

-1 in (‘111

A A A A A A E E E

1s 1s 1s 3s 4s 4s Id Id Id

EZd E E

2d 2d

2!W.

7

1::w

4

494.8 280 2895.8 131% 2 2995.5 1472 821. G 2s8.7 1468.1 1190

There has been much theoretical discussion of torsion-vibration-rotation in ethanelike molecules molecules in the past (Z-6). For the most part, the discussion has been limited to the very high or very low barrier cases. In the limit of a very high barrier case, torsion looks like any other vibration and has no discernable effects on the spectrum. Most of the discussion (46) has dealt with the low barrier case in which internal rotation has large effects even on a medium resolution spectrum as exhibited by dimethyl acetylene (7). Kirtman (3) claims to discuss the intermediate case. According to his treatment however, ethane is put into the high barrier limit and Kirtman does not consider the appropriate factors affecting the high resolution infrared spectra. In this paper, we will consider the effects of torsion-vibration-rotation interaction in ethane and high barrier ethanelike molecules. The theory will then be applied to the analysis of the high resolution infrared spectra of the ~11+ v4 band of ethane in order to get the barrier to internal rotation in the excited state ~11of ethane. II. THEORY We will use the formalism developed by torsionally nonrigid ethanelike molecules that one must use the full group of feasible atomic nuclei, G36+,to completely take into rotation in these molecules. Table I shows the frequencies (8) of the symmetries as described in both the usual cule and the full Gs~+group which contains

Hougen (4) and Bunker (5) for low-barrier in dealing with ethane. They have shown permutation-inversion operators of identical account the effects of torsion, vibration, and fundamental

vibrations

of ethane and their

D-dpoint group of the rigid staggered molea subgroup isomorphic to Dad.Note that E

TOR-VI&ROT-INTERACTION

IN ETHANE

3

levels generally have a close lying A level of like parity which can interact via xy Coriolis interaction. We will treat this explicitly later. Hougen and Bunker have shown that torsional Coriolis interaction in low barrier molecules such as dimethyl acetylene can cause considerable mixing of nearly degenerate ISld and Ezd vibrations forming essentially quadruply degenerate Gd type vibrations. In high barrier ethane, however, torsional Coriolis interaction is essentially off-diagonal in the torsional quantum number y4 and such coupling does not exist. The Hamiltonian we will use is essentially identical to that developed in other works {see, e.g., [7. Eq. (l)]). H = A[(J’z - Pz)’ + (Py - P,)“] + B[(Pz

- P,)” + (P, - P,J*I + k’(r)

+ I$ (J’k* + Xk~k’)j+ C Xk(~)qr~’+ higher terms k = Rut f

Htor + Hvib + HI, + Hwr f

(1)

H’

where H,ot = AP,2 + B[P,2 + P,21, Htor = APT2+ Hvib

=

c

(pk2

V(y) = APv2+ +

(V3/2)(1

- cos6y),

hkqk2),

k

(2)

Ht, = q h,&)qi? = q 1’,1” H,,, = - 2[AP,p,

Sk2 COS 6-r,

+ B(PzPz + PuP,)l - 2AP,P,

where PQ = C Tzj”(PiQj- YiPj),

ii

and H’ includes y dependent and y independent anharmonic potential terms and the dependence of the inverse inertial tensor on the vibrational coordinates. The problem is set up in the direct product representation of eigenfunctions of Hvit, + Hrot + Ht,,. To first order the energy of any state jJk’z’(~a) is

where ~z’,,~(J, R, 4) includes first order vibration Coriolis interaction. The torsional wave functions, #,I< are eigenfunctions of H,,, = .4Py2 + (L’/2)(1

-

cos 67).

Their eigenvalues are easily obtained with the aid of a computer. Hto, is set up in the free rotor basis set, cos my and sin my, and factors into blocks with m = 0, fl, f2, and 3 mod 6 and further into cosine and sine blocks. These can be classified under the

SUSSKIND

4

EIGENVALUESOFTHETORSlONALHAiWILTONIAN C2D6 v3=

State Symmetry

-900

cm1

v3=-1024 cIn-1

-1 v3=-1100 cnl

v3=-1024 m-1

Energy

split

Energy

Split

Energy

A 1s

140.7631

-0.0074

150.6189

-."O%

156.3201

-0.0023

107.8279

0.0000

to,1 E 1 E3d

140.1674

-0.0037

150.6207

-.0018

156.3219

-0.0011

107.8279

0.0000

b,o

A1

Energy

Spilt

sput

'ro,2 E2

%

140.7747

+O.OWG

150.6243

1.0018

156.3242

+0.0012

107.8279

0.0000

io.3 B2

A

140.7784

+0.0073

150.6261

+.0038

156.3254

+0.0024

107.8279

".M,OO

til,3 =1

A 3s

401.9964

-0.2786

437.8503

-.149O

455.0933

-0.1023

316.8856 -0.0009

8 E2 1,2

E3s

408.4344

-0.1396

437.9248

-.0745

455.1444

-0.0512

316.8861 -0.0"04

"1.1 E:

