J. Phys. Chem. B 2008, 112, 433-440
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Theoretical Study of Temperature and Solvent Dependence of the Free-Energy Surface of the Intramolecular Electron-Transfer Based on the RISM-SCF Theory: Application to the 1,3-Dinitrobenzene Radical Anion in Acetonitrile and Methanol† Norio Yoshida,* Tateki Ishida, and Fumio Hirata* Department of Theoretical Study, Institute for Molecular Science, Okazaki 444-8585, Japan ReceiVed: August 3, 2007; In Final Form: NoVember 15, 2007
The free-energy surfaces along the intramolecular electron-transfer reaction path of the 1,3-dinitrobenzene radical anion in acetonitrile and methanol are investigated with the reference interaction site model selfconsistent field theory. Although acetonitrile and methanol have similar values of the dielectric constant, the free-energy profiles are quite different. In the methanol solution, the charge is strongly localized on one of the nitrile substituents due to a strong hydrogen bond between 1,3-dinitrobenzene and the solvent, while the polarization is not so large in the case of acetonitrile. The temperature dependence of the reorganization energy, the coupling strength, and the activation barrier is evaluated in both acetonitrile and methanol. The reorganization energy and the activation barrier decrease with increasing temperature for both cases. The electronic coupling strength also shows a similar tendency in the temperature dependence; it increases with increasing temperature in both solvents but with different rates. The behavior is explained in terms of the strong polarization induced by the hydrogen bond between the solute and solvent in the methanol solution.
1. Introduction Electron-transfer (ET) reactions are among the most fundamental chemical reactions. An epoch making theory was proposed for the reaction by R. A. Marcus few decades ago.1-9 The most important contribution made by Marcus is that he took account of the effect of solvent fluctuation upon the quantum process or ET. The theory has been applied for explaining many experimental results and has made a great contribution in clarifying the reactions. Although the Marcus theory is of such importance in science, it is essentially a phenomenological theory, and two problems have to be solved in order to make it into a molecular theory. First, the theory employs the dielectric continuum model for describing solvent fluctuation. This is a defect that can be fatal for describing a chemical reaction, which, of course, concerns the specificity of molecules and atoms. The other problem inherent in the Marcus theory is that it is limited to the nonadiabatic ET in weak electronic coupling or the adiabatic ET in the very strong electronic coupling. The approaches which cover from the weak to strong electronic coupling regimes still meet challenges. One of the limits in the Marcus theory is the theory employing the dielectric continuum model for describing solvent fluctuation. The solute-solvent local interactions, such as the hydrogen bond, which are ignored in the dielectric continuum model, play an important role in determining the property of ET reaction.10 The reference interaction site model self-consistent field (RISMSCF) theory, which is the statistical mechanical integral equation for the theory of liquids combined with ab initio molecular orbital theory, has been successfully applied to reproduce the solvent effect on the solute electronic structure even for hydrogen-bonding systems11-13 Chong et al. proposed the †
Part of the “James T. (Casey) Hynes Festschrift”. * To whom correspondence should be addressed. E-Mail: noriwo@ ims.ac.jp (N.Y.);
[email protected] (F.H.).
Figure 1. Diagrammatic illustration of the method to calculate the nonequilibrium free energy.
method to evaluate the nonequilibrium free-energy profile due to the solvent fluctuation, which has been applied to the ET reaction.14 Sato et al. have extended Chong’s method to quantum mechanical systems to describe the solvent reorganization process.15-18 The importance of temperature dependence of the properties of ET reaction, such as reorganization energy, the ET matrix element, and the activation energy, was addressed by both the experimental and theoretical communities.19-29 It is known that the temperature dependence of the reorganization energy of an ET reaction may not be described well by Marcus theory since the theory employs the cavity size as a boundary condition.28,30-34 Chong’s method has no such problem because the method is based on the RISM theory, which is free from ambiguous parameters such as cavity size. In the present paper, we propose a procedure similar to Chong and Sato’s method to investigate the intramolecular electron-
10.1021/jp076219i CCC: $40.75 © 2008 American Chemical Society Published on Web 12/06/2007
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Yoshida et al. TABLE 2: The Coupling Strength, Reorganization Energy, and Activation Barrier; Units are kcal mol-1. The Values of Density, G, for Each Temperature Were Estimated from the Experimental Value by the Least-Square Fitting Manner52 solvent
T [K]
F [g cm-3 ]
HAB
λ
∆Gq
acetonitrile
230 270 298 330 350 270 298 330 350
0.8526 0.8078 0.7762 0.7407 0.7183 0.8134 0.7867 0.7565 0.7374
1.36 1.48 1.56 1.65 1.70 0.84 1.07 1.30 1.44
43.42 42.57 42.02 41.40 41.05 69.52 68.24 66.72 65.74
15.35 14.84 14.49 14.12 13.91 28.05 26.89 25.69 24.95
methanol
Figure 2. The scheme of the intramolecular electron-transfer reaction of the 1,3-dinitrobenzene radical anion.
