Reorganization energies and rate constants for electron reactions in glass-forming media and proteins

Photogenerated Spin-Correlated Radical Pairs: From Photosynthetic Energy Transduction to Quantum Information Science

More than a half century ago, the NMR spectra of diamagnetic products resulting from radical pair reactions were observed to have strongly enhanced absorptive and emissive resonances. At the same time, photogenerated radical pairs were discovered to exhibit unusual electron paramagnetic resonance spectra that also had such resonances. These non-Boltzmann, spin-polarized spectra were observed in both chemical systems as well as in photosynthetic reaction center proteins following photodriven charge separation. Subsequent studies of these phenomena led to a variety of chemical electron donor-acceptor model systems that provided a broad understanding of the spin dynamics responsible for these spectra. When the distance between the two radicals is restricted, these observations result from the formation of spin-correlated radical pairs (SCRPs) in which the spin-spin exchange and dipolar interactions between the two unpaired spins play an important role in the spin dynamics. Early on, it was recognized that SCRPs photogenerated by ultrafast electron transfer are entangled spin pairs created in a well-defined spin state. These SCRPs can serve as spin qubit pairs (SQPs), whose spin dynamics can be manipulated to study a wide variety of quantum phenomena intrinsic to the field of quantum information science. This Perspective highlights the role of SCRPs as SQPs, gives examples of possible quantum manipulations using SQPs, and provides some thoughts on future directions.

Quantum coherence in ultrafast photo-driven charge separation

Coherent interactions are prevalent in photodriven processes, ranging from photosynthetic energy transfer to superexchange-mediated electron transfer, resulting in numerous studies aimed towards identifying and understanding these interactions. A key motivator of this interest is the non-statistical scaling laws that result from coherently traversing multiple pathways due to quantum interference. To that end, we employed ultrafast transient absorption spectroscopy to measure electron transfer in two donor-acceptor molecular systems comprising a p-(9-anthryl)-N,N-dimethylaniline chromophore/electron donor and either one or two equivalent naphthalene-1,8:4,5-bis(dicarboximide) electron acceptors at both ambient and cryogenic temperatures. The two-acceptor compound shows a statistical factor of 2.1 ± 0.2 rate enhancement at room temperature and a non-statistical factor of 2.6 ± 0.2 rate enhancement at cryogenic temperatures, suggesting correlated interactions between the two acceptors with the donor and with the bath modes. Comparing the charge recombination rates indicates that the electron is delocalized over both acceptors at low temperature but localized on a single acceptor at room temperature. These results highlight the importance of shielding the system from bath fluctuations to preserve and ultimately exploit the coherent interactions.

Fractional Viscosity Dependence of Reaction Kinetics in Glass-Forming Liquids

The diffusion of molecules in complex systems such as glasses and cell cytoplasm is slow, heterogeneous, and sometimes nonergodic. The effects of such intriguing diffusion on the kinetics of chemical and biological reactions remain elusive. In this Letter, we report that the kinetics of the polymer loop formation reaction in a Kob-Andersen (KA) glass forming liquid is influenced significantly by the dynamic heterogeneity. The diffusion coefficient D of a KA liquid deviates from the Stokes-Einstein relation at low temperatures and D shows a fractional dependence on the solvent viscosity η_{s}, i.e., D∼η_{s}^{-ξ_{D}} with ξ_{D}=0.85. The dynamic heterogeneity of a KA liquid affects the rate constant k_{rxn} of the loop formation and leads to the identical fractional dependence of k_{rxn} on η_{s} with k_{rxn}∼η_{s}^{-ξ} and ξ=ξ_{D}, contrary to reactions in dynamically homogeneous solutions where k_{rxn}∼η_{s}^{-1}.

Microscopic Simulations of Charge Transport in Disordered Organic Semiconductors

Charge carrier dynamics in an organic semiconductor can often be described in terms of charge hopping between localized states. The hopping rates depend on electronic coupling elements, reorganization energies, and driving forces, which vary as a function of position and orientation of the molecules. The exact evaluation of these contributions in a molecular assembly is computationally prohibitive. Various, often semiempirical, approximations are employed instead. In this work, we review some of these approaches and introduce a software toolkit which implements them. The purpose of the toolkit is to simplify the workflow for charge transport simulations, provide a uniform error control for the methods and a flexible platform for their development, and eventually allow in silico prescreening of organic semiconductors for specific applications. All implemented methods are illustrated by studying charge transport in amorphous films of tris-(8-hydroxyquinoline)aluminum, a common organic semiconductor.