E3d

408.4137

+a.1397

438.0739

t.0144

455.2469

10.0513

316.8870 'O.OUU5

$

* ld

408.5541

+0.2801

438.1481

+.14!a

455.2982

'0.1026

316.8874 to.0009

639.847

-4.018

691.940

-2.445

721.141

-1.795

511. 9121 -0.0247

-0. 910

511.9244 -0.0124

o A2

$2.0 Al

*

$,I

3d

16

E1

E3d

641.190

-2.075

693.139

-1.246

722.626

"2.2 E2

E3s

645.%9

+2.014

695.631

tl.246

124.446

-10. 910

511.9492 to.0124

$3

B2

A3d

648.170

13.306

696.928

+2.543

725.383

Cl.847

511.9615 to.0247

$3.3 =I

A3S

813.725

892.317

Y37.611

690.4192 -0.3971

$92

E2

E39

823.370

899.796

943.965

690.6766 -0.1997

$1

E1

E3d

847.139

918.067

958.912

691.0750 +a.1997

$3,0 *2

Aid

866.599

930.458

968.395

691.2160 +0.4007

Cav group of Eltor (which is isomorphic to a subgroup of GBB~)as AI, EI, Es, B1 for the cosine functions and AZ, El, Ez, Bz for the sine functions. Table II shows eigenvalues of the first torsional states for a variety of barrier heights in the region of that expected for ethane. The ground state A value of 2.671 cm-’ (9) is used for the reduced mass in Ht,,. The last column shows eigenvalues for CzDs with a barrier corresponding to its ground vibrational state. The ground state A value of 1.342 cm-’ (10) was used for C2Ds.

It is seen that the energy levels come in groups of four with small torsional splittings between them. These groups are numbered by the torsional quantum number r and the sublevels are further classified by u, a symmetry number given by m mod 6. The symmetries of the eigenfunctions, as classified under the customary Cc0 subgroup and the full G& group of the molecule are shown. Vnlike the rest of the table, the symmetry of +r, is dependent on the sign of V3. According to its definition, y = 0 corresponds to eclipsed ethane while y = 2?r/12 corresponds to a staggered configuration. Therefore, we expect V3 negative for ethane with a staggered minimum. For positive Vat all BI and B2 (Aw and Ass) states must be interchanged. Inspection of the energies E~,(T, a) within a group of same T, as shown in Table II, shows that for low 7, the splittings are of the form x, 2x, x. If we define Et,,(r) as the

TOR-VI&ROT-INTERACTION Table

IN ETHANE

5

III

TORSION-VIBRATION-ROTATIOK

SELECTlOP*

RULES

average energy of Etor(r, a), then for low r, &or(r,

g) = Et~x(r) +

Esplit.(~,

u)

=

ztor(T)

+

NP~-X~

(1)

where N,, equals - 2, - 1, 1, or 2 depending on u and on whether T is even or odd. X, is a splitting parameter depending on barrier heights and torsional quantum number. This splitting parameter is very barrier dependent. We now write e(J,

K, v, 4, 7, U) = Lt(J,

R, 4) +

gvib(u,

l,

7)

+

iv7,X7,0

(5)

I&(7, u) + ztor(7), and X,,, is the splitting where &b(v, 4, 7) = &ib,tor(v, p, 7, u) parameter for torsional state 7 with a barrier given by that in vibrational state V. According to the zeroth order Hamiltonian in Eq. (l), there is no coupling of torsion and rotation. This approximation is actually quite good and we will not discuss torsion rotation operators in H’. Nevertheless, as was pointed out by Howard (3), there is a restriction between the allowed quantum numbers K and m (or u) such that K - u must be even. This is a result of the necessity for single-valuedness of the overall wavefunction upon rotation of one top by 360”. This fact is demonstrated by the nuclear spin statistics for ethane as calculated by Wilson (11) and Bunker (12). These show that for a given JK rotational level, only two of the four torsional sublevels for a given r are populated. Except for the special case K = 0, the relative population of these levels is 2: 1 or 4: 1 if K is a multiple of three. Moreover, for K even or odd, dijerent torsional levels, having different energies, are populated. The dipole selection rules for torsion-vibration transitions have been determined by Bunker (5b) for the low barrier case. The high barrier case differs only in the use of different torsional quantum numbers and the absence of G type vibrations. Table III contains Bunkers Table IX in which he gives the low barrier torsion-vibration selection

SUSSKIND

23f4&lO C5,7J 1 0,6,12

b

0

FIG. 1. Allowed

transitions

between

torsional

c

sublevels.

