12158
J. Phys. Chem. 1994, 98, 12158-12168
Stochastic Model for Solvent-Assisted Intramolecular Charge-Transfer Antonino Polimeno, Antonio Barbon, and Pier Luigi Nordio* Department of Physical Chemistry, University of Padova, via Loredan 2, 35131 Padova, Italy
Wolfgang Rettigt Iwan N. Stranski Institute for Physical and Theoretical Chemistry, Technical University of Berlin, Strasse des 17. Juni 112, 0-1062 Berlin, Germany Received: February 2, 1994; In Final Form: August 4, 1994@
(N,N-Dimethy1amino)benzonitrile(DMABN) and related aromatic donor-acceptor compounds show dual fluorescence emission in polar solvents. The static and dynamic features of the spectra are strongly affected by the polarity and viscosity of the medium. A successful model was first proposed by Grabowski et al., by using a phenomenologic kinetic scheme. According to this interpretation, the excited singlet state undergoes an adiabatic intramolecular electron transfer. Two metastable states are assumed to interconvert by a torsional motion, which provides a natural reaction coordinate for the electron transfer (ET) process. In this work we discuss a stochastic model which extends the simple kinetic picture to a continuous description. The dynamics of interconversion is described as a diffusional process coupled to a solvent polarization coordinate. Decay to the ground state is included in the form of a sink term depending upon instantaneous conformation. The model provides a satisfactory description of all static and dynamic fluorescence spectral features available from experiments. The Grabowski scheme is derived from the continuous model in the case of a relatively high barrier between interconverting metastable states. Agreement between theoretical simulations and observed experimental spectra supports the original hypothesis based on intramolecular electron-transfer involving distinct conformers. 1. Introduction Polar excited states are extremely sensitive to the static and dynamic properties of the solvents. The solvation dynamics of polar molecules may be investigated by dynamic Stokes shift or time-resolved fluorescence experiments. The creation of charged intermediate states is often the result of the excitation of photosensitive molecules. In nature, lightinduced intramolecular charge-transfer (CT) reactions have the role of switching on a complex pattern of reactions, as in the case of photosynthesis. Light-based technologies, from photonics to molecular electronics, are similarly dependent on the creation of CT states from excited polar states, and the role of the surrounding medium, which may be a collection of vibrational modes for large systems like proteins, or a suitable set of solvent coordinates in the liquid phase, is of the utmost relevance. The dynamic evolution of polar molecules switching to intramolecular CT states exhibits interesting features. Recent experiments show that flexible molecules can fold into structurally distinct forms in order to minimize the large dipole moment resulting from the CT process, over a time on the order of nanoseconds or less.’ One major driving force of the structural change is the polarity of the solvent, since in the case of nonpolar solvents the small solvation energy does not favor extensive charge separations on the probe molecule. In the same way, an effective charge transfer between the two anthracene moieties of the neutral bianthryl molecule can be induced in the first excited state, by polarization fluctuations of the surrounding polar molecules.2 Polarization fluctuations cause an instanta-
* To whom correspondence should be addressed. Present address: W. Nemst Institute for Physical and Theoretical Chemistry, Humboldt-Universityof Berlin, Bunsenstrasse 1,D-10117 Berlin, Germany. @Abstractpublished in Advance ACS Abstracts, October 15, 1994.
neous break in the symmetry of the degenerate ionic structures, appropriate to describe electronic conformations of excited states in the isolated molecule. In normal solvents, the kinetics of the CT process is controlled by the relaxation time of the solvent reaction field. In anisotropic media of micellar type, characterized by distinct nonpolar aliphatic regions and strongly polar aqueous regions, CT kinetics is controlled by the translational diffusion of the bianthryl m ~ l e c u l e . ~ (N,N-Dimethy1amino)benzonitrile(DMABN) has been the subject of numerous experimental and theoretical studies in the last few years, since it provides the prototype system for a large class of chemical compounds which undergo the phenomenon of ‘twisted intramolecular charge transfer’ (TICT)!-26 Dual fluorescence emission in solution in conjunction with model compound studies5s6is considered as the main evidence of TICT. In the majority of cases, both static and dynamic experimental data seem to corroborate the original interpretation, first advanced by Grabowski and co-workers. The observed features of the emission signal are qualitatively accounted for by a simple kinetic scheme: the excited state SI in the planar conformation (locally excited, LE, or B* state) is converted to a twisted charge-transfer conformation (CT or A* state). The intramolecular charge-transfer process is characterized by a twist of the dimethylamino group, bringing it perpendicular to the phenyl ring. Although the kinetic approach has been remarkably successful in providing a general interpretation for dual fluorescence emission, it is obviously a phenomenologic model which can not account for microscopic details of the CT process. The solvent role is confined to determining the actual values of interconversion rates between A* and B* states. Moreover, the kinetic scheme implies well-defined states; as a consequence, they must be separated by a relatively high barrier, and so the interconversion process must necessarily be activated. There
0022-365419412098-12158$04.50/0 0 1994 American Chemical Society
