What is the nature of intramolecular coupling responsible for internal conversion in large molecules?

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CHEMICAL PHYSICS LETTEIZS

WHAT’TS:THE NATURE OF INTRAMOLECULAR COUPLING RESPONSIBLE FOR EUTERNAL CONVERSION IN LA&GE MOLECULES? ’ A. NETZAN

Department of Cbnistry,

and-J. JOKTNER Tel-A viv Univertity, Tef-A viv, &aeI

Received9 August 1971

Wediscuss the &&a !arge mokcu.ks.

for the specification of a proper basis set for describing electronicrelaxation

processes in

It is no’cywei! established &at the decaying ele~troni~~y excited states of large molecules shoufd be described in terms of resonance (compound) states f l-3 1. To assert whether the physical phenomenon of the decay of. an _ excited elect+& state of a “statistical” !arge molecule can be described as ~VO independent processes of excitation followed by decay, or as a sin& quantum-mechakal process, one has to speciry the duration of the excitatio:~ time relative to the reciprocal width of the resonance. When the temporal duration of the exciting photon field is short relative to the recipr,ocal width of the metastable state it is possible to separate the excitation and decay processes and to consider radiative and non-radiative decay times. On the other hand, when the ex~~~g photon field is characterized by high energy resolution (being switched on for long periods) resonance scattering involves a single quankm-mechanical process and one can then consider the determination of optical line shapes, cross sections for, resonance fltiorescence and emission quantum yields. The lifetime is then obtained from the reciprocal resonance. width. The physical information derived from “short time” and “long time” excitation pro-. cases is equivaknt. I The basic physical model for a single mol&ular resonance in the statistical limit [I-3: rests on the fo~ow~g assumptions: (a) A single zero order state 1s)carries oscillator strength from the ground state IO>. (b) Is) is intramoleculariy coupled to a zero order quasicontinuum (III). (c) The quasicontinuum is optically inactive. (d),Other optically active zero ,order states Jb) (which correspond to different vibronic components of the same eltktronic state as is> or to other electronic states) are well separated from is> relative to fkek total widths, Conditibns (a) and (c) pertain to the excitation process of the resonance both in “short time” and “long time” experiments. Condition (3) implies the existence of a finite width of the resonance in “Iong time” experiments ‘and intramolecuiar decay for a “short time” experiment. Conditions (a)dd) were. taken to imply that the line shape is lore&an (and the decay is exponential); thus if(c) does not hokl, a Fano=type line shape ,[4] will result, while -k(d) is violated, interference effects in the line shape and in the decay with be exhibited. Several authors Lve implied t@t .when conditions (a)-(d) a&satisfied the resonance wid@ is given by the Fkrmi Golden Rule I’ = 2ntQ[t’@t2pr (where the total molecular hamiltonian iS separated i&o EZ=Ho f V, atid where,p[ is the density of states (In)). When radiative decay is also considered, the radiation field provides an additional independent de. : cay width r’, the total width being rt = F i.rR. If molecular systems would adhere to conditions (a)-(d) fife ;vould. be$mpIe, However, if? real.sstems the role of off-resonance cdupling with @her states lb>(sek fig; 1) is crucial, @d tie simple Gofden Ru~~,e~re~on should be mo@fied; This point; &l&h was not realized &‘Several %cent ‘Works [7,8] leads to tinsiderakle tionf&ion concerning the choice of [email protected] .s#t for deeding ,elec~oni~ relaxation prdketis.