numerical application of the theory to the 1,3-dinitrobenzene radical anion in acetonitrile solution. Concluding remarks are given in section 4. 2. Theory
Figure 3. The electron density of the SOMO in the 1,3-dinitrobenzene radical anion in the gas phase is described. The numbers in the figure are the bond lengths optimized by the ROHF/6-31+G* method in the gas phase (top), in acetonitrile (middle), and in methanol (bottom).
TABLE 1: Solute and Solvent Parameters for the RISM Calculation σ/Å
/kcal mol-1
q/e
solute H C N O
2.420 3.550 3.250 2.960
0.030 0.070 0.120 0.170
acetonitrile CH3 C N rCH3-C/Å rC-N/Å
3.775 3.650 3.200 1.458 1.157
0.207 0.150 0.170
0.150 0.280 -0.430
methanol CH3 O H rCH3-O/Å rO-H/Å ∠COH/°
3.775 3.070 1.000 1.425 0.945 108.5
0.207 0.170 0.056
0.265 -0.700 0.435
transfer reaction. The intramolecular electron transfer in a 1,3dinitrobenzene radical in acetonitrile and methanol solutions is considered as the first application of the approaches. Since acetonitrile and methanol have a similar value for the dielectric constant, the molecular theory which can describe the molecular aspect of the solvent is required to evaluate the appropriate solvent effect on the solute electronic structure. The system seems to be the best test example of the approaches since there are many experimental reports with which to be compared.35-41 In this report, we put special attention on the temperature dependence of the free-energy profiles of the intramolecular electron-transfer reaction. This paper is organized as follows. In section 2, the theory and the computational details are summarized. Section 3 presents
2.1. Nonequilibrium Free Energy. In order to evaluate the adiabatic free-energy profile along the reaction coordinates, we propose the extension of the theory, which is the method proposed by Sato et al. based on Chong’s method to evaluate the diabatic nonequilibrium free-energy surface.14-18 The free energy of the system is changed due to the fluctuations of the solvent distribution and those of the solute structure. The free energy can be expressed as a function of the solute coordinate and solvent distribution.42 The basic idea is to identify a solvent fluctuation, or the solvent distribution which is not in equilibrium with a solute state, as the solvent distribution which is in equilibrium with a hypothetical state of the solute. Then, using the hypothetical charge and structure of the solute as an order parameter, one can produce a fluctuating solvent distribution from an equilibrium solvation free energy. It is different from the case of a diabatic surface, in the sense that the solute electronic structure is assumed to relax to the state corresponding to the nonequilibrium solvent distribution. The deviations of free energy at the equilibrium point can be expressed as follows
∆F(Q,s) ) ∆E(Q,s) + ∆Evac(Q,s) + ∆µ(Q,s) ∆µ(Qr,sr) - ∆Ereorg(Qr,sr) (1) where Q and s denote a set of the coordinates of solute atoms and solvent coordinates corresponding to solvent fluctuation, respectively. ∆E and ∆µ are the differences of the energy and the excess chemical potential. The details of these quantities are mentioned below. The diagrammatic descriptions of the energy components are shown in Figure 1. We assume the five states of solute molecule. The state I denotes the equilibrium state of solute in solution. In this state, the solute conformation, Qr, as well as the electronic structure, Ψ(Qr,sr), and the solvent distribution, sr, are completely relaxed. The state V is the target state to consider the nonequilibrium free energy. In this state, the solute electronic structure is relaxed corresponding to the solvent distribution and solute conformational change. In order to calculate the free-energy change between the state I and V, we assume three expedient states. The state II is the equilibrium state in the gas phase with solute conformation Qr. The state III is a hypothetical state with the hypothetical wave functions, Ψ ˜ (Q,s) in the gas phase. The state IV has the same conformation and electronic wave functions as those of the state III, but it is in solution. The solvent distribution in the state IV is relaxed to the hypothetical solute electronic structure and conforma-
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Figure 4. Free-energy profile of the intramolecular electron-transfer reaction of DNB- at 298 K in (a) acetonitrile and in (b) methanol. Squares and triangles are the free energy of the ground and first excited state, respectively. The reaction coordinate corresponds to ξ defined in eqs 11 to 12.
