Dispersion and resonance terms in exciton—phonon coupling: absorption band profiles in molecular crystals

Chemical Physics 14 (1976) 89-110 0 North-Holland Publishing Company

DISPERSION AND RESONANCE TERMS IN EXCITON-PHONON COUPLING: ABSORPTION BAND PROFiLES IN MOLECULAR CRYSTALS D.P. CRAIG and L.A. DISSADO Research School of Chemistry, The AustralianNational University,Box 4, P.O., Canbem, A.C.T. 2600, Ausfralia Received 4 August 1975

Absorption band protiles in the exciton spectra of molecular crystals depend on exciton-phonon coupling. The combined influence of dispersion and resonance tens in the couplttg is analysed by the Green function method and the proRles discussed as a function of temperature. Ihe profiies depend on the broadening of individual transitions by phonon damping, and on the spread of intensity over various phonon sidebands forming spectral progressions. The nature of the profiles is shown by representative calcule tions and applications indicated.

1. Introduction The band profiles of singlet absorption spectra, which can be studied over a wide temperature range, contain much valuable information [I ,2]. in particular, because they depend on exciton-phonon coupling, they can in principle be exploited to bear on questions of exciton trapping and subsequent photochemical processes. This is the point of view adopted here; we refer to previous work [3-6,131 on somewhat different aspects, and specially to calculations by one of us [7,8] on the band profiles in the spectra of phenanthrene and anthracene. The starting point is the hamiltonian H=H,+H’,

H’=kqq

1I

HO = Fe(k) V%,q)

&k)

+x,(q)1 Bi(k +qPW

W)

+ c m,(q) P;(q) =A

b,(q) ++I,

(1)

P,(d+b;(-911N”’+k7q x&I &W(k) P,(q)+&q)l Id2 1,

e(k)is the energy of au exciton of wavevector k, w,(q) that of a phonon in the s branch of wavevector q and St, B, bi and b are creation and annihilation operators for excitons and phonons. The q sum in the final term of H’ is evidently restricted by symmetry to modes of which the equilibrium is shifted by exciton creation. This restriction is denoted by a bar over the summation sign. In the text a barred sign is used wherever theie is a restricted sum over such x&)-coupled terms; in the same expressions the sum is unrestricted othetiise. The coupling functions Fl(k, q) and x,(q) are defined as in Davydov [S] as follows: (24 0)

_9Q1

‘D.P. Craig, LA. Dissado/Dispersion and resonance terms in biton-phonon

coupling

In these expressions the index I refers to a Cartesian component, m to a molecule site and X to a translational or rotational displacement. d is the component of a unit vector, f, is the mass coefficient (mass or moment of inertia) for the motion of which as(q) is the frequency: o,(q) is thus the corresponding rms amplitude of motion. Mam is the resonance interaction between site 0 and m (4) where p is the molecular transition moment, R the intermolecular separation and & = 6~ - 3fiihp Do, is the difference in dispersion interaction between ground and excited states, the ground state interaction being

where the sum is over all excited states of the molecule. The transfer coupling, namely the first term of H’, has components from both exciton [5] and phonon [13] transfer included in the composite transfer coupling function (3). It depends on the degrees to which the interactions (4) and (5) are changed by lattice motion. For applications where exciton transfer is fast, so that transitions occur within a lattice little disturbed from translational symmetry, the exciton transfer part (2a) is usually large, and the dispersive part (2b) can be considered as a correction to it over the limited range of exciton wave vector important hi actual cases. That is the viewpoint adopted in this paper in which the complete transfer coupling F,(k, q) defined in (3) is used. It is straightforward to treat the dispersive part separately if necessaj, as it is in certain situations (such as very weak resonance coupling) not treated in this paper. The form of the operator shows that the total wavevector k t q is conserved among the coupled states so that, if we use the selection rule Ak = 0 for pure exciton transitions, we are now concerned with the selection rule iU,k + q) = 0. At low enough temperatures the initial states belong to k = q = 0, and the upper states all belong to k + q = 0. Thus exciton levels of k f0 participate only with phonon levels q = -k. At h&Her temperatures ground state phonon levels are populated belonging to q f 0 and additional, temperature dependent, transitions occur. This is a principal source of the temperature variation of crystal band profdes as discussed in [7] and in earlier references. In addition, where there is more than one molecule in the unit cell and therefore more than one optically allowed crystal transition belonging to: a given free molecule transition there is an additional contribution affecting the profile of the higher transitions evep at 0 K. This comes from coupling of phonons belonging to the lower exciton branches with tie upper exciton band again within conservation of total wave vector k + q. This was one of the particular problems dealt with in [7] in the framework of purely resonance zzupling. In this paper we treat in addition the influence of dispersion coupling, and adopt a different approach to the underlying Green functions used in the calculations. The role of dispersion coupling depends on the relative magnitudes (i) Ground state dispersion interactions of resonance and dispersion energies as they occur in.the following terms: which together with intermolecular repulsions determine the ground state lattice dynamics. We note that strong intermolecular interactions give wide acoustic bands and high librational frequencies. (ii) Changes in dispersion interactions on electronic excitation; these changes appear in the scale factor for the coupling of excitons to lattice motion. There are contributions to both terms of H'as seen in (l), the first associatedwith phonon transfer and the second @h displacement of the equilibrium position of ceitain lattice modes. Compatibly with this, the q sum is confined in the second case to modes of which the origin is displaced. (ii) Excitation resonance interactions in the electronically excited crystal, determining the spread of exciton levels belonging to one electronic transition of the free molecule. (iv) Dependence of the excitation resonance interaction on lattice displacement, governing the strength of exciton-lattice coupling by the resonance mechanism. -. The us@ classification of exciton-phonon coupling into weak and strong couplingacruallyincludescertain sub-

