Semiclassical treatment of thermally activated electron transfer in the intermediate to strong electronic coupling regime under the fast dielectric relaxation

The generalized nonadiabatic transition-state theory (NA-TST) (Zhao, Y.; et al. J. Chem. Phys. 2004, 121, 8854) is used to study electron transfer with use of the Zhu-Nakamura (ZN) formulas of nonadiabatic transition in the case of fast dielectric relaxation. The rate constant is expressed as a product of the well-known Marcus formula and a coefficient which represents the correction due to the strong electronic coupling. In the case of general multidimensional systems, the Monte Carlo approach is utilized to evaluate the rate by taking into account the multidimensionality of the crossing seam surface. Numerical demonstration is made by using a model system of a collection of harmonic oscillators in the Marcus normal region. The results are naturally coincident with the perturbation theory in the weak electronic coupling limit; while in the intermediate to strong electronic coupling regime where the perturbation theory breaks down the present results are in good agreement with those from the quantum mechanical flux-flux correlation function within the model of effective one-dimensional mode.     

Semiclassical treatment of thermally activated electron transfer in the inverted region under the fast dielectric relaxation

The previously formulated semiclassical theory (Zhao, Liang, and Nakamura, J. Phys. Chem. A 2006, 110, 8204) is used to study electron transfer in the Marcus inverted case by considering multidimensional potential energy surfaces of donor and acceptor. The Zhu-Nakamura formulas of nonadiabatic transition in the case of Landau-Zener type are incorporated into the approach. The theory properly takes into account the nonadiabatic transition coupled with the nuclear tunneling and can cover the whole range from weak to strong coupling regime uniformly under the assumption of fast solvent relaxation. The numerical calculations are performed for the 12-dimensional model of shifted harmonic oscillators and demonstrate that the reaction rate with respect to the electronic coupling shows a maximum, confirming the adiabatic suppression in the strong coupling limit. The adiabatic suppression is dramatically reduced by the effect of nuclear tunneling compared to the case that the Landau-Zener formula is used. The possible extension and applications to the case of the slow solvent dynamics are discussed.     

反转区电子转移速率的理论研究

利用级联方程方法研究了反转区的电子转移速率与电子态耦合常数以及溶剂弛豫时间的关系.结果表随着电子态间耦合强度的增加,反应速率先增加然后减小,而随着溶剂弛豫时间的减小反应速率增加.我们将这些结果与近似理论费米黄金规则、Landau-Zener公式进行了比较,由于费米金规则基于一阶微扰论,所以只适用于耦合常数较小的情况.而Landau-Zener公式假设电子在跃迁区域弹道运动,在电子态耦合大的情况下也存在问题.     

Photoinduced electron transfer and geminate recombination in liquids on short time scales: Experiments and theory

The coupled processes of intermolecular photoinduced forward electron transfer and geminate recombination between the (hole) donor (Rhodamine 3B) and (hole) acceptors (N,N-dimethylaniline) are studied in three molecular liquids: acetonitrile, butyronitrile, and benzonitrile. Two color pump-probe experiments on time scales from approximately 100 fs to hundreds of picoseconds give information about the depletion of the donor excited state due to forward electron transfer and the survival kinetics of the radicals produced by forward electron transfer. The data are analyzed with a model presented previously that includes distance dependent forward and back electron transfer rates, donor and acceptor diffusion, solvent structure, and the hydrodynamic effect in a mean-field theory of through solvent electron transfer. The forward electron transfer is in the normal regime, and the Marcus equation for the distance dependence of the transfer rate is used. The forward electron transfer data for several concentrations in the three solvents are fitted to the theory with a single adjustable parameter, the electronic coupling matrix element Jf at contact. Within experimental error all concentrations in all three solvents are fitted with the same value of Jf. The geminate recombination (back transfer) is in the inverted region, and semiclassical treatment developed by Jortner [J. Chem. Phys. 64, 4860 (1976)] is used to describe the distance dependence of the back electron transfer. The data are fitted with the single adjustable parameter Jb. It is found that the value of Jb decreases as the solvent viscosity increases. Possible explanations are discussed.     

