Marcus Formula Research Articles

Non-local and overscreening effects in the kinetics of heterogeneous charge transfer reactions

The effects of solvent spatial dispersion including the overscreening effect and field penetration into a metal on the kinetic parameters of heterogeneous charge-transfer reactions are studied. The calculations are based on the exactly solvable ‘sharp-boundary model’ of the interface. The analytical expressions derived using the mean-field theory for the phenomenological Hamiltonian of the liquid are adopted for the dielectric function of the solvent. The parameters entering these expressions are chosen to reproduce the results of molecular dynamics computer simulations presented in the literature. We consider the (Cp) 2Co +/(Cp) 2Co electrochemical reaction and acetonitrile (AN) and dimethyl sulfoxide (DMSO) as the solvents. The smeared Born sphere (SBS) model of an ion is used for calculations of the reorganization Gibbs energies at infinite electrode–reactant separation. The extent of the smearing of the ion excess charge is fitted to obtain the electrostatic part of the solvation Gibbs energy of an ion in the bulk of the solvent. The effective Born sphere (EBS) and renormalized smeared Born sphere (RSBS) models of an ion are suggested for calculations of the reorganization and activation Gibbs energies near an electrode. It is shown that the reorganization Gibbs energy and the activation Gibbs energy decrease with decrease of the electrode–reactant separation in accordance with Marcus theory (in which the non-local effects are neglected). However, the reorganization and activation Gibbs energies are found to be larger than those given by the Marcus formula and agree rather well with the estimates obtained from the experimental data.

Non-local effects in the kinetics of heterogeneous charge transfer reactions

Abstract The effect of the solvent spatial dispersion and the field penetration into a metal on the kinetic parameters of the heterogeneous charge-transfer reactions is studied. The calculations are based on the exactly solvable ‘sharp boundary model’ of the interface, the Born sphere model of an ion and the Lorentzian form for the dielectric function of the solvent. It is shown that the reorganization Gibbs energy and the activation Gibbs energy decrease with decrease of the electrode–reactant separation in accordance with Marcus theory (in which the non-local effects are neglected) and in contrast with the results of Dzhavakhidze, Kornyshev and Krishtalik [J. Electroanal. Chem. 228 (1987) 329]. The reorganization Gibbs energy is found to be smaller than that given by the Marcus formula. The effect of the overscreening phenomenon on the results obtained is discussed.

Estimation of Self-Exchange Electron Transfer Rate Constants for Organic Compounds from Stopped-Flow Studies

Second-order rate constants k12(obsd) measured at 25 °C in acetonitrile by stopped-flow for 47 electron transfer (ET) reactions among ten tetraalkylhydrazines, four ferrocene derivatives, and three p-phenylenediamine derivatives are discussed. Marcus's adiabatic cross rate formula k12(calcd) = (k11 k22 k12 f12)1/2, ln f12 = (ln K12)2/4 ln(k11k22/Z2) works well to correlate these data. When all k12(obsd) values are simultaneously fitted to this relationship, best-fit self-exchange rate constants, kii(fit), are obtained that allow remarkably accurate calculation of k12(obsd); k12(obsd)/k12‘(calcd) is in the range of 0.55−1.94 for all 47 reactions. The average ΔΔGij⧧ between observed activation free energy and that calculated using kii(fit) is 0.13 kcal/mol. Simulations using Jortner vibronic coupling theory to calculate k12 using parameters which produce the wide range of kii values observed predict that Marcus's formula should be followed even when V is as low as 0.1 kcal/mol, in the weakly nonadiabatic region. Tetracyclohexylhydrazine has a higher kii than tetraisopropylhydrazine by a factor of ca. 10. Replacing the dimethylamino groups of tetramethyl-p-phenylenediamine by 9-azabicyclo[3.3.1]nonyl groups has little effect on kii, demonstrating that conformations which have high intermolecular aromatic ring overlap are not necessary for large ET rate constants. Replacing a γ CH2 group of a 9-azabicyclo[3.3.1]nonyl group by a carbonyl group lowers kii by a factor of 17 for the doubly substituted hydrazine and by considerably less for the doubly substituted p-phenylenediamine.

