7

Chapter 2. Background

Overview :

The investigations of stilbene-like donor-acceptor molecules in the literature are numerous, and the interpretation of the results is far from being unanimous. This chapter tries to provide a summary of the spectroscopic characteristics of DCM and some structurally related molecules. The relevant pathways for their relaxation after photoexcitation are outlined. On the subpicosecond timescale, electron transfer, solvent reorientation and vibrational deactivation are the mechanisms prevailing in the discussion. Several models of these processes and their applications are described.

2.1. Donor-acceptor substituted stilbenes

2.1.1. Relaxation pathways for stilbenes after photoexcitation

1,2-diarylethylenes (stilbenes) have been found to be subject to at least seven different types of competing photoreactions (see the excellent review given by Görner and Kuhn [Görn 95]).

The most investigated path is probably twisting about the central C=C double bond, or trans-cis photoisomerization. Owing to the larger dihedral angle of cis-stilbene, the energy of the cis ground state is higher by 10-20 kJ/mol [Görn 95] than that of trans-stilbene. Schematic potential energy curves as a function of the twist about the double bond are displayed in Figure 2.1-1 . The planar trans configuration is denoted with "t", the 180° twisted cis configuration with "c", and "p" marks a hypothetical (perpendicular or phantom) state at a 90° twist of the ethylene bond. The barrier for trans-cis isomerization is much lower for the photoinduced reaction in the first singlet or triplet excited state. In the electronic ground state, the barrier for trans-cis as well as for cis-trans isomerization was

8

Figure 2.1-1: Trans-cis isomerization scheme of stilbene.

Intersystem crossing (ISC) is a relaxation mechanism that may also be involved in the isomerization reaction. The intersystem crossing quantum yield is of the order of for stilbene and cyanostilbenes; Meyer et al. give an upper limit of for the of DCM in methanol [Mey 90]. It is strongly enhanced upon substitution by a 4-nitro or 4-bromo group, and for 4-dimethylamino-4'-nitro stilbene has been determined to be 0.4 in cyclohexane and 0.04 in more polar or polarizable solvents [Görn 87]. Correspondingly, for excited nitrostilbenes the trans-cis and a fraction of the cis-trans isomerization evolve along an intermediate in the first excited triplet state (see Figure 2.1-1 ), while for stilbene, fluoro-, chloro- and cyanostilbenes the singlet mechanism dominates.

In solution at room temperature, cis-stilbene may also deactivate via photocyclization to dihydrophenanthrene (DHP). In the absence of oxygen, DHP relaxes thermally back to cis-stilbene by ring-opening, whereas phenanthrene is formed in the presence of oxygen. Higher vibrational levels of excited cis-stilbene are involved in the cyclization process [Güst 68]. The quantum yield of photocyclization is given as 0.22 for stilbene in n-pentane [Görn 95], and 0.046 in cyclohexane [Jung 68]. It is greatly reduced for nitrostilbenes (<0.001) and estimated to >0.003 [Jung 68] for cyanostilbenes.

Photoreduction by hydrogen-donating solvents is an important relaxation pathway for

9

Radical anions or cations of stilbene and its substituted derivatives are produced by photoinduced intermolecular electron transfer in the presence of electron-donating or accepting substances, such as amines [Hub 84] or cyanoanthracenes [Görn 95].

On a timescale of up to a few picoseconds, photoinduced intramolecular electron or charge transfer is the primary reaction channel discussed for donor-acceptor substituted stilbenes and will be further discussed in 2.1.3. and 2.2.. It is accompanied or followed by dielectric relaxation of the surrounding solvent. In the case of vibronic excitation, intramolecular vibrational energy redistribution and vibrational energy transfer to the solvent molecules must be taken into consideration.

Internal conversion of the excited trans-form directly to its ground state has been assumed to explain the reduction in trans-cis quantum yield with solvent polarity for nitrostilbenes [Görn 78, Gruen 89].

2.1.2. Photophysical properties of some donor-acceptor substituted stilbenes

4,4'-Donor-acceptor substituted stilbenes are known to exhibit a large, polarity dependent Stokes shift between the main UV/visible absorption and fluorescence bands. Exemplary absorption and emission maxima are presented in Table 2.1.1 for 4-{(E)-2-[4-(dimethylamino)phenyl]-1-ethenyl}benzonitrile (trans-DCS), 4-[(E)-2-(2,3,6,7-tetrahydro-1H,5H-pyrido[3,2,1-ij]quinolin-9-yl)-1-ethenyl]benzonitrile (trans-JCS), the stilbene-derivative trans-DCM and N,N-dimethyl-N-{4-[(E)-2-(4-nitrophenyl)-1-ethenyl]aniline (trans-DANS) ( Figure 2.1-2 ). Since only the trans conformers are considered here, the prefix will be omitted in the following.

The Stokes shift for nonpolar solvents is largest for DCM, whereas the solvent polarity dependence of the emission maximum is strongest for DANS. The red shift of the emission maximum in dipolar solvents is ascribed to charge transfer interactions between the donor and acceptor groups in the excited state, and stabilization of the polar compounds by solute-solvent interaction. The differences in polarity dependence of the Stokes shift can be related to a stronger donor character of the julolidinamine group compared to the dimethylamino

10

Figure 2.1-2: Structure of 4,4’-donor-acceptor substituted stilbenes.