Properties of nearly one-electron molecules. I. An iterative Green function approach to calculating the reaction matrix

An ab initio R-matrix method for determining the molecular reaction matrix of scattering theory is introduced. The method makes use of a principal-value Green function to compute the collision channel wave functions for the scattered electron, in combination with the Kohn variational scheme for the evaluation of R-matrix eigenvalues on a spherical boundary surface at short range. This technique permits the size of the bounded volume in the variational calculation to be reduced, making the computations fast and efficient. The reaction matrix is determined in a form that minimizes its energy dependence. Thus the procedure does not require modification or an increase in the computational effort to study the electronic structure and dynamics in Rydberg molecules with extremely polar ion cores. The analysis is specialized to examine the bound-state and free-electron scattering properties of nearly one-electron molecular systems, which are characterized by a Rydberg/scattering electron incident on a closed-shell ion core. However, it is shown that the treatment is compatible with all-electron/ab initio representations of open-shell and nonlinear polyatomic ion cores, emphasizing its generality. The introduced approach is used to calculate the electronic spectrum of the calcium monofluoride molecule, which has the extremely polar (Ca+2F-)+e- closed-shell ion-core configuration. The calculation utilizes an effective single-electron potential determined by M. Arif, C. Jungen, and A. L. Roche [J. Chem. Phys. 106, 4102 (1997)] previously. Close agreement with experimental data is obtained. The results demonstrate the practical utility of this method as a viable alternative to the standard variational approaches.     

Effects of Configurational Fluctuation on Electronic Coupling for Charge Transfer Dynamics

We advance a theory for the effects of bridge configurational fluctuations on the electronic coupling for electron transfer reactions in donor-bridge-acceptor systems. The theory of radiationless transitions was applied for activationless electron transfer, where the nuclear Franck–Condon constraints are minimized, with the initial vibronic state interacting directly with the final vibronic manifold, without the need for thermal activation. Invoking the assumption of energy-independent coupling, the time-dependent initial state population probability was analyzed in terms of a cumulant expansion. Two limiting situations were distinguished, i.e. the fast configurational fluctuation limit, where the electron transfer rate is given in terms of the configurational average of me squared electronic coupling, and the slow configurational fluctuation limit, where the dynamics is determined by a configurational averaging over a static distribution of electron transfer probability densities. The correlation times for configurational fluctuations of the electronic coupling will be obtained from the analysis of molecular dynamics, in conjunction with quantum mechanical calculations of the electronic coupling, to establish the appropriate limit for electron transfer dynamics.     

Influences of molecular packing on the charge mobility of organic semiconductors: from quantum charge transfer rate theory beyond the first-order perturbation

The electronic coupling between adjacent molecules is an important parameter for the charge transport properties of organic semiconductors. In a previous paper, a semiclassical generalized nonadiabatic transition state theory was used to investigate the nonperturbative effect of the electronic coupling on the charge transport properties, but it is not applicable at low temperatures due to the presence of high-frequency modes from the intramolecular conjugated carbon-carbon stretching vibrations [G. J. Nan et al., J. Chem. Phys., 2009, 130, 024704]. In the present paper, we apply a quantum charge transfer rate formula based on the imaginary-time flux-flux correlation function without the weak electronic coupling approximation. The imaginary-time flux-flux correlation function is then expressed in terms of the vibrational-mode path average and is evaluated by the path integral approach. All parameters are computed by quantum chemical approaches, and the mobility is obtained by kinetic Monte-Carlo simulation. We evaluate the intra-layer mobility of sexithiophene crystal structures in high- and low-temperature phases for a wide range of temperatures. In the case of strong coupling, the quantum charge transfer rates were found to be significantly smaller than those calculated using the weak electronic coupling approximation, which leads to reduced mobility especially at low temperatures. As a consequence, the mobility becomes less dependent on temperature when the molecular packing leads to strong electronic coupling in some charge transport directions. The temperature-independent charge mobility in organic thin-film transistors from experimental measurements may be explained from the present model with the grain boundaries considered. In addition, we point out that the widely used Marcus equation is invalid in calculating charge carrier transfer rates in sexithiophene crystals.     

