kenneth fox- vibration-rotation interactions in infrared active overtone levels of spherical top...

8/3/2019 Kenneth Fox- Vibration-Rotation Interactions in Infrared Active Overtone Levels of Spherical Top Molecules; 2-nu-3 a

1/40

J OURNAL OF MOLECULAR S P ECTROS COP Y 9 , 381120 (1962)

Vibration-Rota tion Interac tions in Infrared Ac tive OvertoneLevels of Spherica l Top Mo lec ules; 2~ and 2v, of CH,,

2~3 of CD,*KENNETH Foxt

Th e Harrison M. Randall Laboratory of Physics, The Universityof Michigan, Ann Arbor, Michigan

The theory of th e vibration-rotat ion lines of th e first overtones of th e infra-red active fundament als of tetr ahedr al molecules has been re-examin ed. Theorypredicts an overton e spectru m consist ing of five P, Q, an d R bra nches of rough lycomparable intensities provided th at th e vibrat ional angular momentu mquan tu m nu mber e is approximately a good quant um nu mber for the completevibration-rotation Ham iltonian. In this case th e separat ion of th e E and P?vibrat ional suhst ates of th e G = 2 vibrat ional state must he small compared withth e splittin gs which ar ise from t he BBc(P.1) t,erm . The ban d 2va of CHa is sh ownto be consistent with this appr oximation. If however th e separat ion betweenthe E an d Fz vibrat ional substates is very large th eory predicts an overtonespectru m consist ing of single strong P , Q, an d R branches with P and R branchspacings of app roximat ely 2R(l + 0. Th ese P, Q, and R lines are associatedwith th e Fz vibrat ional suhst ate, an d have relative intensities mu ch lar gerthan the lines of the E vibrat ional substat e. The hands 2~;~ f both CHa and CD4are shown to be accounted for by this limit.

The detailed calculations exploit th e spherical tensor forma lism. In th e firstcase a conventional angular momentum coupled represent ation, an extensionof Hechts work on th e funda men tal Ye , is used in the calculations. In th esecond case a new represent ation is intr oduced which forma lly has man y ofthe mat hemat ical properties of th e conventional represent ation for an ! = 1vihrational state.

The tetr ahedr al splittings in th e vibration-rotat ion levels of 2v,


2/40

-0.050 f: 0.004 cm-. Fr om th e Q bra nch , B - Bo = -0.062 f 0.002 cm-l. In2~~ of CH, th e tetr ahedr al splittings are quite small, mak ing a quan titat ivefit more difficult . However, th e best fit is obtain ed with Dt = 4.5 X 1OP cn-I,P,, = -1.25 X 10-d CIK~, an d 73f = -5.0 X 1OP cm-l. Also, from t,h e P an d Nbr an ches, B + B. + Z(R(a) = 10 .76 Ifi:0.02 cm-, B - Bo = -0.063 f 0.004 cn-I;from t he Q bra nch, U - Ho = -0.058 f 0.002 cm+. The spectru m of 2~4 ofCH, is extremely complex as a result of th e tet,rah edral splittings an d th eoverlappin g of th e five P, Q, and R bran ches. It is not possible to make definiteassignments for the observed lines at this time.

I. ISTRO DCCTION AKD SUMMARYA re-examination of the theory of spherical top molecules in recent years has

led to a thorough understanding of the tetrahedral fine structure of the vibration-rotation lines for the infrared active fundamentals of the methane molecule (1,Z).The explanation of the first overtones of the infrared active fundamentals, how-ever, has presented considerable difficulty. These have recently been studiedexperimentally in methane with very high resolution (S-5). The bands of interestare identified as 2~~ of CHI (at about 6000 cm-), 2~4 of CH, (at about 2600cm-), and 2v3 of CD, (at, about GO0 cm-j. Each of these bands arises from thepure overtone of a triply degenerate normal vibration of the molecule. Thereforewe would expect the spectra to appear qualitatively similar. However, they aremarkedly different. The band 2v3 of CH, (4) consists of single P, Q, and Rbranches, t,he structure we would expect to find in a fundamental. Similarly,2v3 of CD4 (5) consists of single P, Q, and R branches; however the tet,rahedralfine struct,ure splittings are observed to be large in contrast to t,he exbremely smallsplittings in the analogous band of CH, . On the other hand, 2vq of CH, ( 3) con-tains over 400 lines in a 200-cm- region with no apparent regular pattern; thenumber of lines in 2~4 camlot be accounted for by t#he number of fine st,ructureromponent,s for a single P-, Q-, R-band structure.

The paradoxical differences in the appearance of these spectra should be empha-sized. In the quantum mechanical Hamiltonian every term which contributes tothe energy of 2v3 has a counterpart which contributes to the energy of 2~~ . Themolecular parameters which enter as coefficients of these terms are different.However, this should produce only quantitative differences in the spectra of2~3 and 2~4 . Thus, if 2~:~consists of a single P, Q, and R branch only, so should2~4 . The splitting of each P-, Q-, and R-branch line into fine structure compo-nents may be too small to be observed in one case, while it may be very large andeasily observed in the other. However, one would still not expect t,he markedqualitat,ive difference in the appearance of the spectra of 2v4 and 2~:~ which isactually observed in CH4 .

Despite the apparent, paradox it has been possible to explain these spectra.It is the purpose of this article to give a full theoretical account of the overtonespectra.

The most detailed theoretical treatment of XY, molecules of t,et.rahedral


3/40

OVERTONES OF SPHERICAL TOP MOLECULES 383symmetr y ha s been given by Sha ffer et al. (6). They calculated th e ma tr ix ele-men ts of th e full vibrat ion-rotation Ha miltonian to second order of appr oxima -tion in pert ur bation th eory with out , however, applying th eir resu lts to th e ex-periment al spectr a of meth an e. Th ey also calculat ed th e relative int ensities forthe infrared active transitions. Their formulas for the relative intensities of thefirst overtones of th e tr iply degenera te fun dam enta ls ha ve been revised by Lou&(7). The energy an d inten sity calculat ions lead to a very comp licat ed th eoret icalovertone spectrum consisting of five different P, Q, nd R branches of differentspa cings but of rough ly comp ar able inten sity. Su ch complexity would be neededto account for th e observed spectr um of 2vq of CH, . On the other hand, we noteth at John ston an d Dennison (8) in th eir tr eatm ent of symmetr ical moleculespredicted a simple P, Q, str uctur e for th e overtones 2v3 an d 2v4 of tetr ah edra lX Yd molecules.

J ah n was th e first to ma ke full use of th e symmetr y an d group properties in ath eoretical att ack on th e vibrat ion-rota tion term s in CH, (9). In recent paper sHecht ha s exploited th e sph erical tensor forma lism to great ly simplify t he calcu-lations an d give a th orough th eoretical accoun t of th e fun dam enta l v3 of CH, .The present work is par tly an extension of this work to the first overtones of th einfra red active fun da men ta ls. However, it involves also modificat ions which canaccount for the seemingly pa ra doxical na tu re of th e experimen ta lly observedovertone spectra.In Section II is described th e basic form alism an d notation used. In Section IIIth e energy levels an d th e relative intensities for th e first overtones of th e infraredactive fun dam enta ls ar e comput ed in a way which is a na tu ra l extension of th etechniques employed in Ref. 1 for the fun dment als. The repr esenta tion which isused, denoted as th e conventional repr esenta tion (Rl) , serves as a good approxi-mation if the vibrat ional angular momentum quant um nu mber P is approxi-mat ely a good qua nt um nu mber for th e complete vibrat ion-rotation Ha miltonian .In this case the theory predicts five distinct P, Q, nd R branches of roughlycompa ra ble intensity. It is shown tha t th is th eory can accoun t for 2~ of CH,but definitely not for 2~3 of either CHq or CD4 .