Figure 5. Temperature dependence of the coupling strength in acetonitrile (square) and in methanol (circle).
tion. The definitions of the each energy component are as follows
∆Evac(Q,s) ) 〈Ψ ˜ (Q,s)|Hiso|Ψ ˜ (Q,s)〉 〈Ψvac(Qr)|Hiso|Ψvac(Qr)〉 (2) ∆E(Q,s) ) 〈Ψ(Q,s)|Hiso + Vˆ (Q,s)|Ψ(Q,s)〉 ˜ (Q,s)〉 (3) 〈Ψ ˜ (Q,s)|Hiso + Vˆ (Q,s)|Ψ ∆Ereorg(Qr,sr) ) 〈Ψ(Qr,sr)|Hiso|Ψ(Qr,sr)〉 〈Ψvac(Qr)|Hiso|Ψvac(Qr)〉 (4) ∆µ(Q,s) )
4π β
F
∫ r2dr ∑ Rγ
[
1
2
2 2 (Q,s;r) - cRγ (r) hRγ
1 2
]
hRγ(Q,s;r)cRγ(Q,s;r) (5)
where Ψ, Hiso, and Vˆ denote the electronic wave function of the solute molecule, the Hamiltonian of the isolated system, and solvent electronic potential, respectively. Ψvac(Qr) denotes the electronic wave function of a molecule in the gas phase with solute conformation Qr; hRγ(Q,s;r) and cRγ(Q,s;r) are the total and direct correlation functions between solute site R and solvent site γ with distance r, and F is the bulk density of the solvent. The correlation functions, hRγ(Q,s;r), cRγ(Q,s;r), and gRγ(Q,s;r) are evaluated by the RISM integral
Figure 6. Effective charge on each atom of DNB- in acetonitrile solution. The numbers in parentheses are those in methanol solution. The sum of the charges on the charged group is -0.87 in acetonitrile and -1.07 in methanol, and those on the uncharged group are -0.25 in acetonitrile and -0.22 in methanol.
equation.43 In the present study, we employed the hyper-netted chain (HNC)-like closure to solve the RISM equation as follows
gRγ(Q,s;r) ) exp[-βuRγ(Q,s;r) + hRγ(Q,s;r) cRγ(Q,s;r)] (6) where the solute-solvent interaction potential uRγ(Q,s;r) is the sum of the Coulomb and Lennard-Jones interaction potentials and β ) 1/kBT where kB is the Boltzmann constant and T the temperature. We introduced the hypothetical wave function Ψ ˜ , which reproduces the set of the hypothetical charges q˜ . A hypothetical charge and wave functions are related by
q˜ (s)‚V(Q,s) )
∑R q˜ R ∑γ ∫
qγ r
gRγ(Q,s;r)dr
) 〈Ψ ˜ (Q,s)|Vˆ (Q,s)|Ψ ˜ (Q,s)〉 ) where
∑R 〈Ψ˜ (Q,s)|bˆ R|Ψ˜ (Q,s)〉 ∑γ ∫
qγ r
gRγ(Q,s;r)dr
(7)
436 J. Phys. Chem. B, Vol. 112, No. 2, 2008
Vˆ (Q,s) )
∑R ∑γ ∫ bˆ R
Yoshida et al.
