Equilibrium structure and potential function of sulfur dioxide from the microwave spectrum in the excited vibrational state

JOI’RSAL OF MOLECULAR SPECTROSCOPY 13, 95-118

Equilibrium Dioxide

(196-1)

Structure and Potential Function of Sulfur from the Microwave Spectrum in the Excited Vibrational State

The microwave spectra of the SOS molecule were measured in the first escited states of all the normal vibrations as well as in the second exrited state of the Y 2 vibration. The quadratic force constants are calculated from the inertia defects in the excited vibrational states, and the cubic and quartir constants are detrrmined through the changes of the rotationxl constants by the escit:ttion of t he vibrat,ions, combined with the anharmonicity constants of t,he vihr:ttional frequencies, ;ciJ’s, reported by Kielsrn et (I(. The I’, structure was determined definitely and cc~lnlparedwithvariollsst,rll~furry such as r-, I’*, and r,, INTl:OI)l:CTI( dct8crluination

The the

st,udy

of

tential

has

terms

have

ware

spectra

of the iutramolecular

t,he nlolecular

been

st,udied

hardly

detcrnliuatiou

from

by

beeu

of molecules

the

The which

use

iufrared

we

the

ohtaiued, structure.

of the

however, have

Krait~chmau’s

almost

Theoretical

t,hr vihratioll-rotatiou

spectra, of

their

vibrational This

paper

of

sulfur

function

spect,roscopic

nlolecular

part

the

cubic

of

will

dioxide data,

quartic ukxo-

us with

the

up

po-

The

provide

prcscutj

ill

the

and

importance. states

coufiguratiou

must, be the first the

been

r,,-structuw

proposed.

cyuatiou

It is, however, alld

excited

t,hc microwave

call t,he t’,-structure,

of devices

vihratiolls.

in spite

is fundanlcntal

quadratic

to

ccnlhiued

result

ill-

of t,hc

the

quartit

wit811 results

spectra.

I-sually,

lumber

potential

the

of vibratioual

obtaiucd

in the

potentmial fw&iou

hlthough

for this purpose.

the of

detemklatmiou

function.

of

of

the

structure.

in t,hc field

ever

forrnatmiou illdispensablc

terms

)N

(2)

difficult

impossible discussions iuteract,ions,

iu

to give have and

receutly

95

difficult

the

effect

the accuracy physical been

a st,ructure

poteutial study

poiut’ed

(1)

miuimizing

a clear

of the

is very

Costaiu

to estimate

at the

step

obt’ain the

of

and

a

usefuhwss

t’he zero-point

of the

developed

named

to out

nleauiug

nliuitnunl,

of the poteutial

structure to the

for

the

I’~ or T’:~,.was

t’hus

derived effect8 of

proposed

96

MORINO,

ET AL.

(3, 4)) which designates the structure averaged for some particular vibrational state. It should, however, be pointed out that the I-,-structure can not be determined uniquely for molecules with more than three structural parameters (more than two for the planar molecules). However, since the SO* molecule has only two structural parameters, it is possible to calculate various struct,ures from the observed data and to compare them with the r,-structure which will he determined by observing vibrational satellites. It is also shown t,hat the inertia defect, not only for the ground state but also for the excited stat,e, is much more readily used for the determination of the quadratic potential constants than the centrifugal distortion constants obtained by a complex analysis of high-J rotational lines.

A number of extensive studies in the microwave spectroscopy have been made for sulfur dioxide (&--IO) since Dailey, Golden, and Wilson (11) first, reported the n~iero~~avespectrum of this molecule. Polo and Wilson ( 12) carried out the normal coordinate calculation to obtain the force eon&ants from the ~~ibratioI~a1 frequencies and Kivelson (7) determined t,he general quadratic force constants by using the four centrifugal distortion constants. Shelton, Nielsen, and Fletcher (IS) analyzed the vibration-rotation spectra and obtained normal frequencies ws and anharmonicit,y constants :c,,t , which appear in the vibrational energy expression

where v, denotes the quantum number of the 8th vibration. Recently, after the present study was started, Van Riet (I’klC;) published a number of papers on the rotational spectra of this molecule, not only in the ground state, but also in the first excited state of vgvibration. His interest seems to be concentrated on the determinat,ion of cemrifugal distortion const,ants based on the analysis of rather high-J transitions, whereas the purpose of the present i~lvestigat,io~~is to det.erlnille the accurate molecular structure and the potential function of the molecule. I. ESPEILIMEICTAL

The study was started by a rapid scanning of the microwave spectrum in the region from 15,000 to 35,000 Mc/sec with a sinusoidal Stark modulation spectrometer in order to get the general survey of the spectrum. About two hundred lines were detected both at room temperature and at dry-ice t,emperature. In Table I are shown the frequencies and their relative iIltel~sit,ieswith t,he assignments for some of the lines. The frequencies listed in that table are rough values because the lines were recorded under the condition of Stark de voltage of about 600 V/cm with sinusoidal modulation amplitude of 200 V/cm, without extrapolation to zero field. The relative int,ensities are less reliable because the

Table I. Obeervedfrequencies(in Mc/eec)and intensitiesof 'therotationaltransitionsof sulfurdioxide -msN--

___*zmaEsseP frequency

(intensity) assignment

IYID~) temp.dry-icetemp. m13

frequency(intensity) ~~i(plment mmEl teap. dry-iceteap.

17637 (3) 17637 (8)

(1.5)

$32) 7113r59-70X4,56 (d)

17970 (20) 17970 (50) SC%) 5*,4-61,5(a)

15275 (9)

18ox (15)

152m (14)

lW96 (12) O(18) (j)

S(34)v2 153 15-174,u(i) t

15410 (6) 15470 (15)

18262 (3) 18262 (25) O(B) 81,7-?26(k)

15575 (4)

2 81,7%,6(k) 18570 (10) 18570 (15) S(32f2Y lS@O (3) 18600 (15) O(M) (1)

15740 (4) 158% (2)

S(32) V3 92,7-101,10 (10

1wO

(3)

15950 (4)

16760 (18)

15934 (2)

la346 (5) lSg26 (15) o(B) 92 7-1OI lO(k)

15996 (41

1,

16050 (20)

S(32) v2 254 22-2451Y(e)

16126 (IO)

Sf32)"2 ~02‘S-111;+k)

16372 (3)

S(32)2v2'72'@&(k) t p6634 (lo) s(33) 12 3,9-132,12(d) 16653(15)\

16682 (250) 16682 (?)M)S(32) 327,25-336%(d) t l6754 (20) 16765

(ZOO)

S(33) la3r15-174,L4(d)

P6764 (15)i (4)

19095 (4) lYQ95 (15) 19114 (3) 19m

(15) 19232 (15) 192% (25)

19252 (2) 19252 (20) 193o6 (8) 19547 (7) 19547 (25) 195W2)

o(B) (j)

19637 (50) 19657 (50) S(32) 378,x-W7,31 (d) 19680 (25) 19680 (NM) S(32) g2 142 l2-15l,15(e)

16808 (3)

am

16928 (5)

16928 (20)

203104(8) 20104 (2)

17w3?(5)

170481(5)

20261 (10) 20261 (5)

,

17342 (lo) 17342 (20)

203350100)20335(>50)S(32) 123pl32,12(b)

(d) 17540 (3O) 17540 050) S(32) 45s,3e-449,35

203% (10) 20384 (3)

Sf32)J3 162 l5-15,,12(k)

,

MORINO,

98

Table I.