Solvent-Assisted Intramolecular Charge-Transfer is, however, evidence that the activation energy, for most solvents, is on the order of one kBT unit. Multiexponential decays are then expected and effectively observed, so that timedependent interconversion rates must be invoked in order to fit the experimental b e h a ~ i o r . ~ , ~ ~ , ~ ~ It is evident that the discrete nature of the simple kinetic scheme is the major obstacle to the correct interpretation of interconversion dynamics. More advanced theoretical treatments must consider in detail the dynamics of the internal coordinate for the molecular system undergoing the TICT process, and the coupling with the solvent. Sumi and Marcusz7 have analyzed the role of solvent polarization fluctuations in activated electron-transferprocesses. A two-dimensional potential was constructed by intersecting two paraboloid surfaces and identifying reactant and product states as a function of an internal coordinate and a solvent coordinate X. The time evolution of the probability density P(X,t) was assumed to obey a diffusion equation, including a sink term k(X), which represents the solvent-dependent rate for the reactive process. Su and Simonz8 have adapted this treatment to the specific case of the DMABN system. Kim and H y n e have ~ ~ ~given a quantum description of the two-state solute system coupled to the solvent in the presence of solvent orientational polarization, to investigate the rate-determining processes of ionic dissociation reactions. An approximate procedure based on Kim-Hynes concepts was used to interpret solvation dynamics in photoexcited bianthryl by Kang, Jarzeba, Barbara, and Fonseca.z In this case, the LE state is coupled to the CT state by an effective Hamiltonian, and the solvent coordinate is assumed to obey a generalized Langevin equation of motion. More recently Fonseca, Kim, and Hynes30 have studied TICT dynamics in polar solvents by applying to DMABN the KimHynes quantum approach. Two valence-bond states, related to LE and CT states, are coupled by an angle-dependent resonance integral and interact with a nonequilibrium solvent polarization. The reaction path and transition state theory rate constant are then determined, but dissipative dynamics effects are not considered. In the last few years, some of us have developed a stochastic a p p r ~ a c h ~ lto- ~describe ~ the time evolution of the solutesolvent system on multidimensional free energy surfaces. The stochastic model is based on a classical description of the system, considered to be appropriate for molecular conformation changes and solvent rearrangements occurring in dense liquids. The model is able to interpret dielectric friction effects, timedependent Stokes shifts, and TICT dynamic^.^^,^^ In the simplest case, the charge-transfer process in the excited state undergoing conformational transitions in polar media can be interpreted by means of a two-dimensional diffusive equation, defined by the internal rotation angle 8, which gives the instantaneous mutual orientation of the donor and acceptor sites, and by the solvent polarization coordinate X. The adiabatic potential energy of the SI state is represented by a doubleminimum surface in the two-dimensional (8, X ) configuration space. One of the minima represents the slightly polar locally excited (LE) state of the solvated molecule, with a conformation similar to that of the ground state and low polarization of the medium, while the other minimum identifies the CT state, structurally defined by a twisted conformation and large solvent polarization. Schenter and Duke35 have attacked the TICT problem by using a two-variable Langevin approach, in which trajectories are numerically integrated in the large friction limit. In some respects, this method appears to be related to our treatment,
J. Phys. Chem., Vol. 98, No. 47, I994 12159 because a two-dimensional Langevin equation is formally equivalent to a two-dimensional diffusion equation. Actually, only the internal coordinate, and not the solvent one, was treated by these authors as a stochastic variable. Non-Markovian effects on the torsional motions were therefore taken into account only by introducing ad hoc a time-dependent friction term. In our approach, the dissipative dynamics of both variables is treated in a consistent way. With a relatively small number of physical parameters, our model is able to interpret the relevant aspects of the solvent role in charge-transfer state formation, for DMABN and related systems. Since the model mimics all relevant relaxation mechanisms which are supposedly present in the system, by adopting a reasonable potential function obtained from quantummechanics calculations, satisfactory comparison with the experimental results should be taken as support for the TICT hypothesis. Indeed, the Grabowski kinetic scheme is easily derived as a particular case that is valid when the energy barrier is high. The continuous model, however, can quite easily treat situations in which the barrier is negligible, so that the dynamics is nonactivated, and nonexponential decays are obtained. The present paper is therefore organized as follows. Section 2 is devoted to a summary of the original Grabowski model. In section 3 we present the continuous stochastic model. The time evolution equation is set up and numerically solved for the timedependent distribution of the populations of the excited state. Fluorescence emission spectra are then evaluated by integration of the transition dipole moment over the reaction coordinates at each emission frequency. In section 4 we discuss the predicted temperature dependence of stationary spectra and the time evolution of the emission signals. In section 5 the Grabowski kinetic scheme is recovered by means of a rigorous derivation from the continuous model, and its range of validity is discussed.
2. Grabowski Kinetic Scheme Let us recall the original Grabowski hypothesis for the interpretation of the dual fluorescence of DMABN. It is presumed that two species are responsible for the observed dual fluorescence signal, corresponding to different excited singlet states of the parent molecule. State B* (LE) is assumed to be planar, with a low electric dipole moment; state A* (TICT) has a twisted conformation and a high dipole moment. The two states interconvert according to a first-order reversible scheme; kb, is the specific rate for the forward reaction (from LE to TICT); kab is the backward reaction rate (from TICT to LE). To include the long time decay to the ground state SO, the kinetic scheme is completed by adding decay constants kb (from LE) and k, (from TICT). Note that kb and k, take into account the cumulative effect of radiative and nonradiative decays, kb = and k, = k', kr. This distinction will be more deeply discussed in the following section, in the context of the continuous model. A solution is readily obtained for the resulting system of firstorder differential equations. If b(t) and a(t) are the relative populations at time t of B* and A*, respectively, the kinetic equations are
+e
+
b = I, - K,b
+ kaba
a = kbab - K,a
with
K , = k,
+ k,
(3)
Polimeno et al.