Table 2.1.1: Maxima of the absorption and emission spectra of donor-acceptor substituted stilbenes. Structured spectra are denoted by asterisks.

	Abs / nm		Flu / nm
Solute / Solvent	Alkanes	Acetonitrile	Alkanes	Acetonitrile
DCS	375*	390	430	540
JCS	399	407	461	559
DCM	455*	464	582*	637
DANS	417	435	470	>850

11

Figure 2.1-3: Stationary spectra of DCM. The fluorescence spectra have been converted to emission cross section.

The dipole moments in the ground and excited state as published for DCS, DCM and DANS are summed up in Table 2.1.2 . In accordance with the order of the Stokes shift in highly polar solvents, DANS is presumed to undergo the largest change in dipole moment upon excitation, followed by DCM and DCS. Quantum chemical calculations of isolated molecules obviously tend to underestimate the excited state dipole moment in solution, possibly because of their neglect of the molecular environment. The solvent dependence of the excited state dipole moment has been demonstrated for DANS by [Bau 92] with electrooptical investigations. They obtained 25.4 ± 0.1 D in cyclohexane, 27.0 ± 0.2 D in fluorobenzene and 27.8 ± 0.3 D in dioxane for the dipole moment of DANS in its first excited singlet state.

12

Table 2.1.2: Dipole moments of donor-acceptor substituted stilbenes. Franck-Condon excited states are indicated by brackets. Values from electrooptical measurements are denoted by asterisks; values from semiempirical calculations are marked by double asterisks. All other dipole moments are derived from the Stokes shift between absorption and fluorescence/stimulated emission bands.

	DCS	DCM	DANS
µ (S0) / D	7*	6.1* 8.5** 10.2	9.47* (10.0)* 7.7* 10**
µ (S1) / D	21* 22.3 (20) (13)	12.4**	31.8* (32.3)* 25.4* 20-22**
µ (S2) / D		26.3 14.3**
Reference	[1] [2] [3] [4]	[5] [6] [7]	[8] [9] [10]

[1] [Lip 63], [2] [Kaws 77], [3] [Eil 96], [4] [Il'ich 96], [5] [Mey 87], [6] [Marg 92], [7] [Moy 96], [8] [Bau 77 ], [9] [Bau 92], [10] [Farz 99]

Quantum yields of fluorescence and of trans-cis photoisomerization and fluorescence lifetimes for 25°C are compared in Table 2.1.3 for the compounds listed above in different solvents. The fluorescence quantum yield increases with solvent polarity for all compounds except for DANS. DANS shows a maximum of _Flu in benzene; in acetonitrile its emission is strongly reduced. Internal conversion in polar solvents was held responsible for the latter effect, since the quantum yield for intersystem crossing in polar solvents is also very low [Görn 78, Gruen 89].

While for the other compounds _transcis decreases with solvent polarity, the isomerization quantum yield was found to be nearly constant for DCS. The low double-bond twisting efficiency for DCM in polar solvents was explained by the energetic stabilization of the emitting (charge transfer) state relative to the 90° twisted configuration that should act as intermediate in the isomerization reaction [Mey 90, Rett 89].

Given the relation , where is the radiative decay rate, is the natural fluorescence lifetime and is the total depopulation rate of the fluorescent state, should be directly proportional to _Flu. For JCS and DCM, the fluorescence quantum yield changes by a larger amount than the emission lifetime,

13

(next page: )

a) toluene b) MTHF c) see 4.2.1. d) relative _trans-cis with respect to that in chloroform

20°C f) slower component of biexponential fit from [Mey 89]

g) from decay of stimulated emission and excited state absorption

[1] [Gruen 83], [2] [Recht 96], [3] [Les 84], [4] [Gruen 89], [5] [Mey 90], [6] [Mial 93], [7] [Abr 97], [8][Il'ich 96], [9] [Rett 89], [10] [Bau 77], [11] [Shor 58], [12] [Kov 99b].

14

Table 2.1.3: Quantum yields of fluorescence and trans cis-isomerization and fluorescence decay times for donor-acceptor substituted stilbenes in different solvents.

Solvent	DCS	JCS	DCM	DANS
		???Flu)
Cyclohexane	0.03	0.04	0.007 c)	0.33
Benzene	0.03 a)	0.12		0.53
THF	0.06 b)		0.49	0.11
Chloroform	0.05		0.35	0.018
Ethanol	0.07
Methanol			0.43
Acetonitrile	0.13	0.41	0.44	<0.002
DMSO		0.47	0.8
Reference	[1]	[2]	[3]	[4]
		????(trans-cis)
Cyclohexane	0.45			0.28
Benzene	0.45 a)			0.02
THF	0.4 b)		0.5 d)	0.004 b)
Chloroform			1 d) 0.28
Ethanol	0.5			<0.001
Methanol			0.07 d)
Acetonitrile	0.4		0.05 d) 0.022
DMSO			0.04 d)
Reference	[1]		[5] [6]	[4]
		???(Flu) / ns
Cyclohexane	0.085	0.46		1.05
Benzene		0.61		3.3
THF			1.24
Chloroform			1.38 0.74 3.3 e)	0.08
Ethanol			0.67 e)
Methanol			1.36 1.31
Acetonitrile	0.51	1.36	1.93 1.91
DMSO		1.82	2.24 2.18
Reference	[7] [8]	[2]	[5] [3] [9]	[10] [11] [12]

