Coupled oscillators and radiationless transitions

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It is shown that the kincticcncrgy coupI& betweenvibrational modesof the z&umonic local-mode category (e&&I strctdkg modes in aromztic hydnxarbo ns) and the harmonicnod-mode category(e.g. CC skdetalmodesin arcmkic hydzocxbons)providesan cqlanaticmof the cqxthaltauy obssvcd qxrgy gap law for radiatiozlkss tmnsitioIlsin aromatic makcd~-rhea*kno~ illsssitivity of ndiariodus transition rate amsttts to diffinmo&uIarsize. h-n number of vibrational mokuk modes, etc.. is rationalized in tans of geometry changes to wbicb k&tidy coupled vibrational mokcdar moda arc subjected upon ~&ctronic excitation.

l_ln~on The process of radiationless relaxation of an electronically excited molecule usually terminates in highly excited vibrational levels of the final electronic state Hence, finding an appropriate description of the vibrational motion within the molecule is the fundamental problem one is faced with when dealing with the theoretical treatment of electronic radiationless relaxation_ In recent years the results of overtone spectroscopy [l] and of multiphoton infrared experiments [2] have brought about a systematically increasing amount of evidence for very efficient mixing of the different highly excited vibrational modes of the mokcule To the best of our knowledge however, all theoretical descriptions of radiationless tram% tions in polyatomic molecules developed up to now have been based on the assumption that different vibrational modes of the molecule are not coupled to each other_ Hence, the vibrational wavefunctions of the molecule were always accepted in the form of products of wavefunctions of one-dimensional oscilktors (for a review see ref. [3] and references therein). Even in the %ommtmicatiug stat&’ model which Was Originauy intrO&ced by Ftier and schlag [4] to dezxrii the dependence of radiatiotikss transitions on the excess of excitation _energy, CAkulation of thermaliy averaged radiationks rate constants has been

perforked iu the abovementioned staudard way, i.e. all the interactions between vibrational modes of the final ektronic state (high vibrational quanta) which are off-diagonal iu potential energy and momenta have been neglected [5,6]. The corkept of local &odes [7] which has already been used iu the theory of radiationless transitions [II-lo] seems to offer a more correct zero&order picture of +brational modes in polyatomic molecules as compared to the normal-mode description and especially for a high vibrational energy content in a molecule In a tit paper [ll] we have shown that the kinetic-energy coupling between the local &nd normal modes in a polyatomic molecule may be responsible for the observation of continuous background absorption in the overtone ~spectra_In this paper we investigate how such a coupling would influence the caku.lated rate qonstant for radiationless transitions in a polyatomic molecule. Z Intermode c6ttpEttg and merhod of calculation In the statistical limit and with-the %&d&-d assumption& of the Condon approximation for matrix coupling elements* and the crude-adia* Itisawdi~lishdfaczthattbcnon-<=ondmtcomctions

O3O1-ORt4/85/SO330 Q EIsevier Science Publishers B-V. (North-Holland Physics Publishing Division)

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batic (CBO) appro_ximation for molecuIar wavefunctions. the rate constant for radiationless decay of the doorway vibrationless dectronic state Is9OS> to the manifold of vibrational levels of the final electronic state { 11. 1.7~)) is written in the form [lo]

where C,$ is an electronic coupling matrix element by the promoting mode p of frequency ho, and u; denotes the vibrational quantum numbers of the main accepting modes- p,(o;) which corresponds to a density-weighted Franck-Condon factor for the mmaining vibrationai modes incorporates energy conservation and is chosen to have the form of a gaussian [13]: induced

P,(Q)==

-lr-T-‘erp[--T-‘(E,,--E,,--~~)‘]_ (2)

Here &., and EI, are vibronic energies and the r parameter is taken to be of the order of the smallest viimtionaI frequency among the best accepting modes u’ under con&e-ration. chosen in order to obtain a compIeteIy structurehzss specram of k, versus electronic energy gap. AE,, = E,,, - EI,_ which is one of the most basic assumptions for the applicability of the golden rule_ In the framework of the non-interacting osciIIatom (NIO) model the Franck-Condon factor entering eq_ (1) is recast into the form

and ti; denote the quantum number of the best accepting modes of the system_ UsuaIIy these are the highest-frequency vibrational (IocaI and/or normaI) mods that undergo significant changes of their potentiaI+tmrgy surfaces upon electronic excitation [3-fTQLlOj_ It has been shown that the experimentaIly observed energy gap Iaw for radiationIess rate constants in aromatic hydnxarbons can be quite weII reproduced by assuming only the two best accepting modes (groups of modes), Le the CH stretching local mode and the CC totaUy symmetric normal mode, to be active in those systems [lo]_ ObviousIy these modes are in turn