(continued)

MOm temp. dry-icetemp.

20548 (50) 20548 (50) S(34) 164,12-173,15 (d) 20606 (3)

22905 (25) 22905 (50) S(32)V2 225 17-234 ,(a) 22928 (25) 22928 (50) S(32)V2 305'25-296'24(e)

S(3)) 52,4-61,5(a)

23035 (30) 23035 (>lOO)S(32)398,32&7,33id)

I

20611 (10)

S(32)2v2lo2 gill ll(k) . .

23100 (6)

20651 (4)

23194 (1) 23208 (30) 23208 (20)

20676 (2)

23246 (2)

20690 (4) 23336 (3)

20699 (30) 20699 (30) S(34) 82,6-9l,g(a) 20785 (7)

-

frequency(intensity) asjignment

frequency(intensity) aasiwent room temp. dry-icetemp.

20651 (2)

--

~===vB=~=--_

----=_-ezzz

20606 (3) 20610 (4)

ET AL.

20785 (2)

o(18) (3) 23342 (5)

20836?(3)

23402 (10)

21000?(6)

23414 050) 23414 (>lOO)S(32)52,4-6;,5(b,c;

S(32) V3 132 11-123lo(k)

23490 (5)

21265 (15) 21265 (10) (Gk) s(34) V2 52,4-61,5

23630 (5)

21299 (2)

21299 (4)

21508 (5)

21508 (3)

23@4 (3)

21@8 (3)

23734 050) 23734 050) S(34) 1B3 l5-l74,l4(d) * 237W (15) 237tB (10)

21760 (20) S(32)V1 123,g-132,12(k)

21768 (30)

23970 (20) 23970 (25) S(33) 317 23-32626(t?)

22066 050) 22066 (50) S(32)V2 81,7-72,6(esk)

24040 050) 24040 (>lOO)S(32)215;17-224;18(b)

22220 (30) 22220 (20) S(32) V2 276,22-a5,23(a)

24083 (>100)24083(>lOO)S(32)82,6-9l,g(b)

22314 (5)

22314 (3)

22316 (5)

22316 (3)

24165 (20) 24165 (50) S(33) 266 2o-275,23(g) . 24194 (2)

22325 (5)

22325 (5)

24275 (20) 24275 (4)

22327 (5)

22327 (5)i

24301 (30) 24301 (10) S(32)Vl e1 7-72 6(k)

S(33) 82,6-91,g(gvk)

224d3 (>lOO)22483(>50)S(32) 244,20-235,1g (b)

S(32)Vl 52,4-6l,5(k)

24320 050) 24320 (XOO)S(32) 438 3;-42g;33(d)

(e,k) 24345 (15) 24345 (25) O(l8) (j)' 22734 (25) 22734 050) S(32) v2172,16-163,13 24430 (5)

22%4 (7) _-

---~

-__

-=v==-P__

--__

the theoretical estimate; the maxinuuu absorption coefficient, y,,,, is to be under constant pressure. proportional to Tp7’” exp [- (E,,t + E,i,)/kT] The I80 enriched sample was prepared by heating sulfur powder to about 180°C under the low pressure of oxygen gas which was obtained by the electrolysis of water containing 0.8% of 180. It was used for eliminating the lines due to the isotopic species from the lines to be assigned to the vibrational sat’ellites. II. RESULTS

The 32S’602 molecule has the symmetry of Ci2L,and three uorrnal vibratious : a symmetric stretching v1 , a deformation Y?, aud an antisymmet~ric stretching

frequency(intensity)

frequency(intensity)

assignment

24677 (15) 24677 (10) 3(34) v2 e2 6-91 g(i,k) 24799 (30) 24799 (30) o(la) V2 (j; 2488t1(10) 2W

(3)

assignment

room temp. dry-icetamp.

mo. temp. dry-icetemp.

26706 (2)

S(32) Vl 82,6-91,g(k)

o(18) (j) 26768 (20)

'

26777 (30) 26777 (>100)S(32)254 22-2451g(b)

24890 (10) 24890 (3)

26850 (25) 26850 (30) s(32) V2 52,4-61,5 . ' (h,k)

24916 (20) 24916 (20)

26398 (15) 26893 (8)

25012 (7)

26977 (8)

25012 (20) o(ls) (3)

25049 050) 25049 (>loO)S(32)35b,30-347,27(b)

o(lB) (2)

27296 (7)

S(34) V2 172 16-163l3(i)

(10) 27X8 (3)

s(32) $a l72;16-16 3:13(k)

25393 (7100)25393(xOO)S(32) 81,7-72,6(W)

27W

25400 (4)

27398 (25) 27398 (30) S(N) (j) 27452 (5)

25446 (4) 25504 (7)

25504 (6)

'

27120 (8) 27296 (8)

25364 (2)

S(32) V3 103'a'1;,g(k)

27018 (4)

25171 (20) 25171 050) S(X) l63,l3-154,12(d) 25365 (2)

S(32) 2V2 4. 4-31 3(k)

26977 (3)

27452 (6) a562 (3)

O(l8) (3)

25594 (2)

27600 (2)

27600 (3)

o(l8) 52,4-61,5 (k)

25680 (6)

256KI (15)

27633 (4)

27633 (2)

S(34) V2 266,a-275,23(i)

25702 (7)

25702 (15) O(lB) (3)

27656 (10) 27656 (6)

25884 (a)

2w34 (15)

27723 (3)

27723 (2)

25913 (7)

25913 (20) o(m) 40,4-31,3(k)

27756 (6)

27756 (7)

26040 (25) 26040 (30) S(34) 295,25-286,22 (i)

27983 (4)

26098 (5) 26253 (3)

26253 (7)

O(le) G2,6-91,g(k)

S(34) V2 817-72 6(&k)

27933 (20) 27933 (35) S(32) 252,2;-243;21(f) 27983 (8)

