Letter pubs.acs.org/JPCL
Non-Ergodic Electron Transfer in Mixed-Valence Charge-Transfer Complexes Dmitry V. Matyushov* Center for Biological Physics, Arizona State University, P.O. Box 871504, Tempe, Arizona 85287-1504, United States ABSTRACT: Theories of activated transitions traditionally separate the dynamics and statistics of the thermal bath in the reaction rate into the preexponential frequency factor for the dynamics and a Boltzmann factor for the statistics. When the reaction rate is comparable to relaxation frequencies of the medium, the statistics loses ergodicity and the activation barrier becomes dependent on the medium dynamics. This scenario is realized for mixed-valence self-exchange electron transfer at temperatures near the point of solvent crystallization. These complexes, studied by Kubiak and coworkers, display anti-Arrhenius temperature dependence on lowering temperature when approaching crystallization; that is, the reaction rate increases nonlinearly in Arrhenius coordinates. Accordingly, the solvent relaxation slows down following a power temperature law. With this functional form for the relaxation time, nonergodic reaction kinetics accounts well for the observations. SECTION: Liquids; Chemical and Dynamical Processes in Solution ctivated transitions are driven by nuclear fluctuations moving the system over the barrier separating the reactants and products. Both the dynamic and static aspects of the nuclear fluctuations contribute to the observed rate. Their contributions are typically separated in the rate constant formulated in either the transition-state1 or Kramers2 framework. In either case, the rate is a product of the nuclear frequency, ωn, incorporating the nuclear dynamics, and a Boltzmann factor, incorporating the nuclear statistics
A
barrier-crossing formalism but requiring medium modes to project onto the reaction coordinate not only according to their coupling to it but also according to their characteristic relaxation times. This is achieved by postulating that only degrees of freedom faster than the rate contribute to the observed transitions.10 The rate of the process then cuts the phase space, that is, the collection of the coordinates and momenta describing the system, into a fast subspace available for statistical averaging and a slow, dynamically frozen subspace. The ergodicity, even if understood in its restricted meaning of separate time scales,4 is broken. Whereas the Hamiltonian of the system is not affected, the rules of how to use it to calculate statistical averages are affected. The consequence of this rather trivial observation becomes significant any time a reacting system interacts with many medium modes characterized by a distribution of relaxation times. When the reaction is driven by a fast mode from this dynamical spectrum, it potentially makes a slow mode nonergodic, with a consequence that this slow mode does not equilibrate and the free energy barrier becomes a function of the reaction rate. The statistics and dynamics of the nuclear degrees of freedom do not factor anymore in the rate constant into the pre-exponent and a dynamics-independent activation barrier. This physical reality can be mathematically cast10 into the requirement of solving a self-consistent equation for the reaction rate k
†
k = ωn e−β ΔG
(1)
Here ΔG† is the Gibbs activation free energy and β = 1/(kBT) is the inverse temperature. The frequency ωn is a mechanical frequency assigned to nuclear motions in the transition-state formulation, whereas dissipation (friction) is associated with ωn in the Kramers’ kinetics. The use of the Gibbs free energy in eq 1 is bound to the assumption that all nuclear degrees of freedom contributing to the barrier height are fast compared with the reaction; that is, the system is able to sample its entire phase space before moving to the products state.3 This assumption requires a certain separation of time scales, which is always understood to be the case when the rules of statistical mechanics are applied: “all the ‘fast’ things have happened and all the ‘slow’ things have not”.4 The problem clearly becomes more complex when the separation of time scales is not possible, which is always the case when the reaction rate falls into the relaxation spectrum of the thermal bath. The rate should, in principle, be calculated from solving the dynamic equations in this case.5−9 However, the mathematics involved in solving multidimensional dynamic equations quickly becomes impractical with growing dimensionality. The problem can be circumvented by maintaining a © 2012 American Chemical Society
†
k = ωne−β ΔG (k)
(2)
Received: May 17, 2012 Accepted: June 1, 2012 Published: June 2, 2012 1644
