Approximate description of Stokes shifts in ICT fluorescence emission

26 July 1996

CHEMICAL PHYSICS LETTERS ELSEVIER

Chemical Physics Letters 257 ( 1996) 38 l-385

Approximate description of Stokes shifts in ICT fluorescence emission Giacomo Saielli, David Braun, Antonino Polimeno, Pier Luigi Nordio Depariment of Physical Chemistry, University of Padova, via Loredan 2,35131

*

Padova. Italy

Received 12 April 1996

Abstract The time-resolved emission spectrum of a dual fluorescent prototype system like DMABN is associated with an intramolecular adiabatic charge-transfer reaction and the simultaneous relaxation of the polarization coordinate describing the dynamic behaviour of the polar solvent. The dynamic Stokes shift of the frequency maximum of the long-wavelength emission band related to the charge-transfer (CT) state towards the red region is interpreted as a consequence of a kinetic pathway which deviates from steepest descent to the CT state, the rate-determining step being the solvent relaxation. The present stochastic treatment is based on the assumption that internal and solvent coordinates could be described separately, neglecting coupling elements in the case of slow solvent relaxation.

1. Introduction This Letter is concerned with the interpretation of the experimental observation of a Stokes shift towards the red for the CT (charge transfer) band of (N,N-dimethylamino)benzonitrile (DMABN) and the like, which show the well studied phenomenon of dual fluorescence emission [ 11. Traditionally, these systems have been described according to a kinetic scheme due to Grabowski 121, in which the excited molecule relaxes to the ground state from two possible states: the LE (locally excited) state and the highly polar CT state. The emission from the CT state is supposed to be much weaker than from the LE, but the large Boltzmann factor in its favour acts as a counterbalance, so that two bands in the emission spectra are observed in polar solvents thet stabi-

* Corresponding author. OW9-2614/%/$12.00

lize the charge transfer form. The two metastable states are related by a dynamical equilibrium. In the twisted intramolecular charge transfer (TICT) hypothesis, the two states correspond to different configurations of the S, state of the molecule, the LE being planar and the other characterized by a perpendicular twist of acceptor and donor sites. For solvents whose Debye times are significantly longer than the rate of internal conversion, the broad emission band due to the charge transfer state may exhibit a dynamical shift towards the red, which has been experimentally observed in time-resolved fluorescence measurements [3,4]. A natural extension of the Grabowski picture consists in a description of a molecular system which shows dual emission fluorescence by a single internal degree of freedom coupled to a mesoscopic solvent coordinate. According to previous work [3,5], we shall assume that the internal coordinate for a

Copyright 0 1996 Elsevier Science B.V. All rights reserved.

PII SOOOS-2614(96)00563-S

G. SaieNi et al./ Chemical Physics Letters 257 (1996) 381-385

382

TICT system is defined as the torsional angle 8 between the donor and acceptor sites of the molecule. The main purpose of this Letter is to show that provided a timescale separation of the internal conversion process and solvent relaxation can be assumed, it is indeed possible to recover a simple picture of the bidimensional dynamics which explains all the known experimental facts. Moreover, an interpretation of the bidimensional model is given in terms of the superposition of two effective one-dimensional processes, one being related to a fast motion (along 0) at fixed solvent configuration, and another to a slow motion (along X) obtained after averaging over the internal coordinate.

2. The model The fluorescence emission spectrum is given by the double integral in 8 and X, which has the meaning of a solvent polarization coordinate: I( o,t) = w3emketi( o,T) ,

(1)

where k, is the sum of the radiative and nonradiative depopulation rate constant of the excited state S ,, and i(o,t) is: i(~,r)=(~~,(e)g[w-0~(e,X)]~,(e,x,r)). (2) Here p,(e)

is the transition dipole moment, h,(e,x) = E&8,x) - E,(e, Xl is the transition frequency between the S, and S, state at given point (0,X) and P,(e,X,t> is the dynamical population of S, at time 2. The time evolution of P(e,x,t) is dictated by a coupled Smoluchowski equation: -$qe,x,r)

= -~P(~,xJ)

+s(e,x,t),

(3)

where P is the diffusion operator for the Brownian dynamics of 0 and X, given by

(4) and Da and D, are the diffusion coefficients for the two coordinates; R is the index of properties related

to the internal mode, S to the solvent mode. The functions E,,, are given as: 40,

X) = “W(e)

- &&(~)