Charge transport in metal oxides: a theoretical study of hematite alpha-Fe2O3

Transport of conduction electrons and holes through the lattice of alpha-Fe(2)O(3) (hematite) is modeled as a valence alternation of iron cations using ab initio electronic structure calculations and electron transfer theory. Experimental studies have shown that the conductivity along the (001) basal plane is four orders of magnitude larger than the conductivity along the [001] direction. In the context of the small polaron model, a cluster approach was used to compute quantities controlling the mobility of localized electrons and holes, i.e., the reorganization energy and the electronic coupling matrix element that enter Marcus' theory. The calculation of the electronic coupling followed the generalized Mulliken-Hush approach using the complete active space self-consistent field method. Our findings demonstrate an approximately three orders of magnitude anisotropy in both electron and hole mobility between directions perpendicular and parallel to the c axis, in good accord with experimental data. The anisotropy arises from the slowness of both electron and hole mobilities across basal oxygen planes relative to that within iron bilayers between basal oxygen planes. Interestingly, for elementary reaction steps along either of the directions considered, there is only less than one order of magnitude difference in mobility between electrons and holes, in contrast to accepted classical arguments. Our findings indicate that the most important quantity underlying mobility differences is the electronic coupling, albeit the reorganization energy contributes as well. The large values computed for the electronic coupling suggest that charge transport reactions in hematite are adiabatic in nature. The electronic coupling is found to depend on both the superexchange interaction through the bridging oxygen atoms and the d-shell electron spin coupling within the Fe-Fe donor-acceptor pair, while the reorganization energy is essentially independent of the electron spin coupling.     

Non-adiabatic molecular dynamics simulation of ultrafast solar cell electron transfer

A non-adiabatic (NA) molecular dynamics (MD) simulation of the photoinduced electron transfer (ET) from a molecular electron donor to the TiO₂ acceptor is carried out. The system under study is typical of the dye sensitized semiconductor nanomaterials used in solar cells, photocatalysis and photoelectrolysis. The electronic structure of the dye-semiconductor system and the adiabatic dynamics are simulated by ab initio density functional theory MD, while the NA effects are incorporated by the quantum-classical mean-field approach. A novel procedure separating the NA and adiabatic ET pathways is developed. The simulation provides a detailed picture of the ET process. For the specific system under study, ET occurs on a 30 fs time scale, in agreement with the ultrafast experimental data. Both adiabatic and NA pathways for the ET are observed. The NA transfer entirely dominates at short times and can occur due to strong localized avoided crossing as well as extended regions of weaker NA coupling. Although the adiabatic ET contribution accumulates more slowly, it approaches that of the NA ET pathway asymptotically. The electron acceptor states are formed by the d-orbital of Ti atoms of the semiconductor and are localized within the first 3–4 layers of the surface. About 20% of the acceptor state density is localized on a single Ti atom of the first surface layer. The simulation predicts a complex non-single-exponential time dependence of the ET process.     

Electronic coupling matrix elements from charge constrained density functional theory calculations using a plane wave basis set

We present a plane wave basis set implementation for the calculation of electronic coupling matrix elements of electron transfer reactions within the framework of constrained density functional theory (CDFT). Following the work of Wu and Van Voorhis [J. Chem. Phys. 125, 164105 (2006)], the diabatic wavefunctions are approximated by the Kohn-Sham determinants obtained from CDFT calculations, and the coupling matrix element calculated by an efficient integration scheme. Our results for intermolecular electron transfer in small systems agree very well with high-level ab initio calculations based on generalized Mulliken-Hush theory, and with previous local basis set CDFT calculations. The effect of thermal fluctuations on the coupling matrix element is demonstrated for intramolecular electron transfer in the tetrathiafulvalene-diquinone (Q-TTF-Q(-)) anion. Sampling the electronic coupling along density functional based molecular dynamics trajectories, we find that thermal fluctuations, in particular the slow bending motion of the molecule, can lead to changes in the instantaneous electron transfer rate by more than an order of magnitude. The thermal average, (<|H(ab)|(2)>)(1/2)=6.7 mH, is significantly higher than the value obtained for the minimum energy structure, |H(ab)|=3.8 mH. While CDFT in combination with generalized gradient approximation (GGA) functionals describes the intermolecular electron transfer in the studied systems well, exact exchange is required for Q-TTF-Q(-) in order to obtain coupling matrix elements in agreement with experiment (3.9 mH). The implementation presented opens up the possibility to compute electronic coupling matrix elements for extended systems where donor, acceptor, and the environment are treated at the quantum mechanical (QM) level.     