D.P. Craig,L.A. Disado/Dispersion and resonance terms in excifon-pkonon coupling

91

cases, depending on the relative magnitudes of (i)-(iv). In case A where the exciton bandwidth, namely the energy span of exciton !evels over the full spectrum of wave vectors k, is greater than any lattice fundamental, excitation causes no local lattice distortion. Changes in lattice structure and dimensions are spread over a zone of crystal related to the region within which the exciton is delocalized; this is ill-defined, depending on boundaries and crystal imperfections as wel: as excitonTphoton dynamics. Transitions are to states of this zone which cannot in any strict sense be related to single wave vectors of the infinite crystal. They may be represented as linear combinations of the inftite crystal states. The lattice normal modes of which the origins are most shifted, are, for large zones, near Q * 0, though a considerable number of modes may be involved. For very large zones possible only in perfect crystals the hypothetical limit is that a very small number of modes, all very close to Q = 0, would be affected. If the exciton bandwidth is greater than any intramolecular frequency the molecular transition contributes the whole bandgroup oscillator strength. Alternatively if the bandwidth, while greater than any lattice fundamental, is less than spectrally active intramoledular fundamentals (e.g., the breathing vibration of an aromatic) the free molecule transition contributes only that part of its oscillator strength that belongs to one particular vibronic component. If in this situation there is no dispersion coupling via mechanism (ii), the only exciton-phonon coupling is through (iv), the dependence of the resonance interaction on lattice displacements. This is the case of weak coupling, characterised by the absence of lattice vibrational structure built on the origin transition, which is, however, troadened by the phonons acting as a damping field. If in this situation of large intermolecular resonance interaction, the interaction were independent of lattice displacements we would have a second limiting case in which the dispersion coupling under (ii) provided the only exciton-phonon interaction. We shall see that phonons then appear in progressions and the spectrum is broadened into an envelope defined by phonon lines each broadened to some degree by the dispersive contribution to the transfer term in H’. Where both dispersive and resonance coupling to phonons is present, the spectrum consists of the pure electronic transition accompanied by phonon sidebands, all broadened by phonon damping. In case B there is a limit in which the resonance bandwidth (k bandwidth) calculated for the rigid lattice, is small and comparable to the lower acoustic frequencies, Here, a fortiori resonance coupling (see (iv)) is negligible and the dispersion coupling causes lattice relaxation about a particular electronically excited molecule, and the exciton is immobile. This is the strong coupling limit. In an intermediate situation the resonance bandwidth is of the same order as lattice frequencies, greater than the acoustic modes and less than the translational optic modes and librational modes. The resonance coupling is here almost certainly negligible. On excitation the molecule is relaxed with respect to internal vibrations, and the lattice relaxes with respect to librational displacements but not with respect to translations. Thus in this rather special sub-case the molecules are on lattice sites which have the fuIl symmetry of the Bravais lattice, but their orientations are relaxed. The exciton is therefore clothed with librational phonons but not with translational phonons. We expect to observe phonon p:ogressions determined by Franck-Condon overlaps between initial and fmal states, related to the displacement of the various origins caused by the electronic transition. The situation is closely analogous to the activity of totally symmetrical vibrations in the spectrum of a molecule with shape (covering group symmetry) unchanged on electronic excitation. The effect on the band profne of including dispersive coupling between excitons and phonons can be further described in the following way. Instead of a single peak at the energy of the optical exciton (k = 0), there is a series corresponding to two-particle transitions of an exciton and a phonon. When T > CiK temperature dependent transi. tions appear below the optical exciton ievel (hot bands). Each peak will be broadened by phonon scattering, intensity building up near the pure exciton levels from transitions to exciton-phonon combination levels. If the exciton component in these mixed states lies in the same exciton band this is intraband scattering. It belongs to a different exciton band the process is interband scattering. If the scattering baud width is less than the spectrally active phonon frequency a series of phonon peaks is observed; for larger scattering widths an envelope is observed without phonon structure. The shape and temperature dependence of the width of this envelope will be shown to depend upon the relative sizes of dispersion and resonance exciton-phonon COUpbIg. If the crystal lattice is deformed on excitation, the A(k f q) = 0 selection rule is weakened or altogether lost de-

.:$2 ..._I

: : al? eaig, LA. D&ado/Dispersionand resonancem&in hxciron-phononcbupiing

:-.I

,.-

p&i@g’?fi the degree of deformation, and-changes & excitdri wave vectdrs are possible under the dispersive as Well as the reSonance coupling..The two cases of translational synimetry retain* or lost are loosely those of weak. tid strong exciton-ShoDon coupling.

2 I. The Green function

Following Davydov [5] we proceed to calculate the band profile for crystal absorption from the imaginary part of the dielectric constant, related to the crystal Green function as follows (% = c = l), G(&)

- Gi(k, -w) = - [ VW/(~&)]

[~(k, w) - E,J,

(6)

where the Green functions are Fourier components of the retarded time G.reen functions for excitons (bosons) and hole propagation defmed as Gr(k,t-

Q=iLJ(t

- f*> (([BP, 0,5%

@I%

G$k, t - to) = - i B(t - to) <([Lit(k, t), 5(k, to)] )>.

(7)

Vis the volume of the unit cell, pi a component of the transition moment in the cry&l andE the energy of this transition in the free molecule, E(k, w) is the transverse dielectric constant associated with the particular transition, and ~0 the background transverse dielectric constant contributed by all other molecular transitions. Eq. (6) applies to light-propagating homogeneously [5] in the crystal with electric vector directions for the k = 0 Davydov components. The Green ftinctions in eq. (6) are found by Iguchi’s method [6]. The causal time functions so found are then F&i&ransformed to give the functions required in (6). The method throws light on earlier results [3-61 and allows thi various approximations to be compared. Tfie theory will at fast be piven for crystals with one molecule in the unjt cell. The generalisation to two molecules per cell, which is a common situation in molecular crystais, is discussed in section 2.8. The causal Green.function for an exciton at time t is .G(k, f) =.-- i (0: .;av(qs)W (5(&, t) U(t, 0) Bf (k, 0)) IO: emu(qs)>.

(8)

The state IO: .a-u(@ has ~(4, s) quanta of phonon mode Q in the s branch, ali q and s being occupied according to Boltzmann statistics. Its exciton occupation is zero, appropriately for cases to which the theory is to apply, where the excitation energies e(k) to levels in the exciton band are thermally inaccessible at ordinary temperatures. In this limit the causal and retarded exciton Green functions are the same, and G, alone contributes to the lefthand-side of eq. (6). h (8) Tis the time ordering operator, and U(t, 0) is the Dyson function (see [9]):

G felt&-i

i V(t’) dt’), .o

.where V(t) isthe perturbatiotii?

..

in interaction represeptation, i.e.,

: ‘V(i)= exp (tio$H’ ixp (-iHot). -. .: -:... -.. WA&g :?u_=_$(k)[b;..i @s)) .and ROAN y :.

-..'

