Intramolecular vibrational relaxation of CH overtones in benzene

5 Norlxnbcr

CHEhllCAL PHYSKS LETTCRS

Volume 91. nunlbcr 5

MTRAMOLECULAR

VIBRATIONAL

RELAXATION

OF CH OVERTONES

IN BENZENE

1982

’

Edwin L. SIBERT III. Wrlham P. REINHARDT Deporrmetrrof Clwwrr_v. UIIU ersrry oJ Colorado. nnd Jo~rrrhslsrrnue/or Lobor;rro~ .~lm~ropb_vrtcs. Unwersrryorcolorado and Nallrronol BUEW~of Srandards. Bordder. Cohizdo 60309. US4

and James T. HYNES Deparrmenr of Clremfsrr). Unr~ersr~~ o/Colorado. Bouldt-r, Recwcd

21 Scprrmbcr

80309

tiS

1

1981

1. Introduction

2 Theoretical

The highly viiraUonaUy excircd CH overton states benzene habe received Intense experimental [ I,2 j and theoretical (I-61 attention in the quest to reveal and undersrand intramolecular energy flow. Extensive references to previous condensed phase work on benzene are given in ref. [I] Experimentally, the imtially prepared overtone states of gas-phase benzene are well appro.xunated by pure CH local mode states: the broad homogeneous overtone hneshapes correspond to rapid (~0.1 ps) mtramolecular flow of CH bond energy into the remaining molecular degrees of freedom. In this letter, we describe a realistic molecular of gas-phase

model for the ongin. path and tunescnlc for thIsenergy

decay, and present theassociated

theoretical hneshapes.

The coupling responsibk for the relsxarlonare esplicitly Identified. The resultmg picture agrees qualitatively

Colorado

and quantitatively

wth

both

expenment.

We

theoretical work [ I61 on the benzene overtone problem.

mode and overtone

lineshlpes

Our model iocuses on the couplmg bciwcn ular vlbratlons oscdlator.

The first Ingredxm

whose pxamrters

molcc-

is a smgle CH Morse

reproduce

the

obsencd

[I ,?] overtone spacings. w(cm-i)=3157.1(u+4)-57

l(u+~)?.

(1)

We assume thsl the CH overtones and CH combinatron stales do not mrs, due to small couphngs and Inrgc energy mamatch. Tlus ISm accord wrth the !indmgs of prevrous dynxmrcal studies ]3,6]. Only a smgle CH oscdlator nrsd to be consldercd, smcc cvtdencc sug gests that the overtone states are degnerate [7]. An exceptIan OCe”rs i-or ” = 1 since the ” = 0 state IS nondegenerate, but thrs wdl not concern us hers. The

secondmodelingredwnlISthe couplmgof Ihe

CH bond to the remaining degrees of ireedom. In our model,

the CH oscxllator

IS anharmonically

coupled

contrast our model with earlter

vta a Fer,rd resotlatlce

* Supported III pxt b) NW grants CHEIO-I 14 12 .md PHY7904928 IO WPR and CHFBI-13240 to JTH.

plane wag (see below). The latter’s frequency (= 1300 cm-l) is approemately one half the anharmomc CH stretch frequency (X30-2 190 cm-l) m the energy regime (u = 4-8) where fast decay is observed expen? mentally. (Note the importanr anharmonic suppressron of the CH stretch frequency from Its harmonc value at a3157 cm-l.) Because of this I 2 ratio of CCH

0 0092614~8~~00LKMO00~/5 02.75 0 1982 North-Holland

rype mteractron to the CCH in-

455

wag and CH stretch frequencies, this key anharmonic couphng must be hear UIa CWvariableand bllmear in CCHvanzbles. Puiay et al. IS] (whose force field we use) show that anharmonic pore~iaf couphng of this form ISne~glb~e in the bond-an&e coordinates used here, the present work demonstrates that m these coordinates the kketic energy couphng IS considerable, decistve, and ~~yt~c~~ calculable. The kmetic coupling arises m an exact internal coordmate perspective from the dependence of effective masses on internal coordmates. in parttcubr, the effective massg;’ of the CCHin-planewag oschtion depends upon the corresponding CH stretch coordinate s in a known fashion [9]. Here g,\, 8s a Wilson G-matrix element [9] The effecuve mass of the wag mcreases as the CH bond IS extended, the stretch and the wag arc thereby ~~ettc~y coupled. Tius coupling is made exphclt by a Taylor series ehpanston ofg,\ (s) through lirst order tn s about the molecular eqtnhbnum coniiguW ratton. Tfus yields the kinetic enera h~rltonian of the tn-plane wag m the form

