Radiationless transitions. Quantitative interpretation of some experiments.

chemical Physics 14 (1976) 441-453 Q North-Holland Publishing Company

RADIATIONLESSTRANSITIONS. QUANTITAW INTERPRETATIONOF SOMEEXPERIMENTS.

v. YAKHOT Departmentsof QremicalPhysicsand Structural C&em&y, Weizmunn Institute of Science, Rehovot, Israel Received 16 October 1975 Revised manuscript received 30 January 1976 The Green function method is used to calculate the rate of radiationless transitions. The obtained formula is applied for quantitative interpretation of the fluorescence quenching of antbacene, pyrene, methylanthracene in various solvents. The vibrational relaxation rates of small molecules substituted to the noble gas crystalline hosts are evaluated. The agreement with experiment is satisfactory.

I. Introduction In this work we consider the radiationlessdecay of an electronically excited atom or molecule which interacts with the phonons of a crystalline host matrix. We consider also multiphonon vibrational relaxation of a guest molecule in a crystalIinehost. From the mathematical point of view the theory of orbit-lattice relaxation is equivalent to that of the radiationlesstransitions in large molecules, so that the results we here obtain can be applied directly to the calculation of the probabilities of such transitions. These problems have attracted attention . of a number of investigatorsand a lot of theoretical results have been published [l-8]. 1.1. Formulationof the problem

Consideran atom in a crystalline matrix, with the index .$denoting the crystal-lattice site. One can write the hamiltonian Jc = “, + T

V(r, R&

(1)

where I$ and r describe the lattice and electronic coordinates, respectively;JCeis the hamiltonian ofthe free atom and V(r, RF) is the potential of the atom-lattice particle interaction. Let @Jr) be the wave function for the free atom; thus

One can introduce the operatcrs Bi and B,, of creation and annihilation of the nth atomic state, and then write in second quantization representation:

where

V. Yakhot/Quantitotive interpretotionof some experimentson radiationlesstransitions

442

Y‘ = V,, is the interaction of the atom in nth state with lattice and Vnnpare the matrix elements which couple thydifferent electronic states. Let us assumethat the Vnnt are small,i.e., IV,/ >I&,+ Then one can omit the V,,e from (3) and take them into account as perturbation which cause the transitions between the electronic levelsof the impurity atom. Note that {Rp} the totality of equilibriumpositions of the lattice sites when the impurity atom is in the nth electronic state can be found from the dynamic equations:

c av (Ron) 0. au, {Ron}= II

E

E

(4)

E

Weplace the origin of the coordinate system at the reference atom and expand VJR,) in (3) in powers of displacements UEfrom some chosen equilibrium positions, {RF? for example:

(5) One can introduce the phonon creation and annihilation operators bi, bP and rewrite (3) (tr z 1):

where (7) with&f the massof the crystal particle and 5@, 8) are the eigenvahes of the &jnamic matrix of the crystal. Note that (79 Cne can introduce the electron-phonon interaction operator:

3f&=

c qp=c y (b++bp)5;B,, n,p *p p n

(8)

where X’& =-c 7 (b++bP)+$,. p nP P

(8’)

It is possibleto see from the deftition (7) that the electron-phonon coupling constants 7,, can serve as a measure of the energy of lattice deformation caused by 0 + n transitionof a guest atom. It fohows that drastic changes of the atomic states lead to big changesin the atom-crystal interaction, or in other words, to a stronger electronphonon coupling in comparison with the case in which these deformations are small.The energy dependence of _r,, can be very complicated because it includes the totality of parameters describingboth impurity and host par-. titles. Cne can notice, however, the obvious relations:

443

K Yakhotf Quantitative @uepretazion of 5ome experimentson rodintionless transitions

Top = 0,

Ir,,l > Irnfpi

if

n >n’,

IQ

=

Ir,,i,

(9

if n and n’ correspond to states of different principal quantum numbers while all the other quantum numbers of these states are the same;rz3 I corresponds to the state of the ionized atom. The wave function of the system atom + lattice can be written: (0, M = 4n,N, whereN denotes the totality @,}of the phonon occupation numbers. Let us calculate the probability of Pn,n_l of the transition: In.N-+In-1,N’). It isknown [9] that %l-1=