qγ r
gRγ(Q,s;r)dr
(8)
and bˆ R denotes the operator to reproduce the partial charge on the solute site R and qγ is the partial charge of solvent site γ. By using these definitions, eq 1 is reduced to
∆F(Q,s) ) 〈Ψ(Q,s)|Hiso|Ψ(Q,s)〉 〈Ψ(Qr,sr)|Hiso|Ψ(Qr,sr)〉 + ∆µ(Q,s) ∆µ(Qr,sr) + (q(Q,s) - q˜ (s))‚V(Q,s) (9) Eventually, the hypothetical wave functions disappear from the equation. The hypothetical charge, q˜ , is simply defined as follows14
1-s s+1 q˜ R(s) ) qR(-1) qR(1) 2 2
(10)
where the solvation coordinates s ) sr ) -1 and s ) sp ) 1 denote reactant and product states, respectively. The nonequilibrium radial distribution function, g(Q,s;r), as well as the total and direct correlation functions, h(Q,s;r) and c(Q,s;r), are evaluated by the RISM integral equation with respect to the hypothetical charge, q˜ , and the solute coordinate, Q. The nonequilibrium wave function of solute molecule, Ψ(Q,s), should be calculated with g(Q,s;r), which is fixed until the calculation. The partial charges, q(Q,s), are calculated from Ψ(Q,s). 2.2. Computational Details. The theory mentioned above is applied to the intramolecular electron-transfer reaction of the 1,3-dinitrobenzene radical anion (DNB-) in acetonitrile and methanol solutions. The scheme of the electron-transfer reaction of DNB- is shown in Figure 2, which is a symmetrical intramolecular reaction. The geometry optimization of the reactant with Cs symmetry was carried out by the restricted open-shell Hartree-Fock method with the 6-31+G* basis set.44 The mirror image of the optimized geometry of the reactant was employed as those of the product. The equilibrium free energy of DNB- in solution was evaluated by the RISM-SCF/ MCSCF method.11-13 In the present study, the ground and first excited state of DNB- should be considered. In order to calculate the ground and excited state, the state-averaged (SA) complete active space (CAS) SCF wave functions45-48 with the 6-31+G* basis set were employed. In the RISM-SCF/MCSCF formalism derived by Sato et al., the free energy of solvated molecules was analytically expressed as a functional of solute-solvent correlation functions and solute electronic wave functions. However, in the case of SA-CASSCF, the free energy of solvated molecules could not be formulated rigorously. If the excess chemical potentials of all of the states are equal, the free energy of solvated molecules can be evaluated and the SACASSCF and RISM calculations can be carried out simultaneously. Therefore, in the present study, we assume the excess chemical potentials of all of the states are equal. The partial charges on solute atoms were evaluated by the ground-state density matrix, which was obtained from the SACASSCF with the solvated Fock matrix. The solvated Fock matrix was calculated from the RDFs and solute-solvent pair potential with the partial charges mentioned above. The CASSCF wave functions were constructed by distributing five electrons in the six active orbitals, that is, two nitrogen π and two carbon π and π * orbitals. The state averaging was performed over two valence states with equal weights. In the present study, we assume the simple one-dimensional reaction
coordinate and the two state model for simplicity. The multidimensional reaction coordinate was reduced to a onedimensional coordinate, ξ, by
s)ξ QR(ξ) )
1-ξ ξ+1 QR(-1) QR(1) 2 2
(11) (12)
where QR(-1) and QR(1) are coordinates of the reactant and solvent, respectively. Thus, the hypothetical charge, q˜ , are defined by a one-dimensional coordinate, ξ, as follows
q˜ R(ξ) )
1-ξ ξ+1 qR(-1) qR(1) 2 2
(13)
where qR(-1) and qR(1) are equilibrium charges of the reactant and solvent, respectively. The free energy of each state is evaluated by
∆FX(Q,s) ) 〈ΨX(Q,s)|Hiso|ΨX(Q,s)〉 〈Ψ(Qr,sr)|Hiso|Ψ(Qr,sr)〉 + ∆µ(Q,s) ∆µ(Qr,sr) + (qX(Q,s) - q˜ (s))‚V(Q,s) (14) where subscript X denotes the ground (I) and excited (II) states. We implemented the RISM-SCF/MCSCF method on the general atomic and molecular electronic structure system (GAMESS) code.49 The RISM calculations were carried out with the parameters given in Table 1. The parameters of solvent and solute molecules were taken from the OPLS parameter set.50,51 The solvent density which we employed is summarized in Table 2. All RISM calculations were performed on a grid of 2048 points with the grid spacing of 0.05 Å. The details of the procedure of the RISM calculation can be found in ref 43. 3. Results and Discussions In this section, we present the results of free-energy calculations of the intramolecular electron-transfer reaction of DNBboth in acetonitrile and methanol, following the brief note about the geometry of the DNB- molecule. The discussions are focused on the solvent effects and the temperature dependence of the coupling strength, reorganization energy, and activation barrier. The solvent distribution around DNB- is examined to find the microscopic origin of the solvent effects and the temperature dependence. 