28139 (40) al39 050) S(32) U2 4. 4-3, 3(h.k) 28174 (30) 28174 (35) S(32) V2 "2:6-91;g(h.k)

26262 (3)

26412 (25) 26412 (30) ~(32) V2 123 9-132 12(i,k) 28179 (30) 28179 (35) , * 28204 (5) 28204 (12) 26488 (4) 26505 (10) 26505 (4)

28270 (7)

26567 (8)

28276 (7)

26567 (15) O(l8) (j)

2827o (15) S(33) a1 7-72 6(h,k) , 9 I

-___IIv:( Tlwir t’recluencies arc llcii.60 cK1, 526.27 m-l, arid 1380.91 cn-‘, wspc’c( 1.j 1. The defomlation Y:! is comparatively low in frocluency so t)llat the

tively

microwave ahsorpt.ion easily. The assignment the tcmpcrature

lines in the v2 excited state are strong enough to dctcct of the vibrational satellites was first, made hy ohservitlg

dcpcndencc

of the lint intensity.

The int,ensities

of the spectral

lines in the I’? = 1 state are expected to relllain unchanged for wide tenlperatuw range, from dry-ice temperature t,o 25O”C, whereas those of the lines belonging to other

states

are much affected

by the temperature

change.

Thus

t’he rot,atiollal

lines in the first excit’ed state of v2 were easily distinguished. Tlw ~‘IW~IIP~C~shift by t,he St#ark effect gives another clue for t,he axsiglurwllt,

100

MORINO,

ET AL.

Table I. (continued)

frequency(intensity)

frequency(intensity)

assignment

28678 (lo) 2~3678(7)

S(34) "2 l64,l2-l73,l5(i) 31011 (X0) 31011 (50) S(34) 4o,4-3l,3(a)

28858 050) 28858 (>loo)S(32)l72,l6-l63,l3(b) S(32) vl 40,4-313(k) 29030 (3)

290% (3)

assignment

rmm temp. dry-icetemp.

room temp. dry-icetemp.

0(18) (j)

31067 (3)

31067 (?)

31090 (30) 31090 (18) S(32) 283,25-292,28(f)

'

29055 (35) 29055 (20) S(32) V2 213,19-204,11 (h)

31164 (20) 31164 (20) S(33) 172,16-16j,13 (k)

292-72(3)

31169 31162 (20) 31169 31162 (25) (20)I

29321 (X00)29321 (>loO)S(32)40,4-31,3(b,c)

31268 (7)

29862 (25)

31603 (8)

29364 (15) 29864 (3)

S(34) V2 40,4-31,3(isk)

31603 (2)

31653 050) 31653 (30) S(33) l54,l2-l63,l3(j)

29872 (7)

31680 (7)

29896 (25) 29896 (3)

31799 050) 31799 (25) S(33) 244 a-235,1g(j)

30036 (7)

31881 (10) Jl@a (2)

30195 (5)

30195 (5)

30196 (3)

30196 (3)

x197 (5)

x197 (5)

30198 (5)

30198 (3)i

,

31922 (>loo)31922(>loo)S(32)164 l2-l73,l5(f) S(33) 4 0,4-3l,3(a)

319% (3) 31942 (3)

o(le) V2 (j)

31944 (6)

30204 (30) 30204 (40) S(32) 162,14-171,17 (fh)

31994 (15)

30218 (20) 30218 (15)

32074 (4) 32083 (3)

30268 (5) 30455 (30) 30455 (3)

~(33) l61:l4-l7l,l7(j) 31942 (2)

S(32) V3 72,5-81,8@)

32083 (2) 32136 (4)

S(32) 2V2 52,4-6l,5(k)

304= (30) 30488 (4)

32198 (30) 32198 (20) S(32) 5211,41-5310,44 m

30700 (5)

32216 (x)) 32216 (25)

X950 (25) 30950 (3)

32274 (30) 32274 (25)

30965 (15) 30965 (15) o(lS) (j)

32382 (7)

x975 (>uo)m975 050) S(34) 8l,7-72,6(a)

32420 (20) 32421 (3)

S(32) 2W2 S2,6-9l,g(k)

and for some of the low-J lines a definite assignment is made by observing a partially resolved Stark pattern. It is difficult to observe and identify the spectral lines in the z+excited state. It is not only because the lines are weak by the small Boltzmann factor for this &ate, but also because the lines corresponding to the transitions which are allowed in the first excited state of the ~3mode are forbidden in the ground state, so that no prototype of the line pattern of the rotational spectrum of molecules in the u3 = 1 state can be found in the spectrum of the ground stat,e of 32S160~ molecule. The spectrum of 32S160180in the ground state gave a convenient reference, because the transitions which are forbidden in the case of t’he symmetric

Table I. (continued)

frepuencr (intensity) assignment room tesp. dry-ice temp.

32437 (8) 32753 (4) 3m%

O(18) (j)

34376 (5)

(4)

34394 050:

3285R(4)

34425 (8)

33144 (10) 33144 (2)

34427 (20)

33213 (40) 33213 (SO) S(34) 172,16-~63,13(k)

34464 (lo)

33224 (6)

34486 (25)

333% (3) 33364 (5)

W8)

(3)

008) 6)

34530 (50) 33%

(4)

34560 (4)

33440 (35) 33440 (9)

Iwl1. (30)

33504 (3) 33595 (3) 33659 (25) 336e9 (25) 33731 03)

O@)?(j)

34790 (20)

33484 (25) 33484 (4)

34850 (3)

O@)

34972 0%)

s(32) Q2 fj)

(.i)

35068 (4) s(32)v3 1x3 s-l22 Ilfk) ‘,

33734 (3) 33996 (3)

assignment

34346 (3)

32753 (6)

(20) 32m

frequency (intensity) roomtesp. dry-ice temp.

35@4 (251 35126 (30)

33956 (3)

35236 (10)

33w

35296 (lo)

O(W

34098 050) 34098 050) ~(32) 266,a-T15,23(f)

35449 050)

S(32)v2 (2)

34145 (3)

35666 (25)

s(34)W

34178 (25)

(5)

SCM) (9)

3178 (4)

35790 (30)

34294 (‘5)

35908 (25)

34254 (5) 34294 (4)

72,5-Q@

35856 (30)

34344 (3)

Referencesto Table I (a) w. R. smith,Au&al. J* Pbys.2, 109 (1959) (b) 1. H. Sirvetz,J. Chem.Phys.9, 938 (1951) J. Chsa.Phys.2, 502 (1951) (c) G. F. Crableand W. Y. snulth, (d) &I.de Hemptinne,F. Gre&le, and R. Van Riet, Bull.Acad. Boy. Belg.48, 397 (1962) (8) R. Van Riet, Bull.Acad. Roy. Be%. 48, 731 (1962) (f)

1. K. Ccra, privatecomwnication

(g)

R. Van Riat,AnnalsSOC. sci. Bmxelles 76, 56 (1962)

fh) R. Van

Eiet, BUN.. Acad.