Self-organized electron transfer

As other energy fluxes can be largely enhanced under non-linear conditions which permit local entropy export, such as heat flow under the ordered Benard convection, this should also be possible on a microscopic scale for electron transfer. As a precondition an autocatalytic molecular loop must exist, which transfers a small fraction of the energy, which drives the reaction, through the surrounding medium to assist the electron transfer in a non-linear way. Such an autocatalytic side loop, which may be facilitated by transfer of phonon energy, leads to microscopic self-organization, ie to a temporary dissipative structure. It may significantly improve electron transfer through a non-equilibrium electron distribution and a decrease of the activation barrier. The problem is theoretically treated as a Kramers approach, a Brownian motion in a non-uniform force field, in which the friction term may become active ( eg through phonon assisted autocatalytic energy input leading to a dissipative activated complex). A formula for “self-organized” electron transfer is obtained which yields—for finite activation barriers—significantly higher rates than the Marcus formula. The derived new type of electron transfer mechanism is analyzed. It may help to explain how biological catalysts have evolved the ability of generating favourable energy barriers for multi-electron transfer near the thermodynamic overall potential

Nonlocal dielectric response of water and reorganization energies for outer sphere electron transfer reactions

New results on the nonlocal dielectric function of liquid water (Ph. A. Bopp, A. A. Kornyshev, G. Sutmann, Phys. Rev. Let. 76, 1280 (1996)) are discussed and used to calculate reorganization free energies for electron exchange reactions between solute ions. The energies are shown to be several times larger than those given by the classical Marcus formula for the born sphere model of an ion. However, they are reduced to Marcus or even lower values for modified charge density models, which take into account the delocalization of the ion excess charge distribution.

Self‐organized electron transfer as a possible element in the photosynthetic process

AbstractAs energy fluxes in several systems far from thermodynamic equilibrium can be largely enhanced under nonlinear conditions permitting local entropy export, this should also be possible on a more microscopic scale for biological electron transfer processes. As a precondition a molecular feedback loop must exist, which transfers a small fraction of the energy, which drives the reaction, through the surrounding medium to assist the electron transfer in a non‐linear way. This can lead to microscopic self‐organization, i. e. to a temporary dissipative structure which may significantly improve electron transfer through a non‐equilibrium electron distribution and a decrease of the activation barrier. The problem is theoretically modelled as a Kramers approach in which the friction term may become active. A formula for “self‐organized” electron transfer is obtained which yields — for finite activation barriers — significantly higher rates than the Marcus formula. The presented model may help to explain how biological catalysts have evolved the ability of generating favourable energy barriers for multi‐electron transfer near the thermodynamic overall potential.

How accurate is the Rice–Ramsperger–Kassel–Marcus theory? The case of H+3

The classical Rice–Ramsperger–Kassel–Marcus formula is tested at an accuracy level of a few percent by comparing results of numerical phase space integration with lifetimes deduced from trajectory calculations. The test object is HD+2; the calculation has been done for total energies of 0.5, 1.0, and 1.5 eV above dissociation, and for total angular momenta of 0–60ℏ. Presupposing that the trajectory calculations show the true classical dynamics, we find systematic deviations of up to 40% of the RRKM results. They can be fully explained by the influence of ‘‘direct trajectories,’’ a special kind of nonergodic behavior of the system. After correction for this phenomenon, both methods agree to within the accuracy of the calculations, which is about 3%. We also verified that the discrepancy vanishes when the energy approaches the dissociation energy.

On the role of medium electrons in the outer-sphere electron transfer reactions

We consider the nonlinear Schrödinger equation arising when an explicit separation of the medium electron polarization is performed in the Hamiltonian describing the outer-sphere electron transfer (ET) reaction. The corresponding classical activation energy is determined by the point of intersection of two diabatic energy curves rather than by the stationary saddle point of the free energy functional, in agreement with the earlier treatment of Kim and Hynes. Analytical expressions for the free energy terms as functions of the ET coordinate are derived. The relative energy of this cross point is the same as the traditional activation energy of the Et process as given by the Marcus formula. In order to find the interaction matrix elements which promote the nonadiabatic transition and avoided crossing, an alternative quantum-mechanical description of the two-level ET model is given in terms of the wavefunction which allows independent reorganization of electrons belonging to the medium and to the chemical substrate. Explicit expression for matrix elements are presented and different kinetic regimes generated by this refined model are discussed.

Linearized microscopic theories of nonequilibrium solvation

Linearized microscopic theories of equilibrium solvation can be straightforwardly extended to treat the dynamic situation. A particular example, the solvation of a newly formed ion, is developed within the mean spherical approximation. The MSA exhibits significant deviations from the Born–Marcus formula for nonequilibrium solvation energies. The solvation dynamics proceeds on several time scales. The solvation structure first forms distant from the ion, in harmony with a remark of Onsager and with previous numerical calculations of Calef and Wolynes [J. Chem. Phys. 78, 4145 (1983)].