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The kinetics defined by this equation is labeled as nonergodic to distinguish it from the more common case when the standard rules of canonical ensemble apply (eq 1). Equation 2 assumes that the rate k describes mostly single-exponential reaction kinetics. When this is not the case, for instance, for small barriers, the kinetics is nonexponential and eq 2 is replaced by a self-consistent solution for the dynamics of the reactants’ population.11 Radiationless electronic transitions is a specific case in this general framework. They are driven by nuclear fluctuations creating conditions of energy resonance for electronic tunneling between the localized donor and acceptor electronic states.12−14 Dipolar nuclear polarization is often the leading mode driving the transition, and the (longitudinal) polarization relaxation15 and reorganization free energy12 of the polar solvent are two significant solvent parameters affecting the rate. Internal motions of the donor and acceptor molecules affect energies of the electronic states as well, resulting in intramolecular vibrational dynamics and intramolecular reorganization.16 Solvent reorganization energy of electron transfer is a decreasing function of temperature.17,18 Qualitatively, this is clear from the notion that the polarity of the solvent increases as it cools, although quantitative access to the slope of the temperature dependence requires a separate consideration of the solvent’s dipolar rotations and translations.19,20 Kubiak and coworkers21,22 have recently reported kinetic data for mixed-valency complexes at the borderline between localized (class II) and delocalized (class III)23 configurations of the electronic mixed-valence state. According to Hush,24 delocalization is controlled by the relative magnitudes of the nuclear reorganization energy λ and the donor−acceptor electronic coupling Hab: the electron is localized in either donor or acceptor state at 2Hab < λ and delocalized when this inequality is inverted. Mixed-valence dimers of trinuclear ruthenium complexes {[Ru3O(OAc)6(CO)(L)]2BL}− were studied experimentally. Here BL is the pyrazine, and changing the ligand L in a series of complexes allows one to control the extent of electron delocalization. The rates of electron selfexchange were extracted from band-shapes of the IR spectra of ruthenium carbonyl ligands based on the extent of coalescence of two CO sub-bands, which depends on the self-exchange rate.21,22 The remarkable observation of Kubiak and coworkers was an anti-Arrhenius increase in the exchange rate when the temperature was decreased to the point of solvent crystallization. Below the crystallization temperature, the rate stayed constant. Because only nuclear solvent reorganization is expected to depend on temperature, the increase in the rate required a decrease of in the reorganization energy with cooling when analyzed in the standard models of mixed-valence electron transfer.22 The measured rates of self-exchange are in the ∼1 ps−1 range.22 This fast rate cuts off a significant portion of the polarization relaxation spectrum, and the reaction is nonergodic. Equation 2, instead of eq 1, should therefore be used to interpret the experimental data. Within the nonergodic formalism of electron transfer reactions, the solvent reorganization energy is not a thermodynamic free energy but, instead, is a statistical average depending on the reaction rate, k. The nonergodic solvent reorganization energy λs(k) is calculated by integrating the spectrum of the energy gap fluctuations over the frequencies exceeding the rate constant25−27
λ s (k ) ∝
∫k
∞
χX″ (ω)(dω/ω)
(3)
Here χ″X(ω) is the loss function calculated from the correlation function of the energy gap X between the donor and acceptor energy levels, known in spectroscopy as the Stokes-shift correlation function.28 The Stokes-shift loss function has a generic form, involving two relaxation peaks, for many molecular liquids. The long-time peak corresponds to collective relaxation of the dipolar polarization, mostly linked to the longitudinal polarization relaxation of the homogeneous solvent.29 The short-time peak is caused by single-particle ballistic dynamics of the solvent molecules closest to the donor−acceptor complex.30 To illustrate the main conceptual ideas of the problem at hand, Figure 1 shows the loss spectrum of the Stokes-shift
Figure 1. Loss function of the Stokes-shift dynamics (arbitrary units) of p-nitroaniline in SPC/E water from MD simulations at two temperatures indicated in the plot.25,26 The vertical dashed line illustrates the frequency cutoff used to calculate the nonergodic reorganization energy in eq 3. The inset shows the solvent reorganization energy calculated from the simulation trajectories on the fixed 1 ns observation window. The reorganization energy is ergodic (thermodynamic) at high temperatures and increases with lowering temperature, as expected from liquid-state theories of solvation.17,18 It turns to a nonergodic reorganization energy at the crossover (glass-transition) temperature shown by the vertical dashed line in the inset. The decrease in the reorganization energy in the nonergodic regime is typically more dramatic than its increase in the ergodic regime. The solvent remains in its liquid state in the entire range of temperatures studied by simulations.