- P0,lP)X

+ 92x2

(5)

where g and B are quantities related to the dielectric constant of the polar medium and the volume of the Onsager cavity [5]. The excited state energy has two minima, one in (8,, , pLEE2) (corresponding to the locally excited state; for DMABN 8,, = 0”) and the other in (0,) pcT 8 2> (the charged state, for DMABN 8 = 90”) The potential functions V,,,( 8, X) are obtained from ab initio quantum mechanical calculations [7]. The source term s(e, X, t) represents the optical excitation from S, to S,. To complete the parametrization of the total energy of the solutesolvent system, a simple, monotonically increasing dependence on 0 of the molecular dipole moment p(0) is assumed: ~,(e)=CLLE+(CLCT-~LE)Sin2(e).

(6)

dipole moment for the ground state p0 is taken to be constant and equal to pLE. Definitions of D,, D,, g( w - w,) and p&e> are given elsewhere [3]. The time evolution operator can be rewritten as: The

A

r=

a -DR,e~eq(erx)

-D,

$&(B,X)

a zPeq(e,x) &

-I

p,,(e,x)-’,

(7)

P, being the Boltzmann distribution with respect to E,(e, Xl. Our purpose is to discuss the form assumed by P(0, X, t> when D, B D, (slow solvent). It is known from numerical calculations made for DMABN-like molecules [3,6] that the coupled stochastic model is able to reproduce the transient Stokes shift of the CT band as a consequence of the diffusion anisotropy: the emission fluorescence dynamical behaviour is a direct consequence of the time evolution of P(0, X, 2). A simple method is available to describe the time evolution of distribution functions in multidimensional stochastic systems whenever a clear separation of time scales between two sets of variables is realized [8]. The formal

383

G. Saieili et al./ Chemical Physics Letters 257 (1996) 381-385

justification of such an approach is based on a Born-Oppenheimer type approximation. Here we shall assume as our starting point a result of Appendix B of Ref. [8], which states, when applied to the system investigated here: P(B,X,r)

= 6(X-

/.&r&2)

x [ PR( X,&,l&f) +p,(

xle)Ps(

- P*( X10)] II~LEz*lxX;t).

(8)

The total time distribution is separated into two contributions, one ‘fast’ (D,) and the other ‘slow’ CD,); each one contains a dynamical factor which is obtained after solving a one-dimensional problem. The following definitions hold: p,(X)

= V,,(e,X)L

PR( Xle) = pe,(e,X)/ps(

X) .

(10)

= exp( -fkr)a(

8- e,,)

(11)

while the slowly relaxing part is given by: ps( /-%a*

IX,r)=exp(-T,~)cS(X-~~~Z*). (12)

Eq. (8), Eq. (11) and Eq. (12) are based on the assumption He,x,o)=s(e-e,,)s(xp.,,E*). The one-dimensional operators f, and Ts are defined as:

$&(xIe)-19(‘3)

f,=

-D,$P,(Xie)

i;,=

-0,-&(X)-$,(x)-1.

w- 0+wq]

xp,(xIqp,(

~.+B*lx;+

(‘7)

The initial twodimensional problem is reduced to the solution of two dynamic equations in one dimension. In both cases, it is convenient to symmetrize the time evolution operator and to expand in a suitable set of basis functions, such as complex exponentials for the fast process and Hermite functions for the slow process. The eigenproblem of linear algebra is then solved using the same techniques employed as in Refs. [3,5,6]. Once the timedependent populations are determined, the calculation of i,( o, r) and i&w, r> requires a simple integration.

(9)

The former defines an effective distribution for the slow variable only and the latter specifies the conditional probability of the fast variable at a given value of the slow one. The time evolution of the fast relaxing population is: p,(X,e,,ie,t)

is( w,t) = (ko)g[

3. Results and discussion To be consistent with our previous work [4], the parameter set used here is the same as for the comparison with the experimental data of a DMABN-like molecule in glycerol triacetate (GTA), done with the full stochastic model which takes into account coupling elements between the two coordinates. Two aspects must be emphasized, i> the chosen set of parameters implies an effective potential which shows no significant activation barrier and ii) the solvent relaxation time (D, ’> is twice as long as the characteristic relaxation time (D; ’) of the internal process. From the complete two-dimensional calculation [4], we drew the conclusion, summarized in the schematic Fig. 1, that the complete dynamic behaviour of such systems can be divided into two main events: the short time range is characterized by a fast motion along the torsional coordinate, building

(‘4)

We are now in a position to evaluate the total emission intensity itJo, t>: i,,( o,t)

= iR( w,t) + i,( o,t),

(15)

i,(w,t)=(y,(e)g[o-WO(e.LLLE~*)] x [ pR( pLEB*,e,,lw) -pR(

~LEH*l~)])o,

(16)

Fig. 1. A schematic diagram showing the time evolution of the population of the excited state S, for a TICT molecule coupled to a slow solvent polarization variable.