Solid line is even parity

transition.

Dashed

is odd.

rules between two vibrational states of symmetries I’, and IU,. In the absence of G type normal modes, Ta @ TV*cannot contain Ebd, Ed*, G,, or Gd and these transitions are not considered in the remainder of the table. The third column gives the torsional function contain A la, the totally symmetric of appropriate symmetry to have TV @ Itors 8 +. The fourth column gives the values of AT for which the torsional representation of GOB matrix elements are largest. For a positive 113,(eclipsed ethane), A7 for cos 37 and sin 3a interchange. The fifth column gives the parity of the torsional operator under the operation y --+ y 4 2?r/6. Even parity operators are diagonal in (J while odd parity operators take g into f (u + 3) mod 6. The last column gives examples of possible bands of appropriate symmetry. Figure 1 shows examples of allowed transitions between (a) 2 even torsional levels (as exemplified by a fundamental) (b) an even and an odd torsional level (a torsional combination band), and (c) 2 odd torsional levels (a torsionally excited hot band). The solid lines indicate even parity transitions, the dashed lines, odd parity transitions. Also shown are those values of K having the greater nuclear spin factors in each torsional sublevel, assuming a totally symmetric vibrational state, and the values of M,,X,,, for each state. In the absence of interactions, the torsional contribution to the frequency of a transition from 1JKvh) to / J’K’v’4’~‘d) is then given by N,r,JX,j,f - N,,X,,. As shown in the figure, U’ = (r for even parity transitions and f (u + 3) mod 6 for odd parity transitions. Then, as seen from Table II and shown in Fig. 1, N,, = f N,J,J for an allowed transition with an odd AT and an odd parity each introducing factors of - 1. The frequency of a transition from 1JKv&u) to ]J’K’v’C’r’u’) is then r&pole

F 1J'K'v't'~'a') / JKvPru)

= ABrot + AEvib + N,rrr(K)X

(6)

where X = X,,,, F X,, and it is understood that u depends on K. Table IV shows Nr,(K) X and No.(K) X for the strong and weak components of transitions arising from K in a totally symmetric vibrational state. Ml,(K) is applicable to a combination band such as err + v4 or a hot band such as vg + v4 - v4. For a fundamental, or difference band such as ~11- v4, one must use MO,@).

TOR-VIB-ROT-INTERACTION

1N ETHANE

Table I”

TORSIONAL CONTRIBUTION

NL_6) i

s 2X

x

TO SUBBAND ORIGINS

NU_(W

X

s

w

-X

-2X

X

-2X

-X

w

-2X

s

2x

2X

-X

x 2X

-2X

12X

X

2X

-X

X

x -X

-2X

-X -2X 2X

-X

Thus, every transition is doubled into a strong and weak component with splitting 3X. In practice, the weak component may be too weak to be seen and the only observable effects of internal rotation will be oscillatory behavior in the subband origins of the Yk and Rk series. The spectrum of VII + vd was so dense, for example, that no series of weak components could be found without considerable overlapping from other stronger lines. Henceforth, N,,(K) refers to the strong component of u associated with ground state K only. We can readily estimate the values of X expected for different types of bands from Eq. (6). In a fundamental such as v8, AT = 0 and the parity is even. _V,l,l = :V,, and X = X0,“* - XO,gr. There is a cancellation effect of numbers which, as shown in Table II, are very small to begin with. XO,~~= 0.002 cm-‘. X0,“* = 0.001 cm-l and 0.004 cm-’ for barriers of 1100 cm-’ and 900 cm-’ in VS, respectively. These give X = - 0.001 cm-’ and +0.002 cm-’ for the two cases. Each of these is quite small and not observable. For a hot band arising from the first torsional state, Xl,gr = 0.075 cn+. From Table II, one gets values of X of -0.024 cm-’ and +0.065 cn+ for barriers of 1100 cm-’ and 900 cm-‘, respectively, in the upper state. These are significant effects. Of course, if the upper state barrier is very close to the ground state value, cancellation will occur and no splitting will be observed. In combination and difference bands involving torsion, the splitting parameters are additive in the two states for even parity and subtract for odd parity. In any event, the T = 0 splitting in one vibrational state is always negligible compared to the 7 = 1 splitting in the other. For ~11 + v4, we expect X = 0.051 + 0.002 = 0.053 cm-l for a barrier of 1100 cm-’ in vll and 0.140 + 0.002 = 0.142 cm-’ for a barrier of 900 cm-’ in ~11. For ~11- v+ we expect X = 0.077 cm-’ for a ground state barrier of 1024 cm-’ and a typical barrier height in ~11.An experimental X value for this or other such difference band would give an accurate ground state barrier relatively independent of the barrier in ~11. i2s shown in the last column of Table II, significant splittings in &DC first show up in transitions involving the second and third excited torsional states. The theory as