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2.1.3. The TICT concept

The photoreaction of p-dimethyl-aminobenzonitrile (DMABN) in solution led to the concept of twisted intramolecular charge transfer (TICT) [Grab 79]. Within this model, the excited state photoreaction of DMABN should involve an (approximately) 90° torsion of the dimethylamino group, producing a polar, charge-separated state. The internal twist is presumed to decouple the donor and acceptor centers in the molecule electronically. The concept was transferred to other molecules with substituents of electron-donating or -accepting character linked by flexible chemical bonds [Lipp 87, Reviews by Rett 86, 92], so-called "TICT-compounds", among which belong DCM and substituted stilbenes such as DCS or JCS. As the fluorescence exhibited in the charge transfer state is red shifted with increasing solvent polarity and does not appear when the molecule is isolated, it is termed 'anomalous' fluorescence; if emission from the primarily excited precursor state can be recorded as well, one speaks of "dual emission". The latter has been reported for DCS [Eil 96], but the primarily emissive state was found already to have strong charge transfer character. Temperature and solvent dependent investigations of fluorescence lifetimes and fluorescence quantum yields let the following picture emerge for donor-acceptor substituted stilbenes. Competition of the charge transfer relaxation channel with the trans-cis-isomerization channel leads to an increased quantum yield and a longer emission lifetime in polar solvents for temperatures above the activation energy / k_B of the TICT reaction [Rett 89, 92]. Gruen and Görner excluded the involvement of amino group rotation in the charge transfer reaction of DCS by investigating bridged compounds [Gruen 83, 89]. Differences in the optical spectra for high concentrations (> 1 mM) of DCS and high excitation energies compared to dilute solutions and moderate excitation conditions were reported by Gilabert et al. and Lapouyade et al. [Gil 91, Lap 92]. They were explained as being due to a complex formed by two photoexcited twisted substituted stilbenes, termed "bicimer". The risetimes of JCS transients at 616 nm after UV excitation were found to increase with pressure up to 500 MPa, while for a compound similar to DCS, but with bridged single bounds connecting to the ethylene group, the risetimes were constant above 290 MPa [Rett 94]. This was interpreted as the effect of large-amplitude motion related to bicimer formation, which should not be present in the bridged compound. For DCS consequently a rotation of the dimethyl-anilino group around its single bond to the ethylene moiety was proposed to be involved in the bicimer as well as in the TICT mechanism [Abr 97].

16

For DCM, Marguet et al. calculated a large increase in dipole moment (from 14.3 to 22.5 D) upon a 90° twist of the dimethylamino group in S₂ [Marg 92]. In contrast to the idea of a competition between the dimethylamino group rotation and the rotation around the central double bond, these processes were characterized as being independent of each other. No dipole moment increase was observed for other single bond rotations.

17

2.2. Electron transfer

Electron transfer is not restricted to molecules with flexibly bound subgroups of electron-donating or -accepting properties such as in 2.1.3., and it has been extensively treated in experiment and theory over the last decades (for reviews, see [Mar 89, Heit 93, Yosh 95]). One differentiates between "outer-sphere" intramolecular electron transfer, which designates a charge separation between a donor and acceptor site on one molecule separated by a rigid structure, intramolecular electron transfer involving bond rotation and intermolecular electron transfer, for example between solute and solvent molecules or weakly bound complexes.

According to the coupling strength of the reactant and product free energy surfaces, they are characterized as diabatic or adiabatic ( Figure 2.2-1 ). The idea of the diabatic description is that the total Hamiltonian H of the system (solute and solvent) can be partitioned into a zeroth-order part H₀ of the isolated molecule and a weak perturbation V due to the solvent. The reactant or product states are eigenfunctions of H₀, with the electron localized at either the donor or the acceptor site. Movement on either free energy surface does not change the electronic state; electron transfer is induced by the perturbation V coupling reactant states and product states . Depending on the strength of that coupling given by the matrix element V_el = < |V| >, the perturbation might not be treated as weak . The electronic states of the system are then eigenfunctions of the total Hamiltonian H, not of H₀, and the reaction proceeds on the adiabatic free energy surfaces from reactant to product configuration.

Figure 2.2-1 : Diabatic (a) and adiabatic (b) reactant and product free energy surfaces.

18

, (2.1)

where l_LZ is the Landau-Zener length and F is the difference of the slopes of the reactant and product potential surfaces at the crossing point. v is the velocity with which the system moves through the Landau-Zener region around the crossing point. Thus the more steeply the potential surfaces intersect, the smaller will be the Landau-Zener length and the adiabaticity parameter.