(IO)

.(ll)

93

D.P. Craig, LA DissadolDispersionand resonance terms in exciton-phonon coupling

B(k, t) = E(k) exp [i e(k) t] we have G(k, r) = - i exp [- i e(k) t] (*iU(t, O)]*)

(12) (13)

ushg(9). Then, with Iguchi’s symbolic notation, G(k, r) = - i exp [- ie(k)t] 2

$-$

n=O

<[W]"i,

(14)

in which [W]’ is the term of degree n in the series (13). Because the phonon operators b!(q) and b,(q) appear lmearly in the interaction hamiltonian H’, alI odd orders in W are zero. Again following Iguchi orders higher than two are dissolved into contractions as for example

+
Vet,)),.

w

This implies treating all second intermediate states as identical to the’initial state, and avoids dealing with comphcated ranges of energy denominators in the final expressions. in the expansion of the right hand side of (15) only those terms which can be expressed as powers of ([WI 2,appear. We find, with 0 = ([W] “j/2!, <[W]4>= 3 <[WI2P = 3 x 22 x &,

(16)

and generally

WI 2n++(-j,

:[w]*>/@I>!=W/i!.

(17)

The Green function then becomes i G(k, t) exp [i e(k) t] = n$O (- lr @Y/n!

(18)

and G(k, t) = -i exp [-i e(k) t - CR], = 0,

t>o r-co.

(19)

The energy Green function is the Fourier transform (20), and an expression for 0 is ah that is needed to get the band profile through (6). i G(k, CIJ)= (i/Z?r) j 0

G(k, r) exp (iwr) dr.

(20)

94

.-

,’

D.P. &I$, LA ,?Dissado/Dispersion &d resonance t&s

1.

in exciton-phonon coupling

.2,2. The coupling_hamiltonian H’ -In order to deal with both transfer and dispersion-shift parts of the exciton-phonon H’ =.N- Ii2 Z,vs(k,

4 j Bt (k + 4) I

The required expression for 0

.+x,(4) 8; (k) I)

l

N+?)

[b,(q) + bj (- Q)I.

(21)

is found by a straightfo%ard calculation to be

I (/q=z --ixst~Pit+,s~[ex~t~qW+q;s

interaction we write

(

- Dlqs B(sq) [exp (i$&t)

11+ w[exp(-iw,(q)f)-

- 11 - Daqs IT(W) +

11 [exp(- ifi;qsO- 11 , I

lj)

(22)

where

=w,(q)- e(k+ 9) f e(k), %qs

a& =w,(q)+ etk+ d - e(k),

(23)

and D Iqs = IF&k, ~)12/Nsllr,,)2,

Dzqs = IF,(k,~)12P’(~~qs)2.

N is the number of unit cells and P(gs) is the meanphonon

(24)

occupation number-at temperature T for the qs phonon,

where k, is the Boltzmann constant. The cross-term in 07 linear in F,(k, g) and x,(q) can be dropped. The reason is that the dispersive part of H’ is diagonal in the exciton wave vector k, and the only intermediate states giving contributions are those belonging to the initial k and therefore (on account of the restriction A(k + q) = 0 on the resonance interation) to Q = 0. The sum over 4 thus consists of one term per branch only, and the contribution being of order N-’ may be neglected. 2.3, &re transfercoupling in the dampingappr,oximabon .. In order to illustrate the procedure we first take the limit of zero dispersion coupling, namely X,(IZ)= 0, and cal-. &late the Green function. Two solutions will be discussed: the fast leads directly to the band profile through clsmplng theory. The second gives an explicit account of the underlying structure of the band itself. Bearing in mind that the band profile is to be found from the imaginary part of the Green function, we fmt split the_:resonance terms in (n in (22) into real and imaginary parts,

-. . ..~.

D.P. Craig, LA

Disad~fDispersion and resonance terms in exciton-phonon

coupling

9.5

We now consider the limit t % [o,(q)] -1 and use the zeta-function method [lo], with the help of the .following representations of the Dirac 6 and principal value P/x:

S(x) = 7r-l Lt(r -+ “) (1 - cos xt)/x%,

(27)

Applying (27) and (28) to expressions (26) we find the time Green function for large t G(k, f) = -i exp {-i [e(k) t A(k)] t) exp I-@)

t] ,

(29)

with A(k)=P

IF,& q)12a (4s) c - N --~, q,s %qs

y(k) =$

I

B (qs) f 1

Gqr

IF&k,GJ)I* ,,, {iqqs) S(c-kqs) + [F(qs)+ l] s@2+krls)).

The energy Green function is immediateiy found from (29), G(k, o) = [w - e(k) - A(k) t i r(k)] -‘.

(30)

Eq. (30) is identical with that found by Davydov [S] using a different method of dealing with the perturbation expansion. In this approximation the profue is lorentzian; the damping y is made up of two component scattering parts, both for energy-conserving processes. The first part is contributed by upward scattering in the exciton band with phonon destruction and the second downward scattering. The full width at half maximum (FWHM) is given by 2r(k). The absorption maximum is at the pole frequency, w=e(k)+C

IF&k, q)I* B(qs) --N ( %qs q,s

rr(qs) + 1 %qs

.

(31)

)

The application of (30) to the anthracene crystal absorption profile has been given by one of

US [7],

2.4. Alternative procedure We may get a more explicit account of the band profde by avoiding the initial approximation for t + m, keeping the exponential time dependence, and expanding part of the CF) exponential in the time Green function. CF)is first separated according to Cn’Ol

+(Fj,,

(324

96

..

:-

Ok. &&i.A.

.._.

Dimdo!Dispersion and resonance rerins in exciton-phogon

coupling

:

T. cn,=-~~D,,,B(qs)exp(i~~ QP Then,~~pandingexp(-(F)z),

kqSt) +DzqS t%s)

+ 11 exp t-iQ&,0].

.

Wb>

- ~~SQ&S)f]

(33)

we have for the time Green function (19),

i G(k,-t) = exp [i e(k) t] exp (-WI) / 0

xnc c -

(Dlqsfqs v’(q~)~@ (Dzqs)'@

[T(qs) + l]*qs exp ti(r,&~,

‘qs! uqs!

qs rq,=o. Uqs’O

= exp [-i E(k) t] exp -c

4s

{Dlq,F(qs)

f D2qs

m (34) where

and

Wb) The energy Green function follows, G(k, w t iA) = (in)-’

x

5

exp (-

...