8

SUSSKIND

given is not applicable to asymmetrical molecules such as CHSCDZ or CH$iH: because rotation and torsion do not separate. S&H6 and GezHs are other molecules in which the discussed effects should be observed. III. ROTATIONAL

ENERGY

LEVELS-q

CORIOLIS INTERACTION

As is well known, because of off-diagonal vibration-rotation operators in the Hamiltonian of Eq. (i), it is necessary to talk about effective rotational Hamiltonians, Hrotr, in different vibrational states. It is customary to write the rotational energies in an E level, with I = f 1, to third order as &ot(J,

K, 6 v) = (J, K, 4, v[Hrot j J, K, 4, v) = B,J(J

+ 1) + (A8 - B,)K2 - Z(A&Kt + TJIC”K~~ + rj.y’J(J + l)Kl - DdK4 - D.,dJ(J + 1)K2 - DyJ2(J + 1)“.

(7)

To second order, D, = DO for all vibrational states and Q = 0. The breakdown in the second order approximation arises primarily from a rotational dependence on the energy differences of interacting vibration-rotation states. This dependence may be particularly significant in the case of close lying vibrational states exhibiting xy Coriolis interaction. xy Coriolis interaction between A and E levels in C’s”and D3d molecules has been discussed extensively by DiLauro and Mills (13) primarily from the point of view of intensity perturbations to the spectrum. Sarka (14) has discussed the effect of 3cy Coriolis interaction on the A levels as exhibited by parallel bands and has shown the contributions of this interaction to anamolous B, and D, values. We will demonstrate the effects on the E levels. As mentioned previously, v11+ v4 interacts with nearby v2 + v4 via an xy Coriolis interaction. The rotational energy levels of ~11+ v4 can be treated fairly well by a thirdorder rotational Hamiltonian with rather large third-order constants due primarily to the interaction. Alternatively, the interaction can be taken into account expressly as shown in Eqs. (3841) of Ref. (13). We find the second method more satisfying, both because of the use of a smaller number of parameters and because of the easier physical interpretation of the parameters used. In the presence of internal rotation, we replace v7 in Eq. 38 of Ref. (13) by Evib(v, 7) energy expression for + XTT,b(K)XI,“. We also use a pseudo second-order rotational

F,(J, K) F,(J,

K) = B,J(J+

1) + (A, - B,)K2 - Df’J”(J+

1)”

- D&‘J(J

+ 1)K” - DgK4.

(8)

This differs from a strict second order unperturbed Hamiltonian in that we allow different D values in different vibrational states. In order to estimate the effects of xy Coriolis interaction on the rotational levels of VII+ ~4, a model calculation was done using the following nonzero parameters in eq. 38 : VB= 1757.98

A 8 = 2.6576

R, = .6614

f*” = -

vr = 1685.0

A, = 2.671

R, = .6631

{r,szY = 528

The perturbed

energy levels E(J, K, t) in vll+

v4 were calculated

.3344

for each value of Kt‘Z

TOR-VIB-ROT-INTERACTION

FIG. 2. Spectrum of C*He from 1791 to 1784 cm-‘. temperature.

IN ETHANE

9

Pressure is 50 Torr. Path length is 1.5 m room

from - 10 to 10 with J going from K to 25. They were then fit to a power series in K8 and J. The special case case Kl = 1 which gets a large effective &type doubling constant was omitted from this calculation. Effective 11~ and 7~ values of 3.66 X lo-” and -2.79 X lO+ cm-’ were found to be induced by the interaction. Also, AD values of 1.56 X 10-j, -2.71 X 1w5, and 3.13 X lo-’ u-r-r were induced for DK, D JK, and DJ, respectively. Small higher order terms not covered by the third order Hamiltonian were induced as well. They are relatively insignificant except at very high J. It will be shown in VII+ vq that if xy Coriolis interaction is not expressly taken into account very large effective third order constants will be obtained as suggested by the above. When expressly taken into account, the second order unperturbed Hamiltonian is a good one with only very small AD values. IV. EXPERIMENTAL

All spectra were run at the Molecular Spectroscopy Laboratory at the University of Minnesota using the 2.5 m Littrow-McCubbin spectrometer built by John Overend et al. and previously described (15’). The spectrometer is completely evacuated except for a small space in front of the detector which was continuously flushed with dr? nitrogen. There was no evidence of any Hz0 absorption in the spectra. The ethane sample was obtained from Phillips Petroleum Company and was stated to be 99.9% pure. The spectra were run at room temperature at a pressure of 50 Torr in a White cell $ m in length set at 15 m. The detector was Cu doped Ge cooled to liquid He temperature. The spectra were run in 10th and 11th order of a 30 grooves/mm grating blazed at 63”. The spectra were calibrated using the fundamental of CO (16) in 12th and 13th order. The effective resolution of the spectra is ~0.04 cn-’ and frequencies are good to 0.01 cm-‘. The spectra are quite dense and many lines are blended. Figure 2 shows sample spectra from 1791 cm-’ to 1784 cm-‘. The two most prominent features are RQr, and RQ4. Those lines which have been assigned are indicated. Many of the unassigned lines may be due to hot bands such as ~11 + 2~4 - v4 which should have appreciable intensity.