(i) R. Van Riet,AnnalsSOC. (j)

Roy. Belg.48, 1291 (1962) Sci.

Br~~elles 'J7,18 (1963)

Assignedby Van Riet, R. Van Riet,Bull.Acad Roy. Be&

(k) Presentinvestigation

to be published

102

MORINO.

ET AL.

SO, molecule are allowed due to lack of the CZv symmetry in the 3”S’“O’80 molecule. The corrections for t’he centrifugal distortion were calculated by IGveIsollWilson’s met~hod (17) with the values of four 7’s given by Kivelson (7).’ The number of the assigned t’ransitions was uot sufficient to determine the centrifugal distortion constants simultaneously with the rotational constants. A. v?-EXCITED STATE We st,udied the microwave spectra of the %1602 molecule in the first and second excited states of the v? vibration. As the spectra in the first excited st,ate have already been reported by Vau Riet (IL?), our procedure and result will be dcscribed briefly. Two lines in the up = 1 state, 40,4 + 31,3 and j,,, t 61,5 , were first identified by their characteristic Stark patterns and temperature dependence of their intensities. The frequencies of other possible transitions were estimated with the rough values of the rot,ational constants, Aol’, B”” and Co” computed from the two frequencies and the calculated value of the inertia defect, A = I, - ~I0 - Ia , for the ~‘2= 1 state. The inertia defect for the 1~~= 1 sta,te consists of two parts : A”lo = AO+ A, (see Section III) : the first term which comes from the zero-point vibration was experimentally giveu by Kivelsou (7) aud the second term A2 , which conies from the vz-vibration, was estimated by a calculatiou (19). The rotjational constants were determined from the frequencies of four lowJ transitions 11,1 + 20,2 , 21,1 + 30.2 , 4u,4 + X1,3 , aud 41,a - 40.r . The values .J”“’ = 61,954.69 Mc/sec, B”‘O = 10,320.28 lUc/sec, and Co” = 8783.96 Mc,isec are slithtly different from A”’ = 61,956.0999 Mc/sec, B”” = 10,320.5663 nIc;sec, and Co” = 8784.1207 MC/see obtained by Van Riet,, the differences being -1.41, -0.29, aud -0.16 I\Ic/scc, respectively. The differences may come from the fact that Van Riet used relatively high-J transit’ions, whereas the present authors employed relatively low-J transitions, some of which were uot measured by Van Riet. Also, Van Riet used revised values of the centrifugal distortion constants in the analysis, which were considerably different from those assumed in the present paper. Table II shows the observed and the calculated frequencies with the centrifugal corrections. In higher-J transitions the agreement, between the observed and the calculated frequencies is not good, which indicates that the distortion constants used should be revised. Several liues in the second excited state of Ya-vibration were identified as listed in Table III. Their positions were first estimated from the frequencies of the correspondiug lilies in the first Yz-excited state. The lines with reasonable intensities were easily found near the predicted frequencies. The rotational constantjs thus RIc/sec, Bozo = 10,322.22 Mc/sec, and obtained are A”’ = 63,185.20 Co” = 8767.90 Rlc/sec. 1Recently, Pillai and Curl (18) corrected the values reported by Kivelson. crepancies are too small to affect the results obtained in the present paper.

But the dis-

lo:i

MICROWAVE SPECTRUM OF SLXF‘TR DIOXIDE Table

II

Rotational

Transition

lines

Cent .corr.

of

32S160a in the

1, talc.

v,=l

state

(Nc/sec)

y’obs.

1 111

+ooso

-2.41

70735.24

70735.92a

1 171

4-2092

-2.18

13457.55

13457.92

2 191

*2,,2

-1.77

54739.06

54739.4ga

4 014

+-319s

-0.38

28138.27

28138.55

4 l.,3

+40,4

-0.38

~04g9.01

604g8.77a

5214

+6*,5

-22.18

2sr853.90

26850.45

8 3.17

C7296

1.71

22062.80

22065.76

8 2~6

+9199

-12.88

28177.21

28173.65

c-ll.,ll

-12.05

16130.70

16127.00

12 319 +13,,le

-53.70

26427.53

26411.75

21 .go

22717.53

22734.14

10 2Y8

172,16+163,13

a _Prlvat

e communication

from K. Takagi .

The .&.a +- (i1,5 transition was first assigned based on the ten~peraturc dcpcndence of the intensity and its characteristic Stark patt’ern which was rcsolved into four conlponents. The search to find the 4,,,4 +- 31.2 line was continued because that line was expected to show a definite Stark effect by which an walnbiguous assignnlent could be established for the v&ate. Unfortjunately no such lilw has been observed. I;or the 81,7 + 72,6 transition two lines are available at 21,797.16 i\lcisec and at 33,301.59 NC/see, but it cannot be deternlined which of t#hetwo should be assigned to the transition. When the calculated value of the inertia defect for the ~1 = 1 st’ate is conlbined mit,h t)he observed frequencies, X,Z75.-Hi ;\Ic ‘set for the 52,4 - 61.6 transition and H,797.16 11~ ‘set for the 81.7 +- ‘72.6transition, Set ( A 1 of the rotational constands is obtained, whereas Set ( H) is drrived by assigning the X,301.59 Nc/sec line t,o the 81,~ +- 7z,li transition: for Set (Aj ~2~“~= 61,3-11 :\Ic/sec, &O” = 10,372 I\lc, see, alIt ( i”‘i’ = 8841 _Vc/sec, and for Set (B) 21”” = 60,815 JlIc/sec, R’O” = lO,%iU 11~ wc, alld f?“” = 8760 NC/see. The frec~ur~rcy of the &,c + %,a transition was calculated for both sets. So

104

MORINO, ET AL.

Table III

Rotational lines of ssS160s in the vg=2 state (Mc/sec)

Transition

Cent.corr,.

2, talc

.

I/ohs.