‘1, ill =o

2 {DlqS fW , ...

fDzqs

]fW) + 11,)

nq,s B(rgs* 1(4s)

‘q’S’,u*‘S’=o w + ix - E(k) + xq,r (‘s&@

(36) - ~J&)

‘l’he poles of the Green function (36) are at the frequencies: w

=c(k)-

c, (rqsC-i-kqs qr

~~$2;~~ ) = E(k) - FS {(rqS - uqS) w,(q) + (rqS + uqS) tc(W - e(k + q)1](37) ,

for.integral rqS and uqS. For the small values of P(Qs) applicable to low temperature the series (36) is rapidly convergent and only values of rqs and gqS near zero are important. For rqs = uqS = 0 (all qs) the pole coincides with .&e btid maximum (3 1) found in the damping treatment. This transition is damped by coupling to near-resonant levels which are joined to the level at the band maximum by one-phonon changes, i.e., levels belonging to iqS = 0 = 1 or 0. The limitation of significant contributions to near resonant levels appearsin (36) through Dlqs in B(rqs; u&, which are strongly peaked at w values satisfying the resonance conditions w,(@ 3 - e(k)]; This gives the basis for the requirement of energy con$ewation.on single phonon scattering in (29) tb give line broadening. If energy conservation is satisfied’for other values of u and r i&en& is added at the

D.P. Craig, L.A. Dissndo/Dispersiorl and resonunce terms in exciton-phonon

97

coupling

optical peak due to multiple scattering. This does not contribute to the damping and thus affects only the shape of the profile. It does not appear in the damping approximation result, though it has been allowed for by Davydov [s]. When the coupling F(k, q) is small as for anthracene [7] the effect is small and can be neglected. Some intensity will also be transferred to peaks at multiples of phonon frequencies above and below the main peak. However unless the exciton energies e(k) and e(k + 4) are nearly equal for a large number of 4 states this effect is small. Conversely for exciton bands narrower than the energy of the strongljjr coupled phonons intensity is transferred from the optical transition region to the multiple phonon peaks. This method, which is based, with some elaboration, on that of Iguchi [6] gives a usefully detailed picture of the effects of exciton-phonon coupling. In the class of situations where single phonon scatter@ with energy conservation dominates the results are the same as those of the damping theory. 2.5.

The complete With

the

interactiott

inclusion of the coupiing by dispersion forces, the partition of 0

Wk,q)12 F(qs) F(qs) @),=C_!!$f-it+-----;j--(q,s

+D

lqs

(

S

%p

+ 1

G4S

gives new expressions (38) and (39),

it 1

F(w) +Dzqs tP(qs)+ 11+ lNsq)12 WJq) + 11 >

(381

(F), = c - Dlqs F (qs) exp (is2,6st) - Dzqs [U(qs) + l] exp (4&t) 49si - IW7)12 {F(w) exp [iw$)tl

+ F(4s) + 1I exp (-iw,(q)rl

(3%

I

where

NW)=xs mw”2ws(q)]* After expansion of the exponentials in W, and evaluation of the energy Green function as before, G(k, w) = (271)-l exp (-

c {Dlqsa(qs)tD2qs[~(qs)$ q.s

[D,,,F(qs)]‘g” N@

uqs’ “@l Q)

=

{D2,#(qs)

+

11 +‘%?)I2

[2r%s)+ll~)

l] ]“‘?~(qs)i2”~s+2wqs

rqs! usa! vqs_’wQs!

I(q$@

[i$qs)

+

llWQf, (41)

and

(42)

1. gg

:..

.-

Dji &.+$,i.k Dis.&o/Di~spersin andresonance

terms 01exciton-phonori

coupling

..’

: The poles of G(k, co) are at energies

Now m addition to the poles for w = u = 0 contributing to the band maximum (27) the&are simiiar sets of poles giving additional peaks for eash of the values of (w - u). Each has similar features to the sintie peak found in the case of pure resonance coupling treated in section 2.4: there is an accumuiation of intensity around the optical levels of each peak contributed by transitions of near resonant energy, and independent of the particular values of w and u. The shape is Iorentzian with width determined solely by the phonon coupling. Thus the effect of disper: sion coupling is to distribute the intensity over a set of maxima belonging to values of (W - u) and to contribute an extra shift term of

Because the contributions by resonance and dispersion terms affect the profile in distinct ways, we now seek to formulate the Green function (40) more simply. At long times the parts of (40) which involve W and !FL*are dominated by C?-, sl” = 0, while those dependent on o,(q) are not. The former, which come from the resonance interaction, are now replaced by their t + - liiits, as in the damning approximation (30), and the dispersion terms retained, giving an approximation to (F) in (4.Q

- Ixs(r1)12

,A@)- %--q,s NQq)

W=

-

if - y(k) f + C IA( q,s

[Z(p)

f 1]

~lMq)12 {BIqs)exp[iw,(q)fl+ F(qs + 11exp[- i~,b7)fll. .

(45)

Then proceeding to the energy Green function as before, and expanding the exponential of the last term in (45), G(k, W) = (2~)~’ exp

- iY&)12

&@)-A(+

-1

%--q,s N”$q)

,

(46)

where

l?(?Jqs, Iv,,) ‘=lA(qs)j2”~f+2wqsF(qs)“qf (F(qs) t l)““Iuqs!

)+I ! qs

and A(&nd y(k) are asdefiied after expression (29). The result (46) includes both resonance and dispersion coupling and can be applied to a range of temperature by calculating the average phonon numbers Y(qs) by expression (25). One sees that the width of the phonon tran..sitions remains the same for successive peaks and, that as they become weaker they flatten and f&ally disappear. If the phouons are assumed dispersionless, w(q) G wo, the ratio of intensities in successive peaks for very low temperatures may be Found, by a little calculation, and putting vqs = 0, to be . .- $I+1 :-=(w+l)-l -L.:.-

&l&)12[I(0)?l],. q : .

(47)

‘.

99

2l.P. Craig, L.A. Dissado/Dispenion and resonance terms in exciton-phonon coupling

and in the link T= 0 K the last

factor becomes unity. The form of (47) is similar to that fo; localized exciton&onon coupling 151. There is a progressive change from the case of a large excited zone, with a small fraction only of modes signik cantly shifted in equilibriumpositions,to that of localized excitation with many modes changed.The ratio (47) is greater, and the intensity more widely spread, the more [oca!izedthe excitation. 2.6. Culculatiun of the Green function after a canonical

trunsfomtion

The change in dispersion forces on electronic excitation causes a change in equilibrium lattice structure, which is shown in the spectrum by the appearance of phonon structure as described in the last section. Those phonon modes become spectrally active for which the origins are shifted by the lattice changes. The dispeision interaction causing the lattice deformation, given in the second term of H’ in (17) is independent of the k values of the excitons, and depends only on the number of excitons and on the 4 values of the phonons. One thus expects that it can be incorporated by a canonical transformation into the phonon term of a new unperturbed hamiltonian, and removed from the interaction, leaving in the interaction only the genuine exciton-phonon coupling. The purpose of what follows is first to transform the hamiltonian (1) to this form, and then to find the Green function for comparison with (46). The canonical transformation P=ee-SHeS=H-

[S,H] t$ [S,[S,H]]

-._

(48)

is generated by S chosen so that (49)

where Hex and HPh are respectively the first and second terms of Ho in expression (1). The required form of S is

PI s = sg tA*(sQ) q(q) - Nsq)q?)l pm ,

@k)

PO)

where as before, M)

= xS(q)/N”2 w,(q).