Intramolecular conversions over low barriers. VII. The aziridine inversion—Intrinsically non-RRKM

The rate of bond inversion at the nitrogen atom in aziridine (dimethylene-imine) was measured in the gas phase for a range of pressures and temperatures. 1H NMR spectra were recorded of samples at 5–1100 Torr, and from 298 to 388 K. Rate constants for inversion were derived by complete line shape analysis of the NMR spectra. Their temperature dependence at the high pressure limit (1100 Torr) gave Ea =15.8±0.4 kcal mol−1; ΔH‡ =15.2±0.4 kcal mol−1; ΔS‡=−5.5±3 eu; and ΔG‡298=16.8±0.5 kcal mol−1. These values are in excellent agreement with previously reported mangitudes obtained for solutions and in the gas phase at high pressures. However, at all temperatures we found no dependence of the first order rate constants on pressure down to 5 Torr, in sharp contrast to the fall-off curves calculated with standard RRKM codes, which indicated that for pressures below 100 Torr this system should be in the bimolecular regime. A less extended set of measurements was made with 2-methyl-aziridine which does show a small but clear fall off in kuni. These results can be rationalized in terms of the regional phase space model, previously proposed for methylnitrite. In systems such as aziridine, where the density of states at the barrier height is sparse (≈10/cm−1) the rate of inversion is limited by the relaxation time for intramolecular vibrational energy redistribution. Our results indicate that many nanoseconds are required to attain relaxation rather than picoseconds, estimated from the Marcus formula for k23(E)=N‡(E+)/hρ(E). In the methyl-substituted analog the barrier height is approximately 3 kcal mol−1 higher, and the density of states at this barrier is higher than in aziridine. Relaxation data for conformational changes in C6H12, H3CONO, SF4, and C5H10O were reanalyzed in terms of the regional phase-space model. At low pressures these also show departures from RRKM predictions.

Theory of electrochemical hydrogen evolution reactions Part II. Model of harmonic proton terms of different frequencies

The theory of simultaneous electron and proton transfer in polar media developed in Part I [1] is extended to the case of harmonic proton terms with different frequencies in the initial, ω 1 , and final, ω 2 , states. The dependence of the transfer probability w on the free energy of the reaction Δ G is characterized by the transfer coefficient α, which decreases from unity to zero as Δ G decreases, i.e. the overvoltage increases. The relations derived in ref. 1 are shown to be unconnected with the special choice of the terms. The dependence of w on Δ G that is obtained reduces to the relations derived in ref. 1 in a certain range of Δ G ; in another range of Δ G it reduces to the generalized Marcus formula with a pre-exponential factor weighted by the Frank-Condon factor of the proton transfer. The latter is valid for the limited range of Δ G lying within the normal discharge region. Outside this region the relations derived in ref. 1 are valid. The results of the theoretical calculations for ω 2 /ω 1 =0.25–0.30 and h ω 2 / kT =3–4 are in good agreement with the experimental data on hydrogen ion discharge on a mercury electrode. The relations derived in ref. 1 are shown to be valid in three discharge regions with Δ G also varying over 150 kJ mol −1 .

Theory of electrochemical hydrogen evolution reactions: Part II. Model of harmonic proton terms of different frequencies

The theory of simultaneous electron and proton transfer in polar media developed in Part I [1] is extended to the case of harmonic proton terms with different frequencies in the initial, ω 1, and final, ω 2, states. The dependence of the transfer probability w on the free energy of the reaction Δ G is characterized by the transfer coefficient α, which decreases from unity to zero as Δ G decreases, i.e. the overvoltage increases. The relations derived in ref. 1 are shown to be unconnected with the special choice of the terms. The dependence of w on Δ G that is obtained reduces to the relations derived in ref. 1 in a certain range of Δ G; in another range of Δ G it reduces to the generalized Marcus formula with a pre-exponential factor weighted by the Frank-Condon factor of the proton transfer. The latter is valid for the limited range of Δ G lying within the normal discharge region. Outside this region the relations derived in ref. 1 are valid. The results of the theoretical calculations for ω 2/ω 1=0.25–0.30 and hω 2/ kT=3–4 are in good agreement with the experimental data on hydrogen ion discharge on a mercury electrode. The relations derived in ref. 1 are shown to be valid in three discharge regions with Δ G also varying over 150 kJ mol −1.