The conventiona l repr esent at ion (Rl) leads to a good firs t app roximat ion onlyprovided th e separ at ion of th e E and Fz ibrat iona l componen ts of th e & = 2vibrat iona l sta te is sma ll compa red with t he splittings which a rise from th e2B,r i(P.li) ter m; i = 3 or 4 for 2~3 or 2~4 , resp ectively. In Section IV we sh owth at 2v3 of both CH, an d CD, can be accoun ted for if we assu me tha t t he sepa-ration of the E and Fz ibrationa l componen ts is very large. In th at case weintr oduce a new representa tion (R2) which serves as a good first approxima tionin th is limit and facilita tes th e calculat ions . The (R2) repr esent at ion form allyha s ma ny of th e ma th emat ical properties of th e (Rl) repr esenta tion for anti = 1 vibrat iona l sta te. The th eory predicts an overtone spectr um which a c-counts very well for the observed spectra of 2~3 .


4/40

A detailed quantitative fit of (RZ) to 213 of CD, is very successful. In thisovertone spectrum the tetrahedral splittings are large, so that effects in higherorders t,han those considered are negligible. The fit to 2~ of CH, is not as success-ful in accounting fully for the observed fine structure. The tetrahedral splittingsin this spectrum are accidentally very small so that contributions from higherthan third-order perturbations seemingly account for a considerable fract,ion ofthe total splitting.

I I . FORMALISMThe quantum mechanical Hamiltonian for a rotating-vibrating tetrahedral

XY4 molecule has been written by Hecht to third order in perturbation theory( 1) . We follow his notation. The dimensionless variables of the Hamiltonian aret,he molecule-fixed components (P, , , , -) f the total angular momentum P;the normal coordinates q1 , (e, f), (x3 , ~3 , za), (x.4 , ~4, z4) appropriate to thet,etrahedral symmetry; and their canonically conjugate momenta. The normalcoordinates correspond t,o t,he normal frequencies w1 wg , w3 w4 , respectively.Associated with the triply degenerate vibrational modes are the internal vibra-tional angular momenta 1, and 14

The zeroth order Hamiltonian is

+ $+d4(p42r42) &P2,where B, and the wi are in units of cm-. The usual contact transformation (10 )was made t,o remove from the first-order Hamiltonian all terms except thosehaving matrix elements diagonal in the total vibrational quantum numbers21 u2 , vLl v4 To third order of approximation, only quantities which are diagonalin these quantum numbers can contribute to the energy. Relevant. terms of theHamiltonian will be written in detail later on as t#hey are required.

A great simplification was made by Herht, by considering the terms in theHamiltonian to be built up from spherical tensors. To third order in the trans-formed Hamiltonian, the only linear combinations of tensor operators which cancont#ribute t)o the energies of 2~3 or 2~4 are the scalar operators T(O0) and aspecific linear combination of fourth rank tensor operators (70)?(-I-O) +.5[2(4 - 4) + T(44)]. In the conventional representation (Rl) scalar operatorscontribute only to the effect,ive B-, (B


5/40

OVERTONES OF SP HERICAL TOP MOLECULES 385

If th e vibrat ion-rotation wave fun ctions ar e also built up by an gular momentu mcoupling, th e Wigner -Ecka rt th eorm greatly redu ces th e work of calculatin gma ny ma tr ix element s to calculating a single redu ced mat rix element a nd manyvector coupling coefficient s wh ich usu ally ar e ta bula ted. (The vector couplingcoefficient in Eq. (I) is rela ted to th e Wigner 3-j symbol, extensive nu mer icaltables of which have recently been published (ll).) Also, when the vibration-rotat ion wave fun ctions ar e classified according to th eir symmetr y un der Td ,th e ten sor oper at ors of type A, can connect only sta tes A1 to sta tes Al , AZ to Ax ,E to E, F, to F, , and Fz to Fz . Thu s th e energy ma trix is always factored inthis way.

We fur th er note th at when t he electr ic dipole moment opera tor is written a s aspher ical t ensor opera tor, th e relative intensities may be calculat ed in th isformalism.

I I I . THE CONVENTIONAL P RES ENTATION (Rl)A. GENERALREMARKS

In th e conventional repr esenta tion (Rl ) we consider P, which gives t he m agn i-tu de of th e total vibra tiona l an gular m oment um 1 = 1, + 14, to be appr oxi-mat ely a good qua nt um nu mber for the complete vibrat ion-rotation Ha milton-ian. This is a good appr oxima tion if th e tetr ah edral splitt ings, including th osewhich ar ise from t he pur e vibrat iona l Ha miltonian , ar e sma ll compar ed to th eseparations due to

H1 = --2B,[t~(P .b) + rO.14)1.The rotation-vibrat ion wave fun ctions for any stat e ma y be formed from linear

combina tions of pr oducts of rota tional with vibrat iona l functions:

Her e fiJ K is an eigenfun ction of P2 an d P, , 4 [,,,is an eigenfun ction of l2 an d 6 ;an d R = P - 1.Matrix elements of the operators T(kq) between states &Kg take the followingform after su ccessive app licat ions of th e Wigner -Ecka rt th eorem:(uCJRK~ 1T&q) j vWRKR) = (RkK,q / RkRK,)[(2k + 1)(2R + 1)112

. ( _ 1 )h+C+i (~8 jj Tvib(JC1))I ~8)(J II7,0t (M I ). (3)

This last equat ion is subject to th e condition th atT,ib(lClq,)* = (-l)*Tvib(h - 91).

Th e factor ( - l)k+L+C in (3) is not usua lly present (id), an d ar ises becau se ofth e complex conju gat e signs in (1) an d (2). (These comp lex conjugat es ar ise


6/40

386 FOXessentially becau se we wish to use th e usu al angular momentu m addition co-efficients in a scheme which involves s ubt ra ction of an gular moment a: R = P - 1;compa re Refs. 1 an d IS.)

We n ow specialize to th e cas e of th e first overtone 2~4 . [2v4 of CH , seems to beth e only one of th e ban ds of int erest which is cons isten t with (RI) .] Then & = 0,while & ma y tak e th e values 0 an d 2. As basis fun ctions for th e irr educible repr e-senta t,ions of Td , th e % = 0 vibrat iona l function belongs to Al while th e % = 2vibrational functions belong to E and FZ .

We ma y write th e vibrat iona l wave fun ctions as pr oducts of one-dimen siona lha rm onic oscillat or fun ctions depen ding on .r4 y4 , and 24with quantum numbers24.z , u4y , an d 214, resp ectively. The functions ar e of th e form 4(~4~24,2)~~)&14&qu4y)+(2142); 214 = v41 + v4y + 04, . In terms of these suitably normalizedfun ctions, th e vibrat iona l an gular moment um eigenfunctions ar e

400 = (3)-%(200) + +(020) + 4(002)1;&a = (2)-l{ +5(llOj + (2)-2[#4200) - f#l(O20)]),421 = (2)-[il#J (oll) - $(lOl)], (4)420 = (6)-12[4(200) + 44020) - 2+(002)1,

and $V--m . ( -l)&% .In (2) and (3) the quan t,um num bers (Pm) become (&m4).The linear com-

binations of *EKE which tr an sform as basis fun ctions for th e irreducible repr e-sentations of Td follow from t he symm etry properties u nder th e full rotat ion.inversion group and its subgroup Td . These linear combina tions were first,worked out by Jahn (9) for R 5 10, an d extended by Hecht (1) t o R 5 13. Theclassification of wave functions according to th eir tet ra hed ra l symm etr y res ult sin th e ma ximu m factorization of th e energy eigenvalue determ inan t, since th eHa miltonian can not connect sta tes tiRKR f different sym metr y.