The reaction coordinate q could be an intramolecular degree of freedom or geometrical parameter, such as in the TICT treatment, but it has been shown [Zus 80, Cal 83, Per 95] that a good choice is to define q as the vertical internal energy gap U between reactant and product state, depending on the solute and solvent nuclear configuration X :

q(X) : = U = U_P(X) -U_R(X). (2.2)

For outer-sphere intramolecular electron transfer, X denotes only the solvent molecules' coordinates. Thus a transformation from nuclear coordinates, some of which might lead to the same value for q, to the energy scale relevant for the reaction can be performed.

In the non-adiabatic case, the rate coefficient of electron transfer k_NA can be written as [Marc 85]:

, (2.3)

Here _S is the solvent reorganization energy, which is the free energy difference of the product and reactant states for a solvent configuration corresponding to the minimum of the product state.

Equation 2.3 presumes the validity of the transition state theory, and thereby the validity of the following assumptions:

• the reactant is kept in quasi-equilibrium with the transition state

  ---

  19

• the system moves uniformly through the transition state region
• after crossing the transition state region, excess energy is disposed rapidly into the solvent heat bath, precluding a back reaction.

Under the assumption of linear response of the dielectric interaction between solute and solvent, the activation energy can be expressed as :

, (2.4)

being the free energy difference between reactant and product states.

The "energy-gap" dependence of the reaction rate for electron transfer can be divided into three regions:

- < _S, the "normal" regime, where the reaction rate increases with - .

- = _S, the fastest case, since no activation barrier is present.

- > _S, the "inverted" regime, where the reaction rate decreases with - .

[Marc 85]

Figure 2.2-2 : Energy-gap dependence of the electron transfer rate coefficient and its relation to the relative position of the free energy surfaces of reactant and product, from [Yosh 95].

20

The rate coefficient k_NA is the sum over all of the individual rate coefficients for different vibronic channels:

(2.5)

is the rate coefficient for the transition from the vibrational ground state of the reactant to the m-th vibrational level of the product. It is determined by inserting the quantized free energy difference for this transition, , into equation 2.4 and including the Franck-Condon overlap of the vibronic states in the electronic matrix element V_el. The intramolecular vibrational reorganization energy for the high-freqency mode is termed .

In the theory of electron transfer presented so far, it has been presumed that during electron transfer the solvent polarization is always in equilibrium. As the polarization relaxation can be as slow as picoseconds, this assumption is not always justified. Several authors tried to incorporate finite polarization response into their treatment [Zus 80, Cal 83, Rips 87]. In the limit of strong (adiabatic) coupling the solvent relaxation may even determine the electron transfer rate. Rips and Jortner [Rips 87] gave the following result for an adiabatic, solvent controlled electron transfer transition between two states coupled to a dielectric continuum:

(2.6)

is the longitudinal dielectric relaxation time of the solvent (see 2.3). As usually _S < 16 k_BT [Yosh 95], this imposes a limit of on the reaction rate coefficient.

The transition to the non-adiabatic case is smooth and governed by the equation:

, (2.7)

21

. (2.8)

It is obvious that the adiabaticity of the reaction depends not only on the electronic coupling element, but also on the solvent reorganization energy and the time scale of solvent relaxation. In the strongly adiabatic case ( ), the reaction dynamics reduce to diffusion over the lower adiabatic potential surface, corresponding to a Kramers-type problem [Kram 40]. This led to a stochastic treatment of electron transfer [Zus 80, Cal 83, Hynes 86], where the rate for the adiabatic case was found proportional to as well.

It should be noted that the adiabaticity parameter can be expressed as a function of the Landau-Zener length l_LZ and the mean free path l_f, demonstrating the relation of to the Landau-Zener adiabaticity parameter _LZ (eq. 2.1). The mean free path is defined by . Here t₀ is the average time interval between collisions, and is the velocity of the reaction coordinate. is related to l_LZ and l_f by:

[Rips 87], (2.9)

where _rot is the rotation frequency of the solvent molecules and is a numerical factor of the order of unity.

The rate coefficient for electron transfer was limited to by equation 2.6. Sumi, Nadler and Marcus developed a model that explained faster rate coefficients as well as nonexponential reaction dynamics [Sum 86, Nad 87]. They partitioned the reaction coordinate into a solvent coordinate X describing diffusive solvent relaxation and an intramolecular coordinate q, characterized by low-frequency vibrational motion, along which the charge reorganization takes place ( Figure 2.2-3 ).

Both dimensionless coordinates are treated classically and the total free energy of the system is given by:

for the reactant, and by

for the product. (2.10)

22

Figure 2.2-3 : Two-dimensional free energy surfaces in electron transfer reaction, from [Sum 86].