%i,n-1

=

27rAv(fV)c

N’

IGr.M Vln- 1,Z’h12S(E,, - En+&,

(10)

of(G) in the initial and final states, respectively. This transition is where I’&JJand En_1JVV are the eigenvahs caused by the coupling between the n and n’ electronic states given in the offdiagonal matrix elements yn,,-t proposed in [l] :

(11)

(n, M f’ln - 1,Iv’, = C,r~_,UrP’~. Further, Enr&V) E ~nl3ClnNvI= (En + ~JfV,,

Jeep= (nl.X&L

(12)

and let us introduce: P$r’(r) = exp(ijCpt)(R$ - 8&) exg (-iXP r).

(13)

Taking into account that Jeep = ZP~p(~~$, t f) one obtains q:‘(f)

(14)

= F $“’ [b+ cxp(iw, t) + b exp(-iw,t)] ,

and the coupling constant $“’ is: (1%

-yp”‘,’ = ‘yn$ - r,,. In the harmonic approximation the expression(7*) for A!$ can be represented as AEn = En - En_1 = E; - E;_1 i- v”,-

q-1 =E; -

E;_l

-_$ I$“-! 12/cap.

(16)

2. Green function method The Green function for an exciton interacting with phonons has been evaluated by Davydov [IO] and Agmnovich [I l] in order to describe the line shape of the excitonic absorption in crystals. We will use their ideas in order to calculate the probability of radiationlesstransitions in a host crystal. Let us use the following definition of the S-function:

V. Yakhot/Quantitutiveinterpretatio’nof some experimentson radiationlesstm~sition~

1444

-.

6(w)= (l/n)Re

Jmexp[i(w+ic)rl

dr,

E-f 0,

E>O.

(17)

0 &bstituting (17) into ilO) and using (12) one can obtain for the 1+ 0 transition for example: 0

p1 o

=

215 ,*I2Re s

dt {A~[(OlB,e-‘“‘lO)tO(e

i”tB$O)l}

=

215 ,oi2 Re J- dt((Br(r)B$

(18)

0

0

Introducing the Green function: G,(t) = -i 0 (r) U,(r)B>J,

(1%

one can write for the probability of radiationlesstransition: 9 1.0 =-21C,,,i2 Im

s Gr(f) dt = -47rlC,,o12 ImGl(w)l~,o. -m

00)

The generalizationof (20) for the case of an arbitrary transiticn II+n - 1 is trivial: one replaces 1,Oby n, n-l and El by En - En_* ; we have chosen En_1 3 0 in this section only to simplify the notations. Let us introduce now: G(r) = -i6(r)e-iac’.

(21)

Usingthe Feynman expansion 1121one can write: G(t)

=

exp (-iJ$, t){G’(t)

s exp (--iJf,c)

+ mzl

=

1

dt, ... d& C”(t - fr) V(fL)GO(tl - fz) ... V(&) GO(&))

-00

(22)

i. = urn,

Weused here the electronic Green function: Go(t) = -i 6(r) exp (-Xef).

(23)

Fourier transform of (22) is thus givenby

-1 ,zt

G,(u) = G;(u) +

2~

f

-0D

L5 dt eiwt j dt, ...dfrrrG’(t-

tl)({Vn(tr)

-

Taking into account that - -iO{t)exp (-iE,t),

&e can represent (24):

and

G”(tm>,

(24)

where

e(t)

GO(tl - tJ ... Vntt,,$

G:(u)

= I/2n(w

- E,., + id,

V. YakhotlQuantitative

- 1 -

G,(w)=G&)+~~~z;;

interpfetation

$ dfeiW’

-0

of someexperiments on mdktionless transitions

y dtl...dt, -ca

445

G~(t-r~)..._G'(t,)U,,

where

u* = i&

urn= iJzi = ‘Vn($)

= 0,

(26)

r$(r*) ... vn(f*i)).