In the next section we calcula te th e ener gies of -3~4 o th ird order usin g (RI j.For fixed J, th e eigenvalu e of HO is 6(2J + 1 )-fold degener at e (excluding th edegener acy from t he space-fixed Z-component of total an gular moment um in th eabsen ce of extern al fields). Th is degenera cy is par tly r emoved by HI. The eigen-value corr espondin g to th e 44= 0 vibrat iona l sta te is ( 3J + 1)-fold degenerate,while the eigenvalues for the P4= 2 sta tes ar e (2J + 5)-, (2J + 3)-, (2J + l)-,(2.~ - l)-, a nd (3J - 3)-fold degen erat e for R = J + 2, J + 1, J, J - I, andJ - 2, resp ectively. The ener gy levels deter min ed by (Ho + HI) are perturbedby the operat ors of (Hz + Hi). Operators of rank 0, i.e., scalars, contributeterm s to th e energy which a re independent of K, and K,. Thus, for fixed P4J, and R, all sta tes regard less of th eir tetra hedr al symmetr y ha ve their energyshifted by the sam e am oun t. The cont ributions of th e scalar opera tors can beinclud ed with th e contr ibut ions from B,P2, -2L3,[4(P.14), an d -D,P4 by de-


7/40

OVERTONES OF SPHE RICSL TOP MOLECULES 387JJ+2

,4e AlzA2,

/ w E,....J,U,=2) / c JJ+I _s.p--- _ JJ ( c\\\ 1, JJ-I\ r\\ , JJ-2 _

HO + Hi + H;+HiF I G . 1. Splittin gs of th e ener gy levels of 2z~(& = 2) du e to successive pert ur bat ionterms in th e Hamiltonian; (RI).

fining effective rotational constants. On the other hand, operators of rank 4cont ribut e term s to th e energy which depend on KR and K,. Therefore, forfixed b , J , and R , sta tes of different t etra hedr al symmetr y will ha ve th eirener gy shifted by different a moun ts. However, th e cent er of gravity of a givenlevel will not be affected by the fourth rank operators. The effects of successivepertu rbat ions described a bove a re sum ma rized schema tically in Fig. 1, where wehave taken % = 2. The r elat ive intensity form ulas for t his case ar e derived inSection C. In th e last section, th e th eoretical predictions ar e compa red with th eexperimentally observed overtone spectra.B. ENERGIES

1. S calar Perturbations and t he Effective R otational Constan tsThe effective rota tional cons ta nt s for 2~4 ar ise from th e term s (I)

H (scala r , J Q) = H o - 2B,{4(P.14) - D,P4 + &,0pp44(scala r )+ F 4,P2(P.14) + MY&12 + 912) + Y&22 + r 22) + Y3(p32 + r32)+ Y4(p42 + r42)]P2 + [M411(p,2 + 412) + M422(p22 + r22) + M43dp 32 + r 32)+ M444(p42 + r42)1P .14),

wh ere 0pp 44(scalar ) = $5[(P.r 4)* + (P.p4) - %P(p4 + rb)]. If we comp ar e0pp44(sca1a r) with Eq. (1) we see th at th is scalar opera tor (k = p = 0) is th econtraction of a vibrat iona l tensor operat or of ra nk 2(k1 = 2) with a rotat iona ltensor operator of rank 2(k2 = 2). The required matrix elements diagonal inl4 ar e given in Table I. The only difficult ma tr ix elemen t is th at of Opp44(sca1ar )

44 84 2(v411JRKR jOppu(scalar) 121484 RK,) = [5(2R + 1)11J J 2r 1R 0 (5)

. (_lp+l4 (~4 -e4 I Tvib (2) 11 ~4 &)(J 1)Trot(z) IIJ>*


8/40

388 FOX

- .3CN


9/40

OVERTONES OF SPH ERICAL TOP MOLECULES 389The 9-j symbol red uces t o a 6-j symbol (IS) ; the reduced matr ix elements a regiven in (1). For & = _e4(u4 2), Eq. (5) gives-7{3[J (J + 1) - R(R + 1) + 4(4 + 111

.[J(J + 1) - WR + 1) + P.44 + 1) - 11 (6)-4J(J + l)&(& + 1))/12(2& - 1)(28~ + 3).

Note that (6) vanishes for L = 0.We also note t ha t 0P P44(scalar ) ha s nonzero ma tr ix element s for 8~ = 0 an d

% = 2(u4 = 2). In th is case Eq . (5) gives-%[(2J - 1)(2J)(2J + 3)(J + l>lS,, ,

where 8J R s a Kronecker delta. This t erm will appea r as an off-diagona l elementin the energy matrices for 2~4 . It does not cont ribu te to th e effective rota tionalconstants.

The eigenvalues of the scalar operat ors ar e written in Table I in a form whichsh ows th eir contr ibut ion to th e effective r otat iona l consta nt s. We denote avibrat ion-rota tion level of 2~4by JR . Tran sitions to such a level from th e ground-sta te rotat iona l levels of total angular m oment um given by J + 1, J, and J - 1give rise to the P, Q, and R bra nches, respectively. The ma tr ix element s ofOppJ r(scalar ) cont ribu te only to th e effective B- an d (B{q)-values. The ma tr ixelement s of P(P.14) contr ibut e, in ad dition, to th e effective D-values. The con-tr ibution of - D,P4 to th e energy difference between th e upp er an d ground sta tesis 4D,( J + 1)3, 0, and -4D,J3 for the P, Q, and R bra nches, resp ectively. Thema tr ix element s of P2(P.14) ar e factored in two ways in Table I from which t hecont ribu tions to th e effective D-, B-, an d ( B14)-values can be seen for t he P- andR-bran ch lines, respectively. The left-over cons ta nt s of Table I can be in-cluded in the pur e vibrat iona l energy.

The effective r otat iona l cons ta nt s a re given in Table II for th e P branch.T h e Q - an d R-bra nch values are related to these in a simple way. For th e Rbranch, the Beff and Deff are the same as those for the P branch except that the

TABLE IIEFFECTIVE ROTATIONAL CONSTANTS FOR 2vl (& = 2); (R l). P-BRANCH VALUES

R B eff (Bldeff Deff


10/40

order of th e R-valu es is rever sed. Th e (B{4)eff ar e common to all bra nches. Forconvenience, we ar bitra rily ta ke Beff for t he Q bran ch to be the sam e as for t heR branch, and add the appropriate J3-dependent term to the energy of the stateinvolved.

In compa ring th ese resu hs with th e observed spectr um 2~4 of CH, , we findt,ha t it is not p ossible t o distinguish am ong th ese effective rotational constan ts,so th at t,he ambiguity in the Q-bra nch constan ts cau ses no difficulty. To get anidea of the differences between effective rotational constants we estimate Z,,and FJ s from their explicit t heoret ical expr essions (1) a nd from th eoret icalestimates of cubic potential constants (14). We find ZdSE -0.03 =t 0.05 cm-and Fq, E 0.003 f 0.003 cm-. The un cert ain ties ar ise from th e am big&y insign of certa in molecular par am eters, a nd are not sta tistical err ors.

2. Tensor Perturbations and the Tetrahedral SplittingsThe tetr ah edr al splitt ings of th e ener gy levels deter min ed by (Ho + H,) ar e

given by matrix elements of tensor operators of the form(70)T(40) + 5[T(4 - 4) + T(i$)]. (7)

For sta tes in which only vibrat iona l qua nt a of vaar e excited, th e split.ting pat ,ternsare determined byH(4, vq) = -D 0 pppp(tensor) + F4tOPPPdtensor) + &OPP44(tensor) (8)+ NM~OP M( ens or) + Td40h 4(ensor j.Ea ch of th ese tensor operat ors is of the form (7)) an d is built u p from a vibra-tiona l ten sor of ra nk k , an d a rotat iona l tensor of ra nk k2 . In the order in whichth ey appea r in (8), th e tens ors ha ve (k,, X-2)equal to (0, 4), (1, 3), (2, 2),(3, 1)) an d (4, 0). Th ese operat ors ar e given explicitly in Ref. 1.

Between sta tes with vibra tiona l an gular momentu m Pq an d P4 ma t,rix ele-men ts of th ese opera tors ar e zero un less %, t4 , and kl satisfy the triangle in-equality of quan tu m vector addition. Thu s in th e ground sta te, with P4 = !d = 0,t)he tensor split t ings are determined by OPPPP nly. In the funda mental v4 withIa = L = 1, kl ma y be 0, 1, or 2. For the overtone 2vq , there are three possi-bilities :

( 1) !4 = % = 2, all five opera tor s give nonzer o ma tr ix elemen ts ;(2) 1)4= P4= 0, only OPP PP ives nonzero ma tr ix element s;(3) C = 0 an d 44 = 2, onIy 0PP 44 ives nonzero m at rix elemen ts.Accord ing to (3), H (4, ~4) can conn ect only st at es &,& an d J /RKRor which

K, - K, = 0, ~4 an d for which R, R, and 3 satisfy the triangle inequalityof qua nt um vector add ition. Also H(4, vk) can conn ect only stat es of th e sam etetr ah edra l symmetr y. Any ma trix element will consist of a linear combinat ionof vector coupling coefficient s, which con ta in th e en tir e KE an d K, dependence,mu ltiplied by a function of J, R, and R (an d of & an d t4 , implicitly). The


11/40

OVERTONES OF SPH ERICAL TOP MOLECULES 391nota tion scheme for these ma tr ix elemen ts follows Ref. 1. The n onzero ma tr ixelement s m ay be writt en as follows:

(2~4, L = 2; JRKR j H(4, IQ) j 2~4, 84 = 2; JRK,) = fm(R, R)x,where

x = (R4KR0 1R4RK,) for KR = K, ,x = (xb)*(R4KE(f4) [ R4R(K, f 4)) for K, = KR A 4.