coupIed to the remaining vibrational and/or rotational modes of the moXecuIe. so eventually their zeroth-order discrete levels merge into an effective continuum [see eqs_ (1) and (2)]_ Ahhough the coupling due to po:entiaI-energy interaction between these two groups of accepting modes seems to be neghgible, their kinetic-energy interaction trough the common carbon atom of CH and CC bonds is by no means negggible especially for high vibrational quantum numbers of CH and CC vibrations [Ill_ The CH stretching local mode is kinetically coupled to each vibrationa.I motion that dispiaam a carbon atom in the direction of a hydrw=en atom The strength of the kinetic-energg coupling for any given polyatomic moIecuIe can be easily eaIeuIated with the aid of the formuIae derived in ref_ ill]_ Here we start with the assumption that the zeroth-order vibraticnal wavefunctions entering eq_ (1) have

the form

where n- denotes the vibrational quantum number of the CH IocaE mode described by a Morse potential of the energy E,=ho(u+i)-x~o(u~i)+,

(5)

where ho and xho are the vibrational frequencand the anharmonic correction. respectively and ccc is the quantum number of the CC “effective” harmonic mode of frequency /io,T&e CH and CC modes are coupled to each other and the reIevant coupling part of the vibrational hamiltonian lras the form [ll]

where qcn and qcc are ditncnsionIess vibrational coordinates of the CH and CC modes mspectivelyIn general the vahte of the kinetic-energy coupling parameter 6 depends on the nature of the CC normal vibrational motion [ll]_ With the “effective” CC normal mode its “effective” kinetic-energy coupzing pammeter, 0,. to the CH local mode may be deflmed as the length of the vector in

the space of vibrational

normal modes

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Here the summa tion nms -over all CC normal modes of the molecule, ej being a kinetic-energy parameter coupling thejth CC normal mode to the CH local mode. For benzene+ with the parameters given in reF. ill], one obtains a value of f?,= 404 cm-‘_ In the case of a somewhat simplified linear arrangement of CC and CH oscillatcxs (“a linear triatomic CCH molecule”) the relevant parameter can be _mcastinto the form 1141

where pctt and pc- are the reduced masses of the CH and CC oscillators, respectively_ With values typical of aromatic hydrocarbons, fief= 3145 cm -t and hue,== 1400 cm-’ (for details see section 3), one obtains B = 411 cm-‘, which is close to the value estimated above for benzene_ This seems to indicate that for aromatic hydrocarbons the kinetic-energy coupling between the CH local mode and the collection of CC skeletal normal modes has a more or less stable valueThe off-diagonal matrix elements of the coupling hamiltonian (6), in the vibrationai basis defined by eq. (4). have the form

Since the analytical forms of the matrix elements of eq_ (9) are availabIe for both the harmonic [lS] and the Morse [16] basis, construction of the I-I,_,. coupling matrix elements is an easy task The “exact” vibrational wavefunctions are defined as a linear combination of the zero&order vibrational wavefunctions:

where the coefficients @(Us* oco) can be determined from diagonalization of the total vibrational hamiltonian in the basis of eq. (4)- Thus the Franck-Condon factors of the “exact” vibrational

wavefunctions, appearing in eq_ (1)~ are given by-,

Here the kinetic-energg interactions within the initial vibrationless electronic state lo,) have been neglected_

3. Results of calculations In what follows we consider explicitly only an anharmonic CH stretching local mode and an “effective*’ harmonic CC skeletal normal mode. Their Franck-Condon factors are defined by eqs. (3) and (11). The remaining vibrational and rotational degg of freedom of the hydrocarbdn molecule will be incorporated into the densityweighted Franck-Condon factors defined by eq. (2) For such a system we adopt typical vibrational parameters of CH and CC modes available from experiments, i.e. fi~,-u = 3145 cm-‘, xhwcu = 56.5 cm-’ 1171, and fro,--= 1400 cm-’ (which is the average of the skeletal ring vibrations [6,X3-20])_ Electronic.excitation of the molecule generally causes a change in the. electronic density distribution over the molecule, which in turn disturbs the potentials of the molecular vibrations. The foilowing relations have been confirmed experimentally: an increase (decrease) of electron density on a particular interatomic bond causes an increase (decrease) of its stretching force constant (vibrational frequency), and leads to a shortening (lengthening) of the interatomic distance fequilibrium bond length) [21-231. The quantities which determine the overlap integrals of the vibrational wavefunctions between the two combining electronic states [9,15] are usually defmed as: anharmonicity a = (2xtio/rii~)‘~. frequency shift f= oJq. and equ.Gbrium geometry change A,= qos - qor_ The arhrmonicity parameter for CH stretching local modes, in the ground electronic state, is almost independent of the molecule for the aromatic hydrocarbons series [1,1724], and a== 0 (by defiition)_ Similar arguments hold