Evaluating(25) with (26) one must remember that the time ordering operation is assumed to be taken into account. Weknow how to averagethe products ofBoseoperators [12,13]: U2 = Wn(rl) V,(Q)) =

C I$“-’ I*exp[-iw,(t,

- Q)l,

P

where (2~ - I)!! = 1.3 - 5 . ... e(Zn- 1). SO, the averageof the 2nth order is the sum of (2n - l)!! terms V,, which are all possibleproducts of the averagedpairs of operators V,(tJ which one can construct from Every term of the expansion (25) can be represented by its Feynman diagram.One can draw all the diagrams usingthe following 5 rules: (1) The thick line with the arrow from left to right corresponds to C(w). (2) Any graph representingthe term of 211thorder of perturbation theory has 2n vertices (points). (3) Thin lines with the arrows from left to right COMeCtiIIg neighbouringvertices corresponds to GE(o.). (4) The dotted (phonon) lines connecting any two points correspond to wPj and to the factor I~P~n- i I*. (5) The conservation law is assumedto be fulfilled and the summation over all phonon wave numoers should be carried out at every vertex. For example at the points (a) and (b) of fig. 1, respectively, w 1 -w2-w

p=O

and

wl-w2+wp=0. Wl

I

9

W?.

.‘. (a)

_% I

-.

,

, 4’

V4

Fii. 1.

Thus one can represent the series in (25) graphicallyas in fig. 2.

-+-+-?y--f-q-+-$(a)

m

.

xr_l__’

,’

Cd)

V. Ya.Rhot/@antitative interpretationof some exprimenn on radiationlesstransitions

446..

Usingthe results described above one can understand that diagram(b) of fig. 2 corresponds to the first-order term:

T (w -’ En +ie)(uJ?:

P

tiE)(w-E+iE)’

Three graphs(c), (d) and (e) represent three terms of the fourth-order expansion (25). Diagram(d) for example stands for:

and so on it is possibleto write any term of expansion (2’5).One can see that ifEn % uD it is possible to omit all the phonon frequencies from the expressionscorrespondingto ail the orders of expansion (25) This procedure is correct if the coupling is not too strong:

(27) whereN = &/o+, andfis the final result. Thus one can omit the phonon frequency if the contribution from “dangerous” diagramofiVth = E,,/wD order is small.To estimate its contribution one should use, to avoid divergency,the exact expression for the free Green function which includes the natural line width Tn. G:(u) = 1/2a(o - E, + ‘7, + ie),

e-+0,

e>O.

One can see that usingthis expression the terms correspo@ing to the “dangerous” diagram:

(28) where No + N, =A! For the typical casesof interest k+, = 50-100 cm-l, E,, = 1000 cm-l and 7, = h/7 = 10-2 cm-l (7 is the lifetime of the state). One can see that in the weak couplingliiit Err I$Jr-f)2/ $ Q 1: ,$J-yp;

2 lo-3--lo”,

Usingthese numbers one can estimate the contribution from the dangerousdiagramto the sum of all the terms of (25) for the multiphonon processesof different orders .%.One can omit this diagram(and all the others of higher orders) if the value of this diagramis small in comparison with the result of summation of ah terms of (25). The numbers are givenin table 1 (see also tables 2-4). Table 1

Dangerous &gram Our fti result f

N=4

IV=5

lo-9-10-13

.10-‘4-lo-‘7

10-7

10-g -1pJ

N= 14 10-39-10-53

lo-‘5-10-‘0

K

Yakhot/Quantit~tive interpretutionof some experimentson radiationlesstransitions

447

One can see that almost in all the cases we have treated (except Xe$Xe system) the contribution from this diagram is negligiblysmall and thus it can be omitted. The validity of this procedure for the strong coupling case was discussedby the others (see ref. [l] for example). Afterthesesimplifications:

Itisknownthatforx>>l: m

eX242&=+ J x

1

+k?_!z+....