dynamics of the charge-transfer transition in p-nitroaniline in SPC/E water25,26 obtained from numerical simulations. As the temperature is lowered, the relaxation time increases and the peaks shift to lower frequencies. The area under the highfrequency portion of χX″ (ω)/ω in eq 3 (note the ω−1 scaling) shrinks, and the reorganization energy decreases (inset in Figure 1). This temperature dependence is opposite to an increase in the thermodynamic (ergodic) reorganization energy with lowering temperature in the range of high temperatures where ergodicity is maintained. There is therefore a crossover from ergodic to nonergodic behavior at which the reorganization energy reaches its peak value. This peak traces other properties calculated as second cumulants of fluctuating physical properties,26 such as the heat capacity, at the point of ergodicity breaking, that is, the glass transition.31 The drop of the reorganization energy with cooling on the nonergodic side of temperatures is typically more dramatic than the corresponding increase in the thermodynamic, ergodic reorganization energy.32,33 The goal of this Letter is to apply these concepts to chargetransfer kinetics reported by Kubiak’s group. In contrast with a constant 1 ns observation window set up in the molecular 1645
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can be obtained in this case by diagonalizing the two-state Hamiltonian, with the activation energy given as23,38 ΔG† = (λ − 2Hab)2/(4λ). When expanded, this equation results in three terms, the last one depending nonlinearly on the reorganization energy. To simplify the problem, we will neglect the dependence on temperature, arising from λs, in that last term and combine the second and the third terms together into H̃ ab. This gives the following form for the activation barrier
dynamics (MD) simulations reported in Figure 1, the observation window of in the kinetic experiments is set up by the rate, which needs to be calculated by solving the selfconsistent equation (eq 2). Here the main nontraditional result reported by experiment, the anti-Arrhenius temperature dependence, follows from the critical slowing down of the collective relaxation of the solvent on approaching the point of crystallization. When the Stokes-shift dynamics is represented by fast (often Gaussian30) decay followed by an exponential collective relaxation, the dependence of the nonergodic solvent reorganization energy on the reaction rate becomes λs(k) = λG + λp(2/π ) arccot[kτp(T )]
ΔG†(k) =
(6)
The rate constant of electron self-exchange becomes k = ωne−βEae−βλ p(k)/4
(4)
The first term in this equation, λG, is caused by ballistic, singlemolecule relaxation assumed to be always faster than the reaction rate. The second term describes polarization (mostly longitudinal29) relaxation of the solvent producing the reorganization energy component λp. The temperature dependence of the collective relaxation time τp(T) affects the magnitude of the nonergodic reorganization energy. The thermodynamic limit is reached when the reaction is much slower than the medium relaxation, kτp(T) ≪ 1, and the thermodynamic (ergodic) solvent reorganization energy becomes λs = λG + λp. The total reorganization energy requires adding the intramolecular component λin to λs, with the result λ = λs + λin. The temperature dependence of the collective relaxation time τp(T) on approach to solvent crystallization is mostly unknown. However, a plausible form of the temperature dependence can be deduced from the Maxwell relation connecting the collective α-relaxation time of the solvent τα(T) to its viscosity η(T): τα(T) = η(T)/G∞. The hightemperature shear modulus G∞ (≃ 2 GPa for organic liquids) does not change significantly with temperature,34 and the temperature change of the relaxation time mostly traces that of viscosity. Because η(T)→∞ on approach to the point of crystallization Tm, τα(T) and τp(T) should diverge as well. Scaling laws are expected to apply to the relaxation time approaching the point of phase transition35 τp(T ) = a(T − Tm)−γ
1 ̃ (λ in + λs(k)) − Hab 4
(7)
In eq 7, the activation energy Ea incorporates, apart from the intramolecular reorganization energy, the Gaussian component λG of the solvent reorganization energy. Because the oneparticle inertial dynamics is likely to remain in the crystalline state, the rate of electron transfer will be affected by the nature of the solvent even below the crystallization temperature, as is indeed observed.21 However, the overall temperature dependence of the rate in the crystal was found to be low, which implies that Ea is likely small. One can therefore neglect the temperature dependence arising from this term and recast the rate constant in the form k = kcryst e−βλ p(k)/4
(8)
