Complex Franck—Condon overlap integral in nonradiative decays

Considering the nuclear coordinate (Q) dependence of the electronic energy denominator appearing in the virbonic coupling matrix element, a complex Franck—Condon overlap integral which is needed in order to evaluate the nonradiative decay rate constant not only in the weak coupling but also in the strong coupling case is derived. The real part of the overlap integral plays an important role in the weak coupling case. The imaginary part is originated from the potential energy surface crossing regions and, consequently, contributes to the nonradiative decay rate constant in the strong coupling case. When the Q-dependence of the electronic energy denominator is neglected, the complex overlap integral leads to the results ontained by using the usual Herzberg—Teller expansion method. It is shown that the complex integral is expressed by the optical Franck—Condon overlap integral multiplied by a correction factor when the nonradiative decay from the vibrationless state is considered.     

Theoretical analysis of proton relays in electrochemical proton-coupled electron transfer

The coupling of long-range electron transfer to proton transport over multiple sites plays a vital role in many biological and chemical processes. Recently the concerted proton-coupled electron transfer (PCET) reaction in a molecule with a hydrogen-bond relay inserted between the proton donor and acceptor sites was studied electrochemically. The standard rate constants and kinetic isotope effects (KIEs) were measured experimentally for this double proton transfer system and a related single proton transfer system. In the present paper, these systems are studied theoretically using vibronically nonadiabatic rate constant expressions for electrochemical PCET. Application of this approach to proton relays requires the calculation of multidimensional proton vibrational wave functions and the incorporation of multiple proton donor-acceptor motions. The decrease in proton donor-acceptor distances due to thermal fluctuations and the contributions from excited electron-proton vibronic states play important roles in these systems. The calculated KIEs and the ratio of the standard rate constants for the single and double proton transfer systems are in agreement with the experimental data. The calculations indicate that the standard PCET rate constant is lower for the double proton transfer system because of the smaller overlap integral between the ground state reduced and oxidized proton vibrational wave functions, resulting in greater contributions from excited electron-proton vibronic states with higher free energy barriers. The theory predicts that this rate constant may be increased by modifying the molecule in a manner that decreases the equilibrium proton donor-acceptor distances or alters the molecular thermal motions to facilitate the concurrent decrease of these distances. These insights may guide the design of more efficient catalysts for energy conversion devices.     

Efficient calculation of vibrational magnetic dipole transition moments and rotational strengths

A priori prediction of vibrational magnetic dipole transition moments and vibrational strengths requires the calculation of the overlap integral of the derivatives of the electronic wavefunction with respect to nuclear displacement and an external magnetic field. The efficient calculation of this integral, using coupled Hartree-Fock theory, is described.     

Proton-coupled electron transfer in a model for tyrosine oxidation in photosystem II

Theoretical calculations of a model for tyrosine oxidation in photosystem II are presented. In this model system, an electron is transferred to ruthenium from tyrosine, which is concurrently deprotonated. This investigation is motivated by experimental measurements of the dependence of the rates on pH and temperature (Sj?din et al. J. Am. Chem. Soc. 2000, 122, 3932). The mechanism is proton-coupled electron transfer (PCET) at pH < 10 when the tyrosine is initially protonated and is single electron transfer (ET) for pH > 10 when the tyrosine is initially deprotonated. The PCET rate increases monotonically with pH, whereas the single ET rate is independent of pH and is 2 orders of magnitude faster than the PCET rate. The calculations reproduce these experimentally observed trends. The pH dependence for the PCET reaction arises from the decrease in the reaction free energies with pH. The calculations indicate that the larger rate for single ET arises from a combination of factors, including the smaller solvent reorganization energy for ET and the averaging of the coupling for PCET over the reactant and product hydrogen vibrational wave functions (i.e., a vibrational overlap factor in the PCET rate expression). The temperature dependence of the rates, the solvent reorganization energies, and the deuterium kinetic isotope effects determined from the calculations are also consistent with the experimental results.     