12160 J. Phys. Chem., Vol. 98, No. 47, I994
& = kb + kba
(4)
If the effect of the source term I,, simulating laser flash excitation from ground to excited state, is taken into account by simply fixing the initial conditions b(0) = 1 and a(0) = 0, we find
phenomenological scheme, remain hidden under the single quantity kba. A more fundamental approach, which takes into account the TICT hypothesis by considering torsional and solvent dynamics explicitly, provides a microscopic interpretation for all parameters entering the theory and is able to reproduce the kinetic behavior manifested in different physical conditions.
3. Stochastic Model
where
A different way of solving the Grabowski scheme is to include a constant source term la f 0 in the expression for b(t) and to apply the stationary state conditions simulating steady state fluorescencespectroscopy. Fluorescencequantum yields (Pa and (Pb for the two emitting states are then easily evaluated. Since (Pa = ~ u s , / l where a, aStis the stationary population in A* (a similar expression holds for a b ) , we obtain gkba
Qa = kaKb
Qb
+ kabkb k 3 a
= kaKb
+ kabkb
(9)
These equations are particularly useful to interpret the dependence of quantum yields upon temperature. In order to do this, the temperature dependence of forward and backward rates kba and kab is assumed to follow Arrhenius-type behavior. The advantages of the Grabowski approach are obvious. It allows easy interpretation of the observed precursor-successor kinetics. In conditions of moderate or high solvent polarity, the forward reaction rate is obtained from the decay of the fluorescence emission signal at wavelengths in the range 300350 nm, characteristic of LE state emission. In suitable conditions, the temperature dependence of quantum yields can be used to evaluate the activation energy for the process, from static fluorescence measurements. However, the simplicity of the kinetic model also contains its major drawback. In many experimental situations, e.g. measurements performed in very polar or protic solvents (alcohols), the observed decay is typically nonexponential,even if the general precursor-successor relation is maintained. Moreover, the role of the solvent remains poorly defined. It should be clear from the TICT hypothesis that polar solvents favor electron-transfer by lowering the energy of the highly dipolar CT state and the energy barrier between LE and CT states. In addition, solvent viscosity determines the preexponential factor of the activated rate, which in the diffusive regime is proportional to total friction sensed by torsional motion. Experiments performed in solutions having the same temperature and viscosity12show that the isomerization rate of DMABN increases exponentially with the solvent polarity parameter Ed30). For highly polar solvents, CT state stabilization is so large that the energy barrier can become negligible or virtually disappears. In this case, an activation energy for the kinetic rate has been observed which is, within experimental error, equal to or smaller than the activation energy of solvent v i s c o ~ i t y . ~All . ~ ~these effects, within the framework of a
It is natural to consider 8, the angle between the amino group and the aromatic plane, as the relevant internal degree of freedom of the excited DMABN molecule. We summarize here the continuous stochastic model for the charge-transferdynamics of state S1 which accompanies torsional motion (adiabatic ET reaction). The internal coordinate is coupled to a solvent polarization coordinateX,defined as the stochastic reaction field in an Onsager cavity. Both coordinates are assumed to be classic and diffusive in nature. Thus, the time evolution equation for population P(€J,X,t) in the configuration space is determined by the following two-dimensional Smoluchowski operator, modified by the inclusion of sink and source terms:
%(e,x,t) = -[f at
+ k ( e , ~ y q e , x , t+) qex)
(io)
where f. is the standard operator for diffusion along the adiabatic energy surface in the excited state and k(8Jr) is the sink term accounting for decay to the ground state. Lastly, S(0,X) is the source term responsible for radiative pumping from the ground state to SI. Let us define the three dynamic contributions in more detail, by comparing them t: the same time with discrete kinetic equations. The term r depends on the potential energy surface El(8Jr) for state SI and on the coefficients D R and D S for diffusional motion along the 8 and X coordinates, respectively. The ( 8 s variables are therefore coupled by the potential function, and so the explicit form of r is given by
Since this term describes diffusion dynamics only, it is the continuous analogue of the kab and kba rate constants employed in the kinetic approach. The collective solvent coordinate X may be defined as the component along the molecular dipole axis of the stochastic cavity reaction field induced in the homogeneous polar medium. The interaction potential vSol between a molecular dipole p(@ and the reaction field X exerted by the solvent is obtained by the standard form originally proposed by Marcus36and normally used to interpret time-dependent Stokes shifts3 1337938
where Vis the cavity volume, EO and E- are the static and optical dielectric constants of the solvent, and px is the instantaneous solvent polarization, related to the purely orientational reaction field X through the equation
J. Phys. Chem., Vol. 98, No. 47, 1994 12161
Solvent-Assisted Intramolecular Charge-Transfer The first term at the right-hand side of eq 12 represents the stabilization energy when the solvent is in equilibrium with the molecular dipole, while the second term defines the amplitude of the solvent orientational polarization fluctuations around the equilibrium value.39 The total energy of the solute-solvent system, when the fluorescent probe is in its ground state SOor in the excited state S1 is expressed by EOand El, respectively, with
where Vo,l(f3) are the functional forms for the potential energies of the SOand S1 states of the isolated molecule and po,l(f3)are the corresponding dipole moments and 1
1
=2
= 2
= 2
Y
Y O
Ym
1 -
4+
160-1 Eo
L-1)
l/*
E,
+ v2
(15)