Table III . f2J (RjR)=P2J (R,R) { Ctij4fiJ k( J ,R,R))

R R- - f2J (R,R)

Jf2 J+2 25+6)(25+7)(25+8)(25+9) J +l) (2Jf2) (2J+3) j2J+4r ] ( $2J)(2J-1)(2J-2)(2J-3)t044-(2J) (2J-1) (2J-2)t134 -(2J ) (2J-1)t224+2(2J) t314+2t404 1

J+l J+2 5(2J+5)(2J+6)(2J+7)(2J+8) {(2J -1)(2J -2)(2J -3)tOQ4J +l) (2J+2) f2J +3) (2J+4)]+3(~J -1)(2J-2)tl34+(2J -1)(J-2)t22~-(3J -2)t3l~-2t~O~~

J J +2 15(2J+5)(25+6)(26+7)J+lJ(2J+2)(ZJ+j)] {3( 25-2) (2J -3) t 044+ & 25-2)(2J-5)t13b-;(2J2-13J+12)t22~+(2J-3)t314+2t404J-3)t31~+=&

J - 1 J +2 [w] ( 6(2J -3)$+,++3(3J-5)t,34-2(J -2)t224-(J -3)t31q%+,,4j

J-2 J+2 [ ,w] i 6t0 44+6t1 34-2t224 -2t314+2t i+04~

J+l J+l 25+5)(2J +6)(25+7)(25-1)J)(2J+1)(2J+2)(2J+4) 18 {( J 2+2J -20) (25-3). (2J-2)t044-2(J+12)(J -2)(2J-2)t134+2(J2-l2J +12)t22L)- +( J-2) t314-8t404]

J J +l

J -1 J + 1

30(25+5)(25+6)(2J -2)_J ) (25+1)(25+2) ] i (2J -5)(J +4)(2J -3)t,/+/++$(2J 3-3J2-53J+84)t134+ $ (2J2+23J -54)t224+( J-5) t314+f2t40/+}

[ (210(25+3)(2J+5)(25-3) J)(25+1)(2J+2) 1 ( 6(3J2+3J-20)t044+3( 3J 2+3J-28) t13&-( J 2+J -18) t224+10t314-4t40& )


12/40

Table III. (concluded)R 5- f2J (R,R)

J-2 J+l [ -1 {6( 2J +5) tO44+3( 3J+8) tl34-2( J +3) t224-( J+4)t314+2f404}

J J (2J +4)(25+5)(25-3)(2J -2) J ) (2J+2) (2Jf3) (2J-1) ] c (4J4+8J 3-l19J 2-123J+630) to442-12( 4J 2+4J-33) t134+(4J +4J-75) t224- 36t314 +12t404 I

J-l J 30(2J+4)(2J-4)t2J-3) {(25+7)(25_3)(25+5)tJ-l)(2J)(2J+2) 1 044+$(2J3+9J2-41J-132)t134- a2J-25)(J+3)t224+(5+6)t314-2t404)

J-2 J [WI+ \ 3(2J+4)(2J +5)t044+ ;(2J+7)(2J ++)t13Q- ;( 2J 2+17J+27) t224-(2J +5) t314+2t&04 )

L2J+3)(2J-5)(2J-4)(2J-3) *J-l J-l J)(25+1)(25+2)(2J-2) 1 fJ2-21)(2J+4)(2J+5)to44+2(J-ll)(J+3)(2J+4)t134+2(J2+l4J+25)t224+8(J+3)t314-8t404}

J-2 J-l 5(2J -6)(2J -5)(2J -4) J)(25+1)(2J-2) ] 1 2J+3) (2J+4) (2J+5)t044+~(J+7)(2J +3)(2J+4jt~34-(J+3)(2J +3)t224-(3J+5)t3~4+2t4o4~

J-2 J-2 L2(2J-7)(2J-6)(2J-5)(2J-4)J ) (2J +l) (2J -1) (.2J-2) ] 25+2)(25+3)(2J+4) .

(2J+5)to44+(2J+2)(2J+3)(2J+4)t134-(2J+2)(2J+3)t224-2(2J+2)tq,4+2t404j

For P4 = 0, f4 = 2 an d & = & = 0 ap pr opriat e qu an tit ies hs J (R, R) an db&R, R j, resp ectively, ar e defined. These ar e t,he an alogs of .fzJ (R, R 1.

It wiI1 be convenient to use th e following nota tion:t&R, R) = y.,,( R, R){ c MijdJ, R, RI\,

&,h i = k, and j = ke ; an d wh ere, t o Ohird order of appr oxima tion toti = --n, ,t134 Fa , tm = Z 4 t , t a 4 = N 4 4 4 t , f 4 0 4 = TA4 The jzJ(R,R) are given in Table III.

In the case of hsJ(R, R), P4 = 0 implies R = .J. The quantit ies hd J, R)are given in Table IV.

Th e quantit ies k?,(R, R) become tjhe single quan tity k.sJ ( , J ) becau se


13/40

OVERTONES OF SPHERICAL TOP MOLECULES

Table IV. h2J ( J ,R) = j2J ( J ,R) { t224h 224( J j J jR)}

FL hzJ(J,R)J +2 [2(25+5),(25+1) ] (-(2J)(2J-l)t224}

J+ 1 [2(2J+3)/(2J+l) ] 1 (J +6) @J -lb,,4 }

J (2)+ [(2J-3) (2J+5)t2,b)

J-l [2(2J -l)k(2J +l)] + ( (J-5)(2J+3)t224j

J-2 [2(2J-3)/(2J+l)] + { -(2J+2)(2J+3)tz243

393

& = & = 0 imp lies R = R = J. Sin ce (v&J II!Zvib(O)ll &) = (244 + 1) is in -dependen t of v4, th e splittings for this stat e ar e th e same as th ose for th e groundstate, i.e.,kz.r(J, J) = %[(2J - 3)(2J - 2)(2J - lj(2Jj

. (2J + 2)(2J + 3)(2J + 4)(2J + 5j]2t044.Tables II I a nd IV an d th e above equat ion give th e J dependence of all possiblemat rix element s. It rem ains to compute th e specific linear combinat ions of

vector coupling coefficients (R4KRq I R4RK,) required by the tetrahedralsymm etr y for th e possible A,, AB , E, FI , and FP substates of each level. Most ofth ese ar e given in Table VIII of Ref. 1. (The quantities actually tabulated therear e th e app ropr iat ,e linear combinat ions of vector coup ling coefficient s mu ltipliedby the functions g&R, Rj, ra ther than g&R, R).) The only linear combina-tions of vector coup ling coefficients deter min ed by symm etr y which, in effect,ha ve not been previous ly calcula ted in Ref. 1 are those arising from matrixelements between states for which R - R = 3 or 4. In Table V are listed thesequa nt ities m ultiplied by th e appr opriat e g&R, R), through J = 6.As an example, t he complete energy subm at rix for t he FZ states of J = 3 isgiven in Table VI. All relevant scalar an d tensor cont ributions ha ve been in-


14/40

394 FOXTABLE V

g, (R',J Z)timesthelinearombinationf vector couplingcoefficientsequiredby yinmetry;' =J-1, -2.>

-Eyy 35 E F1 E2(1) F2(2)32 -G/J21 2o/J212c )2/b 3146 *l *? IT Fl F2(1) F2C2)

43 56/3fl4 -70/3f77 322/9&4 -70/3f7042 100/3~55 65/9/11 25/3&

5, *I E F (1)1 F,(2) F (1)') F2(2)L * L54 6O/r'286 55/H/39 G/i33 10/U/26 210//6006s3 60/11/2 120/111/39 15/r/33 45O/llr'2730 50/7/1430

