Free energy landscape analysis of two-dimensional dipolar solvent model at temperatures below and above the rotational freezing point

Ionic solvation in a polar solvent is modeled by a central charge surrounded by dipolar molecules posted on two-dimensional distorted lattice sites with simple rotational dynamics. Density of states is calculated by applying the Wang-Landau algorithm to both the energy and polarization states. The free energy landscapes of solvent molecules as a function of polarization are depicted to explore the competition between the thermal fluctuation and solvation energy. Without a central charge, for temperatures higher than the energy scale of the dipole-dipole interactions, the energy landscape for the small polarization region exhibits a parabolic shape as predicted by Marcus [Rev. Mod. Phys. 65, 599 (1993)] for electron transfer reaction, while there is an additional quartic contribution to the landscape for the large polarization region. When the temperature drops, the simulated free energy landscapes are no longer smooth due to the presence of multiple local minima arising from the frustrated interaction among the dipoles. The parabolic contribution becomes negligible and the energy landscape becomes quartic in shape. For a strong central charge, the energy landscape exhibits an asymmetric profile due to the contributions of linear and cubic terms that arise from the charge-dipole interactions.     

Polarizable solute in polarizable and flexible solvents: simulation study of electron transfer reaction systems

A polarizable solute model, based on the empirical valence bond approach, is developed and applied to electron transfer (ET) reactions in polarizable and flexible water solvents. The polarization effect is investigated in comparison with a nonpolarizable solute and solvent model. With free energy curves constructed by a molecular dynamics simulation, the activation energy barrier and the reorganization energy related to ET processes are investigated. The present simulation results show that the activation energy barrier becomes larger in the polarizable model than in the nonpolarizable model and that this makes the ET rate slower than that with the nonpolarizable model. It is shown that the effect of the electronic energy difference of solute molecule on free energy profiles is remarkable and that, corresponding to this effect, the reorganization energy is significantly modified. These results indicate that the process of solvent polarization by the polarized solute to enhance the solute-solvent interaction is a key factor and that treating the polarization of both solute and solvent at the same time is essential. Also, the polarization effect on the diffusive motion of the solute molecule in the polarization solvent is studied. The polarized solute molecule shows slower diffusive motion compared with that in the nonpolarizable model.     

Unified interpretation of exciplex formation and marcus electron transfer on the basis of two-dimensional free energy surfaces

The mechanism of exciplex formation proposed in a previous paper has been refined to show how exciplex formation and Marcus electron transfer (ET) in fluorescence quenching are related to each other. This was done by making simple calculations of the free energies of the initial (DA*) and final (D+A-) states of ET. First it was shown that the decrease in D-A distance can induce intermolecular ET even in nonpolar solvents where solvent orientational polarization is absent, and that it leads to exciplex formation. This is consistent with experimental results that exciplex is most often observed in nonpolar solvents. The calculation was then extended to ET in polar solvents where the free energies are functions of both D-A distance and solvent orientational polarization. This enabled us to discuss both exciplex formation and Marcus ET in the same D-A pair and solvent on the basis of 2-dimensional free energy surfaces. The surfaces contain more information about the rates of these reactions, the mechanism of fluorescence quenching by ET, etc., than simple reaction schemes. By changing the parameters such as the free energy change of reaction, solvent dielectric constants, etc., one can construct the free energy surfaces for various systems. The effects of free energy change of reaction and of solvent polarity on the mechanism and relative importance of exciplex formation and Marcus ET in fluorescence quenching can be well explained. The free energy surface will also be useful for discussion of other phenomena related to ET reactions.     

Solvent reorganization energy of electron-transfer reactions in polar solvents

A microscopic theory of solvent reorganization energy in polar molecular solvents is developed. The theory represents the solvent response as a combination of the density and polarization fluctuations of the solvent given in terms of the density and polarization structure factors. A fully analytical formulation of the theory is provided for a solute of arbitrary shape with an arbitrary distribution of charge. A good agreement between the analytical procedure and the results of Monte Carlo simulations of model systems is achieved. The reorganization energy splits into the contributions from density fluctuations and polarization fluctuations. The polarization part is dominated by longitudinal polarization response. The density part is inversely proportional to temperature. The dependence of the solvent reorganization energy on the solvent dipole moment and refractive index is discussed.     