where kcryst refers to the rate in the crystalline state of the solvent and λp(k) = λp
2 arccot[kτp] π
(9)
Figure 2 shows the results of applying eqs 8 and 9 to experimental data collected for two mixed-valence complexes in acetonitrile.21,22 These two complexes differ by the substitution of the pyridyl ligand L in trinuclear ruthenium dimer
(5)
The proportionality constant a in this equation is estimated below from the time τp(295 K) = 0.63 ps obtained from the Stokes-shift dynamics of coumarin-153 in acetonitrile, as reported by Maroncelli and coworkers;28 Tm = 229 K for acetonitrile. The power law in eq 5 is consistent with the literature data on rotational relaxation of liquid acetonitrile. The fit of rotational relaxation times of acetonitrile presented at several temperatures in ref 36 produces the exponent γ = 0.32. However, acetonitrile takes two crystalline forms, a monoclinic α-type and an orthorhombic β-type.37 The monoclinic lattice is transformed into the orthorhombic lattice just below the melting temperature, at 217 K, and it is not entirely clear which form is stabilized when the solution with donor−acceptor complexes is formed. We now proceed to defining the rate of self-exchange. The mixed-valence dimers studied by Kubiak and coworkers have a significant extent of electronic coupling, placing them near the borderline between localized and delocalized electronic states.23,38 The free-energy profile along the electron-transfer reaction coordinate and the corresponding activation barrier
Figure 2. Experimental self-exchange rates for mixed-valence complexes 2 (squares) and 3 (circles) studied in ref 21. The solvent is acetonitrile and γ = 0.55 was used in eq 5. The theoretical solid lines are produced with λp = 0.115 eV for complex 3 and 0.042 eV for complex 2. The vertical dashed line indicates the melting temperature Tm = 229 K of solution in acetonitrile.21 The top panel shows λp(T) from eq 9 with the rate constant k calculated at each temperature by solving the self-consistent eq 8. 1646
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{[Ru3O(OAc)6(CO)(L)]2BL}−. The change of the ligand affects the extent of electronic delocalization such that complex 2 is more delocalized than complex 3.22 Consistent with this assignment, experimental rates for complex 2 are reproduced using smaller λp = 0.042 eV, compared with λp = 0.115 eV used for complex 3. In both cases, the model gives a correct account of the observed shape of the temperature dependence of the rate. The rate constant gains anti-Arrhenius dependence on temperature because of the shrinking phase space accessible to the system with the lowering temperature, as reflected by λp(T) shown in the upper panel in Figure 2. This general issue does not typically cause much trouble when the phase space available to the system does not change much within the range of parameters sampled by experiment. In such cases, one can still use the standard prescriptions of canonical statistical mechanics applied to the available phase space, with the warning that the statistical averages lose their thermodynamic meaning. The problem becomes more acute when the relaxation time of the bath changes substantially within the space of parameters sampled by experiment. The alteration of the available phase space is projected on the statistical averages, and that brings about the dependence of the activation barrier on the medium dynamics. The relaxation time of the medium changes significantly in a narrow range of temperatures when fragile glass-formers approach their glass transition.31 The result is an abrupt change of the activation parameters of redox reactions (Figure 1).27,33 Another area of potential interest is phase transformations of the bath. 11 Phase transitions are often characterized by critical slowing down, and that dynamical effect alters the activation barrier by limiting the phase space available to the system. The mixed-valence electron selfexchange studied here provides a vivid example of this mechanism. The power-law divergence of the relaxation time on approach to the crystallization temperature projects itself into an anti-Arrhenius temperature dependence of the rate. More detailed calculations will be required to get a better hold of the parameters involved, but a simple model presented here successfully represents this physical reality.
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AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected]. Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS This research was supported by the National Science Foundation (CHE-1213288). The author is grateful to Marshall Newton for useful comments on the manuscript.
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REFERENCES
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