Here &!Hand PC are the inverse masses of hydrogen 4 carbon raspcctivcly 3nd p. and R, 3r.zthe CI1 md CCequlhbrium distances respectlvelg. The first term on the t-@-hand side of rq (2) is the wag osc~3tion.s kinetic energy in the smail ~plltude appro.~matIon~ The second term in eq. (Z), F= i (~~,JW,,os~~

3

(3)

1sa Ferrm resonance term. and cattsesettergy t~florv betwmt the CH stretch and CCH rwplatte wag.

Smaller cubic kinetic couplmgs involvmg the CH stretch, CC stretch, and CCHwag are also mcluded m our model for numerical ~curacy, but eq. (3) IS the most tmportant couphng cont~but~on. It is also worth stressing that in the more complex quantum handtonisn 191, F is also the do~na~t coupling. Attention must now be gtven to the Rnal model ingredlent, the rmg modes. In benzene, CCH in-plane wag and CC ring stretches are stron_ely coupled, due to 456

5 Norembcr 1982

CHEFIICALPHYSICSLETTERS

\‘olume 92. number 5

important quadrattc coupling terms and sirmlar frequencas; the wags and stretches combine to form harmonic normal modes [9]. In our picture, once CH energy flows into the wag, it rapidly drssipates among the other CC and CCH oscdiations, which hence act as a “bath”. IIus view is fully supported by classical trajectory calculations we have carried out with the model’s h~ton~n~ these show irreverstbie CH energy decay ~4th decay times ~0.1 ps, m general agreement W&I experiment [I&?]_ The quantmt description of tha decay is facihtated by expressing the CCH in-plane wags and CC stretches in the normal mode terms mentioned above. In this perspective, the CH stretch is now a~a~omc~y coupIed via F to many of these bath modes. These couplings rnLxquantum states in wluch the CH osctiator has/ quanta and a bath state has k quanta, denoted by [lvCH, X.Y~1, with I(] - l)v,., , (k + 2)~~ 1 states of similar energy. Rg. I illustrates the resultmg picture. There a u = 6 CH overtone ISapprecrably coupled to several (=7) “dOw~@_f’ states [5&-Hf ?I$]. The COUpling magnitude IS= 70-30 cm-t. Each doorway state xs III turn coupled to several (-10) @PC-, 41~~1states, and so on. This picture can be converted to overtone lineshapes by c~culatmg overlaps of an initl~y prepared pure CH Morse oscillator overtone state with the exact total system eigenstates. These overlaps can

5VCH.2ug

=

CHEMICAL PHYSICS LETTERS

Volume 91. number 5

r\: WO

“0

w

WO

(cm-‘)

FIN. 2. Ckulxcd CH overtone bncshapcs (3rblrr.q umts) Energieswc, of pure CH slrclch ~12s u = 5.6 zmd 9 xc [ I l 14072.16550 and 23’76 cm-’ rsspcct~rcl) AbscIw IIC mxhs 1aareu=uo= l5Ocm-‘. Thcsc bncshapes cmz.m oi mm)

sppro\lmare molecular c~gcnstatcs(xc tc\I). cdcll ftrcn d no. 1~1143 cm-’ T\\hm (ik [ 3ly3-j1orcrronc i\\hlII [ 1]) The above cnti:rfws wo. which dlikr WI) sl~ghhrl, irom 1110sc oblalnrd b) lh2 hlorse oscillator tit cq. (I). \lcrc urcd III t11c cslculatlons

wluch is m accord with experrment. There are two key features of the mech3msm. (I) the Fern11 resonance between the CH stretch and the CCH m-plane wag and (2) the strong quadratic coupling of the wag to the remaining CCH in-plane wags and CC strctchcs. It IS this second feature which dlstingulshes the Fermi fesonance of the CH overtones m benzene from that III small molecules. In carbon dioxide [ IO,11 1, for example, the symmetric stretch IS in resonance with the bend. Unlke the wag in benzene, the CO, bend IS Itself a normal mode. thcrr is no b3th to which th2 initial stretch energy c3n be lost. Equivalently. a figure (see fig. 1) tILtrating the eifecmely coupled states to a CO, overtone symmetnc stretch would only contam a sin@2 state 31 esch of

