On the Influence of Dielectric Saturation and Medium Granularity on

5302

J. Phys. Chem. B 2002, 106, 5302-5311

On the Influence of Dielectric Saturation and Medium Granularity on Ion-Ion Interaction and Reorganization Energies Associated with Electron-Transfer Processes in Condensed Matter Gunnar Karlstro1 m* Department of Theoretical Chemistry, Chemical Centre, UniVersity of Lund, P.O. Box 124, S-221 00 Lund, Sweden ReceiVed: July 24, 2001; In Final Form: December 20, 2001

The influence of dielectric saturation effects and the granularity of a medium on the effective ion-ion interaction and the reorganization energy for electron-transfer processes have been studied using a medium built from dipoles and polarizabilities that are placed on a lattice. It was found that these effects lead to an increased repulsion between equally charged ions and an increased attraction between oppositely charged ions at short distances. It was also found that the estimate of the outer-sphere reorganization energy contribution to the activation energy for electron-transfer processes, made by Marcus and Hush, is at least a factor of 2 too large for small ions with large charges.

1. Introduction All molecules interact, and a particularly simple form or interaction is that between two charged ions. According to the law of Coulomb, it can be written

Uint )

q1q2 r12

(1)

if atomic units are used. (Atomic units will be used in this work.) In eq 1, U is the interaction energy between the charges qi and rij is the distance between the ions. The equation is of a fundamental nature and applicable to the direct interaction between the ions as long as the electronic clouds associated with the ions do not overlap. Frequently we ask, what is the effective interaction between two ions in a solution or a crystal? This is a much more complicated question. The direct interaction between the two ions can naturally still be calculated using eq 1. However, the interaction between the surrounding molecules, as well as the interaction between the ions and the surrounding molecules, also changes as the interionic distance is changed. Thus, the effective interaction is a result of many opposing interactions. Despite this complicated situation, it can be shown that the effective interaction between two ions in a condensed phase can be written

Uint )

q1q2 r12

(2a)

Equation 2a is similar to eq 1 apart from the constant (the dielectric permittivity of the medium, frequently called the dielectric constant of the medium). The essence of eq 2a is that the field from one of the ions acting on the other ion is damped by a factor of because of the presence of the medium molecules. If eq 2a is to be valid, it is required that the distance between the ions should be sufficiently large so that the granularity of the medium can be ignored. This requirement must, in one way or another, relate to the size of the medium * E-mail: [email protected].

particles. Consequently, when the distance between the ions is much larger than the size of the medium molecules, we can expect eq 2a to hold. There is, however, another type of complication that is possible. This complication occurs when the ionic charge is so large that the dipoles in the medium are not capable of damping the interaction as described by eq 2a. In this case, eq 2a is valid provided that the charges q are small enough but is not valid for larger charges. The interaction between an ion and the particles constituting the medium results in the solvation of the ion, and the solvation energy of an ion in the dielectric medium can be written

Usolv ) -

- 1 q2 2 r

(2b)

In eq 2b, r is a measure of the size of the ion in the medium. The purpose of this work is to study the deviations from the ideal dielectric behavior induced by the particle nature of a medium. To do so, a three-dimensional lattice of polarizabilities and ideal dipoles will be used. To handle the long-range Coulomb interaction, the lattice sites are embedded in a dielectric medium. The studied system is described in Figure 1. A similar approach has previously been used by Warshel and co-workers to study the solvation of larger molecules, as well as for more methodological studies.1-6 The concept of dielectric saturation has previously been studied using a model that is to the one used in this work, in connection with electron transfer in the Fe2+/Fe3+ system.7 The purpose of that study was to model the influence of dielectric saturation on the effective Fe2+-Fe3+ interaction. There are several advantages of the model that we used compared to ordinary simulation techniques. First of all, it is possible to solve the polarization problem numerically exactly. Thus, the energetic estimates provided by the model are numerically exact. Second, all of the influence on the effective interaction between the studied ions due to packing effects and coupling between packing and orientation of the solvent molecules is fully controlled. The drawback of the proposed methods is, of course, that the nonphysical modeling of these effects makes the applicability to real systems more

10.1021/jp012843y CCC: $22.00 © 2002 American Chemical Society Published on Web 04/23/2002

Electron-Transfer Processes in Condensed Matter

Figure 1. Schematic illustrating the studied system.