10

SUSSKIND V. RESULTS

A number of subbands were easily found and identified by the amount of missing lines between the Q branch and first R or P branch line in the series. The Q branches appear normal from “Qa to RQ2. RQr, RQo, and ‘Qr are greatly perturbed as is pQ4. pQ2, PQO, PQb, and pQe appear normal but ‘QG is shifted about 1 cm-’ up from its expected position. The remainder of the Q branches were not observed. The subbands RR, - RRa, pPz, pP~ and pPs were observed and assigned. In addition, possible fragments of series for RR1 and PPa were observed but thevd are quite perturbed and their identification is uncertain. Of the remainder, pP3 shows large deviation of experiment from theory and RR2 and pP2 smaller but significant deviations. The calculations were done using two different models. Either xy Coriolis interaction with ~2 + v4 was calculated explicitly or Eq. (7) was used with effective third order constants in VII + v4. In both cases, the term N,,(K)X was added to the energy levels and X varied as an independent parameter. The main object of the study is X. Hence, it was thought significant to see how sensitive X is to the model. One problem in taking the Coriolis interaction directly into account is that vz, of symmetry A la (A rg in &) is dipole inactive with any torsional function. While vr1 + v4 is made active via torsional Coriolis interaction, v2 + v4 becomes active only in so far as it interacts with vrr + vq via xy Coriolis interaction. The most prominent feature of such a band is expected to be a broad Q branch shifted down from the band center or split on either side of it depending on where 1vll + v4, - 1, J, K - 1) crosses 1va + ~4, 0, _7, K). From Table II, we estimate VP+ v4 to be approximately 80 cm-’ lower than vrr + up. The spectra remain quite dense in the region 1700-1660 cm-’ and no obvious induced Q branch of v2 + v4 could be found. Therefore, a variety of band centers were chosen for the calculation to test the sensitivity of X on this choice. Table V shows the results obtained by fitting the subbands RR~ - RR~ and ‘Ps to a number of models. The first five calculations take xy Coriolis interaction explicitly into account. Equation (7) was used for the last two columns. The subbands RR2, pP~, and pPs were not included in the fit. In all cases, the parameters shown were varied independently. In the first set of calculations, 7J’ and vg’ were always fixed as well as v,., the band center of v2 + v4. 7.r’ and lZ’ are highly correlated and the data are insensitive to 7~‘. Xl,,, was fixed at zero and the rotational constants for v2 + v4 were fixed at the ground state values in the calculations. The vibration-rotation splittings between the interacting levels of VU+ v4 and v2 + v4 are sufficiently large that small energy changes in v2 + v4 levels do not appreciably affect their interaction with vrr + ~4. In calculations 1-3, v,. was set at 1700 cm-‘, 1685 cm-‘, and 1675 cm-‘, respectively. DK’ was fixed at zero as was the ground state value. VJ’ and 7~’ were also fixed at zero. It is seen that the values of D.J’ and D JK’ are always close to the fixed ground state values reported by Cole el al. (17) as shown in the table. There are no appreciable differences in the root mean square deviations of experimental and calculated frequencies in the three cases. The best value of X does vary from 0.0380 to 0.0409 cm-‘. [2,1rZy also varies from 0.400 to 0.614 cm-‘. Based on a normal coordinate calculation using quadratic force constants provided by I. Nakagawa (18), {z,rr”” was calculated to be 0.55 cm-r. This is most closely approximated by the value obtained when vr = 1685 cm-‘. This value was chosen in further calculations for demonstrative purposes but the

TOR-VIR-ROT-INTERACTIOS

.,m~+ (1. csm

lI.52S f". w.,

0*

"e

lli

o*

/I1

(I *

"*

,I*

O."380iO.""OG

"."398+".0""6

".040!J+"."""G

1757.979~".""3

1757.983+"."03

1757.987 ?".""3

0.662230 ~"."00"43

0.661362+".0""058

U.660762 *0.0"006~

1.995G5I'".""""H5

l.!~Y6226+"."""085

LYYG633 i-"."""085 -".88854c"."""27

(1.01 L.031 x l"-G

(1."G+."::,xI"-G

(1.11+.u:i,x I"-6

-ti l4.4!li.J2)Xl"

(4."7i.36,X 10-6

,".54?.3Y,X 10-G

0.0104

O."l"G

0.0108

1757.682

1757.687

1757. GW

2.65788

2.65759 -0.33439

state Values:

Ground

-".H886Gi".O"O~6

-0.33441

A' '11

0.G1.i+",,1"5

"1

-0.88882 f"."""ZG

Vu

11

IS ETHANE

A =2.671

2.657-1" -".33X17

B = 0.6631

DJ = 1.18 x 1"-5

-6 D JK=3.6x10 4

5

1G85*

lG85*

0.528+0.""4

".471~"."21

o*

u*

0+ (1.33+1."l,xl"

-5

0.04351U.0029 1758.006?".017

DK_

0

6 lGcls.*

7 1685'

"' -4* -2.79x 1"

0* e2.YUz.04)~

-4 1"

o*

-_I* 3.Gtix 1"

u*

1.56 x N-5*

"*

".0410~0.0006

".03Y2iO."""6

1757.990z0.003

1757.953+0.003

1757.115920.014

0' o*

".661369~0."00"58

".6GZ24~0.00025

0.6F4972C0.000023

0.664979+0.""0023

LY947ro.o"11

l.Y9738_t".O"O38

1. Y!l2256 t0.00"06?

LYY4"3GtO."OOOYY

-".88945'"."0065 (l.OGi.03)~ lO-6

-".88324+".0"11 (1.33i.15) x 10-G

-0.88H13t0."""16 (0.W sJ.03) x n-6

-0.99219+0.00031 (".9520."3)x I"-G

(2.16t.36)x 10-G

(6.09~1.6")I 10-G

t8.90~0.2Y)xlO-6

(-9.97+0.99)~1"-~

0.0106

0.0487

0.0111

0.01"9

175?.7O!J

1757.665

1767.6!,3

1757.654

2.GXl?

2.65lW

".G57":1

2.65!Io?

-".::,Ih'i

- 0.:!::rl

-0.:::. I,,,

- 0.::::554

12

SUSSKIND Table VI

AND THEORETICAL

EXI'ERIMENTAL

AK

\J

K

J

WI

FREQUENCIES

FUR vll

EXP

(CAIrEXP!,

+ v4

(C&EXP)2

1

1

8

8

1

1618.95"

-0.007

0.044

1

1

R

9

1

1820.267

0.008

0.061

1

1

R

10

1

1821.613

-"."O(i

"."49

1

1

8

11

1

1822.941

-O.""l

u.055

1

1

8

12

1

1824.278

-0.004

0.063

1

1

8

13

1

1825.603

".""l

0.058

1

1

8

14

0

1826.970

-0.U26

0.032

1

1

Y

15

1

1828.282

-0.002

U.056

I

1

8

16

1

1829.614

-"."02

U.060

1

1

8

17

1

1830.%5

"."ll

0.045

1

1

8

18

0

1832,287

0.005

0.059

1

1

8

19

1

1633,623

0.009

0.059

1

1

6

20

1

1834.963

1

1

7

7

0

11112,155

1

1

7

8

1

1

1

7

9

"

",OlO

0.055

-0.025

-"."6G

1813.457

".""6

-0.033

1814.822

-0.026

-0.062

1

1

7

10

1

1816.138

-0.007

-0.041

1

1

7

11

1

1817.472

-0.006

-0.037

1

1

7

12

"

1818.811

-0.009

-0.038

1

1

7

13

1

1820.150

-0.012

-0.038

1

1

7

14

1

1821.470

0.006

-0.018

1

1

7

15

0

1822.807

6.608

-0.015

1

1

7

16

1

1624.161

-0.007

-0.029

1

1

7

1T

1

1825.493

0.002

-0.020

1

1

7

I8

1

1826.834

o.003

-"."2"

1

1

G

G

0

1805.217

0.034

-0.059

1

1

6

7

1

1806.566

0.016

-0.012

1

1

G

6

0

1607.883

0.036

-0.051

1

1

G

9

1

1809.238

0.016

-0.067

1

1

6

10

0

1810.561

0.624

-0.056

1

1

6

11

1

1811.YJO

U.028

-0.048

0

1813.261

0.005

-0.067

1

1

G

12

1

I

0

13

0

1814.6""

0.""6

-0.063

1

1

CI

14

1

1815.934

0.013

-0.054

1

1

fi

15

1

1817.2!rJ

-0.005

-"."69

1

1

G

lti

1

101H.621

0.01"

-0.051

1 UK-V

LK

U-KU

I-IN

w-1

1 i!,KHL

1lUlU

EXP

IN

L 1 HA?\

13

IL

(CAlrEXP).,

1

”

Lbl!J.

1

1

1821.:1”6

1

1

1

0

1 I 1

1

179!LG21

1

”

1

1

1

0

1803.Gll

-O."Gli

1

”

18"4.95"

-0.048

1

1

lY"G.292

-"."46

1

1

1801.F41

-0.049

1

1

1808.98Y

-"."49

1

1

1810.328

-0.041

1

”

1311.688

-"."52

1

”

1813.020

-0.034

1

1

1814.319

-0.042

1

1

1815.716

-0.027

1

1

lS17.083

-0.042

1

1

1818.4:35

-0.040

977

-“.