Closed-form expressions of quantum electron transfer rate based on the stationary-phase approximation

Closed-form rate expressions are derived on the basis of the stationary-phase approximation for the Fermi golden rule expression of the quantum electron-transfer (ET) rate. First, on the basis of approximate solutions of the stationary-phase points near DeltaG = 0, -lambda, and lambda, where DeltaG is the reaction free energy and lambda is the reorganization energy, three closed-form rate expressions are derived, which are respectively valid near each value of DeltaG. Numerical tests for a model Ohmic spectral density with an exponential cutoff demonstrate good performance of the derived expressions in the respective regions of their validity. In particular, the expression near DeltaG = -lambda, which differs from the semiclassical approximation only by a prefactor quadratic in DeltaG, works substantially better than the latter. Then, a unified formula is suggested, which interpolates the three approximate expressions and serves as a good approximation in all three regions. We have also demonstrated that the interpolation formula can serve as a good quantitative means for understanding the temperature dependence of the quantum ET rate.

Energetics of electron-transfer reactions in soft condensed media

The coupling of electronic transitions within molecules to condensed-phase media involves a complex hierarchy of spatial and dynamical scales. Thermodynamics of activation is related to the length scale of microscopic interactions reflected in the non-Arrhenius reaction kinetics. Solvent dynamics make a particularly strong impact on the activation barrier when the time scale of the reaction is comparable to the relaxation time of the solvent, and the reaction barrier becomes nonergodic. Finally, molecular polarizability is responsible for complex nonparabolic free energy surfaces for electron transfer. We discuss the application of these ideas to soft condensed solvents such as supercooled liquids, liquid crystals, and photosynthetic reaction centers.

Dynamical Arrest of Electron Transfer in Liquid Crystalline Solvents

We argue that electron transfer reactions in slowly relaxing solvents proceed in the nonergodic regime, making the reaction activation barrier strongly dependent on the solvent dynamics. For typical dielectric relaxation times of polar nematics, electron transfer reactions in the subnanosecond time scale fall into nonergodic regime in which nuclear solvation energies entering the activation barrier are significantly lower than their thermodynamic values. The transition from isotropic to nematic phase results in weak discontinuities of the solvation energies at the transition point and the appearance of solvation anisotropy weakening with increasing solute size. The theory is applied to analyze experimental kinetic data for the electron transfer kinetics in the isotropic phase of 5CB liquid crystalline solvent. We predict that the energy gap law of electron transfer reactions in slowly relaxing solvents is characterized by regions of fast change of the rate at points where the reaction switches between the ergodic and nonergodic regimes. The dependence of the rate on the donor-acceptor separation may also be affected in a way of producing low values for the exponential falloff parameter.

Solvent reorganization of electron transitions in viscous solvents

We develop a model of electron transfer reactions at conditions of nonergodicity when the time of solvent relaxation crosses the observation time window set up by the reaction rate. Solvent reorganization energy of intramolecular electron transfer in a charge-transfer molecule dissolved in water and acetonitrile is studied by molecular dynamics simulations at varying temperatures. We observe a sharp decrease of the reorganization energy at a temperature identified as the temperature of structural arrest due to cage effect, as discussed by the mode-coupling theory. This temperature also marks the onset of the enhancement of translational diffusion relative to rotational relaxation signaling the breakdown of the Stokes-Einstein relation. The change in the reorganization energy at the transition temperature reflects the dynamical arrest of the slow, collective relaxation of the solvent related to the relaxation of the solvent dipolar polarization. An analytical theory proposed to describe this effect agrees well with both the simulations and experimental Stokes shift data. The theory is applied to the analysis of charge-transfer kinetics in a low-temperature glass former. We show that the reorganization energy is substantially lower than its equilibrium value for the low-temperature portion of the data. The theory predicts the possibility of discontinuous changes in the dependence of the electron transfer rate on the free energy gap when the reaction switches between ergodic and nonergodic regimes.

Dynamical Arrest of Electron Transfer Reorganization in Super-Cooled Water

This paper reports the results of Molecular Dynamics (MD) simulations of the solvent reorganization energy of electron transfer (ET) reactions in low-temperature solvents. Simulations are carried out for a model charge-transfer optical dye (p-nitroaniline) in SPC/E water in a range of temperatures down to the point of solvent ideal glass transition. We show a significant departure of the solvent reorganization energy, measured on a given time window, from its thermodynamic limit obtained by averaging over long simulation trajectories. Our results thus indicate that optical solvatochromism and activation parameters of ET reactions measured in viscous solvents will be significantly affected by the dynamical arrest of nuclear solvation when the experimental time scale becomes comparable to the characteristic relaxation time of the solvent.