true for the f%equeucy change pamnetersfc, and j& [6,18,19]_ The only vibrational parameters that cannot be det ermined preciseIy enough from the experiment are geometty change parameters A, and A,@hiswncemstheir absolutevahxsas weI.I as their signs)_ In our twemode modeI, four cases should be considered: (a)

A,-=O.

A,-cO;

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A,,-cO,

A,=+O;

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Accepting typical Iimits for the geometry changes due to the r-eI+ronic excitation: ]A& c 0.13 ([aQ&
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b

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(d) differing in the sign of A, are tquivaIent because of the harmonSty)_ It is evident from inspection of fig la that the rate determining parameter is the sign of CH local mode displace-ment [25]_ According to the rest& of section 2 we now include the interactions between these two oscihatot-s due to the kinetic-energy operator, eq. (6), taking6=410 cm-‘_ In the determination of the %xact” vibrational wavefunctions, as described by eq_ (10). all zeroth-order VibrationaI states of our two-mode model up to an energy of 25000 cm-’ have been incIuded (the Iimitation of maximum energy is due only to the numerical ability of the available computer). The Franck-Condon factors for this model of kinetica.IIy interacting osciIIatom have been cahxtlated according to eq. (11). The results of the caIcuIatious are plotted in figs. lb and ic_ It is seen from inspection of these figures that when the nucIear dispIacements of the two wupIed modes have the same signs (cases (a) and (d) plotted in fig lb) the inter-mode coupling generaIIy does not change the previous wncIusion drawn on the basis of the NIO model Thus in these cases the *‘sign rule” for the non-radiative process 125) is wnserved; however, a signiftcaut increase of the radiationless rate wnstant for higher energy gaps is observed as compared to the NIO model

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Fig? I_ Dcpaadcnccof thcradiationkssraucoonanls(inuIativctiu)oo lhcma-gygap. viLx4iod pza5nclcrsarcz-=14oocm-‘. l~,~=l_O.fcc=I_O.~dJ~= -00.13.fen= 0.98 (broken lines)(a) NiO mad& @) and (c) -0.13. fa ==1_02(solidlines).Ac;~ -_ luneucanyiat~~to+smoddfor~A~-~dgn~ardsignA~----sign5cH.~~y.

Qualitatively, a new result has been obtained in the case of different displacement signs of the two coupled modes (Cases (b) and (c) plotted in fig. lc). In such cases the radiationless rate constant is mainIy determin ed by the viirational parameters of the CH local modes in the ground electronic state (practically independent of the molecuIe in the aromatic hydrocarbon series) and shows only a minor dependence on the sign of the CH localmode displacement_ It seems that this result pre vides a key for the exphmation of the surprising stability of the energy gap ia\v for radiationless rate constants in aromatic hydrocarbons “; which now can bc ration$izcd in terms of opposite changes of the geometric parameters of kineticalIy coupled CH and CC modes upon z-electronic t%Ai3tiOlL

It must be born in mind, however, that in aromatic hydrocarbons a large uncertainty is connected not only with the sign of the CH displacement, but also with its absolute value, and the possibility of A a = 0 cannot be excluded. A similar situation is also encountered in the case of the aromatic ring vibrations, because 5cc # 0 only for totally symmetric normal modes_ According to resuIts of ret ill], the CC non-totaIIy symmetric normal modes can alsO couple to CH local modes. Thus the extreme cases A,=0 and/or A,=0 should be considered_ The results of calculations including these cases are plotted in fig. 2. It is seen that in the NIO model a variation of 5, from -0.13 to 0.0 (with fmed fa = l-02) and of 5, from 0.0 to 1.0 (with& = 1.0) causes a variation of t.@e radiationI= rate constant by more &an two orders of magnitude for a given energy gap_ On the other hand, the couplcd-osciIIators model set5 the Limits for variation of the radiationless rate constant within one order of magnitude and this