Proton-coupled electron transfer in soybean lipoxygenase

The proton-coupled electron transfer reaction catalyzed by soybean lipoxygenase-1 is studied with a multistate continuum theory that represents the transferring hydrogen nucleus as a quantum mechanical wave function. The inner-sphere reorganization energy of the iron cofactor is calculated with density functional theory, and the outer-sphere reorganization energy of the protein is calculated with the frequency-resolved cavity model for conformations obtained with docking simulations. Both classical and quantum mechanical treatments of the proton donor-acceptor vibrational motion are presented. The temperature dependence of the calculated rates and kinetic isotope effects is in agreement with the experimental data. The weak temperature dependence of the rates is due to the relatively small free energy barrier arising from a balance between the reorganization energy and the reaction free energy. The unusually high deuterium kinetic isotope effect of 81 is due to the small overlap of the reactant and product proton vibrational wave functions and the dominance of the lowest energy reactant and product vibronic states in the tunneling process. The temperature dependence of the kinetic isotope effect is strongly influenced by the proton donor-acceptor distance with the dominant contribution to the overall rate. This dominant proton donor-acceptor distance is significantly smaller than the equilibrium donor-acceptor distance and is determined by a balance between the larger coupling and the smaller Boltzmann probability as the distance decreases. Thus, the proton donor-acceptor vibrational motion plays a vital role in decreasing the dominant donor-acceptor distance relative to its equilibrium value to facilitate the proton-coupled electron transfer reaction.     

On Theoretical Treatments of Electronic Excitation Energy Transfer

In this paper, we review the generalized Forster-Dexter theory to treat photoinduced electronic energy transfer for a system in dense media and for an isolated system (i.e., a system in the collision-free condition). Instead of expressing the rate of energy transfer in terms of spectral overlap, the expression of the energy-transfer rate constant is obtained by evaluating a Fourier integral involved in the energy transfer rate constant using the saddle-point method. In this way, the energy-gap dependence, and the effect of temperature and the isotope effect on the energy transfer can be easily studied. The effect of bridge groups connecting between donor and acceptor chromophores on the intramolecular energy transfer is also studied.     

Radiationless transitions of molecules with large nuclear rearrangement

Radiationless transitions of molecules accompanied by a large nuclear rearrangement were investigated. A general formula which includes interference effects among promoting modes was used. Numerical analysis for the model system revealed that the energy-gap dependence of the transition probability exhibits quite different behavior from the usual gaussian-type dependence.     

A new ab initio potential energy surface for studying vibrational relaxation in NO(v) + NO collisions

A new ab initio potential energy surface for the ground state of the NO-NO system has been calculated within a reduced dimensionality model. We find an unusually large vibrational dependence of the interaction potential which explains previous spectroscopic observations. The potential can be used to model vibrational energy transfer, and here we perform quantum scattering calculations of the vibrational relaxation of NO(v). We show that the vibrational relaxation for v = 1 is 4 orders of magnitude larger than that for the related O(2)(v) + O(2) system without having to invoke nonadiabatic mechanisms as had been suggested in the past. For highly vibrationally excited states, we predict a strong dependence of the rates on the vibrational quantum number as has been observed experimentally, although there remain important quantitative differences. The importance of a chemically bound isomer on the relaxation mechanism is analyzed, and we conclude it does not play a role for the values of v considered in the experiment. Finally, the intriguing negative temperature dependence of the vibrational relaxation rate constants observed in experiments was studied using an statistical model to include the presence of many asymptotically degenerate spin-orbit states.     

Torsionally broken symmetry assists infrared excitation of biomimetic charge-coupled nuclear motions in the electronic ground state

The concerted interplay between reactive nuclear and electronic motions in molecules actuates chemistry. Here, we demonstrate that out-of-plane torsional deformation and vibrational excitation of stretching motions in the electronic ground state modulate the charge-density distribution in a donor-bridge-acceptor molecule in solution. The vibrationally-induced change, visualised by transient absorption spectroscopy with a mid-infrared pump and a visible probe, is mechanistically resolved by ab initio molecular dynamics simulations. Mapping the potential energy landscape attributes the observed charge-coupled coherent nuclear motions to the population of the initial segment of a double-bond isomerization channel, also seen in biological molecules. Our results illustrate the pivotal role of pre-twisted molecular geometries in enhancing the transfer of vibrational energy to specific molecular modes, prior to thermal redistribution. This motivates the search for synthetic strategies towards achieving potentially new infrared-mediated chemistry.