-2.18

14714.32

14713 l99

215% +20,0

-1.77

56003.62

56003

4 014

c-31,s

-0.38

26898.27

26898.38

4 198

+--4094

-0.38

61826.60

61826.60a

-22.19

30461.96

30455.09

1.71

18562.61

18568.30

-12.85

32428.35

32420.08

-11.91

20620.54

20611.38

16338.21

16372.82

1 191

c--20,8

5 as4 +61,5 8 197

+-7896

8816

-1,9

10a 98 t--11~9u l78,16

+166,16

a Private

20.63

communication

from

ma

K. Takagl.

suitable line was found near the frequency predicted by Set (A), whereas for Set (B) the Jo,4 +- 31,Ktransition of the 211= 1 state might be covered by the strong line 172,~ +- lBa,l( in the ground vibrational state. Since t’he rotational constants in the first excited state of ~1 vibration are given by PI,, - 011, B, - (31, and Co - y1 , where CQ, /31and y1 are linear fun&ions of the anharnlonicity constants /ill1 and kll, , as will be described later Isee Eq. !17’)], the frequencies of the rotational lines in that state are approxinlately expressed by t#helinear functions of Llll and k112. In other words, 011 the plane of 1~~~~ and x‘113, they are represented as the lines that are nearly straight with very little curvature, as shown in Fig. 1. It should be noticed that the lines corresponding to different J transitions must cross at a single point on the plane when snlall corrections are nlade for the centrifugal distortion shifts. It is because the rotational lines with various J correlate with the conunon values of the cubic conrtants. It provides us with a convenient lnethod to confirln the assignrnent of the J quantunl nunlber. Figure 1 clearly indicates that the lines at 28,868 ;\lc/‘sec, 24,275 AIc/‘sec, and 24,302 Nc/‘sec are correctly assigned to the 4 0,? + :31,a, 52.4 f- &,s , and 81.7 +- 72,6 transitions, respectively, whereas it is not correct to assign the line at 24,797.16 Mc/sec to the 81.7+- 72.6transition. Front the intersection of the first three lines it was concluded that ICUg -4s The above result indicates that Set (B) is inore and kllz s - 18 cn-‘. Cd likely t’han Set (A).

k lie Frequency (MC)

Assignment

a

24275.46

52.4-61.5

b

24301.59

8i,?‘7,,,

c

24797.16

--so

-60

a

d

k,,, = -I 8.0 cm-' +40

FIG. 1. Assignment

of the spectra in the VI = 1 stat?

After these trials a liue was obserwd at 27,386.X Xc/see, t,he position preof Set (R) . dicted for t,he 172,16+-- 16, .13trausit,ion by the rot,atioual constants The lint showed a Stark pattern very similar to that, of the corresponding ground-state line. In Table IT- are shown the assigned tjransitions toget#her -\lc-it#hthe frequencies calculated by the use of the fiual values of the rotational constants A’“” = 100 60,809&l A\\Ict’+sec,B = lO,M7.96 ,IIcjscc, aud C1’” = 8757.13 Ylc,/sec, whicsh were obtained from the frrqueucies of the three tramit~ions, 11.I +- Zo,y , 5e,4+ (iI .h , and 8, ,; - i,,s . (!. I!:;-~S('ITED STATE; From the preliluiuary values of the cubic constants, /il:j:j = - 170 CC-~ and I<,,;1 = 5 m--l, estimat’ed by the use of a potential function ($0) with rough values of thr paramet’ers, the rotatioual const,ant,s in the first excited state of J+ vihl~ation were cowputed. Rough values of the expected t~rau&ions were 8680 RIG ,‘scc for the 3c.s +-- 21.2 , 31,836 itlc.~sec for the 72.5 + &,a , and 15,711 I\lc’s:et for the 92.i +- 10l,Io tran&ion. We searched for them in the liei~i~bo~l~ood of the predicted fwquencies and found t’he lines which had appropriate intensities, t’elnperature dependence, and Stark shifts. The lines obtained are listled in . . The Table J . The tramtlon 3o,:$ - 31,~ was confirmed by its Stark pattern. rotational constants thus obtained arc =2”“”= 60,158.77 RIc,/sec, I>““’ = 10,28X% 1\ic ,‘sw, and C”‘“1= 8767.08 :\Icisec.

106

MORINO,

ET AL.

Table IV Rotationallines of 3aS160a in the vL=l state (Mc/sec)

Transition

Cent.corr.

ycalc.

Yobs.

-2.19

12522.89

12522.89

-0.38

28856.37

2aa58a

-0.38

59261.27

5g260.g6b

-22.18

24275.45

24275.46

1.71

24300.97

24301.59

-12.89

24896.77

24888.3a

-12.10

12611.33

12597.69

-53.75

21779

22.40

l

93

27363.82

21768.s4 27386.24

a The frequency is not accurate because of a possible overlap with other line. b Private communicationfrom K. Takagi. D. GROUND

VIBRATIONAL

STATE

OF ?Y601s0

Several lines wereidentified for the ground vibrational state of the 3rS’“O’X0 molecule, as shown in Table VI. The 32S160180enriched sample was used to confirm the assignment. It should be noticed that the transitions, 72,5 + 81.8 and 92.7 +- lO,,lo which are forbidden in the symmetric 32S’602, were observed for this molecule which has no Cz axis. The values of 7 assumed in the calculation of the centrifugal correction were Tbbbb = --0.@%5249 hh/SeC, Taabb = +0.378618 Taaaa= -9.27922 ;\k/SW, ;\Ic/sec. They were calculated by taking the ?\Ic/sec, and T&b = -0.0479964 arithmetic mean of the values for 32S’602and those for 32S’80zwhich were estimated from those of 32S1602 by consideration of the mass difference. The values of the rotational constants thus obtained are Aooo = 59,107.72 Mc/sec, Boo0= 9724.61 Me/see, and Coo0= 8331.70 Mc/sec.

MICR@WAVC

Fable V

SPECTHIW

Rotational lines of ssS60s

Transition

Cent.corr.

10i

OF SVLFIIR DIOXIDE

In the v,=l state (M/see)

Y talc.

Vobs.

1.27

8622.80

8622.74

-1.08

55329.64

55329.30

0.11

62945.19

62g44.88a

-16.63

31993.90

31993.92

-1L.08

15848.09

15848.21

-50.20

26977.94

26977.96

-17.90

8088.26

8087.07

-69.93

33726.12

33730.90

-11.73

23413.36

23402.08

22.36

20390.27

20383.86

8 Private communication from K. Takagi. Table VI

Rotational lines of 3~Sl~0130 in the ground state (Mc/sec)

Sransition

Cent.corr.

ycalc.

vobs.

1 111 -%I,2

-2.07

13291.46

J.3291.69

4 OY4 c-31,3

-0.27

25913.14

25912.80

5294 +6x,6

-21.46

27600.58

27600.00

18261.43

15261.82

-12.09

25251.10

25252.48

-13.90

13475.15

J-3477.28

-15.86

35298.17

35296.30

-9.78

18926.45

18925.06

8 137 *?2,6 8 a¶6 +31,9 102,8 +"-=I,11 72,s -81,* ger?? f-10,,**

2.a

108

MORINO, ET AI,.