We introduce the transforms B’(k) and KS(q) of the exciton and phonon operators, for the q modes with origins displaced on excitation,

(51)

In the new operators the exciton hamiltonian retains the same form as in (l), and the phonon hamiltonian becomes (53)

in which the exciton operators appear only in the number operator which, in the excited.state, has the value unity. With respect to the t!ansformed excitons,and phonons the interaction is.given by

lOti,

-: : .,

-DJ! Craig, &A. DissadofDispmion and resonance

terms in &iron-phonon

coupling

(54) which contains o&transfer ‘I citon hamiltonian: ..

coupling operators. The operators (53) and (54) together.with the transformed ex-

are used to calculate a new expression for W. The result is the addition of the terms (56) two orders higher in the coupling to the. expression (22) found from the original hamiltonian with interaction (21),

These cross terms between resonance and dispersion coupling are of order k’ compared to the bilinear cross term mentioned earlier and can be neglected as a correction to A(k) and y(k) unless the latter happen to be exceedingly small. 2 7. The existence ofa gaussian

limiting profile

Several writers have proposed that under somewhat similar conditions to those now discussed the absorption profile can be approximately gaussian, especially at high temperatures [large values of mean populations F(qs)]. Referring to the time Green function (19) and the approximate form (45) for (F) we see that if the damping r(k) contributed by the phonons through the resonance coupliig is large enough, the exponential in (19) decreases ._rapidly with time, and the important contributions to the energy Green function (20) come from t = 0. In particular if the lifetime imposed by 7(k) is less than or comparable to l/o,(q) for the active phonons, we can expand

the exponentiak in (45) and take terms ouly up to those in t2, givingin this approximation the Green function G(k, w)

=$ p

exp i i[w -E’(k)

t i#)]

t -D(k)&

dt,

0 where E’(k) is given by expression (42), and

D(k) =I+

c Ixs(q)12 [2iqqs) + 11.

q*s.

Expression (57) has the same form as given by Davydov [S] for localized excitons in the situation that the damping 7 is purely radiative-and is not, as here, provided by the very much greater.resonance coupling to phonons. Evaluating the integral in (57) we find

(58)

-

‘.

. ..x={-((k)-i[w-E’(kll}

[4D(k)]-112,

and

erfc(z) =-& [

e-Y*dy.

.: : ~nder’~e present restrictions on y(k), erfc(z) varies slc~wlyin the region of the absorption.maximum, so that Im.G(k; W),is nearly gaussian in that region, the pea& be&g shifted away from t$reundarn~ed resonance w.= _I?(~)

D.P. C&g, LA

D&ado/Dispersion and resohance terms in exciton-phonon

coupling

101

by r(k). The full width at half maximum of the peak can be expressed as a series in 7 being FWHM = 4 [O(k) ln2] ‘12,

w

with a first correcting term 2r

{4[D(k) In 21 II2 + rI/4[D(k) In 21‘12.

Although the gaussian form is referred to as the high temperature limit, the description is applicable only when the increase of y with temperature brings it into a range for which the truncated expansion of (45) can be made: for purely radiative damping the required temperatures are probably unattainable. For y(k) corresponding to phonon. damping the gaussian form applies to values large enough for the truncated expansion, but not so large that the profde becomes r-dependent. In general, the occurrence of gaussian properties must be regarded as the result of an unusual combination of circumstances. 2.8. Extension to two molecules per cell The case of two molecules per cell, with resonance coupling between exzitons and phonons, has been discussed by Dissado [7]. In place of the hamiltonian (1) we write (60) applying to two crystallographically equivalent sites labelled 1 and 2,

where

qk)=etk)~

~lltwfw~lw,

IL&?) = $&) +

q,w,

and the matrix elements, TTA, C, F and G are defmed in (61) for the two unit cell sites h, g = 1,~: T&(lc)=

c

,iw&ww

mg#nh

v

mtt?

T e&(q)eik’fmg-nh)~hs(q)

(av,,/a&),

I+9

mg+nh

cgh(k +4,9) =N- “’

5: Itlg#?Ih

e;(4) ei(k+q)m(~-nh)p&)

(a vmnlatg),

(Glc)

102

D.P. Oaig, LA Distidti/Dispersion and resonance terms in~ex&ton-phonon coupling

In (58) V,, is the dipole-dipole potential energy for coupling of the transition moments in the molecules at mg and nh; &(q) is the rms amplitude of molecules in the set h in mode q and branch s, and the differentiation is with respect to displacements Gg of mo!ecuIe mg, either translational or rotational ek(q) is the polarisation vector component for the mode gs of the set h. The diagonalisation of-the pure exciton and phonon parts of the hamIItonian (60) for ah k and Q cannot be done in a general way, essentially because the particle operators themselves are functions of k or Q characteristic for each parti&lar crystal. The expression (62) applies to the diagonal exciton operators, $, (k) = cos x(k) 4 (k) t sin x(k) Bj(k),

S!(k) = - sin x(k) B:(k) f cos x(k) B&c),

(62)

where x(k) are angle parameters found for each k-by soIving secular equations for the exciton energies. There are similar expressions for the phonon operators Q(P) as linear combinations of the b,&q) and b2(q) in (60). However the hamiitonian can be written in a formal sense as in (63) in terms of diagonal exciton and phonon operators, H = icS ,zV + x,0

$4

]aJk) Ii(k) S;(k) + F,(k, S,W

Q) cL(k + 4) S,(k) G,(q) + F>(k, 4 $,(k + 4 Q(k) ti,(q)

IL,(q).+ x;,(q) t;(k) 5Jk) J/,(d + w,(q) Q;(Q) rl,(d

+ t&ms with p and u intercharnged],

(63)

where e,,(k) and e,(k) are the energies of the excitons for the two branches of the exciton spectrum which, at k ~‘-0, become the two Davydov components. The FJk, q), for p = p, v, are intraband resonance coupling functiorrs for the two exciton and twelve phonon branches and the Fk(k, q) interband functions. x,,(q) and x:,(q) are intra- and inter-band dispersion coupling functions. We ‘mow from application of symmetry arguments that expressions (62) become simpler where the group of the wave vector k includes an operation mapping molecules of one set on those of the other. Then cos x(k) = sin x(k) = 2-l12. In EY!I/acrystals Iike naphthalene, anthracene and a number of other aromatics, the well developed cleavage-plane is the (OOl), and spectroscopic measurements can readily be made with radiation of wave vector in the (010) plane, for which the simple form of exciton operators applies. At k = 0 we then get.the two Davydov components, tith electric vectors for the transition along b and in the UCplane. Likewise for phonon wave vectors q in the (010) plane, or along the b monoclinic axis the phonon operators are

Iti,

t~fi&)-

w

For an incident wave perpendicular to the (001) face, with exciton wave vector k in the (010) plane, coupliig with the special phonons (64) is possible for ? parallel to b [i.e., perpendicular to (010) if the scattered exciton has its wave vector outside the (0 10) plane]. Coupling of the phonons (64) for 4 lying in the{0 10) plane gives scattered -e&citons with wave vectors k + q in that plane, and symmetric or antisyinmetric with respect to reflection in that plane. .Bothintra- and inter-bar@ scatteiing is possible in both cases but, apart from these special directions of k .. .. -. :.