complicated. Apart from the fundamental aspects discussed above, the concept of dielectric saturation can be of great importance not only for electron-transfer processes as indicated above but also for the solvation of highly charged ions in general. The purpose of this work is to make the previous study7 of the influence of dielectric saturation on effective ion-ion interaction complete by also studying the effect on the effective interaction between ions with opposite charges but also to investigate the influence of dielectric saturation on the outersphere contribution to the reorganization energies associated with electron-transfer processes. After this introduction, a description of the physical model and the method used to solve the corresponding mathematical problem numerically will be given. This description will be followed by a presentation of the results obtained together with a discussion of the results. Finally, in the last section, some remarks concerning the importance of the results obtained in this work for real systems will be given. 2. Model and Methods There are, in principle, three ways that a molecule can contribute to the dielectric behavior of a medium. The two most well-known are the electronic polarizability and the orientation of the molecular dipole moments. The third mechanism originates from the fact that the vibrational motions of the atoms in the molecules change the molecular dipole moment. These three mechanisms can normally be easily separated from each other through their response to oscillating electric fields. All modes of motion are able to respond to slowly varying fields, but when the frequency of the field becomes larger than 1012 Hz, then the orientational degrees of freedom are no longer able to respond to the oscillations. When the frequency of the field is larger than 1013 Hz, then the vibrational motion starts to become too slow to adapt to the field, and only the electronic degrees of freedom are capable of fully responding to the field oscillations. The contribution to the dielectric behavior from the orientational degrees of freedom is large for molecules with a significant dipole moment; the contribution from the vibrational degrees of freedom is small, and the electronic contribution is normally rather large. Frequently, the contribution from the vibrational motion is ignored when the dielectric behavior is modeled. This assumption will be made in this work. There is a fundamental difference between the orientational and electronic contributions to the dielectric behavior. A dipole

J. Phys. Chem. B, Vol. 106, No. 20, 2002 5303 cannot orient more than parallel with the applied field, and a further increase in the applied field will not significantly increase the induced effective dipole moment. This process can be described using the well-known Langevin model.8 The induced dipole moment from the electronic polarizability remains almost linear with the applied field for much higher field strengths. This behavior can be illustrated by considering a water molecule. The electronic polarizability of the water molecule is close to 8 au, and if we apply an electric field of 0.004 au, a dipole moment of 0.032 au will be induced. The dipole moment of a water molecule is close to 0.8 au. This value corresponds roughly to an effective polarizability of 210 au. If we calculate the induced dipole moment from the orientational degrees using the field discussed above together with this polarizability, we obtain a dipole moment of 0.84 au, a value that is larger than the water dipole moment and consequently impossible. The actual response to the applied field can be obtained from the Langevin model, which is a statistical mechanical description of a dipole in a field. According to the model, the resulting effective dipole moment from a dipole in the direction of the applied field can be obtained from

(

µ ) coth(y) -

1 µE y 0

)

(3)

where y is given by

y)

µ02E 3kBT

(4)

In these equations, E is the applied field, µ0 is the molecular dipole moment, kB is the Boltzmann constant, and T is the absolute temperature. The limiting polarizability corresponding to a dipole at low fields can be obtained from

R)

µ02 3kBT

(5)

For a full derivation of the Langevin equation or for more information, see , for example, ref 8. A drawback of the Langevin model is that it is a mean-field model and that the “dispersive coupling”, which exists even in the absence of permanent electric fields, is not described by the model. A model of this type has previously been used to model solutions by Warshel and Levitt.9 In this work, a dielectric medium will be built up from a set of polarizable particles placed on a set of lattice points. The primitive cubic three-dimensional lattice is embedded in a dielectric medium. The dielectric permittivity of the surrounding (see Figure 1) is taken from the Clausius-Mossotti (CM) equation,10 which relates the dielectric constant to the polarizability density R/V where V is the volume:

- 1 4π R ) +2 3 V

(6)

The polarizable particles have an ideal polarizability (Ri) and a dipole (µo) described by the Langevin model. The response of the ideal polarizability to an external field (E) is given by

µind ) R i E

(7)

and the response of the dipole is described by eq 3. The polarization equations describing the coupling between the lattice sites and the surrounding dielectric medium are solved

5304 J. Phys. Chem. B, Vol. 106, No. 20, 2002

Karlstro¨m

TABLE 1: Solvation Energies for a Charge Positioned at 0, 0, 0 as a Function of dr, Lattice Size, and a Maximum Value of 1 Used for the Expansion of the Charge Distribution of the Dielectric Mediuma lmax dr

size

4/

0.03 0.03 0.03 0.14 0.14 0.14 0.14 0.25 0.25 0.25 0.25

5 6 7 5 6 7 8 5 6 7 8

0.7177709 0.7176538 0.7166497 0.7157418 0.7163137 0.7156538 0.7156745 0.7138459 0.7150208 0.7146906 0.7149359