“lil

-“.

“44

1822.861

-“.

“52

1824."11

-“.

“53

I

-“.

“59

1

-“.

01-1

-“.

“JZ

18"".!U-l

-“.

OGG

1802.276

-0.058

1

”

1819.7GG

-0."17

1

1

1821.126

-0.022

1

1

1822.494

-0."34

1

”

1823.845

-0.028

1

1

1791.204

0.016

1

1

1792.546

0.009

1

0

1793.805

0.032

1

0

1795.125

0.013

1

1

1196.544

u.031

1

”

1797.G8::

"."J3

1

1

179!1.2*1

0."31

1

1

1XO".58"

"."4"

1

1

lRW.2G2

U.058

1

0

1804.626

V.046

1

1

18W.!IG3

O."GB

1

0

lHw.:x"

"."41

1

1

lS"H.IXl

V.067

I

”

HI". ox:

".OG3

14

SUSSKIND AK

K

J

WT

EXP

(CALEXP),

(CAIrEXPj2

1

4

19

1

1811.383

0.013

".U72

1

4

2”

0

1812.754

".""3

".""l

1

4

21

0

1814.127

-".""7

0.049

1

4

22

1

1815.480

0.005

0.059

1

4

23

1

1816.851

0.002

0.052

1

3

3

0

17RLlG2

0.011

0.052

1

:i

4

1

1785.512

-0.005

1

3

5

1

1786.844

0.000

0.047

1

3

6

0

1788.194

-0.012

0.038

1

3

7

0

1789.522

0.000

0.064

1

3

8

1

1790.863

0.001

0.059

1

3

Y

1

1192.225

-0.016

0.044

1

3

1”

0

1793.538

"."I5

"."81

11

0

1794.403

-".""3

"."GY

1

"."3!)

1

3

12

0

1796.241

".""8

U.084

1

3

13

1

1791.609

-".""8

0.071

1

3

14

”

1798.943

0.011

0.094

1

3

15

1

18"".291

O.Ol!,

O.l"ti

1

?

2

0

1177,142

0.043

"."42

1

2

3

1*

1778.486

0.033

0."34

4

1*

1119.817

0.039

0.042

5

0

1181.161

0.013

0.020

1 1

2

1

2

6

0

1782.508

0."27

0.037

1

2

1

0

1783.646

0.032

0.045

1

2

3

1*

1785.210

0.013

0.030

1

2

9

0

1786.567

0.003

0.024

1

2

10

1*

1781.W8

0.012

0.037

1

2

11

1*

1789.252

0.020

0.049

1

2

12

0

1790.618

".0"9

0.042

1

2

13

1*

1791.984

0.001

0.036

1

2

14

1*

1793.348

-0.003

0.036

1

2

15

-1

2

2

-1

2

-1

2

-1

2

-1 -1 -1

3

1*

1794.722

-0.013

0.021

0

1747.457

0."19

0.047

3

1*

1746.143

0.015

0.043

4

0

1744.824

0.020

0.049

5

1*

1743.538

-O.O".l

"."26

7

1*

174".'9-%4

-0.017

0.015

4

I,

173!I. 62 !,

0.""1

0."M

!I

0

,7::Y.::z!I

,I.,,"7

0.,,.t2

TOR-VI%ROT-INTERACTION 6I-c

h$

IN ETHANE CAL-EXPjl

WI

-1

I*

-1

-1

”

-1

-1

1*

-1

-1

,I

-“.

“15

-I

-1

0

-,I.

“A*

-1

-1

I*

-“.“,lj

-1

-1

1*

-“.

-1

15

-“.

0%

“.““I -0. “38

uoz

-1

-1

1*

-0. “41

-1

-1

”

-0. “17

-1

-1

0

-0. u*Ci

-1

-1

1”

-“,

-1

-1

”

-0. “6”

-1

-1

1+

-0, “77

-1

-1

1’

-“.

“W

-I

-1

0

-“.

“,>!I

-1

-1

1*

-“.

I”4

-1

-1

0

-0. “!r7

-1

-1

”

-0.1””

-1

-1

1*

-“.1U3

-I

-1

1*

-“.

-1

-1

1

“. ““9

us7

124

-1

-1

1

0. “0.5

-1

-1

1

“. 0””

-1

-1

1

“. ““1

-1

-1

1

u.014

-1

-1

1

-0. ““0

-1

-1

1

-0.

-1

-1

1

-1

-1

1

-1

-1

0

-1

-1

1

-1

-1

1

““3

“. ““2 -“.

“1”

“. “18 -“.