D.P. Craig, L.A. DissadofDispersiotiand resonance terms in excitor+phonon

103

coupling

and Q, consideration of the general linear combination of G1,(q) and $2s(q) analogous to that of excitons in (62) cannot be avoided in a full analysis. We have not attempted a treatment including diagonalisation for general directions of 4, but illustrate the general character of the profiles with the help of certain approximations to be described in the following section. An important result is that each of the dispersive phonon coupling matrix elements x,(q) and x;,(q) has the same value for the two Davydov (k = 0) components; for all q, i.e.,

x,,(4) = x,(4), In consequence the frequency shifts caused by the dispersive phonon coupling are nearly the same for the two Davydov components. Thus the second spectral moment at low temperatures should be almost the same for each in cases where the dispersive phonon coupling is greater than the resonance phonon coupling. This has been found for anthracene by Morris and Sceats [I]. It follows that the splitting between the components is determined almost entirely by resonance interactions.

3. Calculations and discussion 3.1. Basis of the calculations The calculation of band profiles from expression (46) is possible with a number of approximations, amounting mainly to the neglect of dispersion in the phonon spectrum and in the coupling functions. A singledispersionless phonon branch ws = 50 cm-1 is assumed, a value close to that for the lowest librational branch in representative molecular crystals [ 11,121. FJk, q) and G(q), considered as band averages, are varied as parameters. Representative values, expressed as fractions of the phonon energy of 50 cm-l, are used in table 1 to indicate the range of situations expected in actual examples.

Table 1 Band profiles and dependence of bandwidth on exciton-phonon couplings

IFl/lXl

Phonon emission a) (mainly interband):

Phonon absorption b) (mainly intraband):

lX12

lX12

0.36

0.051

Sharp phonon structure.

No

distinct profile

1.0

2.25

0.36

1.0

2.25

Near gaussian

Near gaussian

Sharp phonon

structure, no

Near gaussian. Some phonon

Near gaussian.

with some phonon structure

distinctprotlle ‘)

structure

structure

<

a.2

1.0

Lorentzian with

Lorentzian with

some asymmetly width > Z-J(&)

some asymmetry width > 27(k)

Lorentzian, width 27(k)

Lorentzian, width 27(k)

Near-Iorentzian width > 2-&)

Lorentzian, width 27(k)

Intermediate.

at

150 K

Intermediate.

Near-lorenatan Near-Iorentztan width > 27(k)

width > 2y(k)

Lorentzian, width 27(k)

Lorentzian, width 2-t(k)

a) From the term r(k) _ U(qs) + 1 in (46). b) From the term flk) - V(qs)in (46). c) Width at 300 K approximately that of corresponding case in left-hand column.

Some phonon at

at< 100K Intermediate. Near-lorentzian

Lorentzian, width 27(k)

@j* ._-

“-:.I

_-

_.

LU! ~ui~..~A.-~i~dof~j~~er~on hiI ~so~~ce termsy, ~xcir~~~pk~n~~ eoupkhg

_,: -.-. 1 ..-,-. -.-_ -The d~~i~~-~~~~ &&ng From-Phon~n scatterkg and defined forcing expression (29) is divtded mto the &.v& types of t&:[&.Cqijj &d the [F(qsj + l] terks. I$thk fkst a.~e~~y excited phpnon (q$is scattered by ‘an op .-’tie& ex&&s&te@ =O)at’or very near the Iower edge of the exciton band into ~‘exciton ieve (k = -q) of the _-.$me exciton branch;-‘fhe exciton is scattered upward in its branch. The process does nbt occur at 0 K [F (qs) = Of and is m&ly res~on~s~~~for the temperature broadening of the tower Davydov component of molecular cr$tak, &ny residuzi width at 0 ~.presum~bly -uses by ceupling to other ~e~troscop~~ transitions at low& energies, or in ~tlzecase that thk Opticallevel is not at the bottom.of the band, by downward scattering into the lower-levels in the ; second type of Process. The chie~impo~~ce of this second type is in scattering by phonon emission from the optical level of an upper branch (upper Davydov component) inio a lower exciton branch Cl,2 f . There is in this situati&ji reaiduai width at 0 K; temperature breaderring in addition is caused by conco~t~t upward scattering by ~tl$mal phonons as in the f!r+ type. In niaking the cr&Iations half of the states are assumed to contribute to the damping r(k) and hatf to the shift A(k), t&n-to be ._. A(k)

:

= IF,@;

i)i*12w,.

WI

The order of &agr&ude of F#, q) for hbrationai phonons is the nearest neighbour resonance interaction multiplied by the zero point amplitude of the vibration [I’] which is about 0.02 rad [l 13. Since the exciton band width is itself in the order of the resonance interaction all values of F,(k, q) in table 1 apply to realistic situations. The dispersion coupling h,(q) is roughly scated by the dispersion interaction times the zero-point amplitude of the vibration, and the &lue 0.6 for Ixf [units of w(q)] 1s * rep resentative for aromatic molecuIa: crystab.. The value of IFI for antbracene has already been found f7] to be about 0.12 units. We emphasise that the estimates are usually based on physical models .of the coupling of excitons to molecular motions which give maximum values rather k&v&es representative of the k and q spectra as a whole. The dependence on the magnitude and directions of these wave vectors is not known. ‘@e band profdes for various values of parameters have been found for several temperatures allowing the variation of the FWHM and the peak position to be estimated.

