JOURNAL OF

LUMINESCENCE ELSEVIER Journal of Luminescence 76&77 (1998) 482-485

Superexchange in porphyrin-quinone complexes’

Michael Schreiber”, Christofer Fuchs, Reinhard Scholz lnstitut ftir Physik. Technische Univemitiit, D-09107 Chemnit:. Germanic

Abstract

The donor-acceptor transfer in the porphyrinheterodimer-quinone system investigated is strongly influenced by energetically high-lying states of the mediating zinc porphyrin. We use a density-matrix approach for the theoretical

investigation of the transfer, including donor, mediator. and acceptor on the same footing. The corresponding vibronic states are assumed to be displaced along a common reaction coordinate and the surrounding solvent is included via a system-heat-bath coupling, where the vibronic excitations of the solvent remain in thermal equilibrium. We solve the equations of motion for the density matrix to obtain the electron transfer rates for two concurrent mechanisms, superexchange and electron transfer after exciton transfer. Each channel is found to be dominant in a part of the temperature range investigated. The results are shown to be in agreement with experimental data, thus corroborating the proposed significance of the superexchange mechanism for the electron transfer in these complexes. (; 1998 Elsevier

Science B.V. All rights reserved.

Keywords: Electron transfer; Superexchange; Density matrix; Porphyrin-quinone complexes

1. Introduction

The investigation of electron transfer in covalent- ly linked porphyrine complexes is of particular interest as these systems can be considered as model systems for the study of dynamic processes in photosynthetic reaction centers, because their electronic properties are closely related to those of

the chlorophylls. One interesting question is, whether the electron transport occurs always se- quentially, or whether mediator or “spectator” states which do not participate directly in the trans-

*Corresponding author. Fax: + 49 371 531 3143; e-mail: schreiber@physik.tu-chemnitz.de.

’ Dedicated to Prof. Giinter Marx on the occasion of his 60th

birthday.

fer can significantly increase the transfer rates by means of the superexchange mechanism [1,2]. The

superexchange, which was originally proposed by Kramers [3] to describe the exchange interaction between two paramagnetic atoms spatially separ- ated by a nonmagnetic atom, always involves ener- getically high-lying states which are unlikely to be

populated for energetic reasons. Here we present numerical results in comparison

with experimental data [4] for the excitation and electron transfer in tetraphenyl porphyrinhetero- dimers linked to a benzoquinone molecule (Fig. 1). In the experiment, the donor chromophor, namely the free base porphyrin (H,P) is optically excited. Measuring the depopulation of this state by time- correlated single-photon counting, not only the life- time 5 0 z 8.5 ns of the intrinsic fluorescence but

0022-2313/98/$19.00 ,i: 1998 Elsevier Science B.V. All rights reserved

PII SOO22-23 13(97)00289-5

M Schrriher et ul. ! Journul of Luminescence 7b& 77 (1998) 482-4X5 3x3

Fig. I. Structure of the heterodimeric porphyrin-quinone com-

pound investigated.

also much faster decay can be observed in depend-

ence on the temperature [4]. For this decay we consider two competing pro-

cesses for the electron transfer from the donor (H2P) via the mediating zinc porphyrin (ZnP) to the electron acceptor benzoquinone (Q). In the first process

H?P* -ZnP-Q ---f H2PpZnP”-Q

--t H2PpZnP+-Q- (1)

the thermally activated excitation transfer to ZnP is

followed by the charge transfer to the acceptor. In the second case

H,P* -ZnP-Q -+ H2P+-ZnP--Q

+ H2P’-ZnPpQp (2)

the charge separation takes place first and the elec- tron is transferred from the donor via the mediator

to the acceptor. The first step in this pathway, however, requires a change in the free energy

‘4G I z 0.5 eV, which is so large that according to

the classical Marcus’ theory the electron transfer is highly unlikely. This has led to the suggestion [4] that the electron transfer from donor to acceptor in Eq. (2) involves the superexchange, i.e. it takes place without a significant population of the medi- ating H2P+~ZnP-~Q state.

2. Theory

The calculations are based on our density matrix approach which has been described in detail else- where [5,6] and applied, e.g. to the computation of electron transfer rates of betain- [7]. As before we distinguish the degrees of freedom of the mo- lecular system by dividing them into a relevant

reaction coordinate and the environment which consists of the remaining vibrational modes of the system and the solvent modes.

We model the relevant system. i.e. the three states

in Eq. (1) or in Eq. (2) by diabatic potentials 111 = 1, 2. 3, which are displaced along the common reac- tion coordinate. The relative positions of the dia-

batic potentials follow from the estimate of the free energy changes and the reorientation energies [4&J. These are based on electrochemical measure-

ments and depend on the reorientation of the sol- vent molecules reacting to the charge transfer or charge separation in the complex. Due to the

temperature dependence of the static dielectric con- stant of the solvent the parameters become temper- ature-dependent. However. the temperature

dependence of the energy levels turned out [S] to be so small that we did not have to take it into ac-

count. For the energy shifts 1~~ and i:_3 with respect to the first potential we use i;L = 0.19 eV and I:3 = - 0.87 eV for the process (1). and ii2 = 0.5 eV and 83 = - 0.5 1 eV for the process (2). The temper- ature-dependent shifts of the diabatic potential along the reaction coordinate are compiled in Table 1. For the vibronic excitations in the model

we assume a frequency of 1000 cm- I. an average of typical vibrations of the carbon backbone [9], The

coupling ham,,, = V,,J,c between the diabatic po-

tentials is given by the FranckkCondon factors Frc(M, N) between the respective vibronic excita- tions M and N (we abbreviate (1~ M) = 11) and the electronic overlap matrix elements l’,,, which have also been estimated in Refs. [4.8]. We use Viz = 3.9 meV and V13 = 2.2 meV for the process

(l), and VI2 = 65 meV and V2, = 7.9 meV for the

process (2). It should be noted that there is a large uncertainty in the estimation of all these para- meters. Nevertheless, they provide a reasonable basis for the comparison of the two competing

processes (1) and (2).