Table VI1

Observed values of the rotational constants (Mc/sec), and inertia defects (amu Ae) for 3eS160,.

A

State

B

0

bobs.

Ground”

60778.79

10318.10

8799.96

0.1348

Lfl

60809.54

10267.96

8757.13

0.1808

VF?

61954.69

10320.28

8783.96

0.4077

Q3

60158.77

10283.25

8767.08

0.0985

2vz

63185.20

10322.22

8767.90

0.68l.2

accuracy a

Ref.

+s.O

LO.5

20.5

LO.005

(7)

III.INERTIA DEFECT AND QIJADRATIC FORCE COXSTAXTS The principal moments of inertia in the ground state and in the excited vibrational states are easily calculated from the observed values of A, 3, and C for each state. The inertia defect derived for each state is given in Table VII. It was already shown (19) that the inertia defect is closely related to the Coriolis constants {{I’ as follows: AO = Lib + Acentf

=:-

h 2X%

c w2

+

wzw3

wz.w3(w2

(2)

Ae~ec,

+

w2 ,

+

(4)

w3)

) Acent =; _7abab $, + L& + S!$ P c e> (

(5)

(7)

A

0,”

A”“’ A

,1x,

and The Arlec was calculated to be -0.00374 anw A’ by the use of the values of Y obtained by Burrus (91 j : gan = -0.606 f 0.008, Ybb= -0.123 f 0.007, aud YCC= -0.074 f 0.005. The value of ({j;‘)” was obtained from the observed values of the inertia defect’s, as shown in Table VIII. It is seen that ({is’ 1” callnot, he precisely determined from the inertia defect in the ground state, because & involves a large term independent of [ iz’, as is seen from E:cl. (4). rortunat,ely, however, it can be detmerminedmore precisely from thr inertia defect,s in t,he excited states. Sow, for the general quadratic force field, it is necessary to have four force

Table VIII

Coriolis coupling constant 5,:"' obtained from the inertia defects (amu Ae).

Ground

a

(0.048+0.081)a

yl.

0.113to

.012

L’z

0 .ogo+o .017

%3

0.086+0.022

2L’a

O.O8g+O .008 ~~ ~-

average

o.og4+0.02

The value

in

parenthesis

was not

included in taking an average,

110

MORINO, ET AL.

constants, FII , Faz, Rz , and Fz3 , but we have only three normal frequencies observed. The addition of the Coriolis coupling constants obtained above will enable us to get the four force constants unambiguously. That is, the constant FII is directly determined by the relation given by Meal and Polo ($2))

FII= (G-')nPu - (XI- XZ)(&))~],

(12)

and then by using the normal frequencies given by Shelton, Nielsen, and Fletcher (13)) all other elements of the F matrix were determined, as shown in Table IX. It would be interesting to compare the result with that given by Kivelson (7), who used the centrifugal distortion constants instead of the Coriolis constants. The agreement is satisfactory. Their values should be corrected for the difference between fundamental frequencies ~0and the normal frequencies V, .

IV.ANHARMONIC

POTEXTIAL CONSTANTS

The changes of the rotational constants by the vibrational motions are given by the following equations if the higher terms are neglected in the expansions: A, = L4, -

c

LtS(VC + .!z),

and

(13)

where 21denotes a set of the quantum numbers of the three normal vibrations: v = (Us) e’,! ) u,,,). The constants (Ye, p. , and ys designate the changes of the rotational constants by the sth vibration. The observed values of the 01,‘s for SO2 are given in Table X. The general expressions for (Y~, & , and y8 have been given by Nielsen (23).

Table

Present

IX

Force

result

F 11

10.41

F 1e

0.32

F22

0.815~0.007

F 33

LO.20 ~0.21

10.251+0.001

constants

Kivelson(1)

of

SO, (ma/A>

Polo and Wilson(l2)

10.030

10.05

0.267

0.28

0 *7933

0.793

9.982

9.99

MICROWAVE

Table X

SPECTRXM

OF SULFIJR

111

DIOXIDE

The changes of the rotational constants by the excitatiqn of vibrations for ssS6Os

a

vibration -

(MC/S&).