3.1. Geometry Optimization. In Figure 3, bond lengths in DNB- optimized by the ROHF/6-31+G* method in the gas phase are shown. Now, we consider the C-N distances of the nitro group with and without an excess electron, respectively. Hereafter, “charged” group denotes the former and “uncharged” the latter (see Figure 6). The C-N distance in the uncharged group is larger than that is the charged one, while the N-O bond is shortened. These results are in accord with the previous UHF and CASSCF calculations.40,53 The single-occupied molecular orbital (SOMO) is also shown in Figure 3. From the analysis of the wave function obtained, the C-N distance was reduced, coinciding with the π-bond character in the charged groups, while the N-O bond length increased due to the π*bond character. We also performed the geometry optimization of DNB- in acetonitrile and methanol solution at several temperatures. The tendency stated above became more significant in the solution. In Figure 3, bond lengths in gas phase and each solution at 298 K were compared. The C-N distance in the uncharged group decreased with solvation. On the other hand, the distance in the charged group increased with solvation.
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Figure 7. The distribution of solvent around the oxygen of the charged and uncharged nitro groups at 298 K. The distributions of nitrogen and the nitrile carbon of acetonitrile are depicted in (a) and (b), and those of hydrogen and oxygen of methanol are in (c) and (d), respectively. The solid and dashed lines are RDFs of charged and uncharged groups, respectively.
3.2. Free-Energy Profile. In Figure 4, the free-energy profiles of intramolecular electron transfer of DNB- in acetonitrile and methanol at 298 K are shown. The curvature of the free energy surface in methanol is much greater than those in acetonitrile, giving rise to the larger reorganization energy, λ, and activation barrier, ∆Gq, of DNB- in methanol compared with those in acetonitrile. These trends are in accord with those in reported experimental results.35,36 Since the reaction coordinate of the free-energy surface corresponds to the degree of the solvent fluctuation, this result can be interpreted in terms of the interaction between solute and solvent. It is easy to understand that the strong interaction in methanol originates from the hydrogen bond between solute DNB- and solvent methanol. The hydrogen bond effects enhance the polarization of the electronic structure of a solute molecule. Thus, this corresponds to raising the solute energy of the product state.54 It should be noted that the reorganization energy includes the energy of the solute electronic structure. Even though the extent of solvent fluctuation is reduced via the hydrogen bond to some extent, the total reorganization energy becomes larger. The details of the solvation structure are discussed in the next subsection. Since the solute electronic structure considering the electronic energy difference between two states related to ET reaction in the solute can play an important role in determining the relative position of free-energy curves, as pointed out in the other study with empirical model,54 we expect that this effect makes significant contribution to the free-energy profiles obtained from an ab initio level calculation. Usually, in most works on ET reactions, this effect has been neglected. However when it is included, the crossing point of the reactant and product state
curves will be shifted, and the activation barrier height will be modified. Also, the polarization effect is likely to make the freeenergy profiles deviate from a parabolic curve.54 These factors will modify the relation between the reorganization energy and the activation barrier. Our results in Table 2 indicate these clearly. In the table, the reorganization energy and activation barrier are summarized. From the viewpoint of the standard Marcus theory, which does not include the solute electronic structure explicitly, the activation free energy is equal to λ/4 for a symmetric ET reaction. However, the barriers for both acetonitrile and methanol solvents are much higher than λ/4. Therefore, it is considered that these differences are caused entirely by the effect of polarization of the solute electronic structure. Also, in Table 2, the temperature dependence of the coupling strength, HAB, the reorganization energy, λ, and the activation barrier, ∆Gq, is summarized. The free-energy profiles for each temperature are estimated in the same way, as shown in Figure 4. For all of the temperatures, the free-energy profiles show a similar tendency, and the difference is relatively small. Therefore, we only show representative free-energy profiles, but the important values are given in the table. Both the reorganization energy and the activation barrier decrease with increasing temperature for both solvents. It is because the solvent fluctuation is promoted by increasing temperature, which makes the curvatures of the free-energy surfaces smaller. The temperature dependence of the reorganization energy and activation barrier in methanol is greater than those in acetonitrile solution. In our formalism, the temperature dependence of the free-energy profiles is included through the prefactor of the solute-solvent interaction potential, β, in eq 4. Therefore, in the case of the
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Yoshida et al.