The equilibrium positions for both coordinates are zero for the reactant and q₀ and X₀ for the product. is the standard free energy of the reaction. The reorganization energies are given by _S = 1/2 X₀² and _vib = 1/2 aq₀², respectively. The transition state is defined as the intersection of the reactant and product free energy surface (curve C in Figure 2.2-3 ). The relaxation time for vibrational motion is assumed to be substantially faster than the timescale of solvent relaxation, so that the distribution along q will be quasi-stationary. Thus a reduced distribution function P(X,t) for the reactant probability at time t and solvent configuration X may be formed by averaging over the quasi-stationary coordinate. A reaction rate coefficient k(X) for each value of the solvent coordinate can then be obtained using Marcus' theory of electron transfer [Marc 85]. The distribution function P(X,t) is taken to satisfy the diffusion-reaction equation:

, (2.11)

where D is the polarization diffusion constant :

. (2.12)

Since the operator is the Smoluchowski operator, the above equation is the Smoluchowski equation for diffusive motion along X, extended by a sink term -k(X)P describing the reaction part. A similar equation was proposed by Bagchi, Fleming and Oxtoby [Bag 83] to account for the dynamics of activationless electron transfer

23

An important quantity, the survival probability for the system in the reactant state Q(t), can be derived from P(X,t):

. (2.13)

Sumi and Marcus [Sum 86] give an exact solution of equation 2.11 for the four following cases :

• The slow reaction limit

If the solvent reorientation is fast compared to the reaction, a thermal equilibrium is maintained for the distribution along X in the course of the reaction. Q(t) should show a single exponential decay with a rate coefficient slower than, and independent of the relaxation time _L of the reorientational fluctuations. For the non-adiabatic case, the result for k_ET coincides with equation 2.3, only that the solvent reorganization energy _S has to be replaced by the sum of the reorganization energies _S and _vib.

• The wide reaction window limit

If the intramolecular reorganization energy _vib is far larger than _S, the reaction may proceed over a range of X values much broader than the thermal equilibrium distribution of X for the reactant state. The rate coefficient can then be approximated by the average over k(X) over the X distribution in the reactant potential, yielding monoexponential dynamics for Q(t) independent of the solvent relaxation time.

• The narrow reaction window limit

If the solvent reorganization energy _S is far larger than _vib, the system has to cross the transition state essentially in X direction. The rate coefficient can be approximated as

, (2.14)

where k₀ is a constant and X_C is the value for X at the intersection of curve C and the X axis in Figure 2.2-3 . The survival probability will exhibit a multiexponential decay, the features of which will also depend on relaxation time _L of the reorientational fluctuations_.

• The nondiffusing limit

If the reaction proceeds so rapidly that the distribution of X does not change during the course of reaction, the solvent motion is effectively "frozen". The reactant population will decrease with a different rate coefficient for each value of the initial configuration X(0),

24

The Sumi-Marcus model was employed to account for non-exponential dynamics of electron transfer and for reaction rates larger than [Su 88, Braun]. Jortner and Bixon [Jort 88] added high-frequency quantized intramolecular vibrations to their treatment of electron transfer including solvent relaxation. Since the reaction rate was viewed as the sum of the reaction rates to different vibrational product levels (see equation 2.5), fast solvent dynamics were implicitly assumed. Thus, the applicability of the Bixon-Jortner model is limited to cases where solvent relaxation is dynamically unimportant.

Walker et al. [Walk 92] extended both theories to a hybrid model allowing for two classically treated low-frequency modes X and q, characterizing the solvent reorientation and the charge reorganization, and a high-frequency quantum mode. Furthermore, they took into account the initial solvent distribution being displaced from thermal equilibrium by laser pulse excitation.

In their model the rate coefficient for the transition from the reactant vibrational ground state of the high-frequency mode to the m-th vibrationally excited level of that mode in the product state is a function of the solvent configuration X; through its activation energy it depends on the low-frequency intramolecular reorganization energy _lf,vib, the solvent reorganization energy _S and the difference in vibrational quanta :

, (2.15)

and (2.16)

(2.17)

where the Franck-Condon overlap of the vibrational quantum states is accounted for by the term . Such as for the previous models, fast vibrational relaxation compared to the electron transfer reaction is presumed. The solvent relaxation is again described by monoexponential dynamics characterized by the longitudinal dielectric relaxation constant _L. Walker et al. derived values for the reorganization energies _lf,vib and _S from fits to

25

Fuchs and Schreiber [Fuchs 96] also described the temperature dependence of betaine-30 electron transfer dynamics. Limiting themselves to a single reaction coordinate, they treated the system and a coupled bath formed by other intramolecular modes and the solvent quantum mechanically. They achieved good agreement with the experimental data and the simulations of Walker et al. for temperatures down to 228 K [Walk 92].

Van der Meulen et al. simulated the solvation dynamics of DCM after photoexcitation inducing instanteneous charge separation and recombination using a Smoluchowski equation approach [vdMeul 98]. They included anharmonic dependencies of the ground and excited state free energy curves on the reaction coordinate and succeeded in qualitatively explaining the frequency shift and the reduction in width of the fluorescence spectra of DCM in ethylene glycol on a timescale of up to 30 ps.

To allow for non-Debye relaxation behaviour of the liquid, including memory and inertial effects, Hynes proposed a generalized Langevin equation for the electron transfer reaction coordinate q [Hynes 86]:

(2.18)

where is the the derivative of the quadratic potential V(q) = , _L being the frequency of oscillation in the diabatic reactant and product potential wells.