The diffusion coefficient DR may in principle be evaluated by a hydrodynamic model, while the diffusion coefficient for the solvent coordinate DS is usually related to the static and optical dielectric constants EO and E , and the solvent relaxation time ~ 3 . Thus, ~ ~ 3the~ time scales of torsional and solvation processes can readily be estimated, provided one is satisfied with macroscopic models for both solvent viscosity, related to DR by the Stokes-Einstein law, and solvent relaxation time, related to Ds by the standard Onsager-Debye treatment. The relevant features of the solvent dynamics can be sketched in the following way. The time correlation function for the spontaneous solvent polarization fluctuations is directly related to the time-dependent Stokes shift, which is given in terms of macroscopic solvent parameters as39940
Eo -- -exp(-t/ts)
+ v2
Eo
E,
+ Y2)
Kat0 and Amatatsulg have performed ab initio calculations of the set of parameters ac) relative to DMABN in vacuum. Similar results have also been obtained by Marguet et al.42As in previous paper^,^^^^^ the Kat0 and Amatatsu parametrization has been adopted throughout this work, and the complication related to the pyramidalization coordinate of the dimethylamino group has been omitted in a first approximation, since the twisting motion is the only intramolecular coordinate subject to large viscous friction by the solvent. The molecular dipole moment is po(f3) = ,DLE in the ground state but can be assumed to depend upon the geometry of the excited state according to the relation24
(16)
This expression is certainly a very simplified description and only an assumption of the angular dependence of the dipole moment. Indeed, one can expect a state mixing induced by the solvent, but this can be important only for intermediate twist angle. In the limiting case (0 = z/Z), however, the CT state can not couple with z-z* states; thus, this equation is an acceptable approximation given the lack of more detailed knowledge. The overall potential for the SI state generally exhibits local minima at the planar and perpendicular conformations of the molecule and a local maximum for an intermediate conformation. Note, however, that the zero-order potential for the nonsolvated molecule increases monotonically from the planar to the perpendicular geometry. The solvation term stabilizes the perpendicular conformation and can produce an effective activation energy. The planar conformation is defined in (0, X ) space by the point of coordinates (0, p&Z2), while the perpendicular conformation is located at (z/Z, p&E2). The solvent stabilization energies in a polar solvent are defined as
The stochastic operator which describes the time evolution of the reaction field, in the absence of coupling with a solute dipole, is given by the second term of eq 11 by putting pl(f3) = 0
+ &)
n
rx=-D--
The solution of eq 17 is analytical,"l and evaluation of the time correlation function of the reaction field gives
A comparison of the two results allows us to recover the relation between zs and the macroscopic parameters
for LE and TICT states, respectively. Experimental evidence indicates a ratio between the dipole moments in the CT and LE conformations close to 3;43 i.e., polv CT is approximately equal to 9Ep:. We adopt q : as the parameter of our model, corresponding to a measure of the degree of polarity of the solvent. The diffusion coefficients DR and DS also have to be evaluated. A simple analysis based on standard continuum models relates the polarization relaxation time TS on the righthand side of eq 19 to the dielectric relaxation time TD of the solvent m ~ l e c u l e : ~ ~ ~ ~
1 - DsE2 S'
kBT
More refined molecular models for the solvent could be invoked for a better understanding of the dynamics, especially when specific interactions of the medium with the probe molecule are expected to be important, e.g. in protic solvents. 3.1. Parameters. Parametrization of the potential energy is a substantial ingredient of the model. Useful approximate functional forms are given by cosine series:
The dielectric relaxation time t~ = '/2Dis related to solvent viscosity 7 by the Stokes-Einstein relation
and a similar equation holds for the torsional relaxation time ZR = ~ / D R for the dimethylamino group rotation in the same solvent:
12162 J. Phys. Chem., Vol. 98, No. 47, 1994 1 -_R ‘
The ratio
TR/ZS
Polimeno et al.
kBT
8Z7b3
is then evaluated to be
The geometric factor b/a is determined by the physical dimension of the rotating dimethylamino group compared with that of the solvent, and the ratio E&, varies between 1 and 20 for the various solvents. Because of the third power in the geometric factor, only the order of magnitude of TR/Q can be reasonably given. We have kept Z R / Z ~ = 1 for most of the solvents and 5 for the highly polar and small-sized acetonitrile. A value of 0.4 x lo-” s was adopted for ZR, for DMABN in n-butyl chloride at 293 K. For other solvents and temperatures we rescaled the diffusion coefficients according to actual viscosity. The k(8x) dynamic term of eq 10 accounts for the continuous depletion of the S1 state population due to nonradiative and radiative decay. In the Grabowski scheme, the decay rates for LE and TICT states were denoted as k b and k,, respectively. In the continuous approach, a functional form for k(8,X) must be assumed. Fluorescence quantum yields and lifetimes of the CT band of DMABN are rather insensitive to solvent and temperature (see e.g. ref 23 and sources cited therein). Furthermore, the transition moment is a strong function of the torsional angle due to the varying orbital overlap, whereas the coupling with the solvent can only introduce changes in the CT character of the state. This effect is considered to be of secondary importance, and so the decay is assumed to be a function of the torsional angle only, in the following form: k(8) = k,
+ k, cos2 8
where In) are normalized Hermite functions defined with respect to the shifted solvent coordinate 3 = EX - p(@/Z. This variable is chosen for mathematical convenience, because the dimensions of the resulting matrices are significantly reduced with respect to the more opvious choice of Hermite functions in X.33,34By representing r and k(8) in matrix form and S ( 8 a in vector form, we are left with the following set of linear differential equations for the components in the chosen basis set of P(B,X,t): P(t) =
(29)
We assume in this way that the radiation pulse stimulates the DMABN molecules instantaneously in the neighborhood of the planar geometry of the excited state, without changing the distribution of the solvent molecules, which is determined by the equilibrium within the ground state potential. Due to a relatively flat ground state twist p ~ t e n t i a l ?the ~ actual fluores-