5

6/

7

no convergence no convergence 0.7166688 0.7158171 0.7163205 0.7156667 0.7156928 0.7138795 0.7150244 0.7147000 0.7149481

8/

TABLE 2: Difference in Solvation Energy Calculated for the Charge in Points 2, 0, 0, and 0, 0, 0 as a Function of lmax and Lattice Size lattice size lmax value

5

6

7

8

4 5 6 7 8 9

0.001677 0.001428 0.000451 0.000258 0.001501 0.000427

0.000480 0.000451 0.000302 0.000216 0.000387 0.000256

0.000295 0.000258 0.000236 0.000233 0.000130 0.000104

0.000134 0.000126 0.000115 0.000114 0.000114 0.000106

9

no convergence no convergence 0.7171406 .7190091 0.7166869 0.7158332 0.7156938 0.7149211 0.7152093 0.7147927 0.7149484

TABLE 3: Difference in Solvation Energy Calculated for the Charge in Different Points for Different Cavity Sizesa lattice size

a

Each lattice site has an ideal polarizability of 0.18 au if it is assumed that the lattice constant is 1 au. This value corresponds to a dielectric permitivity of 10.194. Note that there is no contribution to the solvation energy from odd l values.

iteratively. The procedure converges slowly for strongly coupled systems (systems with high dielectric constants). Nevertheless, it is possible to solve these equations up to a dielectric constant of 11.7 with this technique for the systems studied here. This value is rather close to values where polarizabilities on a primitive lattice show a phase transition into an antiferromagnetic phase.4 The difficulties involved in solving the polarization at dielectric permittivities are obviously linked to the antiferromagnetic phase, which implies a spontaneous polarization of the Langevin dipole lattice for sufficiently high dipole moments. For more details about the link between forming a dielectric medium from dipoles and polarizabilities, see the excellent discussion by Papazyan and Warshel.4 The data that will be presented here covers dielectric permittivities from 1.86 to 11.7, but the trends are clear and there is no reason to presume that the results cannot be extrapolated to higher permittivities. To include the effect of polarization from far-away lattice points, all lattice points farther away from origin than a certain radius r are modeled by a dielectric surrounding. Consequently, the size of the lattice must be so large that no saturation could be expected from lattice sites that are replaced by the medium. The radius of the dielectric sphere is dr lattice units (l.u.) larger than the distance to the set of lattice points that are most distant from origin (see Figure 1). (The shortest distance between two lattice points is 1 l.u.). The dielectric equations for the surrounding dielectric are solved by expanding the charge distribution at the boundary in spherical harmonics and truncating the expansion at some l value. In Table 1, we present test calculations performed to determine optimal values for dr. In these calculation, an effective site polarizability of 0.18 has been used. (This value corresponds to a dielectric permittivity of 10.194.) Only ideal polarizabilities have been used in the lattice sites to remove the influence from dielectric saturation. There are two effects that must be considered when optimal values are determined for dr and l. First of all, it is desirable that the calculated solvation energy should be independent of the size of the lattice. Second, if dr is made too small and l is made large enough, there is a risk that the calculations diverge because of too strong a coupling between the polarizabilities and the dielectric medium. On the basis of the results presented in Table 1, it has been found reasonable to use a value of 0.14 for dr. It is not possible to determine any optimal value for lmax on the basis of data in Table 1 because there is no influence from the dielectric medium of dipolar and other odd 1-value terms

a

lattice point

7

8

1, 0, 0 1, 1, 0 1, 1, 1 2, 0, 0 2, 1, 0 2, 1, 1 2, 2, 0 2, 2, 1 3, 0, 0 3, 1, 0 3, 1, 1 2, 2, 2 3, 2, 0 3, 2, 1 4, 0, 0

0.000089 0.000173 0.000143 0.000104 0.000286 0.000162 0.000244 0.000054 0.000311 0.000964 0.000746 -0.000067 0.000586 0.000063 0.001276

0.000081 0.000153 0.000165 0.000106 0.000245 0.000250 0.000322 0.000316 0.000175 0.000507 0.000499 0.000367 0.000508 0.000474 0.000524

lmax ) 9, and the 0, 0, 0, point has been used as a reference.