26

Another possibility to account for non-exponential solvent dynamics is to introduce a time-dependent solvent polarization diffusion coefficient D(t) in the diffusion-reaction equation 2.11 [Hynes 86] and to solve this so-called generalized Smoluchowski equation directly for the time-dependent probability distribution. This method was applied by Tominaga et al. [Tom 91] to describe the electron transfer dynamics of ADMA. Rasaiah and Zhu [Zhu 92, Ras 93, Ras 94] showed that the survival probability is the solution of an integral equation derived from reaction-diffusion equations. There D(t) is related by:

(2.19)

directly to C_ÄE(t), the time correlation function of the solvent polarization fluctuations (see 2.3).

In the frame of the TICT-model, the twist angle is an obvious choice for a reaction coordinate. The coupling of a second, solvent coordinate to the twisting motion is provided by the dipole moment µ() of the molecule which is assumed to depend parametrically on and enters the expression for the time-dependent solvent electrical field E(t). \|[Agr ]\| Langevin equation for the evolution of was presented by Schenter and Duke, including time-dependent friction [Schent 91]. The propagation of an initial distribution f(,\|[Egr ]\|,t) for DMABN was computed, where the solvent was treated as a dielectric continuum. The dynamics of the trajectories let a picture of three distinct timescales emerge: first, an initial equilibration of , second, low barrier crossing and third, dielectric relaxation of the solvent. Polimeno et al. [Pol 94] solved a two-dimensional Smoluchowski equation with constant diffusion coefficients for the solvent and the twist angle coordinate, given by a Debye and Stokes-Einstein relation, respectively.

Kim and Hynes [Kim 97] constructed and diagonalized the diabatic Hamiltonian for a

27

Two vibrational modes were included in the electron transfer treatment of oxazine 1 in dimethylaniline (DMA) by Wolfseder et al. [Wolf 98]. As Fuchs and Schreiber [Fuchs 96] and Kühn et al. [Kühn 96], they used a reduced density matrix approach. Three electronic states (ground state, primarily excited state and dark charge transfer state) and the two most dominant vibrational modes from the Raman spectrum of oxazine 1 constitute the "system", while the remaining inter- and intramolecular degrees of freedom are combined into a heat bath. The vibrational Hamiltonians h_i are described in the harmonic approximation by annihilation and creation operators, and in the excited states also by the electron-vibrational coupling _, which is related to the intramolecular reorganization energy and the nuclear equilibrium displacement of the mode between electronic ground and excited state. The vibrational frequencies are assumed to be equal for the three electronic states. The system Hamiltonian contains the electronic and vibrational energies for the ground, reactant and product state and the electronic coupling:

(2.20)

g is a constant determining the electronic coupling V_el =

The reduced density matrix is defined as the trace over the bath degrees of freedom of the total statistical operator W(t) applied to the "system" states:

28

satisfies the Liouville equation :

. (2.22)

L() is a relaxation operator or dissipative term; it contains the vibrational relaxation rates representative of the disspation of heat into the bath. The Hamiltonian H_int(t) signifies the interaction energy of the molecular system and the laser pulse fields E(t). It is approximated using the dipole operator µ :

. (2.23)

The Liouville equation was solved numerically for a limited number of vibrational occupation numbers and the polarization obtained after

. (2.24)

From the time-resolved transmission signal was calculated and compared directly to the experimental signals. Oscillations occuring in the simulated as well as in the measured spectrally integrated intensity were interpreted as hint that coherent wave-packet motion could be coupled to the electron transfer reaction. The population probability W_R(t) (different from the survival probability only in that while Q(0) = 1, W_R(0) = 0) was also calculated as the trace over the vibrational states of the density matrix element :

. (2.25)

After an initial rise, it was found to exhibit a biphasic decay on timescales of about 50 and a few hundred femtoseconds. The 50 fs component was viewed as inherent (system-depending), while the slow component was thought to reflect vibrational cooling [Wolf 98].

Pronounced oscillations with a period of approximately 55 fs persisting until around 1000 fs in visible transients of the same system were considered a case of continued wave packet motion in the reaction product [Eng 99]. A biexponential decay with time coefficients of 30 and 80 fs was manifested by spectrally averaged transmission changes in the red and blue spectral regions associated with the absorption of the primarily excited oxazine1⁺/DMA complex and the product oxazine^•/DMA⁺. The shorter component was discussed with regard to the period of the coherently excited vibration(s): if the interaction region between the

29

The importance of higher vibrational states was stressed for electron transfer of coumarin 337 to DMA by Wang et al. [Wang 97]. They measured IR and visible transients and estimated the solvent and intramolecular reorganization energy for C337 after photoexcitation from stationary absorption and fluorescence spectra to 1525 cm^-1 and 300 cm^-1, respectively. Since electrochemical measurements led them to evaluate the free reaction enthalpy for the reaction of the isolated compounds to the far larger value of -6600 cm^-1, electron transfer via nonequilibrium vibrations of the product state was taken into consideration.

30

2.3. Solvation dynamics

The movement of solvent molecules in a liquid induces fluctuations E(t) in the energy gap E between the solute electronic ground and first excited state, causing spectral broadening of the intrinsic stationary absorption and emission bands. The fluctuations are characterized by their normalized equilibrium correlation function :

. (2.26)

Here stands for averaging over a canonical ensemble. The adaptation of solvent molecules to a change in the solute's electric field going along with electronic excitation is termed solvation dynamics. The electronic polarization of the liquid is assumed to respond instanteneously to the perturbation, whereas nuclear reorientation follows more slowly and leads to an energy shift in time for the electronic transition energy E(t). A normalized solvent or spectral response function, also called "non-equilibrium solvation correlation function", is defined as:

, (2.27)

where is the mean transition frequency. (t) can experimentally be obtained by monitoring the mean emission frequency of a chromophore after electronic excitation, the evolution of which is called time-dependent Stokes shift.