3.2. profires for damping iy phonon production Selected band profiles calculated from (46) and their temperature variation are shown in fig. I. Phonon structure is observed only if $k) is less than half the phonon frequency, otherwise the phonon peaks are absorbed into a structureless env$ope even at zero temperature as observed for example in the upper Davydov component of akhracene; the highest phonon peak from (47) becomes the overall maximum of the envelope. Thus although each phonon peak is +ifted by an energy I -

.

the envelope maximum may be.shifted much less. The temperature dependence of the FWHM for the vake’l4/i~\ = 0.2 which is typical of anthracene can be simulated in a r&de1 without dispersion by a [F(qs) + I] temperature dependence for a phonon of slightly higher_fre-

.. .. Fi&‘l. E&d pdiles (Ihft-hnd d&&s)

and plots of temperature dependence of full bandwidth tit half maximum (right-hand ; f&ans) fdr scatterinpwith phonon emission :(A); iFi/[xl = 0.2; I#= 0.36; (B):.IFl/lxl T 0.057, Ix12= 1.0; Q: lFl/lxl = 1.0,1x\* = 0.36.

‘:&ergy i units of 50 cm:t . (a): T= 0 K, (b)i 100 K, (c): 200 K, (d): 300 K. Insert in top diagram: &iation of peak shift with digrams caMat& curys are full lines, loren@n curves (2$k))‘dashed lines,.-and gauss& in:. -. --::tetiptlddashes,. : _.

_. tempera&e. in the‘r&t-lqd ~..,..

__.

‘._,y, ..

: I ,_

.._ . ..

,-

..

.A .

.. :.

::,:

.:

.

,./ ‘.

:’

.. :

.

.:

:

,.

.::._

:.,

‘,

&Ff Ciaig, LA

Dissado/Dispersion and r&onance r&as in exciton-phonon

,

-10

I

-5

5

105

coupling

IO

w-e!k) -

(A)

I

I

w-e(k)-

T-

1

200

loo

300

(B)

I ,-

b

1r-

100

I

_

200

(cl. Fig.1. ‘.

.-106.::.

‘.

0.f. Chig, L.A.DissodolDTspersion and resonance temx in exciton~phonon coupling

.que&y (“55 cm-!) and-an exciton-phonon couphng_higherthan r(k). Thus the dispersioncouphnghas contri.‘.out&d extra width, and an analysisof experimentalresultson the basis of resonancecouplingalone leads to an overestimate-ofthe couphng and phonon frequency, altbougb acceptablein order of magnitude.The.fact ihat a zero-phonon peak does not appear in the upper component of anthraceneeven at zero temperature need only mean .-thatit has been absorbed into a structurelessabsorption envelope. ~&Profiles for dampingivyphonon destruction

.Thisbehairiour is typical of the lower Davydov component. Although the band profues are broadly similar to those for the.upper component, phonon structure may occur as the temperature approacheszero, appearingat progressively’lowertemperatures as y(k) increases.Againwhen 7(k) > OS (in units of phonon frequency) the phonon structure is lost into an envelope with maximum at the position of the highest phonon peak. Representative band.profdes are shown in fig. 2 for temperatures of 50 Ii and upwards, together with the temperature variation of the FWHM. The values Illustrated in the lower diagrams of figs. 1 and 2 iead to an approach to a gaussian curve. at high temperatures, when the phonon structure is lost at higher y(k) values and the FWHM is dominated by the progression of the phonon peaks. At low temperatures the phonon structure is marked and the most intense peaks dominate the profile, giving a FWHM approaching 2y(k). The lower Davydov component of anthracene has a temperature dependence not satisfactorily accounted for by resonance coupling alone [7]. Calculated as an example of an intermediate coupling case with Ir;l/lxl * 0.2 (fig. 2, upper diagrams) the temperature dependence of the FWHM is non-linear at T 2 50 K and at all temperatures exceeds the lorentzian value. The best lorentzian fit is below the calculated curve at T < 100 K and above at T > 200 K. This is the observed behaviour. We note in the calculations. of fig. 2 that low temperatures < 50 K, for values of the dispersion coupling between about one and two phonon energies, the one phonon peak has the greatest intensity. It has been proposed 1131 that the corresponding state should be regarded as a polaron (phonon clothed exciton) which is trapped and moves at a rate governed by a resonance interaction reduced by multiplication by the factor exp (-;

I

lA(sr7P) .

(67)

The interpretatiqn is not straightforward,becausethe exciton can be regardedas trapped only if the changein latticeenergy due to lattice distortion of

is greater than the nearest neigbbour resonance interaction. If the lattice distortion energy is less than the resonance interaction, the exciton rema@ delocalised (band description). The same conclusion was reached by Toyozawa [ 141

for ionic crystals;he also showedthat when either the resonance interaction or lattice distortion were greater than the phonon frequency the two descriptions are mutually exclusive‘and an intermediate region only occurred when both were less than w,(q). ‘Ihe resonance interaction is very small for triplet excitons and the hopping polaron result should apply. For ‘. singlet excitons, bearing in mind the approximate derivation of the parameters, it can be seen that.the band picture --. .-,Fig. 2. B%d profde! (left-hand diig&m$ and plots of temperature dependence of bandwidth (right-hand diagrams) for scattering .!.tiijith,photionabs?rptidn. .’ (A): lfl/lxl,= 0.2,1x1’= 0.3k; (B): v;l/iil =-O-057,lx? = 2.25: (C): ln/lXi = 1.0, [Xi* = 0.36. &erb in Unitsof 50 c#. (a): T= $0 K,(b): 100 K, (c): 200 K, (d): 306K. Insert in top.diiam: WIhfiOtl of peak shift with -tempkrature, III the right-hand diagrams hlculated curves are full tines, lorenthn curves (Zy(k)) dashed !ines, and gaussians interrupted &sly?; .. : ., ..

‘.

: z : .‘I.

. . . ... .

.:

:_. .. -: : _- ...

: .:

:.. ..

107

D.P.Craig,LA Dimdo/Di&rrion and resonanceterbs in exciton-phonon coupling

loo

T-

200

300

I

t -10

8

I

-5

w-&l--5

IlltIll -2-l 0 I 2 3 4

1

IO

T-

t

r Ii I I

I

-10

I

IO

t

300

I

30

I

d-..,..

100

z

I

I

200

phonon peaks

200

-30

100

T-

u-r(k)-

(c) Fig. 2

160~

300

108 --

....

..