G. Saielli et al./ Chemical Physics Letters 257 (1996) 381-385

384

Fig. 2. Population

P,(B, t) as a surface plot versus I3 and t. Fig. 3. Population

up a metastable CT’ state which is in the perpendicular conformation but not in equilibrium with the solvent surrounding. As a second step, the system relaxes towards the equilibrium CT minimum. The approximate description applied here neglects entirely the dynamical coupling between the two steps, and assumes that two uncoupled processes which do not follow the steepest descent energetic pathway, can occur. In Fig. 2 the calculated population P,(8, t) is presented. The very beginning is dominated by a spike centered at 0 = 0” due to transient effects following the excitation pulse. The important feature is the relatively fast depopulation process in the region around 0” and the complementary rise around 90”. This behaviour can be traced back to the interconversion from the directly excited LE to the metastable CTt state, which takes place on a time scale of DR '.Fig. 3 shows the evolution of the population P,(X, I), which starts at a solvent polarization equal to Z+ and reaches at long times equilibrium with the dipole moment per of the charge transfer state. According to Fig. 1, the relaxation towards the CT minimum is controlled by the solvent polarization on the time scale of D, '.From these results it can be concluded that the Stokes shift of the CT band at long times is governed by the dielectric properties of the solvent. Statements about the Stokes shift at shorter times remains problematic, due to the superposition of the LE and CT band immediately after optical excitation. The total time-resolved fluorescence intensity I,,, is equal to I, at longer wavelengths, where it is

P,(X,

t) as a surface plot versus X and

t.

dominated by the CT emission. No significant emission of the internal mode fluorescence I, could be found. The LE emission, on the other hand, shows a more complex behaviour. In fact, the decay curve of the LE emission, shown in Fig. 4, is the sum of two contributions. The characteristic decay time of I, is twice as long as the decay time of I,, in accordance with Fig. 1. Both characteristic times are similar, but not equal in magnitude to the two dominant decay times calculated after solving the full bi-dimensional problem [3]. Only in the limit of a large diffusion anisotropy would the approximate treatment reproduce the same decay times predicted by the exact procedure. As a final comment, we note that biexponential decay is actually observed in ICT systems [9] sup-

t (PSI Fig. 4. Calculated decay curves at 350 nm in the LE emission band: the total intensity 1,0x (solid line), the solvent induced part I, (dot-dashed line) and the internal part IR (dashed line).

G. Saielli et al. / Chemical Physics Letters 257 (1996) 381-385

porting the arguments given here, which identify independent decay channels related to different variables.

Acknowledgements

This work was supported by the Italian Ministry for Universities and Scientific and Technological Research, and in part by the National Research Council through its Centro Studi sugli Stati Molecolari the Committee for Information Science and Technology, and the HCM contract N. ERBCHRXCT930282.

385

References [I] E. Lippert, W. Rettig, V. BonaiX-Koutecky, F. Heisel and J.A. MiehC. Advan. Chem. Phys. 68 (1987) 1. [2] Z.R. Grabowski, K. Rotkiewicz, A. Siemiarczuk, D.J. Cowley and W. Baumamr, Nouv. J. Chim. 3 (1979) 443. [3] D. Braun, P.L. Nordio, A. Polimeno and G. Saielli, Chem. Phys. (1996), in press. [4] D. Braun, Ph. D. Thesis, Humboldt University, Berlin, (KGster Verlag, Berlin, 1995). [5] A. Polimeno, A. Barbon, P.L. Nordio and W. Rettig, J. Phys. Chem. 98 (1994) 12158. [6] A. Polimeno and P.L. Nordio, Mol. Phys. (19%), in press. [7] S. Kato and Y. Amatatsu. J. Chem. Phys. 92 (1990) 7241. [8] A. Polimeno, G.J. Moro and J.H. Freed, J. Chem. Phys. 104 (1996) 1090. [9] F. Heisel and J.A. Miehe, Chem. Phys. 98 (1985) 233.