A phenomenological model of dynamical arrest of electron transfer in solvents in the glass-transition region

A phenomenological model of electron transfer reactions in solvents undergoing glass transition is discussed. The reaction constant cuts off slow polarization modes from the spectrum of nuclear thermal motions active on the observation time scale. The arrest of nuclear solvation in turn affects the reaction activation barrier making it dependent on the rate. The resultant rate constant is sought from a self-consistent equation. The model describes well the sharp change in the solvent Stokes shift of optical lines in the glass-transition region. It is also applied to describe the temperature dependence of primary charge separation and reduction of primary pair in photosynthetic reaction centers. The model shows that a weak dependence of the primary charge separation rate on temperature can be explained by dynamical arrest of nuclear solvation on the picosecond time scale of electron transfer. For reduction of primary pair by cytochrome, the model yields a sharp turnover of the reaction kinetics at the transition temperature when nuclear solvation freezes in.

Electron transfer between cytochrome c and cytochome c peroxidase in single crystals

Cytochrome c (Cc) and cytochrome c peroxidase (CcP) form an important redox pair for understanding interprotein electron transfer (ET). Measurements of ET rates from photoexcited CcP substituted with Zn porphyrin to either yeast Fe(III)Cc or horse Fe(III)Cc in crystals reveal that the molecular associations found in the respective crystal structures determine solution reactivity. Similar forward rates for yeast isozyme-1 Cc (yCc) and yCc homologue horse Cc (hCc), despite different orientations relative to CcP, suggest small-amplitude conformational gating of ET even in the crystalline state; faster back ET in the yCc compared to the hCc complex agrees with the relative coupling between redox sites predicted by the structures.

Interprotein electron transfer in a confined space: uncoupling protein dynamics from electron transfer by sol-gel encapsulation

In this paper, we describe the first observations of photoinitiated interprotein electron transfer (ET) within sol-gels. We have encapsulated three protein-protein complexes, specifically selected because they represent a full range of affinities, are sensitive to different types of dynamic processes, and thus are expected to respond differently to sol-gel encapsulation. The three systems are (i) the [Zn, Fe(3+)L] mixed-metal hemoglobin hybrids, where the alpha(1)-Zn and beta(2)-Fe subunits correspond to a "predocked" protein-protein complex with a crystallographically defined interface (Natan, M. J.; Baxter, W. W.; Kuila, D.; Gingrich, D. J.; Martin, G. S.; Hoffman, B. M. Adv. Chem. Ser. 1991, 228 (Electron-Transfer Inorg., Org., Biol. Syst.), 201-213), (ii) the Zn-cytochrome c peroxidase complex with cytochrome c, [ZnCcP, Fe(3+)Cc], having an intermediate affinity between its partners (Nocek, J. M.; Zhou, J. S.; De Forest, S.; Priyadarshy, S.; Beratan, D. N.; Onuchic, J. N.; Hoffman, B. M. Chem. Rev. 1996, 96, 2459-2489), and (iii) the [Zn-deuteromyoglobin, ferricytochrome b(5)] complex, [ZnDMb, Fe(3+)b(5)], which is loosely bound and highly dynamic (Liang, Z.-X.; Nocek, J.; Huang, K.; Hayes, R. T.; Kurnikov, I. V.; Beratan, D. N.; Hoffman, B. M. J. Am. Chem. Soc. 2002, 124, 6849-6859. Intersubunit ET within the hybrid does not involve second-order processes or subunit rearrangements, and thus is influenced only by perturbations of high-frequency motions coupled to ET. For the latter two complexes, sol-gel encapsulation eliminates second-order processes: protein partners encapsulated as a complex must stay together throughout a photoinitiated ET cycle, while proteins encapsulated alone cannot acquire a partner. It further modulates intracomplex motions of the two partners.