22x3 23x5

(301

24x7

In our case:

IE,/t/BI %1,

where B = ~l$‘?2,

and comparing(29) and (30) one can conclude that

G,(w)=(1/2ti)~

$ eA2-’ dt,

withA

=i&E,/m.

(31)

A

Separatingthe imaginary part of (31) and using(20) we obtain for the rate of radiationlesstransition m

3n,,_1 = (l/277)m

s eBt2 cos(&E,

(32)

t) dr.

This integral can be evaluated exactly [ 141 and we get:

3n,n_I = (ICn,n_r12/W &fQ

exp [-(aE,>?12sl .

(33)

Formula(33) was obtained for T = 0 in order to simplify the notation. Generalization of (33) for the case of an arbitrary temperature is trivial: instead of B = IZPI-yPn+r12one should use B = c l$+112(2Np+ 1). P

(33’)

Thus at low temperatures where NP Q 1, the radiationlesstransition rate does not depend on temperature. At high temperatures (I 5 oD) one obtains:

where k is the

Boltzmannconstant and

Such temperature dependence has been observed in our department experimentally for the rates of S-T transitions of the series of organic molecules.The authors of [15,16] found the relaxation rate to be constant and finite

448

V. Yakhotf Quntitative inttrp,ietation of Some experiments on

radtitionlessfipntitio~

at low temperatures. At high temperatures-they obtained the thermally activated behaviour of the probability of S-T transition with the activation’energyEn = 1000-2500 cm-l. Let us show that this value is reasonable from the point of view of our theory. One can see from the above expressionsfor qn,n_l that En = (AEn)2/4Dn= (E,”- E,“_l- Dn)2. Thus in order to obtain the value of E,, one must know Eg - Ei_l andD,. Typical values of AL?:for S-T transition for the molecules investigatedby Kornstein et al. are cv4000-5000 cm-l. TOestimate Dn let USnotice the following: the energy of the molecular deformation which occurs in the So + SI transition in these molecules is usually - iAS = 4000 cm-1 (M is the Stokes shift). This deformation corresponds to an electronic transition (So-S,) with A,!?:% 20000 cm-1. To get a rough estimate of Dn let us assumea linear dependence of D, on the energy of transition: D,, = c&C;; then the energy gap 4000-5000 cm-f in our S + T process corresponds to Dn = 700-800 cm-t. Usingthese numbers and above formula one gets: Ea = 3000-4000 cm-1 .

Such a similarity between the calculated and observedvalues shows the ability of formula (33). Of course, the linear dependence we assumedabove is not theoretically based but is not unreasonabie and is not in contradiction with the general relations (9). Since the estimate of E, is very sensitiveto the value of Dn used, a more critical comparison of theory and experiment must await a more reliable value for Dn.Similartemperature behaviour has been observed in the we&known experiments on orbit-lattice relaxation of the rare earth ions dissolvedin solid matrices [17-191. We applied the formulae obtained above for the interpretation of radiationlesstransitions of anthracene molecule in various solvents. The temperature dependence of the fluorescence quantum hield from anthracene in different solvents has been studied by many investigators(see ref. [20] for example). One can summarizethe main features of the results as follows: (a) The fluoresecence quantum hield is constant at low temperatures. (b) In the high temperature regimethe yieldof fluorescence decreaseswith increasingof temperature according to the Arrhenius law. An empirical formula for the rate of radiationlessdeactivation which describesthe observations(a) and (b) is R = R, + R, exp(-AE/T).

(34)

(cj The activation energy of radiationlessdeactivation depends strongly upon the nature of solvent, varying in very wide liits [21]. An overall interpretation of the radiationlessprocessesin the anthracene molecule has not yet been achieved. The more-or-lessaccepted point of view is that expression(34) describestwo different transitions. It was assumed that the temperature independent part of (34) originates from the S, + T2 transition while the Arrheniuslaw corresponds to the S, + TJ relaxation pro-ess. The T3 state has not been observed but was assumedto lie a little higher in energy than the S, state to explain the activation behaviour found experimentally. The T2 state is the triplet state discoveredin anthracene by Kellog [22] and lies slightly lower (a 500cm-1) than the S, level (ET2 = 26050 cm-*). Usingthe above formula it became possible [23] to show that in the casesof anthracene in various solvents one can quantitatively explain the high temperature fluorescence quenching consideringthe S1 -+T2 transition only. Thus there is no need to introduce the assumedT, state of a certain energy in order to obtain the consistent picture. l”@eresults are presented in tab!e 2. One can see that this interpretation is valid for pyrene and methylanthra-cene too. The agreement between calculated and measured data is satisfactory.