Channelling vibrational excitation energy to achieve ground-state charge-transfer (CT)-assisted isomerization in a donor-bridge-acceptor molecule in solution.     

Buffer-assisted proton-coupled electron transfer in a model rhenium-tyrosine complex

The mechanism for tyrosyl radical generation in the [Re(P-Y)(phen)(CO)3]PF6 complex is investigated with a multistate continuum theory for proton-coupled electron transfer (PCET) reactions. Both water and the phosphate buffer are considered as potential proton acceptors. The calculations indicate that the model in which the proton acceptor is the phosphate buffer species HPO(4)2- can successfully reproduce the experimentally observed pH dependence of the overall rate and H/D kinetic isotope effect, whereas the model in which the proton acceptor is water is not physically reasonable for this system. The phosphate buffer species HPO4(2-) is favored over water as the proton acceptor in part because the proton donor-acceptor distance is approximately 0.2 A smaller for the phosphate acceptor due to its negative charge. The physical quantities impacting the overall rate constant, including the reorganization energies, reaction free energies, activation free energies, and vibronic couplings for the various pairs of reactant/product vibronic states, are analyzed for both hydrogen and deuterium transfer. The dominant contribution to the rate arises from nonadiabatic transitions between the ground reactant vibronic state and the third product vibronic state for hydrogen transfer and the fourth product vibronic state for deuterium transfer. These contributions dominate over contributions from lower product states because of the larger vibronic coupling, which arises from the greater overlap between the reactant and product vibrational wave functions. These calculations provide insight into the fundamental mechanism of tyrosyl radical generation, which plays an important role in a wide range of biologically important processes.     

Excitation transfer induced spectral diffusion and the influence of structural spectral diffusion

The theory of vibrational excitation transfer, which causes spectral diffusion and is also influenced by structural spectral diffusion, is developed and applied to systems consisting of vibrational chromophores. Excitation transfer induced spectral diffusion is the time-dependent change in vibrational frequency induced by an excitation on an initially excited molecule jumping to other molecules that have different vibrational frequencies within the inhomogeneously broadened vibrational absorption line. The excitation transfer process is modeled as Fo?rster resonant transfer, which depends on the overlap of the homogeneous spectra of the donating and accepting vibrational chromophores. Because the absorption line is inhomogeneously broadened, two molecules in close proximity can have overlaps of their homogeneous lines that range from substantial to very little. In the absence of structural dynamics, the overlap of the homogeneous lines of the donating and accepting vibrational chromophores would be fixed. However, dynamics of the medium that contains the vibrational chromophores, e.g., a liquid solvent or a surrounding protein, produce spectral diffusion. Spectral diffusion causes the position of a molecule's homogeneous line within the inhomogeneous spectrum to change with time. Therefore, the overlap of donating and accepting molecules' homogeneous lines is time dependent, which must be taken into account in the excitation transfer theory. The excitation transfer problem is solved for inhomogeneous lines with fluctuating homogeneous line frequencies. The method allows the simultaneous treatment of both excitation transfer induced spectral diffusion and structural fluctuation induced spectral diffusion. It is found that the excitation transfer process is enhanced by the stochastic fluctuations in frequencies. It is shown how a measurement of spectral diffusion can be separated into the two types of spectral diffusion, which permits the structural spectral diffusion to be determined in the presence of excitation transfer spectral diffusion. Various approximations and computational methodologies are explored.     

Multireference CI calculation of nuclear quadrupole coupling constants of CN+ and CN-: rovibrational dependence

The 14N quadrupole coupling constants of rovibrational levels of the X1sigma+ and c1sigma+ states of CN+, and the ground electronic state of CN- are calculated from molecular wavefunctions which explicitly describe nuclear displacement. From the electronic states considered, the excited 1sigma+ state of CN is predicted to exhibit the strongest N coupling, at least in the ground vibrational state. Compared to the vibrational dependence of the 14N QCC's, which is found to be significant in all cases, the rotational dependence is predicted to be unimportant. Special attention is paid to the assessment of adequacy of the expectation value approach to the evaluation of the electric field gradient tensor within the applied multireference configuration interaction formalism. Spectroscopic constants are derived from corresponding potential energy curves to testify to the quality of the correlated wave functions used.     