tlic

tiers depicted m fig

I _lhs is m obnous contrast to the benzene case. Previous theoretical work on the benzene CH overtone problem may be divided mto two classes.The first [ 1,2,4,5] employs 3 phenomenological appro3ch. Usmg hneshapr rh2ori2s. vrbrational couphng and dephasing parameters are chosen to iit theory to expcnment. The second class [3.6] is more closely related

be found approumately from a secular equatlon mvolving the effectively coupled states. For esample, the secular equation for u = 6 would include the states in fig 1 and (x100) slates of [3r$-H, 6vB] character. Fig 3 displays overtone lineshapes so calculated. Each hneshape consists of many appro.umate molecular elgcnstates which have bren given a naturaJ iwhm oi 43 cm-t, which IS the fwhm of the 3vCH overtone [I] The overall resultant lmcshapes do not depend strongly on this cholcc. The hnewidths agree well both with ehprnment and with the classical decay rates noted above. An Important charactenstlc fcaturc IS the relative KUIOW~II~ at u = 9. TWOcombined ZISPCC~S of our model explain this. First. thr coupling strength between an tnttial CH overtone with u quanta and a doorway state wth u - 1 CH quanta Increases only weakly (~‘1~) wnh u. Second, the density of doorway states is, however, mGmum at u = 4-6 (in complete contrast to the orer~ll density of states) As a result, the IJ= 9 hfetime exceeds that for. say, IJ = 5: at the h@er CH energy. there are fewer (= 4) doorway states with which the overtone can ~LX. Equivalently, the Femu resonance is “detuned”, and the CH energy flow rate dlminahes.

results. We are currently ektendmg the concepts discussed m this paper to include the study oi parktlly dcuterated benzrne molecules. It IS prrhaps worth stressmg here that the kmetic energy coupling mechanism is 3 natural vzlucle for lsolope eficcrs.

3. Concluding remarks

References

We have described a mechamsm ior intramolecular decay of CH overtone enera

in gas-phasebenzene

to our work. and models the benzene system dynam~cally m an attempt to understand possrble decay

mechanisms. In a previous classical trajectory study [3], no appreciable energy decay on 3 prcosecond timescale was observed. This study, however, used 3 model with anomalously high CCH wplane wag force constants. this precludes thr Fcrml resonance obscrvcd in our stud). The decay process has also been modelled quantum mechsnicdly [6) Here the eiiccr of all the CCH in-plane wags and the CC strctchcs was considered to be of secondary importance and only the CH osctiators and the interactions bctwcrn thrm were included in the model. no decay from the CH overtones was observed. This contrasts wnh the present

[ I I ti V Ruddy. D I‘ Hcllrr .md hl J Bcrr). J Clwm. Phls 76 (1981) ‘I314

Volume 91. number 5

CHEMICAL

PHYSICS LEITERS

[Z] R C Bray and M.J. Berry, J. Chem Phys 71 (1979) 13 1 P.J. Nagy snd W L Hase, Chem Phys Lertcrs 54 (1978) 73.58(1978)481 141 M.L. !Gge and J. Jortncr, Chcm. Phys Letters 62 (1979) 451;Adun Chcm. PhyS 47 (1981) 293.

151DE C1~UrruldS.hlubmcl.l.Chen1.Phys 70(1979) 363. 161 P R Smnnxd and W.hl Gelbxr.

458

[7] L HJonen.Chem.

Phys. Letters 87 (1982)

1982

??I.

[S] P. PIAIY. G. Fog;lrasl and J.E. Bogs. I. Chem. Phys

4909.

(1981)

5 Norember

3592.

J Phyr Chcm 85

74 (1981)

3999.

[9l E.B. Wdson Jr, J C. Deems and P.C Cross, hlolccular vlbrarions(McCr3w-HIIl. New York, 1955) ch IO. app. VI.

[lo]E.Fermi,Z.Ph~silc71(1931)?50. 11l] E 1. Hellei, E.B. Stechel and M.J. Dim& J. Chcm. 73(1980)4710.

Phys