Figure 8. Temperature dependence of the distribution of solvent around the oxygen of the nitro group at 270K (dotted lines), 298 K (solid lines), and 330 K (dashed lines). The distribution of nitrogen and the nitrile carbon of acetonitrile are depicted in (a) and (b), and those of hydrogen and oxygen of methanol are in (c) and (d), respectively.
present study, the methanol solution system, which has larger solute-solvent interactions, shows greater temperature dependence. To examine the behavior of the coupling strength, the quantity relative to that at 298 K, ∆HAB(T) ) HAB(T) - HAB(298 K), is plotted against 1/T (see Figure 5). The coupling strength increases with increasing temperature in both solvent systems. In order to clarify this behavior, the dipole moment of DNBat the transition state for the ground and first excited states is considered. Here, a center of mass of solute DNB was employed as the origin of the dipole moment. The magnitude of coupling strength HAB corresponds to the height of the free energy relative to that of ground state at the transition point, ξ ) 0 (see Figure 2). In acetonitrile, the dipole moment of DNB- for the ground state at 298 K is 5.16 Debye, and that for the first excited state is 8.36 Debye. Therefore, the first excited state which has larger polarization is likely to be stabilized by solvation more than the ground state is stabilized. With rising temperature, ∆HAB in acetonitrile increases and approaches that in gas phase, 3.56 kcal mol-1, because the solvent effect decreases. In methanol, the temperature dependence of HAB has same tendency with that in acetonitrile. The dipole moment of DNB- for the ground state at 298 K is 7.77 Debye, and for the first excited state, it is 10.26 Debye in methanol. Since the methanol system shows strong solute-solvent interactions, as mentioned above, the temperature dependence of HAB in methanol is greater than that in acetonitrile solution. 3.3. Solvation Structure. As implied in the previous subsection, the behavior in the electronic structure has its origin in the coupling between intra- and intermolecular processes or the effect of solvent fluctuation on the electronic structure. Par-
ticularly important is the difference between methanol and acetonitrile in polarizing the electronic structure of DNB-. In Figure 6, the partial charges of the equilibrium reactant state on the nitro group, which were determined so as to reproduce the electrostatic potential around the solute DNB-, are shown in methanol and acetonitrile. The difference of the sum of charges on the nitro group between charged and uncharged groups is -0.62 and -0.90 in acetonitrile and methanol at 298 K, respectively. The total charges on the uncharged nitro group in the two solvents are very close to each other. On the other hand, the total charges on the charged nitro group is quite different between the two solvents. It is obvious that this difference in the polarization of DNB- between the two solvents has its origin in the difference of the strength of the temperature dependence of HAB in the two solvents. Then, the next question is from where does the difference in the polarization between the two solvents originate. Depicted in Figure 7 are the radial distribution functions (RDFs) between the oxygen of the solute nitro group and the solvent sites. High and sharp peaks are found in methanol, Figure 7c and d, while only weak peaks are seen in the case of acetonitrile solution, Figure 7a and b. The sharp peaks imply that the oxygen in the nitro group and the hydrogen of MeOH are forming a strong hydrogen bond. Such a hydrogen bond suppresses the solvent fluctuation, and thus, the free energy curve in the case of methanol becomes steeper than that in acetonitrile. Since both oxygen in the nitro groups and nitrogen in acetonitrile have negative charges, the interaction between those atoms is repulsive. Thus, the RDF of the uncharged site has a weak peak at 3.5 Å in Figure 7a, while the peak disappears entirely in the case of the charged site. The interaction between