449

K Yakhod Quantitative interpretaticn ofsome experiments on radhtionless transitions

TabIe 2

Experimentally determined and the calculatedactivation energiesof the S1 +Tz tramLion in the high-temperatureregime (all in cm-‘) Molecule

Solvent

-%

Es,

Antkracene

Ethanol Dioxane PMM

26590 [24] 26445 [24] 26450 [ 251

26505 [24] 26400 [24] 26380 [25]

Pyrene

Light petr.

26900 (261

26850 [ 261

50

27200

800

800 [ 261

9 Meth. Anth.

Cyclohexane

26900 [26]

25550 (261

230

24400

800

750 [26]

3. Multiphonon

&I 85 45 70

=2

AE

26050 26050 26050

380 500 240

w=w)

AE (=d

320 [21a]; 900,500 [21k] 400 (petr. ether) 220 121

vibrationalrelaxationof a guestmoleculein a host crystallinematrix

The theory of multiphonon vibrational relaxation of a guest molecule which is situated in a host crystal has been developed in our laboratory for the case of an exponential molecule-crystaI interaction potential [7]. This theory has been generalizedto the case of an arbitrary potential [8]. Wewiil show very briefly in this section how to get the results of [7] using the procedure of section 2, and severalnew quantitative estimates of vibrationaI relaxation rates wiIIbe made. The method of evaluation of the probability of the muttiphonon vibrational relaxation is very similarto the one developed above for orbit-lattice relaxation and we believe that the agreement with experiment achieved for the former problem can serve as an argument in favour of the correctness of formula (33). It was shown [7] that in the case of an exponential potential vibron-phonon interaction can be written: V a - X@BP exp (-Py),

. (35)

where X describesthe intramolecular vibrations {X is the deviation from the intramolecular equilibrium distance): DB =D exp (-&,) is the molecule-crystal interaction at the equilibrium distance yo; (3-l is the characteristic parameter of the molecule-crystal atom interaction; 17is the number of the nearest neighbours;y describes the

oscillationsof the crystal atoms. Introducing the phonons: y = C (ll2Mr~~)“~ {exp [i@o+wpt)] bi + exp [-i(pa +wpt)] $1 P

(36)

One can take into account in (36) all the local frequencies associated with oscillationsof the relaxing molecule itself and with the lattice deformation caused by this molecule. In order to calculate the relaxation rati! one can repeat the procedure described in section 2 and end up with: 3 M-1

3

-

1

w

7 lZ,?Z-1

where F=

P27 GJ,,/W@$+ 11,

f? = 2p2 G (1/2MCQ(2$ + 1).

(38)

(3%

This result is quantitative; in order to get the relaxation rate one must know the parameters of the moleculecrystal interaction and the vibrational spectrum of the crystalline host. We calculated the vibrational relaxation rates of the Nei excimer in solid Ne matrix and quantitatively explained the anomalousemissionspectrumob-

‘-

-450.:-

V. Yakhot/

Quantita&e

interbretotiorl

of some experim&

on r&ztionless

transttions

served in this system. To achieve this we used a very accurate Ne*-Ne potential [26] and the known phonon spectrum of solid Ne [27]. We made the reasonable assumption that the formation of excimer by the atoms of the crystal does not lead to strong deformations of the surroundinglattice and thus one can use the phonon spectrum of the ideal solid. Lzt us apply the same representations and calculate the rates of vibrational relaxation of Ar\It,Krz and Kez in their host matrices (Ar, Kr and Xe, respectively)using formula (37). in order to estimate the characteristic length of the potential let us use the semiempiricalformula which usually leads to reasonable results 1281: fl= 17.5/R -6 X 108cm-1,