Ultrafast charge-transfer dynamics of donor-substituted truxenones

The photophysics of two donor-substituted truxenone derivatives has been studied by femtosecond time-resolved transient absorption spectroscopy. The systems consist of a central truxenone acceptor with three triarylamine (TARA) branches which act as electron donors. Upon excitation in the visible regime an electron is transferred from the donor to the acceptor, generating a charge-separated state. This state can be probed via the characteristic absorption of the TARA radical cation around 700 nm. A second absorption band around 420 nm exhibits the same kinetics and is assigned to an absorption of the radical anion of the truxenone moiety. The back electron transfer and the recovery of the ground state can be interpreted within the framework of Marcus theory. To study the dependence of the back electron transfer on the electronic coupling, the distance between the donor and the acceptor was adjusted. Two solvents were employed, dimethylsulfoxide and dichloroethane. A biexponential decay of the bands assigned to the charge-separated state was observed, with time constants in the picosecond range. Surprisingly, the rates for electron back transfer do not follow the simple picture of the donor-acceptor distance being the determining factor. The observations are explained within a model that additionally takes steric interactions between the donor and the acceptor into account.     

Theoretical investigation of electronic structures and excitation energies of hexaphyrin and its group 11 transition metal (III) complexes

Density functional theory is carried out to study hexaphyrin and its bis-metal and mixed bis-metal (M = Cu³⁺, Ag³⁺, and Au³⁺) complexes. The electronic structures and bonding situations of them are studied by using natural bond orbital approach and the topological analysis of the electron localization function. Electronic spectra are investigated by using time-dependent density functional theory. The introduction of group 11 transition metals leads to red shifts in the spectra of these metal complexes with respect to that of hexaphyrin. Moreover, it is noteworthy that the spectra of copper contained complexes are mainly derived from combination of ligand-to-metal charge transfer and ligand-to-ligand charge transfer transitions. In addition, the relativistic time-dependent density functional theory with spin-orbit coupling calculations indicate that the effects of spin-orbit coupling on the excitation energies are so small that it is safe enough to neglect spin-orbit coupling for these systems.     

Correlation effects in R-matrix calculations of electron-F2 elastic scattering cross sections

Correlation effects are studied in electron scattering off the fluorine molecule. Fixed-nuclei approximation R-matrix calculations of the elastic collision cross sections are presented for a set of internuclear distances at three levels of correlation. The aim of this work is to study the role of electronic correlation on the properties of the 2Sigmau resonance. The Feshbach-Fano R-matrix method of resonance-background separation is used to study the effect of inclusion of various levels of correlation on the energy and width of the 2Sigmau resonance. Data required for construction of the nonlocal resonance model (construction of a discrete state and its coupling to the continuum) which allows the calculation of inelastic processes such as dissociative electron attachment and vibrational excitation [W. Domcke, Phys. Rep. 208, 97 (1991)] including the correlation are presented.     

Electron transfer in porphyrin—quinone cyclophanes studied on the pico- and femto-second time scale

Electron transfer in porphyrin—quinone cyclophanes is investigated by fluorescence and absorption spectroscopy with pico- and femto-second pulses. In nonpolar solvents, the S₁ state of the porphyrin shows a lifetime from 300 ps up to several nanoseconds, depending upon the number of quinones and upon their electron affinity. Comparative measurements in polar solvents demonstrate very fast electron transfer on a time scale between 1 and 10 ps. The results are analyzed with the aid of quantum-chemical calculations which give the energy of the charge transfer states and the relevant coupling strengths. For nonpolar solvents, theory suggests fluctuation-induced charge separation and/or direct radiationless internal conversion from the porphyrin S₁ to the ground state. In polar solution, the molecules exist in a tilted configuration with strong electronic coupling and charge transfer states well below the S₁ level, resulting in fast electron transfer and subsequent charge recombination within 10–40 ps.     