If the perturbation from the electronic transition is not large, linear response theory provides a relation between the equilibrium fluctuations and the dissipative relaxation dynamics, so that . Molecular dynamics simulations of the neat liquid can be performed under equilibrium conditions and the results can be compared to measurements of the time-dependent Stokes shift. In this way, the liquid response was found to proceed on various timescales. An ultrafast part of sub-100 fs, approximated by a Gaussian function in time, was assigned to inertial motion of the "free streaming" of solvent molecules uncoupled from each other [Stratt 94, Ros 94, Mar 94, Rain 94]. This effect was first reported for solvation dynamics of LDS-750 [Ros 91, Cho 92], although this was

31

Treatments of the stationary Stokes shift with reaction field models were initiated by Lippert [Lipp 57] and McRae [McRae 57] and others based on an idea of Onsager [Ons 36]. The reaction field model pictures the liquid as a dielectric continuum of dielectric constant (continuum model). The solute is represented by a point dipole at a center of a spherical cavity of radius a filled with dielectric material having a dielectric constant . The latter is related to the solute's molecular polarizability through the Clausius-Mosotti equation:

.

The interaction energy of solute and solvent can be written in the form:

, (2.28)

where is the solute dipole moment and is the reaction field on the solute arising from the polarization of the surrounding dielectric by the solute dipole. The reaction field includes the orientational (nuclear) and instanteneous (electronic) polarization of the liquid. McRae also considered the polarization of the solute induced by the reaction field and dispersive forces between solute and solvent molecules. In this work, results from the treatment of Amos and Burrows [Amos 73] are used:

(2.29)

and

. (2.30)

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The second term on the right hand side of equations 2.29 and 2.30 characterizes dipole-dipole interactions, the third describes interactions between the solute dipole and the electronic polarizability of the solvent. The term in angular brackets is the Debye reaction field factor F.

For the Stokes shift =_Abs-_Flu the following equation holds:

. (2.31)

The static reaction field model was extended into the frequency domain by assuming Debye relaxation for () [Bag 84, vdZwan 85]:

. (2.32)

denotes the Debye relaxation time. After Fourier transformation, the reaction field and the solute dipole moment are now time-dependent, with a step function assumed for µ(t). The fluorescence shift (t)( was found to be proportional to [Bag 84], where _F is related to by . D is the rotational diffusion coefficient of the solute and _L is the longitudinal relaxation time of the solvent:

. (2.33)

For a small rotational diffusion coefficient this is approximated as:

, (2.34)

so that

S_ÄE(t) . (2.35)

The model may be extended to cover multiple Debye relaxation times, with () given as

and . (2.36)

It can also be used to extract the Stokes shift dynamics directly from the experimental dielectric frequency response of the liquid, as in [Ruth 98].

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The concept of a spectral density of solvation modes was also used by Cho et al. [Cho 92], Bagchi and Roy [Bag 94] and Yang et al. [Yang 95] to compare results from heterodyne detected optical Kerr effect (OKE) measurements and the time-resolved Stokes shift. Ladanyi and Klein [Lad 96b] confirmed that the response of the liquid to the perturbation induced by changes in the solute electronic structure or by the polarized field of the pump pulse (OKE) proceeds via similar mechanisms.

Experiments on solvation dynamics in nonpolar solvents Gardecki et al. [Gard 95] have shown large time-dependent frequency shifts also for cases where dipolar solvation mechanisms can be excluded because of the small solvent polarity. Gardecki et al. ascribed them to solvation via interaction with large quadrupole and higher electrostatic moments of these solvents. For the solvation dynamics of nonpolar solutes studied by transient hole-burning in nonpolar solvents [Four 93], a subpicosecond, viscosity independent and a slower, viscosity dependent dynamic component were found. Berg [Berg 94] suggested that a change in the solute size on excitation could produce a significant solvent response. A model of the solute as a spherical cavity and the solvent as viscoelastic continuum with time-dependent compression and shear moduli yielded two distinct components for the solvation dynamics: the fast creation of vibrations (phonons) of the instanteneous structure of the liquid, and a slower reorganization of that structure. The model was also successfully applied to the temperature-dependence of the solvation dynamics of the nonpolar s-tetrazine in the highly dipolar propylene carbonate [Ma 95].