(40)

where R is tie distance from the atom of the molecule to the nearest crystal atom. One can check using the known excimer parameters [29] and the rare gas crystal’sgeometry that R = 3 A @ extracted from the very accurate Ne*-Ne potential is 5.8 X IO8cm, in very good agreement with the estimate (40). It was shown that in the rare gas crystals [7]:

c s

w -+ 3 wgiw)do =2.82~,,,

P

(41)

p

where @Dis the maximum frequency of the solid spectrum. Using(40) and (41) one can rewrite (37): 3 = [rf2(DB)2~2/MAE1 me’

exp(-u*lK),

(42)

K= S.64g21i/2M~, = 101.6 X 10-f’/MwD,

(43)

v = m/fi#D )

(44)

= i$SfAE in the harmonic approximation. One can calculate now the rate of vibrational relaxation. and IX&$ We take DB = 500 cm-1 which represents the value of the excimer-atom interaction at the distance - 3 A and the results of calculationswith formula (35) are given in table 3.

Table3 AE (cm-‘)

WDa) (cm-‘)

JGfAs

310 [29]

64

C$fKT

190 [29]

Xe,*fXe

140

50 44

System

.

al A!l the WD given in table 3 are taken

K

d

theory

&JD

(s-r)

from

3&lexper.

c

(s-‘1

1.2 .0.9

2x lO’6

4.8

6 x 108

> 106

1Ol6

3.8

2x 10’0

-> 106

1.2 0.7

0.9

6 x lOl5

3.2

2 x f0”

> I@

0.6

[ 301.

One can see from table 3 that our theory explains the very effective vibrational relaxation of these excimers in their matri&. IN aI these casesthe reIaxation time is much smallerthan the radiative lifetime (X 10e6 S) and thus one can conclude that emissiontakes place from the excimer which is in thermal equilibriumwith the medium:.T& has been observed experimentally [3 I]. At the same time relaxation of Nez in solid Ne is uneffective (AE * 600 cm-r). This is the anomalousemissionspectrum of Nez in crystalline Ne which can also be quantitatively ex&ined by our theory [7]. --, -Let us examme now the relaxation properties of some light diatomics substituted in crystalline rare gas ma. -tick. In this case the guest molecule causesstrains jn the crystal and thus one can assumethat local modes WL~ should be taken into account [32] _Moreover the modes oLz correspomjingto the vibrations of the relaxing

V. Yakhot/Qantitative interpretation of some experimentson radiationless transitions

451

diatomic molecule itself must also be Muded in (37j(39). The value of oL2 has been calculated [33] and determined experimentally [341 for CO in Ar and the value is reported as circa 100 cm-l. So, ~~~=ti~~=~~=lOOcrn-~, and let us take them into account in (37-(39):

whereh$_= [exp(wL/T) - 1I--‘. Here M, and M, are massesof molecule and crystal itom, respectively. The factor 3 which appearsin (45) takes into account three directions of vibrations. Taking (DE) = 1500 cm-l which is reasonablewe calcuiated the relaxation rates in some systems.The results are givenin table 4. Table 4

System

AE

(cm-l)

E;:iAr

1400

NgINe

1400

2F

b.#L?F

cq2, 180 x 10-29

. 245 X lo-”

A

ti

(S-9

Sn-1 talc.

(s-9

%-I

exper. (s-‘)

43.3

- 10’6

2.2

10”

2 < [35]

31.5

- 10’6

3

4x 103

> 0.3 [35]

Wewant to mention that the calculated temperature dependence of all the rates listed above is ve!y weak in the range of T until T= 30 K. This agreeswith the conclusionsof [35J. To conclude this section we givethe result of estimation of
=9,;_, = lo25 s.