Study of the Intramolecular ET based on RISM-SCF the solute oxygen and carbon of MeCN is attractive, and the charged site shows a conspicuous peak at 3.2 Å. In the case of the methanol solution, oxygen in the solute and H in MeOH form the strong hydrogen bond as mentioned above, which is manifested in a well-defined peak at 1.8 Å; see Figure 7c. Although the height is smaller, such a peak that originated through hydrogen bond is also found in the uncharged group. The RDFs between oxygen in the solute and oxygen in MeOH, as seen in Figure 7d, show the same trends as those of the oxygen-hydrogen case mentioned above, in which the RDF of the charged group has a larger first peak, though oxygen in both the solute and solvent has a negative charge. The large first peak resulted from the hydrogen bond between the solute and the solvent, through the intramolecular correlation on the “bond” between oxygen and hydrogen atoms in the solvent. In Figure 8, the RDFs at different temperature are compared. In all RDFs, the peaks become lower with increasing temperature since the solute-solvent interaction is weakened. Decreasing the peak height means that the solvent fluctuates easily; therefore, both the reorganization energy and activation barrier are decreased. The dependence of the RDFs on temperature is quite large in methanol, while that in acetonitrile is small. That explains the reason why the temperature dependence of the reorganization energy and activation barrier in methanol is much higher than those in acetonitrile solution. 4. Conclusion In order to estimate the free-energy profile along the reaction coordinate of the intramolecular electron transfer of DNB-, we employed the method to evaluate the nonequilibrium free energy developed by Sato and Chong et al., with an extension of the theory to evaluate the adiabatic free-energy profiles. The numerical calculations of the profiles along the reactions in acetonitrile and methanol were carried out. The reorganization energy and activation barrier in methanol became larger than those in acetonitrile. The phenomenon is attributed to the hydrogen bond between the solute nitro group and hydrogen of MeOH, which suppresses the solvent fluctuation. The temperature dependence of the free-energy profiles were also considered. Both the reorganization energy and activation barrier were decreased with increasing temperature for both solvents. The results are in accord with those from the experiments. The electronic coupling strength also shows the same behavior in the temperature dependence. It increases with increasing temperature in both solvents. The behavior is also explained by the strong polarization induced by the hydrogen bond between the solute and solvent in the methanol solution. The method proposed in the present paper can provide essentially all of the free-energy properties which are required to calculate the rate constant of the reaction from the theoretical expression proposed by Zhu and Nakamura.55-61 It is of great interest to combine the approach presented in the present paper with the Zhu-Nakamura theory to evaluate the electron-transfer rate which involves the nonadiabatic transition among the adiabatic as well as diabatic free-energy surfaces. For the numerical test in the present study, we employed several assumptions for simplicity. In order to carry out the SACASSCF calculation combined with RISM-SCF formula, we employed an assumption that the excess chemical potentials of ground and excited state are equal each other since the stateaveraged electronic structure with the RISM-SCF has not been formulated, yet. One way to consider such cases is with the linear response free-energy method proposed by Yamazaki et al.62 Their approach is appropriate to tackle the state-averaging.