Electron tunneling in single crystals of Pseudomonas aeruginosa azurins

Rates of reduction of Os(III), Ru(III), and Re(I) by Cu(I) in His83-modified Pseudomonas aeruginosa azurins (M-Cu distance approximately 17 A) have been measured in single crystals, where protein conformation and surface solvation are precisely defined by high-resolution X-ray structure determinations: 1.7(8) x 10(6) s(-1) (298 K), 1.8(8) x 10(6) s(-1) (140 K), [Ru(bpy)2(im)(3+)-]; 3.0(15) x 10(6) s(-1) (298 K), [Ru(tpy)(bpy)(3+)-]; 3.0(15) x 10(6) s(-1) (298 K), [Ru(tpy)(phen)(3+)-]; 9.0(50) x 10(2) s(-1) (298 K), [Os(bpy)2(im)(3+)-]; 4.4(20) x 10(6) s(-1) (298 K), [Re(CO)3(phen)(+)] (bpy = 2,2'-bipyridine; im = imidazole; tpy = 2,2':6',2' '-terpyridine; phen = 1,10-phenanthroline). The time constants for electron tunneling in crystals are roughly the same as those measured in solution, indicating very similar protein structures in the two states. High-resolution structures of the oxidized (1.5 A) and reduced (1.4 A) states of Ru(II)(tpy)(phen)(His83)Az establish that very small changes in copper coordination accompany reduction but reveal a shorter axial interaction between copper and the Gly45 peptide carbonyl oxygen [2.6 A for Cu(II)] than had been recognized previously. Although Ru(bpy)2(im)(His83)Az is less solvated in the crystal, the reorganization energy for Cu(I) --> Ru(III) electron transfer falls in the range (0.6-0.8 eV) determined experimentally for the reaction in solution. Our work suggests that outer-sphere protein reorganization is the dominant activation component required for electron tunneling.

Conformational gating of long distance electron transfer through wire-like bridges in donor-bridge-acceptor molecules

A series of five donor-bridge-acceptor (DBA) molecules in which the donor is tetracene, the acceptor is pyromellitimide, and the bridge molecules are oligo-p-phenylenevinylenes (OPV) of increasing length has been shown to undergo electron transfer (ET) by means of two mechanisms. When the bridge is short, strongly distance dependent superexchange dynamics dominates, whereas when the bridge is longer, bridge-assisted hopping dynamics prevails. The latter mechanism results in relatively soft distance dependence for ET in which the OPV oligomers act effectively as molecular wires. We now report studies on the critical influence that bridge dynamics have on electron transfer through these oligomers. The temperature dependence of the charge separation (CS) rates in all five molecules does not appear to obey the predictions of standard ET theories based upon the Condon approximation. All five molecules show behavior consistent with CS being "gated" by torsional motion between the tetracene donor and the first bridge phenyl ring. This is based on the near equivalence of the CS activation energies measured for all five molecules with the frequency of a known vibrational mode in 5-phenyltetracene. In the molecule containing a trans-stilbene bridge, a competition occurs between the tetracene-phenyl torsional motion and one that occurs between the vinyl group and the phenyls linked to it. This results in complex temperature-dependent CS that exhibits both activated and negatively activated regimes. The charge recombination (CR) reactions within the molecules which have the two shortest bridges, namely phenyl and trans-stilbene, show a weaker dependence on these molecular motions. The three molecules with the longest bridges all display complex temperature dependencies in both their rates of CS and CR, most likely because of the complex torsional motions, which arise from the multiple phenyl-vinyl linkages. The data show that long-distance electron transfer and therefore wire-like behavior within conjugated bridge molecules depend critically on these low-frequency torsional motions. Molecular device designs that utilize such bridges will need to address these issues.

Intramolecular electron transfer between [4Fe-4S] clusters studied by proton magnetic resonance spectroscopy

The rate constants for the intramolecular electron transfer between the two [4Fe-4S] clusters of a series of native and genetically engineered ferredoxins have been determined by proton magnetic resonance (1H NMR) spectroscopy. The measurement relies on the properties of the signals assigned to beta-protons of the coordinating cysteines when the protein is substoichiometrically reduced: these signals include coalesced peaks arising from the fast hopping of an extra electron between the two oxidized clusters of the protein. An upper limit of significantly less than 10(5) M(-1) s(-1) for the intermolecular and an average of the order of 5 x 10(6) s(-1) for the intramolecular electron transfer rate constants of several ferredoxins have been obtained. Owing to the edge-to-edge intercluster distance of approximately 10 A derived from the crystallographic structure of Clostridium acidurici ferredoxin, the rate constant associated with the intramolecular process is as expected for a nonadiabatic redox process, assuming a reasonable value of less than 1 eV for the reorganization energy. The latter could not be determined from the temperature dependence of the rate constant since no variation was observed over the temperature range accessible in these experiments. Structural changes introduced around and between the two [4Fe-4S] clusters in Clostridium pasteurianum ferredoxin by site-directed mutagenesis have been used to probe the potential involvement of dominant electron transfer pathways between the clusters. These changes have no major effect on the value of the intramolecular electron transfer rate constant. From this analysis, no specific amino acid side chain seems to play a central role in the process. The rate constants derived in the present work may serve as a basis for the study of enzymes containing two closely spaced [4Fe-4S] clusters such as found in these ferredoxins.