agrees better witi the variation of experiientally determined rate mnstants for different radiationless processes in aromatic hydrocarbons’(seefig_ 2b). Summing up the final condtio& of this wo& we believe that the observed “stability” of the enew gap law for radiationless deactivation of excited aromatic hydrocarbon molecules is governed by two major facts: (a) the nature of z-eIectr&ic excitation which affects the geometry of .CH 1ocaI modes and CC skeIetaI modes in opposite ways, ad (b) the high anharmonicity of the CH stretching. local modes, whose. value is nearly constant for ali members of the aromatic hydrocarbon series. This latter fact, however, may be subject to further verification due to the proposition that changes in CH anharmonicity [30] could also lead to similar agreement as that due to the IcineticalIy coupled vibrations. Although the results of this work apply to aromatic hydrocarbon molecules, they are expected to be of more general significance; we. expect thai kinetic-energy coupIing wilI be important in radiationless transitions in ti. molecules which possess anharmonic vibrational local modes.

2_ Dependence of the radiationless rate constants (ii reIative units) on the mcrgy gap, cakulatcd for the NIO model (a) 2nd for kinetically int exacting oscillators model (b)_ Vibrational palame tcrsarCA ,--O.13.A,=LO(): 4, = -0-13, A,=O_O (---): A,=O_O. d,=l_O (----): 4,=0-O. A,--=O_O [- -----)_ In all cases I-=1400 cm-‘, fat = I.02 and& = LO have born assumed. ExpeCnmtal rate constants f~rinusyszmcrossingT, -+ s, (open circles) 126-281 and for imernal conversions S, -, g. Tz -+ Tt (black circles) 1291 in a.nxnatic hydmcarbons, axe given for comparison Tlxe internal convaxion rate constants arc multiplied by the spin prohibition factor IO-9

Fig.

* Ir is lmown that for

these compounds uperimentauy determined rate o~nstants for T, --s, ima-sysan crossiq [26-281. as a& as for S, --, S,, and T2 --, T, internal conversio1~5.can be fitted to the energy gap law with an uponc~tial fdon of the form k, = C, cxp(- nAE,). where C, depends only on the orbital nature of the non-radiative process [26,29J_ Moruwu it is also bdiewd that the deviations from an exponential gap law arc qstanatic and dcpead on the ratio of Ihe mImbux of carbon and hydrogen atoms in the mdccule [26].

We thank Professor W_ Siebrasd discussions and cxitical comznmts.

for helpful

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50 (1083) 9x_

tl2I W- Siebmnd and MZ Zgiaski. Ghan ?hys Letters 58 (1978) 8. [13] J. Ihrup and J. Jortner, J. C&m. Php 63 (1975j 4358. 1141 _ . V. Buch, RB_ Gakr and MAX Ramcr. J. Ghan_ Ph\% 76 (1982) 5397. [iSI BR Hem-y and W_ Sidxand. inz Organic molecular photoph_vdcr VoL 1, ah J_B_ Birks (Wiley. New York. X973)_ [16] -ML_ !tkge Chcm Phys 35 (1978) 37X [ll] J-W- Perry and AH_ ZsaiL Chem Ph>= Letters 65 (1979) 31_ [IS] DAM Burland and G-W_ Fbbiia I C&m Ph>x 51 (X%9) 455_ 1191 SF- F&. AL Stanford and EC lim. J. C&cm_ Ph3s_ 61 (1974) 58Z [20] Y. Ftajimura and T_ h’aka$ma. chcro Php Letters 14 (1972) 108. [21] W-E Donath. J. Ghan Phy. 42 (1965) 1X8_ (221 RJ_ Elliot and W-G_ Richards. J. Md Struct 87 (1982) 211_ [23] P-C Mitsura and K_ Jup l-haxc~ Chim Acca 61 (1982) 559_ [24] R Swofford. ME Lang and AC Afixech; J_ Chun Phvr 65 (1976) 17r3_ [25) ES_ rMcdvcdcv_ CXan ph)s 73 (1982) 243. w] Km !?kbrand. I Clmn Ph>% 47 (1967) 2411. [27] RLiandEC~J_CkmPh_ys57(1972)605. [XI] RW. Clarke and lfA Frank. J_ Ghan Php 65 (1976) 39. [29] G-D_ Giiie and EC_ Lim. Ghan Php lxttcrs 63 (1979) 193_ [30] w. Si&r.md. prime communication.