B

r

~--

L’1

-31.05

50.14

42.83

L's

-117zj.go

-2.18

3-6.00

L'S

620.02

34*85

32.88

~~~~~27.31

,,=-0.12 13

&*=-0.03

Their explicit expressions for t,hc three ~~ib~at,iotlsof the SOZ n~olecult are as follows :

aid

112

MORINO, ET AL.

where sin y = a$““‘/22/Ih”)

= a$bb’/22/11e) ,

cos y = aibb’/2z/Ite) = -a4”“‘/2dIp)

.

On the other hand, the potential function is expressed in terms of the dimensionless normal coordinates qs = (X,/G”) 1’4&s: V/he

= 9; c

3

fiX:‘“q,’ + S,Fzsn

k.ss~,.y,y~~y,.+ . . . .

(15)

It should be noticed that 0~~, & , and ye depend only on the cubic constants but not on the quartic constants of the potential function, whereas .rsstwhich appear in the vibrational energy expression of Eq. (1)) are related to the quartic as well as the cubic constants: 3 a11 = - knn 2 3

x33

=

3 k,,,, 2

-

(16)

3 Xl:{ = kn,, - -km

k,,,

-

Wl

X23

=

k2*33 -

-

1

h12

3

-h-222 w2

k233 -

-

1

2wa

Ii233 4wa2

w2

k122 ku3

WI2

2W3

4w3”

Wl

-

-

kiTa

+

c,

w22

z

+

(

2’

(&I

j”.

)

By making use of the F-matrix obtained in the preceding section and the structural parameters to be given in Section V, the explicit forms of (Y~, & , and ys for 32S’602are given by cy1= -261.911

-11.8086

fi1 = - 10.712 -0.988327 y1 = -7.521 o(~= -810.169

x.111-0.922019

-0.973173

x-n1-0.343749

-3.93619

pz = - 17.046 -0.329442 y2 = 12.302 -0.324391

km +15.3594

k~, X-lla, &a ,

i&2 +46.0781 1~222 , 1~12~ -2.76606

kzza,

,&, - 1.03125 I&Z ,

and

(17) a3

=

-77.545

-3.93619

X-1:3, +15.3594

l
fi3 = -13.281

-0.329442

licit -0.922019

, /iz:~$

y3 = -1.7.230

-0.324391

klaz -0.343749

kza:,,

MICROWAVE

SPECTRUM OF SULFUR DIOXIDE

11:s

potential where 01,) PI, and yy are expressed in MC/see and the anharmonic coustants in cn-‘. Substitution of the experimental values of (Y~, & , and ys listed in Table X, leads to the cubic constants shown in Table XI. Accordingly, fro111 Eqs. (16j the quartic constants listed in Table XI are obtained by usitlg t,he values of :r11 = -3.99 tin-‘, .ree = -3.00 cnl-‘, .ra3 = -5.li cl11 ‘, .L’lZ= -2.05 m-l, xl3 = -13.71 cni-‘, and .rZR= -3.90 en-’ given h_v Shrltoll, Tielscn, and Fletcher ( 13). V. RIOLECI‘LARSTRUCTURE

Since(Ye , /3, , and ys have been obt’ained for the first excited states of the three normal vibrations, the equilibrium rotational constants and the nloments of inertia can be obtained, if the higher terms in the expansion of A, R, and (’ arc‘ disregarded, as expressed in Eq. (13 j .The results are as follows: :I,, = (iO,J85.33 MC/see,

B, = lOJ59.51

Mcjsec,

aud C’, = 8845.82 I\lc RC(*

and Id”’ = 8.357!)1 aim A’,

1:“’ = #.7987

aim A”,

and

II.”

= 57.1-191 amu A”.

Th(l morncuts of inertia given above include the contributions from the outof-plane electrons. Wheu t’he contributions of the electrous are subtracted front the moments of inertia by the equation r,4,, = - (,rnjMjZ,g,, \vit,h the g-values already cited in Section originate only from the nuclei arc I,:’ ’ = 83551.5 amu AZ,

1:”

Table XI

IV, the moments

= 48.7954 anlu A’,

\\:c point, out that the requirement t.hat 1:” - 1:’ - 1:‘) = -0.0038

(18‘)

)

and

1:”

of inertia

whit+

= 57.ldG3 amu =\“.

of planarity is nicely satisfied: it is found anlu A”, which is of t.he order of bhe errors ill

Cubic and quartlc constants of the potential function of ~eS160a

k asa=

k .,1=-44.1 k leS= k 1x11= k,lea=

9.3

1.7 -2.6

-6.6

k 199=-159.7

-1 (cm

1

k lla=-18 Baaa=

k aeaa=

-1.7

ksssa=

k 1x88=

15.4

kaasa=

l9 4.8 3.1 -6.1

114

MORINO,

ET AL.

the measurement (see Section VI). Therefore, any pairs among the three moments of inertia listed above lead to the final and identical molecular structure at the equilibrium position. The internuclear distance and the bond angle are given by the expressions r” = (l/m,)

,:?I

+

(

l/2)120)1,(C))

(19)

and tan’ ee = [m8/ (2,~ + 7~,)] ( Z~e’/l~e’).

(20)

The final result of the r,-structure is as follows: r&S-O)

= 1.4308 A f

&(0-s-0)

= 119Y9 VI.

f

0.0002 A, 2’.

DISCUSSION

First of all, let us consider the accuracies of the 00s and the r,-structure obtained above. As the rotational constants are accurate within fl Mc/sec for A and f0.5 Rlc/sec for B and C, accuracies of the I,, Ib , and I, are f0.0002, f0.003, and f0.003 amu A2, respectively, and that of 1’ is f0.00009 A. It should, however, be pointed out that the most serious errors may come from the approximation assumed in Eqs. (13)) that is, the neglect of the higher terms in the expansion of A, B, and C. Fortunately, as the rotational constants in the second excited state have been obtained for the ve vibration, we can test the above assumption for this case. Thus, if we add the next higher terms to Eqs. (13) A, = A, -

c cr,(a, + ?,6) + .z, (~ss,(as+ ?i)(v,, s _(

+ 1/i),

(21)

then CQ= (.yz lUlcor+ 2a22 + 4l(w

+

a23) )

(22)

and A, = ,pcor

+ 3i(N1 +

a22

+

w)

+

>i(a2

+

ff23 +

w31).

(23)

and the corresponding expressions for other quantities. Here Al? and A’?“= denote respectively their values evaluated in the preceding section by disregarding the higher terms. It was found that o(22 = 27.31 I\lc/sec, /3z2 = -0.12 l&/see and yz2 = -0.03 R!lc/sec. Accordingly, the correction to a2 due to the neglect of o(22is about 5%) that to ye is about 0.03%) and hence, considering various sources of errors, the accuracy of the r,-structure is &O.O002A for the S-O distance and f2’ for the OS0 angle. Although we do not know the magnitude of the corrections due to other terms, the above example will show US the order of magnitude of the corrections. We have used Kivelson’s values of 7 and Kivelson-Wilson’s formula for the

~ICRO~A~rE

SPECTRUM

OF SULFUR DIOXIDE

115