because of the geometry. In all of the calculations presented below, a dr value of 0.14 is used. To determine an optimal value for lmax, a series of calculations were made where the charge has been positioned in the point 2, 0, 0, and the energy difference between when the charge is in 0, 0, 0 and this point has been calculated. The differences obtained are presented in Table 2. From this Table, it is clearly seen that a radius of 7 or 8 lattice units and an lmax value of 9 give very small energy differences between the two geometries. The same lattice-site polarizability as that in Table 1 has been used to construct Table 2. It can also be seen from Table 2 that some influence of numerical instability caused by too strong a coupling between the polarizabilities and the dielectric medium can be observed for the smaller lattices (5 and 6) and the largest lmax values (8 and 9). To further investigate how the energy varies when the charges are moved in the lattice, a series of calculations where the charges have been placed in different lattice points have been undertaken. The results obtained for lmax ) 9 and lattice sizes 7 and 8 are presented in Table 3. From Table 3, it can be seen that both of the studied models can describe these energy differences well. Furthermore, the fact that most of the calculated energy differences are positive indicates that the continuous medium does not solvate the charges as well asthe discrete lattice does. This result may be due to the fact that dr is given a value that is somewhat too large or to the fact that the true of the lattice is larger than the used for the continuum. The possible use of too large a dr is a necessary price that has to be paid to avoid the numerical instabilities that occur when the polarizabilities come too close to the continuous medium. On the basis of the results presented in Tables 1-3, it can be concluded that both systems of sizes 7 and 8 are capable of describing the solvation of a charge, provided that the charge does not come closer to the boundary than 3 lattice units and that an lmax value


J. Phys. Chem. B, Vol. 106, No. 20, 2002 5305

TABLE 4: Calculated Difference in Solvation Energy for an Ion Placed in the Center of a Latticea and the Calculated Solvation Energyb,c charge lattice size

1.0

0.6

0.2

0.1

4

0.10596

5

0.104167

6

0.104007

7

0.104087

8

0.104043

0.028318 0.078661 0.027765 0.077157 0.027743 0.077063 0.027767 0.077130 0.027754 0.077095

0.001003 0.025083 0.000980 0.024491 0.000980 0.024498 0.000982 0.024544 0.000981 0.024536

0.000081 0.008145 0.000079 0.007936 0.000079 0.007947 0.000080 0.007968 0.000080 0.007967

a

In the lattice, half of the polarizability originates from an ideal polarizability, and half originates from a dipole modeled by the langevin model. b The solvation energy was calculated using a lattice where all of the polarizability is ideal. c The magnitude of the ionic charge and the size of the lattice have been varied. The total polarizability for each lattice site is the same as that in previous tables. lmax ) 6. Two values are given for each entry except for charge 1. The upper is the calculated difference and the lower is this difference divided by the square of the used charge.

of 9 is used. Unless it is explicitly stated, a lattice size of 8 and an lmax value of 9 will be used below. So far, nothing has been said about dielectric saturation. In Table 4, however, data is presented that shows the difference in solvation energy between a charge of a given magnitude in a medium built up from ideal polarizabilities and in a medium where half of the polarizability originates from an ideal polarizability and half from a dipole. The total lattice-site polarizability is the same as was used in Tables 1-3 (0.18 au at each lattice site), and thus the total dielectric permittivity equals 10.194. Note that the dielectric medium surrounding the discrete part cannot be saturated. In Table 4, we present the difference in calculated solvation energies together with normalized differences. The normalized difference is the total difference divided by the square of the used charge. An lmax value of 6 has been used so that the smaller clusters also could be studied, and the influence of the cavity size has been investigated. To translate the calculations presented in Table 4 to a real system, the size of a particle must be determined. If we choose a lattice unit to be 6 au (close to 3 Å), which is close to the size of a water molecule, we obtain a conversion factor of x6. This means that a charge of 1.0 in the calculations corresponds to a real charge of x6 which is close to 2.4. Table 4 shows that significant effects (more than 5%) of dielectric saturation can be expected for all monoatomic ions and that all lattices studied describe the effect well. The results presented in Table 4 do not show the importance of different regions of space on the dielectric saturation effect, but in Table 5, we present the solvation energies calculated with an ideal polarizability lattice and that obtained with a lattice where half of the polarizability originates from dipolar degrees of freedom. The lattice points are not surrounded by a dielectric continuum. A charge of 1.0 has been used, and the total latticesite polarizability equals 0.18, as in all previous Tables. From Table 5, it is clearly seen that most of the saturation effects, as expected, originate from the first solvation shell, and most of the remaining part comes from the second shell. Only very small contributions are obtained from more distant regions. Finally, before starting to study the effect of dielectric saturation on the effective ion-ion interaction, we will present data showing how the value of different amounts of ideal and