-(r + k)P(t) + S
(32)
Matrices and vectors are truncated to finite dimensions, large enough to achieve convergence. The solution corresponding to the initial conditions P(0) = 0 (excitation) is P(t) = [l - T exp(-At)T’]P,,
(r + k)P,, = S
(28)
The first term is responsible for nonradiative decay and is taken to be also independent of 8, due to lack of knowledge.@ The second term stands for the radiative emission rate, with the condition that emission probability is zero at the CT conformation, for orbital symmetry reasons. In previous papers, we showed that the available data were satisfactorily reproduced by allowing the nonradiative rate constant knrto depend weakly on temperature and by keeping kr constant at -lo7 s-l, for all solvent^.^^.^^ Since the fluorescence decay constant obtained in this way for DMABN emission in n-butyl chloride at 150 K is smaller by a factor of 10 with respect to the experimental value, in the present work we neglect k, in comparison with k, and assume k, to be constant and equal to 2 x lo8 s-l for all temperatures and solvents. This is qualitatively consistent with the experimental observation that the ratio of nonradiative to radiative TICT decay is about 30 for DMABN.44 Lastly, in the continuous model an absorption term is included in the form of a contribution S(e,X), defined as a peaked function around the LE conformation multiplied by a Gaussian distribution with respect to the solvent coordinate:
s(e,x)= s(e) exp(-E2X2/2kBT)/&
cence experimental measurements involve broad distributions along the torsional coordinate. In the simulations, the choice of a narrow distribution along 8 does not alter significantly the predicted dynamical behavior of the system, except for transient times. 3.2. Numerical Treatment. The distribution probability P(O,X,t) can be evaluated by numerical solution of eq 10. It is convenient to represent the time evolution equation on a set of orthonormal functions
(33) (34)
while the solution for P(0) = POand S = 0 (emission) is P(t) = T exp(-At)T’Po
(35)
In these equations T is the unitary matrix which diagonalizes
+ k; Le., P(r+ k)T = A. Thus, we have to solve a standard
eigenvalue problem to obtain the time evolution of the fluorescence spectra, while a linear system determines the stationary population of the excited state. In all the cases treated in the following sections, matrix dimensions were about 150-250 (usually 30 exponential functions multiplied by 5-8 Hermite functions). The dynamic emission spectrum is obtained in analogy with eq 11 of ref 2 by an integration over all the microscopic states (0, x):
q ~= o3JR ) de
J-z
d~,u,(e)~(e~,t)g[o,o,(e~r)i (36)
where function ,uu,(e) is the electronic part of the total transition moment, that we take as essentially proportional to the radiative emission rate. The function g[w - wo(e,X)] is a band shape function centered on the emission frequency:
oo(eJr>= IWX)
- E,~BX)I/~
(37)
When solvent polarity is low, wo is practically constant; thus - 00) is the static fluorescence emission spectrum in apolar solvents.2 A simple Gaussian function, chosen to reproduce
g(w
Solvent-Assisted Intramolecular Charge-Transfer the fluorescence spectrum in hexane at all temperatures, was then adopted for g(w). The stationary signal Zdw) is obtained from the stationary population P(8,X).
4. Simulated Emission Spectra In the previous section, various parameters entering the continuous stochastic model are listed. Two classes of physical parameters are defined, i.e., quantities related to the solute molecule and parameters depending on the solvent. In the first class we include the adiabatic energy surface parameter for an isolated DMABN molecule. Different DMABN derivatives (such as the diethylamino, pyrrolidino, or piperidino compounds DEABN, PYRBN, or PIPBNZ4)should in general have different energy surfaces. The ratio between the dipole moments ~ C and T ,MLE (fixed to 3 for DMABN) is dependent upon the molecule structure. The band shape function g(w wo) is determined by the emission band of the specific molecule in apolar solvents. The diffusion coefficient DR depends upon molecular geometry but also upon solvent viscosity. For the various DMABN derivatives, we also change the dimension of the group undergoing torsional motion. Lastly, the decay constant k is likely to depend strongly upon the electronic structure of the molecule. In the second class we include the diffusion coefficient DS and the solvent stabilization energy The diffusion coefficient DSdepends upon both the molecular size and bulk dielectric properties of the solvent. Note that the assumption of a unique Debye relaxation time for the medium would be incorrect if protic solvents were considered, and a more complex time behavior for the reaction field should then be modeled. It remains to consider the solvent stabilization energy polv LE . Although in principle sophisticated electrostatic solvation models could be employed to evaluate it, we prefer to maintain it as a free parameter, which gives a measure of the degree of polarity of the solvent. Let us summarize. The band shape g(w - 0 0 ) is obtained from the static emission spectrum in a solvent of low polarity (e.g., hexane). The torsional diffusion coefficient DR is estimated from the Stokes-Einstein relation. It was taken as 2.5 x 1Ol1s-l in a reference solvent (n-butyl chloride at room temperature) and then changed according to the viscosity of the actual solvent. The DR value is slightly higher than that used in previous calculations to compensate for the increase in the decay rate adopted here. The nonradiative decay rate k, is kept constant for all solvents and temperatures at a value of 2 x lo8 s-l. The solvent diffusion coefficientDSis obtained from the Onsager-Debye continuum model for the solvent. Lastly, is chosen for each solvent as an adjustable parameter depending upon solvent polarity. Before presenting static and dynamic spectra computed for a series of solvents, it is useful to consider first the shape of the potential when is varied. If solvent relaxation is fast compared to the solute intemal dynamics, the coordinate X can rapidly equilibrate with respect to the instantaneous values assumed by the probe dipole moment. In this case, the dynamics of the torsional angle 8 is subjected to an effective potential obtained by averaging the total distribution function P(O,X),derived from eq 14, over the solvent variable X:
e$.