484 M. Schreihrr rt al. /Journal qf Luminescence 76& 77 (I WY) 482-485

Table 1 probability Relative positions q$“’ and q!:” of the minima of the diabatic

potentials for the states nt = 2 and nt = 3 on the reaction coordi-

nate of the investigated model P3(d = 1 e3~.3dO = 1 - exp(-kd) (4)

M

T(K)

135

143

167

200

250

333

Process (1) Process (2)

&’ q!?’ @’ q\“’

1.0 7.40 5.54 11.16

1.0 7.35 5.48 11.05

1.0 7.25 5.42 10.93

1.0 7. I3 5.33 10.74

1.0 6.98 5.2 1 10.50

I.0 6.82 5.08 IO.25

Starting from the Liouville equation, the equa- tions of motion for the density matrix Q,, can be obtained [5,6]:

of all the vibronic levels M for wz = 3, i.e. when the electron has been transferred to the acceptor. It is important to note that in process (2), the popula- tion P2, which describes the probability of the elec- tron on the mediator molecule zinc porphyrin (m = 2), is negligibly small throughout the simula- tion. This corroborates the notion of the superex- change being responsible for the electron transfer, process (2). In the density matrix formalism this is reflected in the negligible flow of occupation prob- ability between the diagonal elements of the density matrix @i 0. 1 o and @2M,2M. The transfer rather takes place via the offdiagonal elements of the density matrix, thus circumventing the occupation of the energetically unfavorable m = 2 states.

Here E, is the energy of the vibronic state ,u, and R,,,, comprises the dissipative part of the electron transfer due to transitions between vibronic states which are accompanied by injection of quanta from the heat bath into the relevant system or tlice wrsu. This dissipative part stems from the system-envi- ronment coupling and is characterized by damping functions ;’ = 27ca (1 + n) which are determined by the spectral density a of the heat bath modes and the thermal distribution function n. The depend- ence of the spectral density on temperature has been obtained [S] from the temperature depend- ence of the static dielectric constant and the visco- sity of the solvent.

The results are presented in Fig. 2 in comparison with the experimental data [4]. Clearly, two differ- ent temperature regimes can be distinguished: For high temperatures the electron transfer after excita- tion transfer dominates the rates, while for low temperatures the superexchange mechanism is far more effective than the other process.

Y 20

E

18 -

3. Results

The transfer processes (1) and (2) have been sep- arately simulated, starting with an initial popula- tion of the vibronic ground state of H2P*-ZnP-Q, i.e. m = 1 and M = 0. The transfer rates km have been determined from the total occupation

Fig. 2. Electron transfer rates for the depopulation of the op-

tically excited H2P*-ZnP-Q. Numerical results for process (1).

electron transfer after excitation transfer(O), and for process (2).

superexchange (O), with fits (thin solid lines) and total rates (0).

in compartson with experimental rates (0, thick solid line) with fits (broken lines) assuming two activation energies for the two

competing processes [2].

M. Schreiher et al. 1 Journal of Luminescencr 76di 77 (1998) 482-4X.5 385

The rates for the superexchange process (2) are in good agreement with the experimental data. For this process the straight lines fitted to the numerical results and to the experimental data show nearly the same slopes, i.e. the same effective activation

energy. The simulation of the competing transfer process

(1) yields a much larger temperature dependence of the rates than for the superexchange, in agreement with experiment. However, in the experiment the

temperature dependence is even stronger as dis- played in Fig. 2. This discrepancy may well be attributed to the uncertainty in the determination of the system parameters, in particular the elec-

tronic coupling between the diabatic potentials, and the spectral density of the heat bath modes. Also, the choice of an average frequency of 1000 cm- ’ for the diabatic potentials is a severe approximation. Therefore, it is not surprising that the experimental data cannot be reproduced more quantitatively.

Nevertheless, the results of the simulations cor-

roborate the experimental conclusion [4] that the fluorescence of the optically excited state H2P*- ZnP--Q decays due to two different electron trans- fer processes and that the relatively large rate at

low temperatures can be attributed to a super- exchange mechanism.

Acknowledgements

Financial support of the DFG is gratefully acknowledged.

References

[l] J. Rodriguez. C. Kirmaier. M.R. Johnson, R.A. Frtesner.

D. Holten, J.L. Sessler, J. Am. Chem. Sot. 113 ( lY91)

1652.

[Z] A. Osuka. S. Maruyama, N. Mataga. T. Asahi, I. Yamazaki,

N. Tamai, Y. Nishimura, Chem. Phys. Lett. 181 (IYYI) 313.

[3] H.A. Kramers. Physica 1 (1934) 182.

[4] U. Rempel, B. von Maltzan, C. von Borczyskowskt, Chem.

Phys. Lett. 245 (1995) 253.

[S] V. May, 0. Kuhn, M. Schretber. J. Phys. Chem. 97 (1093)

12591.

[6] 0. Kuhn. V. May, M. Schreiber. J. Chem. Phys. 101 (1994)

10404.

[7] C. Fuchs, M. Schreiber, J. Chem. Phys. 105 (1996) 1023.

[S] C. Fuchs, Doctor Thesis, Technische Untversitlt Chemnitz.

1997, http://archiv.tu-chemnitz,de/pub.;1997/0009:.

[9] C.A. Naleway. L.A. Curt&. J.R. Miller. J. Phys. Chem. 96

(1991) 8434.