D.F. Craig and resoqancetermsin em&m-phbnon : .I LA._ Dissado/Dispersion

coupling

.&plies iind the pa&uniters used here apply to s&let transitions and are consistent with the derivation of the Greeri f&&on’@). The exciton is completely delocalised, even at low temperatures, and its absorption lnten.-,si~~is distributed.ov&r a set of pe&s by the creation of an in&ml number of ~hdno~s.simultaneously with the exbitoii. The’affect of the dispersion interaction is to relax the overall wavevector conservation rule for certain of-the phonons. _. .3.4. Vari&o~ of peak poSition with temperature z -As shown in section 3.1 the shift of the profile maximum is less than one unit of phonon frequency. This has been_f&nd from the b&d profiles and plottedin figs. 1 and 2 where identifiable. Some interesting trends emerge. In all cases the envelope maximum reverts to e(k) above a certain temperature, which is higher (- 200 K) for cases of phonon loss th& for phonon gain. In the latter the peak shifts continuously from 0 K to the temperature at &ich it falls at-e(k); for the former a limiting temperature is reached below which no change occurs. In the case typified.by anthracene the upper component shifts in the range O-100 K and the lower component shifts in the -range 75-150 K; Experimentally [I] .a shift of 24 cm-l is found for the upper component in the region 10-77 K; no shift occurs for the lower component in thii range. Bearing in mind that the shift should be less than one phonon frequency-we think that the large shift (- 100 cm-l) at high temperatures may be partially accounted for by a change in the dispersion interaction caused by changes in lattice constants. After allowing for a similar shift of - 30 cm+’ in the lower component as expected between 77-150 K the large shift is nearly the same for both components. Such a change in the dispersion interaction would net affect the dispersion coupling significantly, and no alteration in the band profile is expected. 3.5. Effect of dispersionin the phonon spectrum We now lift the restriction to dispersionless phonons by assigning a very simpIe.dispersion relation and recalculating the band profdes. The phonons are divided equally into four frequency groups with frequencies 50,SO X 1.06, 50 X 1.12, and 50 X 1.18 cm-l. To each of these groups is assigned the same value of the coupling &&(~)I*, equal to one @titer of the value in the calculation for dispersionless phonons. We note that this distribution takes : no accaunt of the fact that the coupling goes down rapidly with increasing 4. The results are in fig. 3. All profiles show that the height of the zero-phonon peak is unchanged, as expected from (47). The higher phonon peaks are broadened and shifted to higher energies by phonon dispersion. When -the value for 1x1without dispersion is 50 cm -l the 1-phonon transition is strongest; phonon dispersion lowers it below the zero-phondn peak at T = 25 K. Thus the simple method of finding the highest phonon peak used to fur ix12 is sensitive to phonon dispersion. At higher temperatures the effect is less pronounced (fig. 3, upper diagrams). In a band showing no phonon structure the peak height is slightly reduced and shifted (- 0.2), and the asymmetry increased (lower left diagram). Any method of band analysis sensitive to profile asymmetry must be accompanied by allbw&ce for this effect. : ..4. ~onclu$ons ‘. Expression (46) for the exciton Green function &lows for exciton-phonon couplingat any temperature through both resofitice and-dispersion interactions simultaneously. The Green function can readily be computed, and a varletydfpiofdeshas been found using a single dlspersi&less phonon mode. A set of broad conclusions follows: (a) When y(k) > W&J)@ the bahd prpftie shows no phon& structure. .-.&I)JVhip lfl/lxi 5 0.1 tile band profile has a near-gaussiari shape under certain restrictions on r(k). .. .(c) Fe@ Ifljl~lx .i the band prafde hss? lorentzian shape-for v&es of r(k) > w,(q)/2. (d) .%en IFj/lxl = 0.2 the structu+ess b_md profile is intermediate in character_@id essentially a$mmetric. .. ...-. ., ‘, .. _. ‘: .. __. -: ‘.

DP. Craig, LA. Disrada/Dispersion and resonance terms in exciton-phonon cou~liitg

:

.109

Fig. 3. Bandprofdesfor scatteringwith phonon absorptionwith and without allowancefor dispersion.Light line: no phonon disdispersed into four sets (see text). Tap profdes at T= 25 K, bottom profdes at T= 50 K. Top left: IFljlxl = 0.2, lxlz = 1.0; top right lFl/lxl = 0.2,1x1*= 2.25;bottom left: lFl/lxl = 0.2,1xiz = LO;bottoin right: lFI/lxl = 0.2, 1x1”= 0.36.

persion. Heavy line: phonons

Its FWHM at all temperatures is greater than 27(k). It is possible witI& the necessarily approximate approach to analyse experimental band profdes at various tem-

peratures and to derive values for the resonance and dispersion coupling and the dominant phonon frequencies. An investigation of the effect of neglectingphonon dispersion shows that the overall band profde is only slightly changed.There is increased asymmetry, and the relative intensity of the phonon transitions may change. Generalisation to two molecules per cell is formally described but in practice knowledge of the band structure is far too limited to enable the extra complications to be handled in any but the most restricted sense. Generalisation to more than one free molecule transition is also not difficult in a formal way, but cannot yet be exploited numerically.

References 111.G.C. Morris and M.G. Sceats, Chem. Phys. 3 (1974) 332,342. [2] K. Tomi?ka, hL Animato, T. Tomotika and A. MaWi, J. Ckem. Phyr 59 (1973) 4157. [3] S. Fiier and S.A. Rice, J. Chem.Phys, 52 (1970) 2089;

S. Fischer,J. Luminescence1.2 (1970) 747. [4] hl.K. Groverand R. Silbey, I. Chek Phyr 52 (1970) 2099.. [ 51AS. Davydov,Theory of MolecularExcitons (Plenum,NewYork, 1971) chs. IV and V; Phys. Stat. Sol. 20 (1967) 143.

LIP. Cm&,LA. DiCsado&msion ~ndremnance termsin:exciton-phononcoup&g

110

[6J K. Iguchl,J. Chim. Phys. 71(1974) 654.

..

[71 LA. D&ado, Chem. Phys. 8 (1975) 289. [E] LA. Diido, Chem. Phys. Lett. 23 (19’25)57. 191 &T. March, W.H. Young and S. Sampanthar, The Many Body Problem ir.Quantum Mechanics (Cambridge, 1967) p. 78. [IO] W. Heiffer, Quantum Theory of Radiation, 3rd Ed. (Oxford, 1954) p_ 163. [ll] M. Suzulci. T. Yokoyama and M. Ito, Spectrochii. Acta 24A (1968) 1091. [ 121 G.S. Pawley, Phys. Stat. SOl.20 (1967) 347; G.S. Pawley and S.J. Cyvin, J. CItem. Phys. 52 (1970) 4073. [ 131RM. Hochstxasscxind P.N. Pcasad, Excited States 1 (1974) 79. [ 141 Y. Toyozawa, J. Luminescence 1,2 (1970) 732.

-. .:.

: : .’

_’

‘. :.

I