J. Phys. Chem. B, Vol. 112, No. 2, 2008 439 We employed the simple one-dimensional reaction coordinate by using the least linear motion technique. The solvent fluctuation and the coordinates of the solute atoms should be treated separately, and the transition state and reaction coordinate have to be founded by a more appropriate formula. The extension and improvement of our approach along those directions stated above are in progress in our group. Acknowledgment. The authors are grateful to Professor Hiroki Nakamura (Institute for Molecular Science, Japan) and Professor Yi Zhao (University of Science and Technology of China, China) for their helpful discussion. The authors are grateful to one of the anonymous reviewers for suggesting very helpful comments on the importance of geometry change of the solute molecule in the solution phase. This work is supported by the Grant-in Aid for Scientific Research on the Priority Area of “Water and Biomolecules” of the Ministry of Education in Japan, Culture, Sports, Science and Technology (MONBUKAGAKUSHO). We are also supported by the grant from the Next Generation Supercomputing Project, Nanoscience Program of the Ministry. Numerical calculations were partly carried out in the Research Center for Computational Science, Institutes of Natural Science. References and Notes (1) Marcus, R. A. J. Chem. Phys. 1956, 23, 966. (2) Marcus, R. A. J. Chem. Phys. 1956, 23, 979. (3) Marcus, R. A. J. Chem. Phys. 1957, 26, 867. (4) Marcus, R. A. J. Chem. Phys. 1957, 26, 872. (5) Marcus, R. A.; Sutin, N. Biochim. Biophys. Acta 1985, 811, 265. (6) Marcus, R. A. Discuss. Faraday Soc. 1960, 29, 21. (7) Marcus, R. A. J. Chem. Phys. 1965, 43, 679. (8) Marcus, R. A. J. Phys. Chem. 1968, 72, 891. (9) Marcus, R. A. Annu. ReV. Phys. Chem. 1964, 15, 155. (10) Fonseca, T.; Ladanyi, B. M. J. Mol. Liq. 1994, 60, 1. (11) Ten-no, S.; Hirata, F.; Kato, S. Chem. Phys. Lett. 1993, 214, 391. (12) Ten-no, S.; Hirata, F.; Kato, S. J. Chem. Phys. 1994, 100, 7443. (13) Sato, H.; Hirata, F.; Kato, S. J. Chem. Phys. 1996, 105, 1546. (14) Chong, S.; Miura, S.; Basu, G.; Hirata, F. J. Phys. Chem. 1995, 99, 10526. (15) Sato, H.; Kobori, Y.; Tero-Kubota, S.; Hirata, F. J. Chem. Phys. 2003, 119, 2753. (16) Sato, H.; Kobori, Y.; Tero-Kubota, S.; Hirata, F. J. Phys. Chem. B 2004, 108, 11709. (17) Sato, H.; Hirata, F. J. Phys. Chem. A 2002, 106, 2300. (18) Sato, H.; Hirata, F. J. Am. Chem. Soc. 1999, 121, 3460. (19) Finckh, P.; Heitele, H.; Volk, M.; Michel-Beyerle, M. E. J. Phys. Chem. 1988, 92, 6584. (20) Heitele, H.; Finckh, P.; Weeren, S.; Pollinger, F.; Michel-Beyerle, M. E. J. Phys. Chem. 1989, 93, 5173. (21) Liang, N.; Closs, G. L.; Miller, J. R. J. Am. Chem. Soc. 1989, 111, 8740. (22) Hupp, J. T.; Neyhart, G. A.; Meyer, T. J.; Kover, E. M. J. Phys. Chem. 1992, 96, 10820. (23) Asahi, T.; Ohkohchi, M.; Matsusaka, R.; Mataga, N.; Zhang, R. P.; Osuka, A.; Maruyama, K. J. Am. Chem. Soc. 1993, 115, 5665. (24) Kroon, J.; Oevering, H.; Verhoeven, J. W.; Warman, J. M.; Oliver, A. M.; Paddon-Row, M. N. J. Phys. Chem. 1993, 97, 5065. (25) Harriman, A.; Heitz, V.; Sauvage, J.-P. J. Phys. Chem. 1993, 97, 5940. (26) Cortes, J.; Heitele, H.; Jortner, J. J. Phys. Chem. 1994, 98, 2527. (27) Troisi, A.; Ratner, M. A.; Zimmt, M. B. J. Am. Chem. Soc. 2004, 126, 2215. (28) Manjari, S. R.; Kim, H. J. J. Chem. Phys. 2006, 125, 011101. (29) Manjari, S. R.; Kim, H. J. J. Chem. Phys. 2005, 123, 014504. (30) Vath, P.; Zimmt, M. B.; Matyushov, D. V.; Voth, G. A. J. Phys. Chem. B 1999, 103, 9130. (31) Vath, P.; Zimmt, M. B. J. Phys. Chem. A 2000, 104, 2626. (32) Kumar, K.; Kurnikov, I. V.; Beratan, D. N.; Waldeck, D. H.; Zimmt, M. B. J. Phys. Chem. A 1998, 102, 5529. (33) Read, I.; Napper, A.; Zimmt, M. B.; Waldeck, D. H. J. Phys. Chem. 1994, 98, 5152. (34) Manjari, S. R.; Kim, H. J. J. Phys. Chem. B 2006, 110, 494. (35) Grampp, G.; Shohoji, M. C. B. L.; Herold, B. J. Ber. Bunsen-Ges. Phys. Chem. 1989, 93, 580.
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