% Al E(l) E(2) Fl(l) Fl(2) F2(1)

F2(2)

lo/d6 30//210

65(l) -240/1342170 -660/1/26598 -330/13J154 0 15/13/3 -165//5005% (2) 440/13h10 --Llo/J2oo2&4 60/13 9660/13Jj5805 30/'/403 105/13/35 15Nl3 15//22 105/1/2730

Note: a/b/c means a/(bk).

eluded. The term s proport iona l to Fason the matrix diagonal arise only for the Qbra nch, a s explained in Section III.B.l. The pur e vibrat iona l scalar cont ribut ions,except th at from G4J d2, re collected int o a single ter m E. No distin ction ha s beenma de between th e various effective rotational constan ts. The tij4 ha ve been ab-breviated to ti . It is clear that even for low J values, the energy matrices becometoo lar ge to diagonalize except by m ean s of a high -speed au toma tic digita l com-pu ter . Before discussing the actu al nu mer ical calcula tions, we pr oceed to cons iderthe relative intensities.C. RELATIVEINTENSITIESANDSELECTIONRULES

The relative intensities in an infrared a ctive ba nd a re proport iona l to th equantity


15/40

TableI.Energyub-matrixfor2tatesf=3;RI,).

&O 3F p'=2

-%l)

35F2

(2)

35F2

34F2

33F2

31F2

;+12B-144D

+U3Ot,

00/-/910)(-30t2)

-(21/2/7)(-7a2)

+2

E& +12(Br;)-144D-54F4s

t(90t0-120t1-30t2

+G!t2t4)

(5/7/35)(90t0-=otl

-30t2+12t3+2t4)

(10/&0)(60t0+10t1

+5t,-7t3-zt4)

-(10/J14)(36to+6t,

+3t2+3t3+a4)

(5/3)(d6)(6to+6tl

at,-2t3+2t4)

I-=-

E+6G44+12B

+12(B$$(pj + r i)

-I&P* + F3 ,P2(P .13) + C Y&S(pi2 + r i2)P2 + C Mdpi2 + ri? (P.15)+ .&Opp33(scalar) - ~~~~~~~ tens or) + FztOPP P3(ensor)

+ &Op~33( tensor) + N333&333( tensor).Since Opppp(tensor) is independent of vibrat iona l coordina tes, its mat rix

element s a re the same as for v3 Oppp3( ensor) depends on vibrat iona l coordinatesonly th rough componen ts of 13 As a result, its matrix elements are the negativesof those for US

The opera tors 0ppz3 (scalar ) an d 0ppS3 tensor) ar e built up from ~2~ an dsecond-ran k rotat iona l tens ors Pzm ; 0p333 s built up from TQ, 4lq and PI, . Forth e vibrat iona l wave functions of th e fundam enta l we ha ve r2&lm =- (5/3)12(211wn I 211(m f ~)h(~+~) . Ther e is no such simple relation forrzp$hn * Therefore th e ma tr ix element s of th e opera tors in question can not bereadily valuat ed by applicat ion of th e Wigner-Eckar t th eorem. But we candefine an operator p 2 P : p 2 h 2 = ~ - 2 ~ 2 ~ 2 ~1 = - n f l , ~20 = r 2 0 ; and thenP2pdh = +(5/3)12(21pm I2I I(m + P))&+, .Next we express 0ppX3and 0pSS3 n terms of p z P instead of rzp . For the 0~~33operat ors we pr oceed as follows: The ten sor T(kq) is defined by

T(kq) = c ~;aP2@43 I=kq).lzIf we also define

&.33 (sca lar ) = ( 5)12T( 00))


28/40

andQppaa tensor) = (70)2T (40) + 5[T(4 - 4) + T (444)],

we find that

and0ppZ3 scalar ) = I/$ppa a (tensor) - jJQP PBHscalar),

0 pp3 3 ten sor) = f@;l~~~a tensor) + ( 24/5)GpS3 (scala r ).The ma tr ix element s of !G&~ (scalar ) an d c&~ (tensor) will be th e negatives ofth e mat rix element s for vx of Opp33 scalar ) an d 0ppZ3 ten sor), resp ectively.Pr oceeding in a similar way for OP sas ten sor) we define

T(4p) = c vk P@(3W 1314p),and

T (00) = c VT, P ,~(llf&? 11100)o(where

VlkT = c P2&tl21UE ( 21W).If we also define

andxPnsnscalar) = 3(5)12T(00),

xp333 ten sor ) = (70)9+(40) + 5[T(4 - 4) + T/(44)];then

0~333(tensor) = (8/5)x~333 (scalar) + %x~333 (tensor).The m atr ix element s of xpa33 scalar ) ar e simply related t o th e ma tr ix element s of(p: + rt ) (P.1,). The ma tr ix element s of xP3a (tensor) ar e identically zero for(R2) becau se th e tr iangle ineyuality is not sat isfied in th e vibrat iona l par t of th ematrix elements.

It is importan t to note th at th e effect.ive rotat iona l constan ts will now cont ainth e tensor par am eters ZSt an d N,,, , while th e tens or pertu rba tion term s willconta in th e scalar pa ra met er ZZj6 The effective r otat iona l const an ts ar e givenin Table VIII.

The matrix elements which determine the tensor splittings depend on K, andK, only th rou gh th e vector coupling coefficient s (L4K,q 1L4LK,), whereq = 0, +4. The linea r combin at ions of $ LKLgiven by the tetrahedral symmetrydeter min e t he linea r combin at ions of vector coup ling coefficient s which occur.


29/40

OVERTONES OF SPHERICAL TOP MOLECULES

Table VIII. Effective rotational constants for 2g3(..$=2); (R2).

409

Rotational COnStant Effective ValueBeff L=Jtl $++(Y1+2Y2+7Y3+3Y4) -2F3&/15) (Z3s-z4z3t)

Beff L=J Be+B(Yl+2Y2+7Y3+3Y4) - F3s+(2/15)(Z3s-24Z3t)

(B


30/40

410 FOXTable IX. f,J(L',L)=g,J(L',L) {-DtfoQ4( J ,L,L)+F3t f13~(J ,L,L)

+V3$22&( J ,L,L) )

L' L flJ(L',L)-J+l J +l ~2~+7)(2J+6)(2J +5)(2J +4)(2J )(25_1)/(25+1)(2J +2) 1

. {-;(2J 2- 5J +3)Dt+( J -1)F3t ++ r3,}

J J t l +(zJ+~) (2~+5) (2J+4) 2J +3) (25-l) (2J -2)/2(2J +l) (2J+2)] . [(2J -3)Dt+$( J -3)F3t ++r3t }

J-l J +l b(25+5) (2J +4) (2J+3) (25-2) (2J-3)/2(2J +1) 1 fl-3D,+3t+$ r34

J J [(2J+5) (25+4) (2J+3) ( 2J -1) (25-2) (2J -3)/(2J ) (2J+2) 1 . {-( J 2+J -10)Dt+4F3t- 3t ]

J - 1 J -[ 5(2J+4)(2J +3) (25-2) (2J -3) (2J -4)/2(2J ) 1 +.{(2J +5)D,+t(J+4)F3,-3 f3%)

J-l J-l k2J+3)(2J+2)(2J-2)(2J-3)(2J-4)(2J-5)/(2J)(2Jtl)] . [-+(2J2t 9J +10)Dt-( J +2)F3t ++ y3t}

C. RELATIVE IN TENSITIE S AND SELECTIONRULESWe calculate th e relat ive intensities for (R2) by using th e tr an sform ed electr ic

dipole momen t. To second order of app roximat ion, (f 1 L j i) becomes


31/40

OVERTONES OF SPH ERICAL TOP MOLECULES

TABLE Xm e r g y s u b - m a t r i x f o r F 2 states of J = 3; (IQ.).

411

J L 34Fz 33F2 32F2J L 12Beff-144Deff

-6(Bg3)eff34F2 - (390/7) ( -3Dtt 2F3t t+r 3t)

33F 2 150( 7)-3(.3D&63i)

(60/7) (5) s( -3Dt-BF gt , -lClO( 35) -fr( 1lDt 12Beff-144Deff32F2t93t ) +$F3t-frll j t)

te (BJ 3) e f f-(16/7) (-27$Dt -5F3t t @3t

wher e +(OOO) is th e groun d-sta te vibrat iona l wave fun ction. The cons idera tionsof Section III . D sh ow th at th e contr ibution from [IT;, S,] can be expected to benegligible.