Ions in a binary asymmetric dipolar mixture: mole fraction dependent Born energy of solvation and partial solvent polarization structure

Mean spherical approximation (MSA) for electrolyte solution has been extended to investigate the role of partial solvent polarization densities around an ion in a completely asymmetric binary dipolar mixture. The differences in solvent diameters, dipole moments, and ionic size are incorporated systematically within the MSA framework in the present theory for the first time. In addition to the contributions due to difference in dipole moments, the solvent-solvent and ion-solvent size ratios are found to significantly affect the nonideality in binary dipolar mixtures. Subsequently, the theory is used to investigate the role of ion-solvent and solvent-solvent size ratios in determining the nonideality in Born free energy of solvation of a unipositive rigid ion in alcohol-water and dimethyl sulfoxide-acetonitrile mixtures, where the solvent components are represented only by their molecular diameters and dipole moments. Nonideality in Born free energy of solvation in such simplified mixtures is found to be stronger for smaller ions. The slope of the nonideality for smaller alkali metal ions in methanol-water mixture is found to be opposite to that for larger ion, such as quaternary tertiary butyl ammonium ion. For ethanol-water mixtures, the slopes are in the same direction for all the ions studied here. These results are in qualitative agreement with experiments, which is surprising as the present MSA approach does not include the hydrogen bonding and hydrophobic interactions present in the real mixtures. The calculated partial polarization densities around a unipositive ion also show the characteristic deviation from ideality and reveal the microscopic origin of the ion and solvent size dependent preferential solvation. Also, the excess free energy of mixing (in the absence of any ion) for these binary mixtures has been calculated and a good agreement between theory and experiment has been found.     

Reorganization free energies for long-range electron transfer in a porphyrin-binding four-helix bundle protein

To explore the possibility of electron transport in a recently designed four-helix bundle protein (Cochran, F. V.; et al. J. Am. Chem. Soc. 2005, 127, 1346), we have computed the reorganization free energy for (i) oxidation of a single Ru-porphyrin cofactor and (ii) electron self-exchange between two Ru-porphyrin cofactors binding to the solvated protein. Sampling the classical electrostatic energy gap for 20 ns, we find that the fluctuations are well described by Gaussian statistics and obtain reorganization free energies of 0.90 +/- 0.04 eV for oxidation and 1.36 +/- 0.08 eV for self-exchange. The latter is 0.1-0.2 eV higher than the experimental estimate for interprotein electron self-exchange in cytochrome b5. As in natural electron carriers, inner-sphere reorganization is very small, 88 meV for self-exchange between two model cofactors computed at the density functional level of theory. Decomposing the outer-sphere reorganization free energy, we find that the solvent (aqueous ionic solution) is the primary outer-sphere medium for oxidation, contributing 0.60 eV (69%). The protein contributes only 0.27 eV (31%). For self-exchange, the solvent contribution, 0.68 eV (54%), and the protein contribution, 0.59 eV (46%), are almost equally important. The large solvent contribution is due to the slow decay of dipole reorientation of the solvent as a function of distance to the cofactor, implying that the change in the electric field upon electron transfer is not as effectively screened by the four-helix bundle protein. However, ranking the residues according to their free energy contributions, it is suggested that the reorganization free energy can be decreased by about 0.2 eV if two glutamine residues in the vicinity of the cofactor are mutated into less polar amino acids.     

Potential-Barrier Profile for the Adiabatic Electron Transfer Reaction in the Condon Approximation

An effective reaction coordinate is introduced and a self-consistent algorithm is proposed for the calculation of the potential barrier profile of the adiabatic electron transfer reaction in a polar solvent, taking into account electron wave functions of reagents with respect to the media polarization. A change in the functions when moving along the reaction coordinate reduces significantly the activation barrier and increases the reaction rate.     