Resonance energy transfer from a dye molecule to graphene

We study the distance dependence of the rate of resonance energy transfer from the excited state of a dye to the pi system of graphene. Using the tight-binding model for the pi system and the Dirac cone approximation, we obtain the analytic expression for the rate of energy transfer from an electronically excited dye to graphene. While in traditional fluorescence resonance energy transfer, the rate has a (distance)(-6) dependence, we find that the distance dependence in this case is quite different. Our calculation of rate in the case of the two dyes, pyrene and nile blue, shows that the distance dependence is Yukawa type. We have also studied the effect of doping on energy transfer to graphene. Doping does not modify the rate for electronic excitation energy transfer significantly. However, in the case of vibrational transfer, the rate is found to be increased by an order of magnitude due to doping. This can be attributed to the nonzero density of states at the Fermi level that results from doping.     

On the temperature dependence of electronically non-adiabatic vibrational energy transfer in molecule-surface collisions

Here we extend a recently introduced state-to-state kinetic model describing single- and multi-quantum vibrational excitation of molecular beams of NO scattering from a Au(111) metal surface. We derive an analytical expression for the rate of electronically non-adiabatic vibrational energy transfer, which is then employed in the analysis of the temperature dependence of the kinetics of direct overtone and two-step sequential energy transfer mechanisms. We show that the Arrhenius surface temperature dependence for vibrational excitation probability reported in many previous studies emerges as a low temperature limit of a more general solution that describes the approach to thermal equilibrium in the limit of infinite interaction time and that the pre-exponential term of the Arrhenius expression can be used not only to distinguish between the direct overtone and sequential mechanisms, but also to deduce their relative contributions. We also apply the analytical expression for the vibrational energy transfer rates introduced in this work to the full kinetic model and obtain an excellent fit to experimental data, the results of which show how to extract numerical values of the molecule-surface coupling strength and its fundamental properties.     

Protein dynamics and electronic fluctuation effects in electron transfer reactions of membrane-bound proteins and metalloprotein complexes

We have analyzed specific long-range features of the electron transfer reactions in the cytochrome c/cytochrome b₅ complex, and the bacteriopheophytin/quinone and quinone/bacteriochlorophyll special pair cation radical long-range electron transfers. The analysis rests on an electron transfer theory which incorporates vibrational dispersion in the protein and solvent environments, and environmental fluctuation effects on the electronic transmission coefficient.The dispersion represents broadly the “inverse” temperature dependence of the photosynthetic reactions, but an explicit temperature dependence of the rate parameters is needed for quantitative data agreement. The data suggest that the transmission coefficient may exhibit some reaction free energy and temperature dependence caused by fluctuations in the environmental nuclear motion. The data are presently insufficiently diagnostic in this respect, but transmembrane potential induced rate variations over wide temperature ranges could lead to its clarification.     

Angular overlap model parameterisation of Anderson''s superexchange theory. I. A quantification of Goodenough-Kanamori rules

A parameterisation of Anderson's exchange formulas on the basis of an extension of the angular overlap model (AOM) is proposed. Transfer integrals are expressed in terms of the metal-metal and metal-ligand bonding parameters, which can be estimated from independent spectroscopic studies, or calculated using solid state expressions. Analytical expressions for the transfer integrals between various d-orbitals, appropriate for cubic crystal lattices, comprising octahedra sharing common vertices and common edges serve as a quantification of the Goodenough-Kanamori rules. On the basis of the present parameterisation we give an explanation of the “exchange integral versus bond-distance” dependence. Some potential applications of the model are briefly discussed.     

Theoretical investigation on intramolecular electron transfer in polypeptides

Theoretical investigations on the intramolecular electron transfer between the intermediate residues of different secondary structures of an oligopeptide have been carried out. Density functional theory calculations have been performed to calculate the charge transfer integral, spatial overlap integral and site-energies for the optimized secondary structures of the glycine oligopeptide by varying the dihedral angles ( and ψ) along the -carbon atom of amino acid subgroups. The reorganization energy has been calculated in the presence of an excess negative charge. The electron transfer rates for the model peptide have been estimated and the dependence of the rate on secondary structures is discussed.