““1

u. “11

uncertainty in parameters due to the uncertainty in vr is kept. It should be noted that B’, the unperturbed B value in VII+ ~1, is quite different for each Y,, but Beff is identical in all calculations. B’ is important as a measure of the average R rotational constant in a given rotational state. Hc.f gets a big Coriolis contribution.

SUSSKIND

16

Column 4 shows the results when DK’ was allowed to vary as well. DK’, like vK’, is found to be very uncertain and relatively highly correlated with X. A value of AD = 0 is thought to be reasonable when Coriolis interaction is taken into account. Hence, not much weight is given the value 0.0435 f 0.0029 cm-’ obtained for X in this calculation. Column 5 shows the results when X is fixed to zero. The root mean square deviation between experiment and calculation rises to 0.0487 cm-‘. The effect of torsional splittings on the spectrum is clearly a very important one. Table VI shows the values of the esTerimenta1 frequencies and deviation of esperiment and calculation for the lines assigned. The weights indicate whether the lines are heavily blended or not. In the case of RRZ, pP2, and pP~, weights of 0 were actually used in the calculation. The calculated frequencies correspond to the Hamiltonians in columns 2 and 5 of Table V. The oscillatory contribution of Table VI is clearly shown when the effect is not taken into account. The remaining two columns in Table V are calculated from Eq. (7). In column 6, r]~‘, ~g’, and DK’ were fixed at those values given by the model calculation. Hence, xy Coriolis interaction was still implicitly taken into account. In column 7, 11~’was varied but 7~’ and DK’ were fixed at zero. As before, the data are highly insensitive to these parameters. Aside from quite different constants, the results are not much different from those in columns 14. The best value of X remains about the same. From Table V, we choose X = 0.0395 f 0.015 cm-‘. Assuming a ground state barrier of 1024 cm-‘, we get X1,11 = 0.0377 f 0.0015. This corresponds to a barrier in ~11 of 1165 =F 8 cm-’ (3330 ? 23 cal/mol). The barrier to internal rotation is thus found to be roughly 140 cm-’ higher in ~11 than in the ground state or a change of roughly 13%. While this appears large, work on methanol by Woods (19) indicated an increase in barrier height in methanol of 6y0 in the first excited CO stretch and at least 13% in the CHI stretch. Therefore, a change of this magnitude is not uncommon. It’s significance will be discussed in more detail when compared to the barrier in v12 of ethane in a subsequent publication. RECEIVED : August

14,

1972 REFERENCES

1. S. WEISS AND G. E. LEROI, J. Chem. Phys. 48, 962 (1968). 2. (a) J. B. HOWARD, J. Chem. Phys. 5, 442 (1937); (b) 5, 451 (1937). 3. (a) B. KIRTMAN, J. Chm. Phys. 37, 2516 (1962); (b) 41, 775 (1964). 1. (a) J. T. HOUGEN, Canad. J. Phys. 42, 1920 (1964); (b) 43, 935 (1965). 5. (a) P. R. BUNKER, J. Chenz. Phys. 42, 2991 (1965); (b) 47, 718 (1967). 6. D. PAPOUSEK,J. Mol. Spectrosc. 28, 161 (1968). 7. I. M. MILLS AND H. W. THOMSON, Proc. Roy. Sot. (London), Ser. A 226, 306 (1954). 8. D. W. LEPARD, D. E. SHAW, ANL H. L. WELSH, Canad. J. Phys. 44, 2353 (1966). 9. D. E. SHAW, D. W. LEPARD, ANC H. L. WEUH, Canad. J. Phys. 42,3736 (1965). 10. D. W. LEPABD, D. M. C. SWEENEY, AND H. L. WELSH, Canad. J. Phys. 40, 1567 (1962). II. E. B. WILSON, JR., J. Chem. Phys. 6, 740 (1938). 12. P. R. BUNKER, Mol. Phys. 8, 81 (1964). 13. C. DI LAUKO AND I. M. MILLS, J. Mol. Spectrosc. 21, 386 (1966). I-!. K. SARKA, J. Mol. Speclrosc. 41, 233 (1972). 15. J. OVEREND, A. C. GILBY, J. W. RUSSELL, C. W. BROWN, J. BEUTEL, C. W. BJORK, AND H. G. PAULAT, .4ppl. Opt. 6, 457 (1967).

TOR-VIB-ROT-ISTERACTION

IN ETHASE

17

16. N. K. RAO, C. J. HUMPHREYS,ANDD. H. RANK, “Wavelength Standards in the Infrared,” Academic Press, New York, 1966. 17. A. R. H. COLE, W. J. LAFFERTY,ANDR. J. THIBAULT,J. Mol. Spectrosc. 29, 365 (1969). 18. I. NAKAGAWA, private communication. 10. D. WOODS,Ph.D. Dissertation, University of Michigan, 1970.