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CHEMICAL PHYSICS LIBTERS

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Fig. X.A s&~~maticrepre~~~tiDnof the relevant molecular states a& co@ings. Arrows in&care d&o&zcou@ingviathe interaction with the r3diation fieb3.Wavy lines represent intramolecular couptiing. Following classiixl work in molecular and solid state physics 1161it was widely accepted [l-3,sf. that the zero order states Is>and (10) can be taken as Born-Oppenheimer states, whereupon the nuclear kinetic energy operator provides the coupling term for internal conversion. This approach has been cheeped by Burland and Robinson [7f and by Sharf and Silbey [S]: who claimed that the “~er~berg-T~~I~r coupiiq is more eEe&e,in causing radiationless transitions than the breakdown of the Born~ppenhe~er approximation” [7j. In view of this fiv.ly controversy we shall attempt to cohsider the problem of the nature of intramotecular coupling in internal conversion by discussing the general separation of the hamiltonian, the form of the resonance width in a realistic system *andthe general criteria for the choice of a basis set for specifying electronic relaxation processes. Consider the dissection of the molecular hamiltonian H = TR + ?‘r + g(r,B) where r and R represent electronic and nuclear coordinates, T represents kinetic energies, while U(r,R) is the potcntiai energy. Let 5nv>be any complete set of molecular zero order functions where ebctrotic and nuclear motion has been separated artiitrariIy (so that the fast index refers to the ebctronic state, while the second index labels the ~bratio~a~ &ate). We can then = Inu) (nvl. Utilizing the completeness assumption C,;P,, = f , and the trivial ~~~~~ ~~~~~~~~~~we can then define

(4)

where () denotes integration in the electronic I space, while ( ) denotes the integration in the nuclear R space. In the crude adiabatic representation (nu) = t&*(r,Ro)~(R) where the electronic &A arndnuclear xc* wavefunctions

Satisfy

where AU= U(r,_R) - U(r,Ro) andRo is an arbitrary fmed nuciear confguration.

in this representation (6)

EQs. (6) and (7) provide a self-consistent definition of the zero orderhamihonian and the perturbation in the CA basis. The following remarks shouid be made at this point: (1) Both A and CA (unuuncated and compiete) sets are adequate from the formal point of view. (2) The vibrational wavefunctions xA and xc* are different in the two approximations, see eqs. (2) and (5). (3) While the adiabatic potential surfacesE,,{R) are known from theoretical calculations no information is availCA(AU(~,R)I~~). In particular, it should be noticed that in the able concerning the‘CA potential surfaces (cp, limiting case oPR + - the crude adiabatic potential surfaces dissociate to the wrong energy?. Thus the high vibration& wavefunctions and zero order IeveIs in the {In) manifold may- diffe: appreciably in the A and CA representations. (4) Both A and CA basis sets are diagonal within the same electronic configurations, i.e. ((nvfA)j

VA(no’(Ab)

= (d&*)IT,

tAU(r,R)lnu’tCA)>) = 0

for

u + u’ .

This point is well known concerning the A set but was not clearly realized concerning the CA basis. by the application (C) The interstate coupling matrix elements [eq. (4)] were conventionally treated [l-3,5,6J of the Condon approximation: (a) replacing the energy denominator in eq. (4) by the constant electronic energy gap and (b) assuming that the matrix element
: I.

where’E’$ J?+ is the Franok-Condon factor between the adiabatic functions, A&$ is the effective e&actronic energy gap 131 (modi~ed by a promoting mode) and J is an average frequency. This isjust the Condon approx~imation modified by a correction factor 71,which exhibits f9f a linear dependence on ~~~~~ and a weak dependence on the coupling strength. Thus for near resonance coupling, eq. (8) is practically independent of the energy gap, removing the paradox of Burland and Robinson f’7J concerncing the appreciable difference bettieen A tid CA restnance couphng, ht the weak ~oup~ng limit. For off resonance coupling in the A basis (say with state b) in the we*!: coupling limit we have q = 1 so that

Thus, off resonance ~oup~g in the A basis is small, (66)In the CA basis the coupling terms are of the same form for both near res&a.nce and off resonance coupfing

Thus the off resonance matrix elements in the CA basis are relatively large. The ratio of the off resonance coupling terms in the CA and A representation is A&‘/X.5* 10. The last point brings up the major d~f~cult~ associated with the use,of the CA basis, Let us consZder the extensiO3 of ‘the Bixon-former mode1 @‘i. I) where the role of other excited states (which always exist in real life) is included. We shall provide a general expression for the resonance width from the optical line shape <&ho&& any other “short time” or “long time” experiment could be considered). Including the effects of radiative coupling to first order, the line shape &5’) for absa~tion from the ground level 10)is [X0]

wheri: the Green operator is G(Ej= (.!%H+i?$-l

;

y b 0-P

(f2)

and or is the dipole operator. Provided that conditions (2) and (d) hold the line shape {in anjr basis) is