Explicit calculation gives(r $ - ?@, & + ~Zl)~(O~)~J K = ad 2 c d41*~.m,J

where a is defined at th e end of Table VII provided we interchan ge subscripts 3an d 4. Then (20) becomes-adZ(JlMO 1J lJ M) c (lJ ( --q)K 1 J LK,) (lJ ( --q)K 11J'JK)P

= -a&( J lMO 1J l JM)&, ,us ing th e orth onorm ality of th e vector coup ling coefficient s. Fina lly we find theexpression analogous to (15) :

MT I Cf I IL I i> I2 = 2a72J + 1). (21)This relative intensity factor is th e sam e as th at of th e fun dam enta l ~3

The infrar ed selection ru les ar e AJ = 0, f 1; AM = 0, f 1; AL = AK, = 0.These selection ru les ar e ident ical with th ose for v3 if we int ercha nge R and L.As in v3 , there is a single P, &, and R bra nch. Also, tr an sitions ar e allowed onlybetween s ta tes of th e sam e tetr ah edral symmetr y. Moreover, th e selection r uleAK, = 0 implies a fur th er restr iction. That is, if th ere are several initial a nd/or


32/40

412 FOXTABLE XI

OBSERVED FREQUENCIE S OF 2~~ OF CD, CORRECTED FOR THEORETICAL GROUNDAND FINAL STATE TENSOR SPLITTING PERTURBATIONS

FinalObserved frequency state(cm-l) pert.(cm-)

4476.121 0.144475.92 -0.06

.69 -0.274475.98

.98

.96v 4475.98

4469.9G 0.33.76 0.15.64 0.04.25 -0.34

4469.63.Gl.60.59

a v 4469.60

4463.55 0.37.35 0.19

4462.76-0.39.69 -0.45

4463.18.I6.15.14

la v 4463.16

4457.15 0.51.08 0.45

4456.95 0.31.17 -0.46.07 -0.56

4455.96 -0.66

4450.64.63.64.63.63.62

nv 4456.62

4450.73 0.68,631 0.00

4449.611-0.44.47 -0.57.39 -0.66.23 -0.81

-0.01 4450.04-0.01 i .02

0.00 .050.00 i .040.01 .OG0.01 .05

av 4450.04

4444.31 0.92 -0.01 4443.38.22 0.86 -0.01 .35.21 0.84 -0.01 .36

4442.83,-0.55 0.00 .38

-!roundstatepert.(Cm-) Observed frequency(cm-1 j Finalstatepert.(cm-)4442.65 -0.57

.43 -0.95

.40!-o.Q9

4437.74i I .OQ.71 0.97

4436.04 -0.68.OO -0.73

4435.83 -0.84.58i-0.96.50;-1.16.44,-1.25

0.010.010.01

0.02-0.020.000.010.010.000.020.02

0 .03-0.03-0.030.000.000.010.020.020.03

4443.23.39.40

.v 4443.35a

44%. 63.72.72.74.68.54,123.il

v 443li 67

4431.23 1.35.23 1.38.23 1.37

4429.23 -0.84.16 -0.86

4428.83 -1.07.67 -1.24.59 -1.38.51 -1.45

4429. x5.82.834430.07.024429.9193I9.!I!~,v 4429.95a

4500.47 0.00 4500.47

4506.35 0.00 4506.35

4512.12 0.03 4.512.09.08 -0.01 .09

4517.85 0.10.74 0.01.59 -0.13

4517.75.73.72

a 4517.74-

hundstatepert.(cm-)


33/40

OVERTONE S OF SPH ERICAL TOP MOLECULES 413

FinalObstm~dfr~puenc statepert.(cm -)

R(4)AiF,EF?

R(5!,,F2F,EFp

R(6)EFi2A*(1)F2F,A,

B(7)FIF!A-lF2ElF (2)2

R(9)F:F(l)1

4523.56 0.27.45 0.18.37 0.12.05 -0.17

4529.11 0.424528.97 0.32

.45 -0.17

.35 -0.24

4534.47 0.67.47 0.64.33 0.56

4533.63 -0.11.44 -0.28.38 -0.45

4540.10 1.01.06 0.97

4539.18 0.144538.95 -0.06

.80 -0.20

.47 -0.56

4545.57 1.62.57 1.53.57 1.40

4544.36 0.31.03 0.03

4543.53 -0.52.39 -0.68

4550.99 1.93.99 1.92

-c,

_

-

TABLE XI-Continued;roundstatepert.

:Cm-l)

-0.01-0.010.060.000.010.01

-0.010.01

-0.010.00

-0.010.010.01

-0.02-0.02

4523.29.27.25.22

v 4523.25

4528.69.G5.62.59

v 4528.64

4533.80.83.77.74.72.83

v 4533.77

4539.08.08.04.Ol.Ol.04

.v 4539.05

4543.944544.05

.l6

.054543.994544.06

.08IV 4544.05

4549.04.05

Finalstatepert.(cm -)

E(3)F*ASF;?FjA,

Q(1)F1

QWF,E

Q(3)A2F,Fl

Q(4)F?FlE92

Q(5)F(")ElF,(1)F1

Q(6)A2F2(1)FtA1,j?'E

4549.53 0.60.44 0.46.13 0.25

4548.52 -0.47.29 -0.70.lO -0.93

4494.35 0.00

4494.17 0.06.02 -0.07

4494.11 0.374493.81 0.09

.54 -0.17

4493.G9 0.46.69 0.46.27 0.02

4492.68 -0.51

4493.27 0.66.06 0.454492.47 -0.164491.98 -0.63

4493.01 1.104492.95 0.82

.38 0.504491.73 -0.21

.17 -0.74

.04 -0.85

-

statepert.(cm-)0.000.000.010.010.020.02

av

av

av

BV

at

a1

-

4548.93.98.89

4549.00.Ol.05

4549.00

4494.35

4494.11.09

4494.10

4493.74.72.714493.72

4493.23.23.25.19

4493.23

4492.61.61.63.61r 4492.62

4491.914492.134491.88

.94

.91

.89T4491.95


34/40

414 FOXTABLE XI-Continued

FinalObserved frequency state(cm-) pert.(cm-)-

4492.32 1.354491.98 0.96

.73 0.644490.98-0.16

.22 -0.934489.65-1.32

4491.73 1.67.50 1.46.13 1.01

4489.98 -0.214488.81-1.31

.74 -1.37

.49 -1.60

4491.50 2.31

- (

--

-

:roundstatepert.(cm-*)

0.010.010.000.00

-0.01-0.01 Ia

Correctedfor pert.(cm-)

4490.984491.03

.09

.14

.144490.96

, 4491.05

0.01 4490.070.01 .050.01 .130.00 .19

-0.01 .ll-0.01 .lO-0.01 .08

av 4490.10

0.02 4489 * 1

Observed frequency(cm -1)

F2 (2) 4491.13 1.95A:z 4489.98490.72 0.93.57

El's' 4488.9481 -0.220.33FZ1) 4487.55 -1.66F(l) .23 -1.91

4491.13 3.254490.72 2.714489.98 2.074488.34 0.364487.76 -0.18

.3B -0.524485.77 -2.20

.77 -2.17

.61 -2.30

-

-

statepert.(cm-)0.020.000.010.010.00

-0.02-0.02

0.020.030.020.010.000.00

-0.03-0.03-0.03

_-

a

a-

Correctedfor pert.(Cm-)

4489.20.15.06.17.14.19.12

v 4489.16

4487.904488.03-1487.93

.99

.94

.88

.94

.91

.88v 4487.93

final sta tes of th e sam e tetr ah edral symmetr y, only t hose tra nsitions ar e allowedwhich sat isfy AK, = 0.