Dynamic Nuclear Polarization in Suspensions Consisting of Xylene Isomers and Asphaltene Extracted from MC‐800 Liquid Asphalt

Overhauser effect type dynamic nuclear polarization experiments were performed to study suspensions of asphaltene in the xylene isomers (o‐, m‐, p‐) at a low magnetic field of 1.44 mT and three different temperatures (15, 25, and 35°C). The asphaltene was extracted from MC‐800 liquid asphalt. Intermolecular spin‐spin interactions occur between nuclear spins of hydrogen in the solvent medium and the free electron spins in the asphaltene micelles. The electron paramagnetic resonance spectrum of the asphaltene was obtained and the saturation experiments were applied to the samples prepared in vacuum. For all media, the dipole‐dipole interaction is predominant due to the negative signal enhancements. In all temperatures, the ultimate enhancement is the smallest for the p‐xylene solvent medium which has the lowest electrical dipole moment. The normalized low frequency relaxation components were calculated for 25°C, and the behavior of the nuclear‐electron coupling parameter according to this component is in agreement with the other works in the literature.     

Change in a protein's electronic structure induced by an explicit solvent: an ab initio fragment molecular orbital study of ubiquitin

The effect of solvation on the electronic structure of the ubiquitin protein was analyzed using the ab initio fragment molecular orbital (FMO) method. FMO calculations were performed for the protein in vacuo, and the protein was immersed in an explicit solvent shell as thick as 12 A at the HF or MP2 level by using the 6-31G* basis set. The protein's physical properties examined were the net charge, the dipole moment, the internal energy, and the solvent interaction energy. Comparison of the computational results revealed the following changes in the protein upon solvation. First, the positively charged amino acid residues on the protein surface drew electrons from the solvent, while the negatively charged ones transfer electrons to the solvent. Second, the dipole moment of the protein was enhanced as a result of the polarization. Third, the internal energy of the protein was destabilized, but the destabilization was more than compensated for by the generation of a favorable protein-solvent interaction. Finally, the energetic changes were elicited both by the electron correlation effect of the first solvent shell and by the electrostatic effect of more distant solvent molecules. These findings were consistent with the picture of the solvated protein being a polarizable molecule dissolved in a dielectric media.     

Glassy protein dynamics and gigantic solvent reorganization energy of plastocyanin

We report the results of molecular dynamics simulations of electron-transfer activation parameters of plastocyanin metalloprotein involved as an electron carrier in natural photosynthesis. We have discovered that slow, non-ergodic conformational fluctuations of the protein, coupled to hydrating water, result in a very broad distribution of donor-acceptor energy gaps far exceeding those observed for commonly studied inorganic and organic donor-acceptor complexes. The Stokes shift is not affected by these fluctuations and can be calculated from solvation models in terms of the linear response of the solvent dipolar polarization. The non-ergodic character of large-amplitude protein/water mobility breaks the strong link between the Stokes shift and the reorganization energy characteristic of equilibrium (ergodic) theories of electron transfer. This mechanism might be responsible for fast electronic transitions in natural electron-transfer proteins characterized by low reaction free energy.     

Origin of the solvent effects on the barrier to amide isomerization from the combined QM/MM Monte Carlo simulations

The origin of the solvent effects on the free energy of activation for the isomerization of N,N-dimethylformamide in water, CHCl₃ and CCl₄ has been investigated through statistical mechanical simulations using the combined quantum mechanical and molecular mechanical AM1/OPLS potential. The differential solvations between the ground state and transition state in various solvents can be attributed to the differences in molecular dipole moments in solution, and to the solvent polarization effects. In polar solvents, DMF is polarized more favorably in the ground state than in the rotated conformers, leading to greater solvent contributions. The modest solvent effects in CCl₄ are a reflection of its much smaller dielectric constant.     

丙酮分子n→π*跃迁光谱的平均溶剂静电势/分子动力学方法研究

采用量子力学/分子动力学方法研究了具体溶剂分子结构对溶质光谱行为的静电影响. 通过拟合溶质所处外电场和引入溶剂分子极化率, 考虑了溶质溶剂分子之间的相互极化效应, 得到合理的溶质和溶剂分子的电荷分布. 经过严格推导发现, 在传统的显溶剂模型中, 平衡和非平衡溶剂化能表达式均未考虑溶剂分子永久偶极弹簧能的贡献. 因此, 在正确计算永久偶极弹簧能的基础上, 重新建立了溶剂化能的表达式和新的吸收/发射光谱移动公式. 采用修改后的ASEP/MD程序, 计算得到了与实验值比较吻合的丙酮在水溶液中n→π*跃迁的光谱移动值, 验证了新公式的合理性.     