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CHEMICAL PHYSICS LEX-fERS

1 November 1971

.. This regult brings us i0 a basic principle for the choice of the appropriate basis set far des&ibing electronic relaxation processes. In real molecular systems conditions (a)-(d) are necessary but not sufficient for describing the re!axation fate in terms of the Golden Rule and we have in addition to assert that; (e) the coupling between lb) and 1s) and between lb> and Ifi is negligible.

the appropriate basis set has to be chosen to satisfy conditions (a)-(d) and COmhimize the non-diagonal off resonance cqupling terms V&EI=E,j and ybS. From these general considerations we conclude that: (7) The adiabatic basis set is conceptually superior to the CA basis for describing eIectronic relaxation processes, as it invoives much smaller off resonance coupling terms and thus can be described by decay in a two electronic level system. (8) The appreciable contamination of the zero order states IS)and i0 by other 16)states in the CA scheme [ 121 implies that is is metiningless to consider the decay of an initially crude adiabatic state, as pointed out by Lefebvre 1131. (9) The a priori reason for choosing the Born-Oppenheimer basis for describing radiationless transitions is to minimize the off resonance coupling terms. If aU the off resonance states were included, the CA basis is perfectly Thus

r;rlequate. (10) The choke of the basis set is arbitrary

and does not reflect on the physical features of the problem. However, at the present state of our ignorance of the fine detaiis of molecular coupting terms, onIy the adiabatic basis provides a sound physical description of a “two electron level system” where the role of other states is disregarded. Finally, we would like to state one interesting shortcoming of the A basis. It can be demonstrated [14] that the adiabatic (Ifi} q uasicontinuum does in principle carry oscillator strength from the ground state, pal f 0, so that strictly speaking the optical line shape is fanoian. The same result is obtained for the CA muItiIeve1 system [14]. However, the Fano line shape index is [ 143 4 = &Jr, so that in most cases of physical interest 4 & I and the line shape is reduced to eq. (11). The theory of electronic relaxation processes has been fraught with conceptual difficulties concerning the proper choice of the basis set, We hope that the present treatment will remove this confusion.

References 111M. Bivontid J. Jortner, J. Chem. Phys. 48(1968)715;50(1969)3284,4061, [Z] B.R. Henry nnd M. Kasha, Ann. Rev. Phys. Chem. 19 (1968) 161. {3] K. Freed and 3. Jortnci, J. Chem. Phys. 50 (1969) 2916. (41 U. Farm, Phys. Rev. 124 (1961) 1866. [Sj S.H. Lin, J. Chem. Phys. 44 (1966) 3759. [6j R. Kubo, Phys. Rev. 86 (1952) 929. [7] D. Burland and G.W. Robinson, Proc. Natl. Aud. Sci.66 (1970) 257. (81 B. Sharf and R. Siibey, Chem. Phys. Letters 4 (1969) 423; 4 (1970) 561. [9] A. Nitmn and J. Jortner, J. Chem. Phys., submitted for publication. [IO] R.A. Harris, J. Chem. Phys. 39 (1963) 978. [ll] M.L. Goldberger and K.M. Watson, Collision theory (Wiley, New York, 1964). 1121 H.C. Longuet-Higgins, Advan. Spectxy. 2 (1961) 429. [13] R. Lefebvre, Chem. Phys. Letters 8 (1971)‘306. [14]

A. Nitran

and I. Jortner,

to be published.

(151 G. Orlandi and W. Siebrand, Chem. Phys. Letters 8 (1971) 473. 1161 B:Sharf and R. Silbey,Chem. Phys. Letters 9 (1971) 125.

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