This result agreeswith the latest observationsmade in the Legay group. They have found that the multiphonon relaxation has not been detected in their experiments and the temperature dependence of the infra-red emission time was caused by the other reasons [36]. Weunderstand that our results are sensitiveto the parameters we used and serve rather to illustrate the quantitative ability of the expressions(37)-(39). It is clear, however, that using the precise parameters, as was done for Ne;lNe, one can obtain accurate theoretical predictions. Wewould like to emphasizethat without taking into consideration the local modes wL1 and wL2 we could not explain the experimental data listed in table 2 even within tens of orders of magnitude. Our choice wL z=lOO~rn_~was based on the experimental [34] and theoretical [35] data. The value LB = 1500 cm-l we used above is reasonable and the fmaI resultsarenot verysensitiveto the valueused. 4. Discussion In this work we presented a formula for the rate of radiationIes$transitions which was applied for both electronic and vibrationai multiphonon relaxation processes.We obtained a “gausSi” dependence of the rate of vibrational relaxation upon the energy gap and explained a number of experimentalresults. At the s&e time there are questions remained unanswered. First of all, what is the energy dependence of the electron-phonon relaxation process?There is not yet a simple answer to this question. In fact one can write in the one-phonon frequency approximation [see eqs. (33)X33’)and the deftition of D,] that at high temperatures:

K Yak-W/ Quantitative interpretorionofsome experimentson radiationtess mm&ions

452‘

hpn = const:- (AEn)2/4Dnkr,

(46)

SOthat to know the energy dependence of 9, it is necessaryto have the Dn = D&E,). In accordance with (9) D,(O)=O? and we can assume that in some simple cases: lnPn = ccnst. -CY(AE$

(47)

where BC 2. In the particular case of Dn = U, one obtains the “energy gap law” !9n a exp(-a,,). In general (47)should not be valid because as was stressedin the introduction Dn depends not only on Al?,, but on the whole tot&Q of parameters describingthe atom-lattice interaction. It is even possiblethat in generalD,,is not a monotonic function of At& Experimental examplesshowingdifferent behaviour of P+,!T,,) have been reported: thus in [17] there was observed the energy gap law, while in 1371much more complicated dependencieswere reported. One understands that there is a possibiity to influence the radiationlessprocessesin a given species.If one fmds a matrix which is not deformed upon some transition of the guest atom the radiationlesslossesshould be very small.On the other hand if it is necessary to quench the excited state one must look for a matrix in which strong deformations accompany the transition. From this point of view the work [38] in which the rate of radiationless decay of the rare earth ions was investigatedin different glassesis very interesting. Let us consider qualitatively the radiationlesstransitions in big molecules.Wewould like to emphasizethat the energy of deformation we introduced D, = 0: + Di,where DAis the energy of nuciear relaxation in a molecule because of electronic transition and Di describe the variation of molecule-mediuminteraction caused by this transition. So we can expect the probability of radiationlessprocessesto be high in the casesin which transition is accompanied by breaking the molecule-mediumbinding. Accordingto our treatment the radiationlessdeactivation should be rapid for the systems in which the Stokes shift is big. In keeping with this, Korenstein et al. have found for a series of folvenes that as the substituents are varied so that the molecule becomes more non-planarthe Stokes shift increasesas does the fraction of singletexcited moleculescrossingto the triplet level [39]. On the other hand the SI-‘l’ transition in the rigid planar molecules such as antluacene, pyrene and so cn is slow and at the same time the Stokes shift is small. The quantitative theory of D,,isa function of all the parameters of the processis very interesting. From this point of view Warshersapproach to the calculation of the moiecular conformations in excited eiectronic states 130,411promises to enable estimation of deformation energiesaccompanyingthe transitions and hence of rates of radiationlessdeactivation.

Acknowledgement i would like to thank ProfessorsM.D.Cohen and S. Lifson for their constant interest and help in the course of this work. I am grateful to Dr. R. Korenstein and Dr. M Shapiro for very helpful discussions.