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2.4. Vibrational relaxation

As in the case of electronic spectra, the vibrational lines obtained directly by Raman or IR spectroscopy in solution are broadened by energy relaxation processes and frequency modulation due to changes in solvent configuration (pure dephasing). The latter is divided into a part of inhomogeneous broadening, where the solvent modulation is slow on the timescale of the transition and can be pictured as frozen, resulting in a distribution of solvent configurations, and homogeneous broadening by fast solvent movement. The exact form by which the solvent modulation influences the lineshape depends on the model for the solvent fluctuation correlation function (for a review of different models, see [Flem 96]). If the solvent fluctuations are delta-correlated, no memory effect exists (Markovian dynamics), and inhomogeneous broadening induces a Gaussian lineshape. For only homogeneous broadening present, the lineshape is Lorentzian, with its width given by

. (2.37)

Here 1/T₁ is the rate coefficient for vibrational energy relaxation and 1/T₂^* is the rate coefficient for the pure dephasing. (1/T₂ = 1/(2T₁) + 1/T₂^* is the rate coefficient characteristic for the decay of coherently excited vibrations, also termed dephasing).

The loss of vibrational energy of large molecules with a high density of vibrational states has been pictured to involve two temporally distinct steps. Energy initially localized in one or more Franck-Condon active modes is rapidly redistributed among the entire vibrational space of the molecule because of anharmonic coupling between modes (intramolecular vibrational redistribution, IVR). The vibrationally hot molecule subsequently equilibrates thermally with the solvent by vibrational energy transfer between low-frequency solute and solvent modes (vibrational cooling). Evidence for the sequentiality of this process has been obtained by time-resolved transient absorption or emission experiments of organic dyes [Els 91, Hüb 91, Zhong 96, Ang 89, Mok 89]. The relevant timescale for IVR was reported to be sub-60 fs for rhodamine 6G [Ang 89], nile blue, DODCI, cresyl violet and oxazine 725 [Tayl 84], LDS 698, 751, 765 and 821 [Zhong 96] and coumarin 102 [Kov 98] in dipolar protic and aprotic solvents. It was found to be dependent on the excitation wavelength for oxazine 1, increasing from 30-50 fs for visible to 180 fs for UV excitation [Laerm 89] and

35

Vibrational cooling was reported to be at least by an order of magnitude slower than IVR [Zhong 96, Hüb 91, Mok 89, Ang 89]. For azulene in different solvents [Suk 90] and over the complete gas-liquid transition for supercritical fluids [Schwarz 96] it could be described by extensions of the isolated binary collision model.

Several recent experiments suggest that the separability of the timescales for IVR and vibrational energy transfer to the solvent may not be universal. From the delayed rise of the dye IR 125 stimulated emission in ethylene glycol [Hasch 95], an estimate of one picosecond was deduced for internal conversion from higher-lying electronic states S_n to the lowest singlet state S₁, including IVR and energy transfer to the solvent. To explain the opposite temperature dependence of vibrational lifetimes of chromium and wolfram carbonyls in chloroform, Tokmakoff et al. [Tok 94] invoked coupling of the excited vibrational modes to intramolecular and solvent high frequency modes and to low-frequency solvent phonon modes, with temperature-dependent phonon occupation numbers and phonon density of states.

Sension et al. [Sens 93] showed that IVR in the S₁ state of cis-stilbene in solution is not complete on the scale of the isomerization, but continues up to 6 ps. Jean and coworkers have presented time-resolved resonance Raman investigations of trans-stilbene [Qian 93, Qian 95, Schultz 97]. Using the bandwidth of the ethylenic mode at 1565 cm^-1, the vibrational cooling time was determined to 10 ps in alkanes and alcohols [Qian 93]. Biexponential decay of the corresponding anti-Stokes resonance Raman line was observed and the shorter 2 ps-component interpreted as being due to IVR [Schultz 97]. Neither the slow part of the ethylenic band intensity decay, nor that of other modes in the anti-Stokes RR spectrum could be modelled assuming a Boltzmann distribution of the vibrational temperature calculated from the excess energy. They concluded that energy flow to the solvent started before IVR was complete. Nakabayashi et al. [Nak 98] confirmed a nonstatistical distribution of intramolecular vibrational energy for trans-stilbene in 1-butanol picoseconds after photoexcitation.

Vibrational relaxation in liquids has been modelled under the assumption that in analogy to the gas phase an effective collision process between the solute and solvent molecules plays a central role. The isolated binary collision (IBC) model predicts collision frequencies and

36

In this work the comparatively simple model of Montroll and Shuler [Mon 57] will be used to simulate vibrational relaxation. They studied the relaxation of different initial non-equilibrium distributions of harmonic oscillators of frequency contained in a heat bath with constant temperature, formed by harmonic oscillators of the same frequency. The oscillators can exchange energy by radiation and by collision only with the bath, interacting with the vibrational and translational degrees of freedom of the bath oscillators which themselves remain Boltzmann-distributed.

Transitions will take place only between adjacent vibrational levels. The probability for the transition of nn+1 as ii-1 per collision of oscillators initially in states |n> and |i> is assumed to be a linear function of the vibrational excitation of both oscillators:

, (2.38)

where P₁₀ is the transition probability per collision for transitions between the first vibrational excited state and the ground state.

The time-dependent probability fraction of x_n(t) of oscillators in state |n> is then given by a master equation:

(2.39)

37

_vr is the total vibrational relaxation rate of the oscillator for collisional and radiative transitions between levels |1> and |0>.

Equation 2.39 was solved for an initial distribution of x_n:

(2.40)

to obtain:

. (2.41)

Here , and F is the hypergeometric function.

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