e:
The expression for the effective potential is evaluated to be
J. Phys. Chem., Vol. 98, No. 47, 1994 12163
(39)
This effective one-dimensionalpotential is actually meaningful only in fast relaxing solvents, so it gives only an approximate picture for the cases considered here, where ZR/Q is on the order of unity. Even in this case, however, the CT process is strongly affected by the free energy profile along the torsional coordinate, and so representation of the effective potential is useful for a qualitative understanding of the TICT formation. Six different solvents were selected: hexane (hex), n-butyl ether (bob), diethyl ether (eoe), n-butyl chloride (bcl), nbutyronitrile (bcn), and acetonitrile (acn), in order of increasing polarity. Stabilization energy values E$, obtained by optimizing the simulation of the static spectra, are given in Table 1. The corresponding potential curves are shown in Figure 1. It is interesting to note that only for solvents of intermediate polarity does a barrier for LE CT conversion, on the order of one kBT unit at room temperature, actually exist. The potential profile depends upon the functional form of PI(@, which is taken here to obey to the relation given in eq 21. 4.1. Static Spectra. The full dynamical problem, as presented in section 3.2, has been solved to obtain spectra at various solvents and temperatures. As mentioned above, these spectra were simulated to determine the solvent stabilization energy parameter In the simulations, the expression chosen for the emission rate to the ground state is crucial. In order to obtain the best results for both static and dynamic calculations, the electronic transition moment in eq 36 has been taken to be proportional to ky k,' cos2 8. The effect of the constant term k: is that of increasing the radiative emission of the CT state, which otherwise would be strictly forbidden for 8 = d 2 . A ratio ky/k/kf= 0.07 has therefore been adopted, small displacements from this value giving relative intensities for the LE and the CT bands very different from the experimental ones. The values of all other relevant parameters used in the calculations are reported in Table 1. Figure 2 shows solvatochromic effects calculated for the fluorescence emission bands of DMABN, at room temperature. The spectra are normalized; Le., calculated intensity divided by its maximum value is plotted in each case. Obviously, a strong red-shift is obtained for the CT band, but to a minor extent the position of the LE band is also red-shifted when solvent polarity is increased. The calculated spectra in Figure 2 come very close to published experimental fluorescence s p e ~ t r a : ~the . ~ relative intensities of the LE and CT bands and their maximum positions have been reproduced. In Table 1, the energy differences between TICT and LE states, computed from the potential function used in the simulations,are compared with those obtained from electrostatic calculations based on the Onsager Knowing that AJZ = 7 kT mol-' in n-butyl chloride,44by taking the result of a recent dipole moment determination @LE C m43)and an estimated = 22 x C m, p m c ~= 60 x value of 5 8, for the Onsager cavity radius, AJZ in vacuum is calculated to be -48 kJ mol-'. Thus, TICT state formation is endothermic in the gas phase, in agreement with the recent quantum chemical ab initio and AM1 calculation^.^^^^^ In hexane, the endothermicity is still present but reduced (AE = -23 kJ mol-'), and only in s o l h t s of medium and strong polarity does TICT formation become exothermic. The resulting values for AE are in good agreement with those obtained by the stochastic simulation of solvatochromic effects. Figure 3 exemplifies the temperature dependence of the fluorescence emission of DMABN in n-butyl chloride. The
-
e!.
+
Polimeno et al.
12164 J. Phys. Chem., Vol. 98, No. 47, 1994
e:,
TABLE 1: Stabilization Energies Energy Differences between the LE and TICT States [from the Potential Function El(O,X)], Dielectric Constants, Viscosities, and Torsional Correlation Times for DMABN in Different Solvents, at 293 KO ESOIV
AE
LE
tR
solvent (Hmol-') (Hmol-') EO hexane 2.5 -26.8 (-23) 1.88 dibutyl ether 4.24 -9.5 (-9) 3.08 diethyl ether 5.65 2.0(-1) 4.34 n-butyl chloride 6.07 5.3 (7) 7.4 n-butyronitrile 6.82 11.3 (15) 20.3 16.6(17) 37.5 acetonitrile 7.48
7 (cP) 0.31 0.6 0.24 0.45 0.62 0.35
x 10"
(s) 0.27 0.56 0.21 0.4 0.56 0.3
TABLE 2: Stabilization Energies, Dielectric Constants, Viscosities, Torsional Correlation Times, and r& Ratios for DMABN in n-Butyl Chloride at Different Temperatures 293 210 173 145
7.4 9.5 11.1 12.9
6.07 6.34 6.45 6.63
0.45 1.4 3.3 8.5
1.o 1.3 1.5 1.7
0.4 1.81 5.O 14.3
Energy differences from electrostatic calculations are given in parentheses for comparison.
300
400
500
600
fnml Figure 4. Fluorescence spectra calculated for DMABN in n-butyl chloride at 173 K, by assuming the functional form k, = kf k: coszO for the radiative emission rate, and for different values of the ratio
+
k:/k:.
135
0
45
90
e
Figure 1. Averaged torsional potential F ( 0 ) for several solvents at 300 K: (hex) hexane; (bob) n-butyl ether; (eoe) diethyl ether; (bcl) n-butyl chloride; (bcn) n-butyronitrile; (acn) acetonitrile. The graphs have been obtained by eq 38. One should remember that the effective potential is meaningful only under assumption of fast solvent relaxation. I. 00 /,e,
0.50
0.00 300
400
500
600
(nml Figure 2. Calculated solvatochromic effects on the static emission of DMABN at room temperature in the solvent series reported in Figure 1. The calculated spectra reproduce closely the experimental results as reported in Figure 2.10 of ref 4.