References [ll R Engelmanend J. Jortner,Mol.Phys. 18 (1970) 145. 121IC.Freed and I. Jormer,J. Chem.Phys. 52 (197G)6272. 131T. Miyakawa andD.L;Dexter,P&s. Rev.BI (1970) 2961. [4] A.-Ntzan,S. Mukameiand J. Jortnex,J. Chem.Phys 63 (1975) 200. [ 51 D.3.Dies&r,J. Chem.Phys.60 (1974) 2692. [6] S.R. Lin, J. CBem.Phys. 61(1974) 3810.

K Yakhot/Quantitative interpretationof some experimentson radiatfonlesstransitions

453

(7] V. Yakhot, M. Berkowitz and R.B. Gerber, Chem. phys. 10 (1975) 61. [8] V. Y&hot, Mol. Cry% Liq. Crysr. 32 (1976) 91. [9] L. Landau and E. Lifshitz, Quantum mechanics (Pergamon Press, Oxford, 1965). [Xl] AS. Davydov, Theory of molecular excitons (McGraw-Hi& New York, 1962). [ 111 V.M. Agranovitz, Theory Excitonov (Nauka, Moscow, 1968). [ 121 G. Ziman, Advanced quantum theory (Academic Press. Cambridge, 1969). [ 131 A. Abrico~v, L. Gorcov and I. Dzyaloshinsky. Methods of quantum field theory in statistical plzysics (Prentice-Hall, NJ.. 1963). [ 141 LS. Gradstein and LW. Ryzik, Tables of integrals, series and products (Academic Press, New York, 1965). [ 151 R. Korenstein, K. Mu&at, M. Slifkin and E. Fischer, J. Chem. Sot., to be published. [16] R. Korenstein, to be published. [17] M.J. Weber, Phys. Rev. 88 (1973) 54. [ 181 L.A. Riiberg and H.W. Moos, Phys. Rev. 174 (1968) 429. [19] F.K. Fongand MM. Miller, Chem. Phys. Lett. 10 (1971) 408. [20] E.W. Sheag, S. Schneider and S. Fischer, Ann. Rev. Phys. Chem. 22 (1971) 465. [21a] L.J. Bowen and I. Sabu, J. whys. Chem. 63 (1959) 4. [2ib] A. Kearwell and F. Wmnson, J. Chim. Phys. 1970 (1370) 125. [22j R.E. Kellog, J. Chem. Phys. 44 (1966) 411. [23 1 V. Yakhot, Chem. Phys. Lett., in press. [24] M. Nakamizo and Y. Kanda, Spectrosc. Acta 19 (1963) 1235. [25] R.P. Widmanand J.R. Huber, J. Phys. Chem. 76 (1972) 1524. [26] J.B. Birks, Photophysics of aromatic moIecuIes (Wiley. New York, 1970). [U] J.A. Leake, W.B. Daniels, J. Skalya, B.C. Fnzer andG. Shiiane, Phys. Rev. 181 (1969) 1251. [28] J-0. Hirschfelder, C.F. Curtis and R.B. Bird, Molecular theory of gases and liquids (Wiley, New York, 1954). [29] V. Yakhot and R.B. Gerber, Chem. Phys. 9 (1975) 366.

[30] C. Kittel, Introduction to solid state physics (Wiley, New York, 1971). (311 J. Jortner, L. Meyer, S. Rice and E.G. Wilson, J. Chem. Phys. 42 (1965) 4250. [32] H. Friedman, private communication. [33] H. Friedman and S. Kimel, J. Chem. Phys. 43 (1965) 3925. [34] H. Dubost, thesis, University Pa&Sud, Orsay (1975). [35) D.S.TmtiandG.W. Robinson, J. Cbem. Phys.49 (1968) 3229. [36] H. Dubost, Chem. Phys, to be published. [ 371 E.D. Reed Jr. and H.W. hfo~s, Phys. Rev. B8 (1973) 980. (381 R. Reisf’eld,L. Boehm, Y. Eckstein and N. Lieblich, J. Luminesc. 10 (1975) 193. [39] R. Korenstein, to be published. [40] A. Warshel and E. Huler, Chem. Phys. 6 (1974) 463. [41] A. Warshel, E. Huler, D. Rabmovich and Z. Shakked, J. Mol. Structure 23 (1974) 175.