Hydroxylation of methane by non-heme diiron enzymes: molecular orbital analysis of C-H bond activation by reactive intermediate Q

The electronic structures of key species involved in methane hydroxylation performed by the hydroxylase component of soluble methane monooxygenase (sMMO), as proposed previously on the basis of high-level density functional theory, were investigated. The reaction starts with initial approach of methane at one of the bridging oxo atoms in intermediate Q, a di(mu-oxo)diiron(IV) unit. This step is accompanied by a proton-coupled outer-sphere transfer of the first electron from a C-H sigma-bond in methane to one of the metal centers. The second electron transfer, also an outer-sphere electron transfer process, occurs along a two-component reaction pathway. Both redox reactions are strongly coupled to structural distortions of the diiron core. The electronic consequence and driving force of these distortions are intuitively explained by using the computed Kohn-Sham orbitals in the broken-symmetry framework to incorporate the experimentally observed antiferromagnetic coupling of the unpaired electrons at the metal centers. The broken-symmetry orbital scheme is essential for describing the C-H bond activation process in a consistent and complete manner, enabling derivation of both an intuitive and quantitative understanding of the most salient electronic features that govern the details of the hydroxylation.     

Quantum theory of dielectric relaxation in polar solids

A novel approach to the impurity dipole relaxation in polar solids is described in which the dipole reorientational sites are formed by an appropriate electron—phonon coupling interaction. The electronic states used for a two-site formulation are defined as ones pertaining to the impurity and/or the associated intrinsic point defect. The theory takes into account the adiabaticity of the electron transfer in addition to the barrier-controlled ionic tunneling. It compares favorably with available experimental relaxation-time data on alkali halides.     

Dynamics of nonadiabatic chemical reactions

New methods are proposed to treat nonadiabatic chemical dynamics in realistic large molecular systems by using the Zhu-Nakamura (ZN) theory of curve-crossing problems. They include the incorporation of the ZN formulas into the Herman-Kluk type semiclassical wave packet propagation method and the trajectory surface hopping (TSH) method, formulation of the nonadiabatic transition state theory, and its application to the electron transfer problem. Because the nonadiabatic coupling is a vector in multidimensional space, the one-dimensional ZN theory works all right. Even the classically forbidden transitions can be correctly treated by the ZN formulas. In the case of electron transfer, a new formula that can improve the celebrated Marcus theory in the case of normal regime is obtained so that it can work nicely in the intermediate and strong electronic coupling regimes. All these formulations mentioned above are demonstrated to work well in comparison with the exact quantum mechanical numerical solutions and are expected to be applicable to large systems that cannot be treated quantum mechanically numerically exactly. To take into account another quantum mechanical effect, namely, the tunneling effect, an efficient method to detect caustics from which tunneling trajectories emanate is proposed. All the works reported here are the results of recent activities carried out in the author's research group. Finally, the whole set of ZN formulas is presented in Appendix.     

Non-Condon电子转移速率理论与含时波包方法

随着电子转移理论在化学、材料科学、生物医学等领域的广泛应用,人们针对不同体系提出了多种电子转移理论模型。本文主要总结了近年来我们在non-Condon电子转移理论以及含时波包方法等方面的相关工作。首先阐述包含non-Condon效应的电子转移速率理论并用于二噻吩四硫富瓦烯有机半导体迁移率的计算。而后介绍了包含量子相干效应的含时波包方法,并初步用于研究二聚芴分子三三态能量转移过程。另外,本文还阐述了如何采用量子化学计算获得电子转移速率的结构参数。     

Increasing Electron‐Transfer Rates with Increasing Donor–Acceptor Distance

Electron transfer can readily occur over long (≥15 Å) distances. Usually reaction rates decrease with increasing distance between donors and acceptors, but theory predicts a regime in which electron‐transfer rates increase with increasing donor–acceptor separation. This counter‐intuitive behavior can result from the interplay of reorganization energy and electronic coupling, but until now experimental studies have failed to provide unambiguous evidence for this effect. We report here on a homologous series of rigid rodlike donor‐bridge‐acceptor compounds in which the electron‐transfer rate increases by a factor of 8 when the donor–acceptor distance is extended from 22.0 to 30.6 Å, and then it decreases by a factor of 188 when the distance is increased further to 39.2 Å. This effect has important implications for solar energy conversion.     

Charge transfer in organic molecules for solar cells: theoretical perspective

This tutorial review primarily illustrates rate theories for charge transfer and separation in organic molecules for solar cells. Starting from the Fermi's golden rule for weak electronic coupling, we display the microcanonical and canonical rates, as well as the relationship with the Marcus formula. The fluctuation effect of bridges on the rate is further emphasized. Then, several rate approaches beyond the perturbation limit are revealed. Finally, we discuss the electronic structure theory for calculations of the electronic coupling and reorganization energy that are two key parameters in charge transfer, and show several applications.