CT band at decreasing temperatures. One is again referred to Figure 2.1 of ref 4 for comparing the theoretical predictions with experimental results. The change with temperature of the relative intensities of the fluorescence bands and the minimum at 180 K of the LE emission are well reproduced. The parameters entering the simulation of temperaturedependent spectra are given in Table 2. The activation energy for the viscosity has been taken to be equal to 1.7 kJ and the temperature dependence of the dielectric constant for butyl chloride has been assumed to be equal to that of diethyl Because of the temperature dependence of EO, both p oLE l v and ZRIQ vary with temperature. All the simulations are in reasonable agreement with the experimental data. They are essentially identical to those previously obtained with slightly different p a r a m e t r i ~ a t i o n . ~ ~ . ~ ~ In Figure 4 the effect of the &independent term k: is displayed. It can be seen from the figure that the relative intensity of the CT and LE bands is strongly dependent upon the ratio k:lkf and that it can be inverted by assinging the value kylk,' = 0.5. From the comparison of simulated and experimental spectra, we can thus conclude that k:lk,' is indeed small for DMABN, Le. that TICT emission is indeed forbidden for a 90" twist. 4.2. Dynamic Spectra. The solution of the dynamic equation for the time-dependent distribution P(e,X,t) allows a detailed interpretation of the parametrization used in the phenomenologic model. In fact, numerical calculations directly provide decay frequencies in the form of the eigenvalues of the matrix r k, according to the results of section 3.2. When the time-dependent distribution P(O,X,t) is analyzed at a fixed point (0, X), it may be expressed as a sum of exponentials:
+
300
400
500
600
(nml Figure 3. Calculated temperature dependence of DMABN emission spectra in n-butyl chloride. The calculated spectra correspond closely to the experimental behavior exhibited in Figure 2.1 of ref 4. complete calculations show that the emission profile exhibits a minimum around 180-20oX for the LE band, as observed in the experiments in refs 5 and 48. This is a direct consequence of competition between the decay to the ground state and the interconversion to the TICT state. This effect can easily be rationalized in terms of the quantum yield expressions, reported in eqs 8 and 9. Note that a red-shift is observed for the broad
P(ex,t) =
Cwi(e,x) exp(-il,t)
(40)
i
where wi(6,X) determines the contribution of eigenmode Ai. For quantitative analysis of the time dependence predicted by the model, let us look at the fluorescence emission signal Z(r) and at the population P(t). According to eqs 36 and 40, the time dependence of Z(t) is given as a sum of exponential decays, and a similar expression holds for P(t):
Solvent-Assisted Intramolecular Charge-Transfer
The weights A: and Bf are calculated by substituting eq 40 into eq 36 and integrating over the (8, X ) variables, with frequency values OCT = 450 nm and OLE = 350 nm, respectively. One obtains
and a similar equation holds for Bf:
The weights A: and BY for the populations are obtained by simply fixing into eq 40 the coordinates (8 = 0, X = , u ~ d F ) and (0 = d 2 , X = ,uc@), respectively. Eigenvalues with dominant weights in LE and CT time evolution are reported in Table 3 for DMABN in butyl chloride at T = 173 K and different values of ZR/Q.. This solvent has an intermediate polarity, and it ensures a bistable potential along the torsional coordinate. Values of 0.15, 1.5, and 15 were assigned to the ratio Q/ZS for illustrative purposes, with no reference to the real solvent. In order to understand the entries in Table 3, we note that the eigenvalue 21 = 0.2 ns-' always corresponds to the fluorescence decay rate, which is decoupled from internal and solvent motions. The second dominant mode A 2 occurs in the sub-nanosecond region and reflects internal conformational kinetics. The faster modes correspond to diffusive motion within the potential minima, and therefore they contribute to redistributing the population within the LE site, after the exciting pulse has created the initial angular distribution according to eq 29. It can be seen from the table that for ZR/Q 5 1 solvent fluctuations give rise to an additional source of friction, i.e. the dielectric friction, which slows down the rate of the TICT process. The time evolution of the LE and CT bands, calculated after a 10-ps pulse for DMABN in n-butyl chloride at 173 K, is shown in Figure 5. Band intensities are calculated at fixed wavelengths corresponding to the emission maxima. Within the first few hundred picoseconds the decay is essentially monoexponential, and the curves follow the typical precursor-successor pattern, in agreement with the experimental observation that the B*state emission corresponds to the rise time of the A*-state e m i s s i ~ n . ' ~At . ~longer ~ times the signal evolution is dominated by the nonradiative decay rate, slower by 1 order of magnitude. In the case of n-butyl chloride, where a sufficiently high energy barrier exists, we can conclude that LE state decay and rise of the TICT state population are controlled by the interconversion process. Experimentally?0the LE band of DMABN in n-butyl chloride at 173 K exhibits a fast decay component, with weight 298% and rate constant 2.3 x lo9 s-l, corresponding to the rise time of the CT band (440 ps). The decay of the CT band intensity is determined by a slowest component, of rate constant 0.24 x lo9 s-l, the relative weights for nonradiative decay and TICT formation being 0.6 and -0.4, respectively. The theoretical predictions reported in Table 3 for ZR/Q = 1.5 agree reasonably well with these data. The nonradiative decay time of the fluorescent states has been taken as a fixed parameter of the
J. Phys. Chem., Vol. 98, No. 47, 1994 12165 TABLE 3: First Few Relevant Eigenvalues A and Relative Weights A, B for the Time Evolution of Fluorescence Signals (f) and Populations (p), for DMABN in Butyl Chloride at T = 173 K and Different zR/Zs Ratio* tRirS 1 x 10-9 (s-1) B' x 104 A' x 104 BP AP 0.15 0.2 1.3 3.1 3.6 x 1.2 0.8 253.7 -0.7 0.84 -1.5 -1.1 2.9 0.4 3.3 x 0.3 1.5 0.2 1.4 10.6 3.6 x 1.2 4.1 279.7 -9.8 0.83 -1.5 0.5 4.3 x 0.3 27.3 0.2 15 0.2 1.3 3.1 3.6 x 1.2 15.2 256.7 -1.6 0.77 -1.4 0.4 237.8 1.1 -2.1 3.8 x IO-' a Labels A and B refer to CT and LE states, respectively. The values s have been taken for all t$$ = 6.45 kJ mol-[ and t~ = 0.5 x cases.