Since (21) is indep enden t of th e I, an d K, quant um nu mbers, i ts value isindependent of th e tetr ah edral sym metr y of th e initial an d final sta te. The rela-tive inten sity of a single line in a vibrat ion-rota tion ban d su ch as 2v3 for a nytransition J, ---f JJ, is then

g(2J + 1) exp [-B,J(J + l)/kT](2a2),where g is the nuclear spin statistical weight factor.D. COMPARISO N OF (R2) WITH EXPERIMENT1. 2~3 of CD,The ban d of CD4 at 4500 cm- is ident ified as th e overtone 2~9 of th e infra red

active fundamental ~3 The observed tetr ah edral splittings are appr eciable, an dconst itut e a str ong t est of th e th eory.

The ener gy ma tr ices of (R2) ha ve been diagonalized n um erically t,h roughJ = 10. The theoretical tetra hedral split t ing patt erns ar e determined thr ough-out the spectrum by only three parameters: D, , Fat, an d y31 = .16(&, + Z3J.


35/40

OVERTONES OF SPHERICAL TOP MOLECULES 415These are coefficients of terms approximately J4, J, and J2 dependent, respec-tively. We obta in th e best fit by takin g D, = 1.1 X 10e6 cm-l, Fzt = - 1.4 X lo-*cm+ , and ?3t = 1.16 X 1O-2 cm-. The th eoret ical valu e (1) of D, was used. Thisconst an t also determ ines the ground-stat e tensor splittings which amount toonly a few hu nd red th s of a cm- even for J = 10 .

The pr edictions of (R2) ar e compa red with th e observed s pectru m by car-recting th e observed line positions for t he predicted ground an d final sta tesplittin gs. This compar ison is given in Table XI for t he P, R, and & bran ches. Thecorrected position for th e tr an sition 77E + 67E, for examp le, is 4449.39 +.66 + .Ol = 4450.06 cm-. If all th e pr edicted splittin gs were exactly right , th ecor rected lines for a given JL set would all coincide in position.We use th e average values of the weighted P and R branch corrected linepositions to evalua te cert ain linear combinat ions of th e effective r otat iona lconstants. We find

andB + Ba + 2(B13) = 6.00 f 0.02 cm-,

B - B, = -0.050 * 0.004 cm-l,(using D m Do = 2.5 X 10e5 cm -, th e th eoretical value). BO s th e ground -sta te

Fy) F; A, lo (2)F, FY e*I IF(l) FlI 2 E F(3) 9 A, Fa +2 AI 2 2

I I

4 E(I) 7 4 2 I

8E F*) AI 6 F/ F2 A2

FIG. 4. Q branch of 2~~ of CD, _Identification of lines according to (R2)


36/40

416 FOXvalue, B is the effective value for the P and R branches, and (B{s) is the effectivevalue common to all branches. If we further use the value B, = 2.633 f 0.001cm- obtained from the Raman spectrum of CD, ( 17)) we find B = 2.583 f 0.005cm-, and c3 = 0.152 f 0.002 (effective value).

The Q-branch analysis is complicated by the fact that lines belonging to differ-ent J-values overlap because of the comparatively large tetrahedral splittings.Figure 4 gives the identifications through J = 10. From 4494.35 to 4490.98 cm-all observed lines have been accounted for. It is possible that some lines belongingto J Z- 11 also fall into this region. Below 4490.98 cm- there are some Q-branchlines which have not been accounted for; these presumably belong to J >= Il.

The weighted average correct,ed line positions can be fit byQ(J) = const. + (B - B,,)J(J + 1) + (D, - D)J( J + 1).

According to Table VIII, the effective B value for the Q branch may differ fromthat of t,he other branches. We find

B - B, = -0.062 f 0.002 cm-l,and const. = 4494.48 cm-,(using D M Do). Then B = 2.571 f 0.003 cm-.

W. 2~3 of CH,The simple appearance of the band of CH, at 6000 cm-, identified as 2~ , has

been accounted for by (R2). The tetrahedral fine structure, which has beenpartly resolved by Rank et al., should also be accounted for by the theory. How-ever, the splittings in 2v3 of CH, are very small. For example, in P(7) of CH, theoverall splitting is 0.21 cm-l, while in P( 7) of CD4 it is 1.50 cm-.

The best fit to the observed splittings is obtained with D, = 4.5 X low6 cm-(approximately the theoretical value), Fat = - 1.25 X lo-* cm-, andYat = -5.0 X lo-* cm-. Line positions corrected for ground and final statmetensor splitting perturbations are given in Table XII. Since the tensor splittingsare very small, smaller than would be expected from order of magnitude con-siderations, it is evident that fourth and higher order terms in the Hamiltonianmay give relatively sizeable contributions. These contributions increase with J,and contribute to the apparent discrepancies in Table XII for J 2 7. We alsonote that, the value of Fat used to fit 2~ of CH, is different from Fat = 0 used tofit va of CH4 (1). This may be the effect of higher order terms which are notnegligible in 2v3 , but are negligible in v3 where the tensor splittings are larger(for example, t#he overall splitting in P(7) is 0.80 cm-.)

The following combinations of effective rotat,ional constants have been oh-tained: From the P and R branches,

B + Bo + 2(B


37/40

TABLE XIIO B S E R VE D F R E Q U E N C IE S O F 2 ~ O F C H , C O R R E C T E D F O R T H E O R E T IC AL G R O U N D

A N D F IN A L S T AT E T E N S O R S P L IT T IN G P E R T U R B A T ION S

PC4F2EP(3)

A,F,Fz

P(4)F2EF,A1

P(5)(P)FPEF,.2(l)

P(6)A1F1(1)F2A?(2)FSE

P(7)F;EFj?)AlFl1(1)F2

P(8)(2)FPF1

5983.18( 0.002 -0.001 5983.18.18( -0.003 0.001 .18

5972.13(lO(lO(

0.018 0.002 5972.110.003 0.000 .lO

-0.009 -0.001 .llav 5972.10

5960.88(.83!.83!.83!

0.029 0.004 5960.86-0.005-0.001 .84-0.016 -0.002 .85-0.031-0.004 .86

av 5960.85

5949.611,611.55:.52i

0.049 -0.007 5949.560.036 0.006 .58-0.025-0.004 .57-0.049 0.007 .58

av 5949.57

5938.204.17.17.09:.06:.06:

0.094 0.017 5938.130.071 0.013 .120.045-0.014 .12

-0.049 -0.009 .13-0.076 0.008 .15-0.085 -0.015 .13

av 5938.13

5926.675.617.617.573.482.462

0.132 0.028 5926.570.089 0.019 .550.062 0.013 .570.010 0.003 .57

-0.122 -0.026 .58-0.139 -0.029 .57

av 5926.575915.053

,0530.190 / 0.045 5914.910.097 -0.050 ~ .91

F i n a ls t a t ep e r t .(cm-)C o r r e c t e dfor p e r t .(cm-)

EC21 5914.999F, ,916EC, .769A1 ,769Fw .748

P(9)A, 5903.223,p ,223F(?)Al?

.183

.113E .OlOFi?3) 5902.985F(l),:1,

,931.931

PW)F4" 5891.792E (2) .754F(?)s:

.640,6125848

FP .491FAl?

,455,371

EC, ,341Fp ,064

R(O)81 6015.659

R(l)FZ 6026.223R(2)E 6036.652F, ,652

R(3)F? 6046.960F1 ,960

F i n a ls t a t ep e r t .(cm -)C o r r e c t e dfor p e r t .(cm-)

0.169 0.040 5914.870.006 0.005 .92

-0.204 -0.048 .93-0.226 -0.053 .94-0.212 0.024 .98

av 5914.92

0.291 0.075 5903.010.217 0.006 .Ol0.258 0.067 5902.990.118 0.032 5903.03

-0.011 0.000 .020.014 0.057 .03

-0.315 -0.081 .17-0.328 -0.085 .17

av 5903.06

0.384 -0.010 5891.400.332 0.094 .520.285 0.082 .440.146 0.045 .50

-0.050 -0.130 .410.003 0.004 .46

-0.457 -0.128 .70-0.472 -0.133 .68-0.468 0.108 .64

av 5891.51

0.000 6015.66

0.000 6026.22

0.000 0.001 6036.650.000 -0.001 .65

0.004 -0.001 6046.96-0.001 0.000 .96I I

417


38/40

TABLE XII-Continued

-41 6046.949R(4)Al 6057.094

F, .094E ,094F: .077

R(5)FZ 1) 6057.151F, .151FZ2) ,091E ,091

h(6)E 6077.061Ff' .034II ? .034F(l)F:

6076.954.954

d, .930

R(7)F;" 6086.859Fi" .816-11 .775F (2)El

.666,666

F1 ,648

R(8)$U J1 6096.5194488,(2) .399F!"ELil,

.399

.206399,p ,191

R(9)FI" 6106.302

-

-

Final Groundstate s tate Correctedpert .(cm-l) pert .(cm-)

~;w~;t.