Equilibrium solvation in quadrupolar solvents

We present a microscopic theory of equilibrium solvation in solvents with zero dipole moment and nonzero quadrupole moment (quadrupolar solvents). The theory is formulated in terms of autocorrelation functions of the quadrupolar polarization (structure factors). It can be therefore applied to an arbitrary dense quadrupolar solvent for which the structure factors are defined. We formulate a simple analytical perturbation treatment for the structure factors. The solute is described by coordinates, radii, and partial charges of constituent atoms. The theory is tested on Monte Carlo simulations of solvation in model quadrupolar solvents. It is also applied to the calculation of the activation barrier of electron transfer reactions in a cleft-shaped donor-bridge-acceptor complex dissolved in benzene with the structure factors of quadrupolar polarization obtained from molecular-dynamics simulations.     

A modified two-sphere model for solvent reorganization energy in electron transfer

In this work, the solvent reorganization energy is formulated within the framework of classical thermodynamics, by adding some external charges to construct a constrained equilibrium state. The derivation clearly shows that the reorganization energy is exactly the polarization cost for the inertial part of the polarization. We perform our derivation just within the framework of the first law of thermodynamics, and the final form of the reorganization energy is completely the same as that we gave in our recent work by defining a nonequilibrium solvation free energy. With the two-sphere model approximation, our solvent reorganization energy is derived as λ(0) = Δq(2)/2[1/r(D) + 1/r(A) - 2/d][(ε(-1)(op) - ε(-1)(s))/(1 - ε(-1)(s))]. This amends Marcus' model by a factor of (ε(-1)(op) - ε(-1)(s))/(1 - ε(-1)(s)), which is coupled with the solvent polarity. Making use of the modified expression of solvent reorganization energy, two recently reported electron transfer processes are investigated in representative solvents. The results show that our formula can well reproduce the experimental observations.     

Solvent mediated intramolecular photoinduced electron transfer in a fluorene-perylene bisimide derivative

Photoinduced electron transfer from fluorene to perylene bisimide has been studied for 2,7-bis(N-(1-hexylheptyl)-3,4:9,10-perylene-bisimide-N'-yl))-9,9-didodecylfluorene (PFP) in 11 different organic solvents. The intramolecular charge-separated state in PFP is almost isoenergetic with the locally excited state of the perylene bisimide. As a consequence of the small change in free energy for charge separation, the electron transfer rate strongly depends on subtle changes in the medium. The rate constant k(CS) for the electron transfer from fluorene to perylene bisimide moiety in the excited state varies over more than 2 orders of magnitude ( approximately 10(8)-10(10) s(-1)) with the solvent but does not show the familiar increase with polarity. The widely differing rate constants can be successfully explained by considering (1) the contribution of the polarization energy of the dipole moment in the transition state and by (2) the classical Marcus-Jortner model and assuming a spherical cavity for the charge-separated state. Using the first model, we show that lnk(CS) should vary linearly with Deltaf [Deltaf = (epsilon(r) - 1)/(2epsilon(r) + 1) - (n(2) - 1)/(2n(2) + 1), where epsilon(r) and n represent the static dielectric constant and the refractive index of the solvent, respectively], in accordance with experimental results. The second model, where the reorganization energy scales linearly with Deltaf, provides quantitative agreement with experimental rate constants within a factor of 2.     

Invited review solvatochromic shifts: The influence of the medium on the energy of electronic states

The displacement of electronic absorption and luminescence spectra (solvatochromic shifts) are related to the solute—medium interactions. These interactions can be non-specific (dielectric interactions) when they depend only on multiple and polarizability properties of the solute and solvent molecules; but specific associations such as hydrogen bonding can also be important.

A number of examples of solvatochromic shifts are shown and discussed according to the various solute—medium interactions. The properties of solvent mixtures and those of rigid media are considered, as well as the “thermochromic shifts” which result from the change in the temperature of the medium.

The use of solvatochromic shifts for the determination of the dipole moment and of the polarizability of electronically excited molecules has been important for an understanding of electron distribution changes in such states; examples of such determinations are given, together with references to the original literature.

In the final section some limitations of the theories of solvent shifts and possible improvements are discussed.     

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.