5d
0
20
40
60
60
100
t (Ps) Figure 5. Precursor-successor behavior for the predicted time evolution of DMABN fluorescence emission, in n-butyl chloride at 173 K at fixed wavelengths 350 and 450 nm (the sharp peak in the 350 nm curve is an artifact due to the stepwise excitation assumed in the calculations). At longer times, both signals decay because of the intrinsic fluorescence lifetime km-' = 5 ns. calculation, but its contribution to the decay process relative to the interconversion kinetics is determined by the dynamical model. Some interesting conclusions can be reached by comparing the results of Table 3 with the kinetic treatment. According to the Grabowski scheme, if the time dependence of the LE and CT populations is written as
PLE(t) = b, exp(-l,t)
P,--(t) = a, exp(-A,t)
+ b, exp(-A,t) + a, exp(-A,t)
(45) (46)
definite relations hold, for eigenvalues and corresponding weights, under the simplifying assumption k, = kb = k. From section 2, one obtains for the eigenvalues
A,=k
(47)
A, = k + kab + kba
(48)
and for the weights
These relations are only approximately obeyed by the continuous model. Note that the coefficients A: should sum up to zero, because no appreciable CT population is created after the 10ps pulse used in our simulated experiment. This is not always apparent from Table 3, because only the first few dominant
12166 J. Phys. Chem., Vol. 98, No. 47, 1994
Polimeno et al.
eigenvalues are reported, but other fast modes with significant weights are actually present.
Kramers expressions for the kinetic decay constants kba and kab are recovered?*
5. From the Continuous Model to the Kinetic Scheme The necessary condition for the Grabowski scheme to be applicable is the existence of well defined potential minima for the LE and TICT states. However, only under the condition of time scale separation for the processes involving solvent and internal coordinate is it possible to recover from the stochastic model a simple interpretation of the phenomenologic parameters entering the kinetic scheme. This is done by using a projection operator procedure51 for averaging the diffusion equation over the fast relaxing coordinate. Ln the case of fast relaxing solvents, the time evolution equation for the reduced distribution is given by
where $0) is the source term averaged over the equilibrium Gaussian distribution of the solvent; the averaged operator is simply defined as
f- = -D -Peq(6)-Pe,(6)a a
1
a6
(53)
whereas the fluorescence decay rates are simply kb x kl and ka x k2. In eqs 62 and 63 are square roots of the second derivatives of the effective potential Vff(6),calculated in correspondence with the LE and CT minima and with the saddle point, respectively. The activation energy @E is the CT, while EET is the banier for barrier for the process LE the inverse process CT LE. The superscript R denotes that the reaction coordinate is given by the torsional angle 6. The overall temperature dependence of the kinetic eigenvalue includes the temperature dependence of the viscosity 7, which enters the definition of DR = ~ / Z Raccording to eq 26 and which follows the Arrhenius-type law q = 70exp(E,,/kBT). In the case of polar solvents EET >> E:E, barrier recrossing can be neglected and the kinetic eigenvalue reduces to
QzT, Qt,
--
where
Pe,(6) = exp[ - vff(6)/kBr]/(exp[- Vff(6)/kBr]) (54) is the (normalized) equilibrium distribution for torsional angles, defined with respect to the averaged potential given in eq 39. After defining a set of localized functions g,(6),52 with m = 1 or 2, referring to the LE and CT minima, respectively, we can write a projection operator in the form: = '
CPeqlgm)Qm(gm
I
(55)
m
where Qm = (gmlPeq)are equilibrium site populations and the brackets denote integration over the 6 variable. The projected equation of motion becomes
?&P at
=
-&(f + k)&P + &s
(56)
If the population in the mth minimum is denoted by Pm(t),we then obtain the master equation for state populations in the form
P(t) = -(1T where
+ ii)P(t) + s
(57)
The eigenvalue in the fast relaxing solvent limit comes out to be
A;
S
(58) kba
im
= (gmlkPeqlgm)/Qm
S m = (8, I$
= -kab,
r21
= -kba,
r 1 1 = kba, r 2 2
Qmax D S K P ? E
exp(-'$E/kBT)
(66)
(59) (60)
where the matrix k is diagonal. Equation 57 is similar to eq 32, with the important difference that in the latter equation vectors and matrices have in principle infinite dimensions, while now we are dealing with a two-by-two problem. Since is a transition matrix:' the following identities hold rlz
(65)
The Kramers eigenvalue obtained with this formula for ZR = 5 x lo-" s as used in Table 3 is equal to 1.6 x 1O'O s-l, which should be compared with the exact numerical result 22 = 1.5 x 1O'O s-l reported in the table for Q/ZS = 15. In the opposite condition TR/ZS