PIPl 6046.96

I i.012-0.004 6057.080.006 -0.002 .080.002 -0.001 .OQ

-0.011 0.004 .OQBV 6057.09

0.019 0.007 6067.140.011-0.004 .14

-0.019 -0.007 .lO-0.015 0.006 .11

av 6067.12

0.033 -0.015 6077.010.030 0.008 .Ol0.022-0.009 .oo

-0.017 -0.014 6076.96-0.022 0.013 .99-0.036 0.017 .98

av 6076.99

0.052-0.029 6086.780.049 -0.026 .74

-0.005 0.003 .78-0.021 0.013 .70-0.031 0.019 .72-0.048 0.028 .72

av 6086.74

0.082 -0.053 6096.380.077 -0.048 .320.079 0.024 .34

-0.005 0.005 .41-0.028 -0.050 .38-0.057 0.040 .30-0.065 0.045 .30

av 6096.35

0.115 -0.085I I 6106.10

,:I' 6106.250E .077FPA; ,062062FL(?) ,062F? 6105.657A1 ,643

Q( l )F l 6004.842QWF , 6004.634E .617

Q(3)F2 6004.302F1 .302A? ,275

Q(4)E 6003.861Fz ,861A? ,861F, .825

Q(5)E'(2) 6003.285242FZ ,242Fl ,228

Q(6)FA:

6002.586.534

F, ,534-41 ,534E ,534F() .512

I-

ITl-

Finals tatepert.(cm-)

6106.060.005 0.000 .07

-0.002 0.057 .12-0.025 0.032 .12-0.066 0.006 .13-0.079 0.067 6105.80-0.091 0.075 .81

av 6106.04

0.000 6004.84

o.@OO -0.001 6004 .630.001 0.001 .62

,av 6004.63/ I

0.001 0.000 tiOO4.30-0.001 -0.001 .3O0.000 0.002 .28

av 6004.30

0.001-0.001 6003.53-0.001-0.002 .86-0.002 -0.004 .860.002 0.004 .83

,av 6003.85

6003.270.004 0.006 .w-0.001-0.004 .24

-0.005 0.007 .24av 6003.25

O.OC6 -0.014/ 6002.570.013 0.017! .540.010 0.013 .54

-0.003 -0.009 .53-0.008 -0.015, .53-0.009 0.008 .53

itv 6002.54

418


39/40

OVERTONE S OF SPH ERICAL TOP MOLECULES 419

Observed frequency(cm-l)

Q (7)F1() 6001.736Fp .736E .681F:2 .681As .681F(2) .655

Q( 8 ),!I' 6000.743FP .666E (2) ,666FI .666E(1) .666-42 .666FS 2) ,638

Q( 9 )F(2)1 5999.626

TABLE XII-Continued

-0.023 -0.029 6001.73-0.017 -0.026 .73

0.016 0.019 .680.012 0.013 .68

-0.001 0.003 .690.023 0.028 .66

av 6001.70

0.025 -0.050 6000.670.039 0.045 .670.035 0.040 .s7

-0.001 0.005 .67-0.033 -0.048 .65-0.040 -0.053 .65-0.036 0.024 .70

av 6000.67

0.052 0.006 5999.58

Observqd ffepuencycm

FP 5999.626F(l) .626AZ .518F22) .518A1 .518E .518FY .482

Q( ; $Fl 5998.437Ff .437Al .320E(l) .320E(Z) .280FZ2) .280FZ1) .280A2 .238F? .w1

-0.060-0.081 5999.61-0.064 -0.085 .61

0.068 0.075 .530.061 0.067 .520.035 0.032 .52

-0.006 0.006 .520.000 0.057 .54

av 5999.56

0.100 -0.010 5998.33-0.016-0.130 .32-0.097 -0.128 .29-0.098 -0.133 .28

0.117 0.094 .260.077 0.082 .29

-0.001 0.664 .290.627 0.045 .26

-0.102 0.108 .42av 5998.31

* The nu mber of observed lines in P(10) is one greater th an th e nu mber predicted by(R2).and

I3 - B. = -0.063 f 0.004 cm-;from th e Q bran ch,

and B - B0 = -0.058 f 0.002 cm-,const . = 6004.96 cm-.

Using th e ground-stat e value B. = 5.240 cm-, deter min ed from th e infrar edactive fund am ent al v3 (I), we find B = 5.177 f 0.004 cm- (P and R branches),B = 5.182 f 0.002 cm-l (Q bra nch), an d {3 z 0.0327. These may be compa redwith the corr esponding constan ts determ ined in Ref. 1 for v3 :

B = 5.201 cm-l (P an d R), B = 5.191 (Q), and l3 = 0.0547.ACKNOWLEDGMENTS

The au th or wishes to tha nk Prof. Kar l T. Hecht for his suggestion of this work as a thesisproblem, for his guidance, an d for his generous per sonal interest. Detailed informa tion


40/40

42 0 FOXon th e spectra of CH, ma de available by Pr of. D. H. Ran k, an d by Dr. E. K. Plyler wasgreatly appreciated. Special tha nks must be given to Prof. C. W. Peters and Mr. Robert E.Meredith for their work on CD, .RECEIVED: May 10, 1962

REFERENCFS i1. K. T. HECHT, J. Mol. Spectroscopy 6, 355 and 390 (1960).9. J . MORET-BAILLEY, (Thesis), Cah iers phys. 16, 237 (1961).3. IL K. PLYLER, E. D. TIDU~ELL, AND L. R. BLAINE, J . Research Nat l. Bur . Sta nda rds

64A. 201 (1960).4. D. H. RANK, D. P. EASTMAN, G. SKORI NBO, AND T. A. WIGGINS , .I. Mol. Spectroscopy 5,

78 (1960) .5. K. Fox, K. T. HECHT? R. E. ME REDITH, AND C. W. P ETERS, J. Chen l. Phys. 36, 3135(1962).

6. W. H. SHAFFE R, H. H. NIE LSEN, AND L. H. THOMAS, Phys. Rev. 66,895 an d 1051 (1939).7. J . I). L~)I:cK, th esis, The Ohio Stat e Un iversity, Colum bus, Ohio (1958). Also see Ii. T.

HECHT, dppl. Spectroscopy 11 , 203 (1957); th is is th e abst ract of an invited pap erpresent ed at th e Symposium on Molecular Stru ctu re and Spectr oscopy, The OhioSta te Univer sity, Colum bus, Ohio (1957).

8. M. J OHNSTOX AND D. M. DENN ISON, Phys. Rev. 48,868 (1935).9. H. A. JAHN, Proc. Roy. Sot. A168, 469 an d 495 (1938); A171,450 (1939);

W. H. J . CHI I,DS AND H. A. JAHN , Pr oc. Roy Sot. A169.451 (1939).10. See, for exam ple, H. H. NIE LSE N, En cyclopedia of Ph ysics, S. Flt igge, ed., Vol. 37.

Pa rt . 1. Spr inger-Verlag, Berlin, 1959.11. M. ROTE NBERG, R. BIVINS, N. METROPOLIS, .~ND J . K. WOOTEN, J R., The 3-j an d S-j

Symbols. The Technology Pr ess, M. I. T., Cambr idge, Mass., 1959.f.2. A. R. EDMOKDS, Angular Moment um in Qua nt um Mecha nics. Pr inceton Un iv. Pr ess,

Princeton, 1957.IS. Ii. Fr x, t.hesis, Un iversity of Michigan (1961).14. K. KCCHITSI- (privat e comm un icat ion, 1960).15. W. H. SHAFFE R ANI) J . D. LOUCK, J . Mol. Spertroscopy 3, 123 (1959).f6. M. E. I

kenneth fox- vibration-rotation interactions in infrared active overtone levels of spherical top...

Documents