correction of the centrifugal distort.ion. It may directly give some uncertainties to the values of the rotational constants, but we observed rather low-J transitions so that, the effect would be of t~heorder of the errors of the l~~eas~lrel~~~Ilt, &imated above. Heretofore the effect of the Fermi resonance has not been considered at all. The cubic constants obtaimzd above indicate that the resonance between tht? L)Y~and pl stat,es t~hrough the t-erm of X*122and that between 2~ and vl st~ates t,hrough kiss might be serious. However, simple calculation shows that t.he ef’fect, is about r3 Rfcisee to A, 0.1 RIc/scc t,o 23, and 0.0’2 Mci’sec to C, smaller t.hau the ~ol~t,ributio~lsof the higher t,erms iu t,hc expansion of the rotational constant,s iii IQ. ( 13 1. Sow, it would he of interest. to compare the r~z-stt*ucturewith the y,-structur(x obtained above. The I*,-structure was first pro~ose(~ for the ground ~ribra~io~lal state ( 3, 4 1 and it was made clear ( $4 ) that t,he l*? is the average distance in t2hr ground state. It’ can, however, be extended to the excited vibrational stat’es, if we assume the same treatment t,o the observed 1~lo~~~eIlts of inertia in t,hc excited st.at,es: for instance, ( I*)fa’, ~Zh)f(~,and (Z,)‘;“” are o&ained by subtracting thtx harluonic eont,ributiolls of the vibrations

1 from t,he effective Illo~nents of inertia (In)“‘, ( Za)luo, and ( ZC)lM1, respectively. Here A iza’ designates t,he coeRcient of the t.erm Qse in the expansion of I, ($3). The r,-struct,ures for the excited states are readily obtained by any pairs among t,he t,hree moments of inert,ia, ( I, ) t”” = pmcfm,/ (‘2,,lQ+ m,)] (I’, CO8I” & j2, (~~)~~ = 2m~(rz sill !$@,)‘, and

(“5) i I, ) t”” = [2?n0nz,/(Z~m0+ in,) ]rz3( 1 + 2( nzo,/nk,)sin” > ,fJl),

where fi is the S-O bond length and 0, the angle of OS0 in the state. The structures thus obtained are shown in Table XII. B’or comparison, in Table XIII are shown the ~~-structures obtained from Z”‘s t,hcmselves. It is seen in Tables

116

MORINO,

Table

State

XII

ET AI,.

r,-structures

of

S*S160,

(I;, $1

(I;,

1.434908

1.434gOA

1.4349 08

1.434gOA

11g021.1’

119021 .l’

11g021 .l’

11g021 .l’

1.438818

1.438878

1.438848

1.438848

119O23.9’

11gO23.4

11g”23.g’

11gO23.7’

1.43461A

1.434668

1.4346 3A

1.43463A

119O32.4’

11g031.9’

11gO32.5’

11gO32.3’

1.43956A

1.439608

1.439588

1.439 58A

llg”lo

11g010.3’

11g010,6’

llg”lo

1.434188

1.43441A

1.43429A

1.4342gA

11g”45.0’

119O43.1’

11gO45.3’

11gO44.4’

(Ii,

1;)

.6’

1:)

average

.5’

XII and XIII that the consistency among the three structures is more satisfactory for rz than for TO . Table XII indicates that the bond length and bond angle increase by the excitation of the ~1and v2vibrations, respectively. On the other hand the three r.O-structures,which were calculated by selecting any two among the three components of the moments of inertia, 1:, 1:, and 18, show large discrepancies among them and do not indicate whether or not the bond angle is increased by the vibrations. The r,-structure is also calculated, for comparison, by combining the moments of inertia in the ground state of the isotopic molecules. The first set is obtained from the combination of 32S’fi02, 32S’60180,and 33S1602, and the second one from 3W602 ) 3%‘60180, and 34S’60a, by the use of the equations ’ = (A&%)[1 a.3

+ AJ:/(la”

-

&?I,

and

(26) b,2 = (Ar&)[l

-

AI&‘(I,o

-

I:)],

where a, and b, are the a- and b-coordinates of the atom, for which an isotopic substitution is made, and the a-coordinate of the sulfur atom is assumed to be zero. p = Ic!fAm/(dl + Am>, where M denotes the total mass of the original

Table XIII

Ib)

(Ib’ ‘~1

(Ic, Ia)

ground r

1.4322A

1.435lA

1.43368

e

119'32'

119" 8'

llgO36'

r

1.4347A

1.43868

1.43668

9

119O40'

119" 8'

119O46'

1.4286A

1.43758

1.433lA

120" 0'

118O48'

120°12'

r

1.43588

1.4380A

1.43698

0

119022'

119" 4'

119'24'

v,=2 r

1.4250A

1.43998

1.4325A

12OO30'

118'28'

120°50'

(Ia, -

ro structures of seS160,

--

VI--1 v,=l

r 0

vg=l

l9 Table XIV

Comparison of various structures in the ground state of SOz

Structure

r

e

r_JI)

1.43188

1190211

rs(IU

1.4305A

119'38'

1.4336A

119O25'

1.4349A

119021'

1.4308A

119"19'

rO

?z re

118

MORIKO,

molecule and Am the increase moments

of inertia

of the

of

original

mass

ET AL. on

molecule

the and

isotopic AZ0 its

substitution, increase

I” is the on

the

sub-

stitution. The first and the second sets give the structure listed in Table XIV, in which the comparison of the various structures is shown. It is true that the r,-structure is the nearest to the restructure among the three approximate structures, as frequently mentioned, but the deviation of the atomic distance between two choices is appreciable. ACKNOWLEDGMENT The authors would like to thank Professor Kozo Hirota, Osaka University, for his kind offer of the ia0 enriched sample of SOZ. The authors are indebted to Mr. Kojiro Takagi for the measurement in the millimeter region and also to Dr. R. Van Riet, Universite deLouvain for informing us of his recent data cited in Table I. This work was carried out with the financial support of the Fund of Toyo Rayon Foundation for the promotion of Science and Technology, for which the authors are grateful. RECEIVED

August 5, 1963 REFERENCES

1. C. C. COSTAIN, J. Cherrl. Phys. 29, 864 (1958). 2. J. KRAITCHMAN, Am. J. Phys. 21, 17 (1953). S. T. OKA, J. Phys. Sot. Japan 16, 2274 (1960). 4. D. R. HERSCHBACHAND V. W. LAURIE, Bull. Am. Phys. Sot. [2], 6, 500 (1960). 5. G. F. CRABLE AND W. V. SMITH, J. Chem. Phys. 19, 502 (1951). 6. M. H. SIRVETZ, J. Chem. Phys. 19, 938 (1951). 7. D. KIVELSON, J. Chem. Phys. 22, 904 (1954). 8. G. R. BIRD AND C. H. TOWNES, Phys. Rev. 94, 1203 (1954). 9. R. WERTHEIMER AND M. CLOURD, Corn@. Rend. 246,1793 (1957). 10. W. E. SMITH, Australian J. Phys. 12, 109 (1959). 11. B. P. DAILEY, S. GOLDEN, AND E. B. WILSON, JR., Phys. Rev. 72, 872 (1947). 12. S. R. POLO AND M. K. WILSON, J. Chem. Phys. 22, 900 (1954). 15. R. D. SHELTON, A. H. NIELSEN, AND W. H. FLETCHER, J. Chew. Phys. 21, 2178 (1953); 22, 1791 (19s). 14. M. DE HEMPTINNE, F. GREINDL, AND R. VAN RIET, Bull. Acad. Roy. Belg. 48, 397 (1962). 16. R. VAN RIET, Bull. Acad. Roy. Belg. 48, 659, 731 (1962). 16. R. VAN RIET, 872%. Sot. Sci. BruueEles 77, 18 (1963). 17. D. KIVELSON AND E. B. WILSON, JR., J. Chem. Phys. 20,1575 (1952). 18. M. G. KRISHNA PILLAI AND R. F. CURL, JR., J. Chem. Phys. 37, 2921 (1962). 19. T. OKA AND Y. MORINO, J. Mol. Spectroscopy, 8, 9 (1962). 20. Y. MORINO AND K. KUCHITSU (to be published). 21. C. A. BURRUS, J. Chem. Phys. 30, 976 (1959). 22. J. H. MEAL AND S. R. POLO, J. Chem. Phys. 24,1119, 1126 (1956). 25. H. H. NIELSEN, Revs. Modern Phys. 23. 90 (1951). 24. M. TOYAMA, T. OKA, AND Y. MORINO, J. Mol. Spectroscopy (to be published).