A Parameter-Free Quantum-Mechanical Approach for Calculating Electron-Transfer Rates for Large Systems in Solution

Theoretical calculation of reorganization energy for electron self-exchange reaction by constrained density functional theory and constrained equilibrium thermodynamics

Within the framework of constrained density functional theory (CDFT), the diabatic or charge localized states of electron transfer (ET) have been constructed. Based on the diabatic states, inner reorganization energy λin has been directly calculated. For solvent reorganization energy λs, a novel and reasonable nonequilibrium solvation model is established by introducing a constrained equilibrium manipulation, and a new expression of λs has been formulated. It is found that λs is actually the cost of maintaining the residual polarization, which equilibrates with the extra electric field. On the basis of diabatic states constructed by CDFT, a numerical algorithm using the new formulations with the dielectric polarizable continuum model (D-PCM) has been implemented. As typical test cases, self-exchange ET reactions between tetracyanoethylene (TCNE) and tetrathiafulvalene (TTF) and their corresponding ionic radicals in acetonitrile are investigated. The calculated reorganization energies λ are 7293 cm(-1) for TCNE/TCNE(-) and 5939 cm(-1) for TTF/TTF(+) reactions, agreeing well with available experimental results of 7250 cm(-1) and 5810 cm(-1), respectively.

Fragment-based Quantum Mechanical/Molecular Mechanical Simulations of Thermodynamic and Kinetic Process of the Ru2+-Ru3+ Self-Exchange Electron Transfer

A fragment-based fractional number of electron (FNE) approach, is developed to study entire electron transfer (ET) processes from the electron donor region to the acceptor region in condensed phase. Both regions are described by the density-fragment interaction (DFI) method while FNE as an efficient ET order parameter is applied to simulate the electron transfer process. In association with the QM/MM energy expression, the DFI-FNE method is demonstrated to describe ET processes robustly with the Ru2+-Ru3+ self-exchange ET as a proof-of-concept example. This method allows for systematic calculations of redox free energies, reorganization energies, and electronic couplings, and the absolute ET rate constants within the Marcus regime.

Mapping protein electron transfer pathways with QM/MM methods

Mixed quantum mechanics/molecular mechanics (QM/MM) methods offer a valuable computational tool for understanding the electron transfer pathway in protein-substrate interactions and protein-protein complexes. These hybrid methods are capable of solving the Schrödinger equation on a small subset of the protein, the quantum region, describing its electronic structure under the polarization effects of the remainder of the protein. By selectively turning on and off different residues in the quantum region, we are able to obtain the electron pathway for short- and large-range interactions. Here, we summarize recent studies involving the protein-substrate interaction in cytochrome P450 camphor, ascorbate peroxidase and cytochrome c peroxidase, and propose a novel approach for the long-range protein-protein electron transfer. The results on ascorbate peroxidase and cytochrome c peroxidase reveal the importance of the propionate groups in the electron transfer pathway. The long-range protein-protein electron transfer has been studied on the cytochrome c peroxidase-cytochrome c complex. The results indicate the importance of Phe82 and Cys81 on cytochrome c, and of Asn196, Ala194, Ala176 and His175 on cytochrome c peroxidase.

Two-state model based on the block-localized wave function method

The block-localized wave function (BLW) method is a variant of ab initio valence bond method but retains the efficiency of molecular orbital methods. It can derive the wave function for a diabatic (resonance) state self-consistently and is available at the Hartree-Fock (HF) and density functional theory (DFT) levels. In this work we present a two-state model based on the BLW method. Although numerous empirical and semiempirical two-state models, such as the Marcus-Hush two-state model, have been proposed to describe a chemical reaction process, the advantage of this BLW-based two-state model is that no empirical parameter is required. Important quantities such as the electronic coupling energy, structural weights of two diabatic states, and excitation energy can be uniquely derived from the energies of two diabatic states and the adiabatic state at the same HF or DFT level. Two simple examples of formamide and thioformamide in the gas phase and aqueous solution were presented and discussed. The solvation of formamide and thioformamide was studied with the combined ab initio quantum mechanical and molecular mechanical Monte Carlo simulations, together with the BLW-DFT calculations and analyses. Due to the favorable solute-solvent electrostatic interaction, the contribution of the ionic resonance structure to the ground state of formamide and thioformamide significantly increases, and for thioformamide the ionic form is even more stable than the covalent form. Thus, thioformamide in aqueous solution is essentially ionic rather than covalent. Although our two-state model in general underestimates the electronic excitation energies, it can predict relative solvatochromic shifts well. For instance, the intense pi-->pi* transition for formamide upon solvation undergoes a redshift of 0.3 eV, compared with the experimental data (0.40-0.5 eV).

Molecular Simulation of Solvent-Induced Stokes Shift in Absorption/Emission Spectra of Organic Chromophores

The values of steady-state solvatochromic Stokes shifts (SS) in absorption/emission electronic spectra of organic chromophores are studied theoretically in the framework of the Hush-Marcus model. Charge distributions for chromophore solutes in their S0 and S1 states are found by means of conventional quantum-chemical methods combined with the continuum PCM approach for treating solvation effects. The solvent reorganization energies, which are expected to correlate with the solvent-induced part of 1/2 SS, are found in a molecular dynamics (MD) simulation which invokes a novel method for separation of the inertial piece of the electrostatic response (Vener, et al. J. Phys. Chem. B 2006, 110, 14950). Computations, performed in two solvents (acetonitrile and benzene), consider three organic dyes: coumarin 153 as a benchmark system and two other chromophores, for which experimental spectra are also reported. The results are found to be in reasonable agreement with the experiment. A consistent treatment of nonlinear effect in the solvent response, promoted by the polarizability of solutes and contributing to the solvent reorganization energies (Ingrosso, et al. J. Phys. Chem. B 2005, 109, 3553), improves the results of computations.

Coupling coherence distinguishes structure sensitivity in protein electron transfer

Quantum mechanical analysis of electron tunneling in nine thermally fluctuating cytochrome b562 derivatives reveals two distinct protein-mediated coupling limits. A structure-insensitive regime arises for redox partners coupled through dynamically averaged multiple-coupling pathways (in seven of the nine derivatives) where heme-edge coupling leads to the multiple-pathway regime. A structure-dependent limit governs redox partners coupled through a dominant pathway (in two of the nine derivatives) where axial-ligand coupling generates the single-pathway limit and slower rates. This two-regime paradigm provides a unified description of electron transfer rates in 26 ruthenium-modified heme and blue-copper proteins, as well as in numerous photosynthetic proteins.