TABLE 5: Solvation Energies and Solvation Energy Differencies between an Ideal Polarizability Lattice and a Lattice Where 50% of the Polarizability Is Ideal and 50% Has a Dipolar Origina lattice size

ideal polarizability

50% dipolar polarizability

difference

1 2 3 4 5 6

0.3785 0.5003 0.5731 0.6046 0.6278 0.6429

0.2826 0.4003 0.4704 0.5007 0.5237 0.5388

0.0958 0.1000 0.1027 0.1038 0.1041 0.1041

a No dielectric surrounding is used. Other parameters are the same as those used before. A charge of 1.0 is used.

TABLE 6: Calculated Solvation Energy as Function of the Lattice-Site Polarizability and Dipole Moment for a Charge of 0.4a dielectric permitivity total optical polarizability 1.697 1.697 1.697 2.815 2.815 2.815 10.194 10.194 10.194 2.007 2.203 2.926 4.904 8.125 10.264 11.442 11.778

1.697 1.312 1.00 2.815 1.697 1.00 10.194 2.815 1.00 2.007 2.007 2.007 2.007 2.007 2.007 2.007 2.007

0.045 0.0225 0.00 0.090 0.045 0.00 0.180 0.090 0.00 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.06

dipole moment

solvation energy

effective radius

0.00 0.008216 0.011619 0.00 0.011619 0.016432 0.00 0.016432 0.023238 0.00 0.005 0.010 0.015 0.018 0.019 0.0194 0.0195

-0.045132 -0.043613 -0.040053 -0.073750 -0.070466 -0.063199 -0.114511 -0.105629 -0.091607 -0.055893 -0.061087 -0.073581 -0.088864 -0.098319 -0.101493 -0.102771 -0.103092

0.7280 0.7534 0.8204 0.6994 0.7320 0.8162 0.6299 0.6831 0.7876 0.7182 0.7151 0.7157 0.7167 0.7135 0.7114 0.7104 0.7101

a The effective radius calculated according to eq 2b is also shown. The permitivities presented in the Table are obtained from the Claussius-Mosotti equation.

dipolar polarizabilities influence the calculated solvation energies for different sizes of the solvated ions. In Table 6, we present the calculated solvation energies together with effective particle radii calculated from these solvation energies and eq 2b using a CM estimate of the dielectric permittivity. In Table 6, a charge of 0.4 has been used because this value corresponds almost to a real charge of 1.0. As expected, the dielectric saturation, when present, manifests itself as a larger effective radius for the solvated ion. There is also another trend that can easily be seen from Table 6. Larger dielectric permittivities result in a smaller effective radius, again indicating that the dielectric permittivity of the lattice is larger than that sugested by the CM equation. The purpose of the first part of this work is to study the effective interaction between two ions as a function of their charge and their relative positions in a primitive cubic lattice. We will use a practical definition of this interaction by using eq 2a and by representing the interaction with the effective dielectric constant that makes this equation valid. As was seen above, the energy of an ion will slightly depend on the position of the ion in the lattice (i.e., where it is positioned relative to the boundary). Thus, it is convenient to write the measure of the effect of the medium on the interaction according to

eff )

q1q2 r12(E(q1, r1, q2, r2) - E(q1, r1) - E(q2, r2))

(8)

It should be noted that eff is not a local dielectric constant but only a measure of the effect of the lattice on the ion-ion


Karlstro¨m

Figure 2. Effective permittivity calculated according to eq 8 for a lattice-site polarizability of 0.06 l.u. and a lattice-site dipole moment of 0.015 l.u. The charge that was used is 0.001, and the bulk dielectric permittivity is 4.90. The upper and lower curves corresponds to the + - and the + + situation with no dipole assigned to the sites with ions. The curve between corresponds to both the + + and the + - situation when the charged lattice sites have a dipole.

interaction. eff will approach the dielectric permittivity when the distance (r12) between the charges (q1 and q2) becomes much larger than the lattice constant. E(r, q) is the energy calculated when a charge q is placed in r, and in the same way, E(q1, rl, q2, r2) is the energy that is calculated when a charge ql is placed in rl and another charge q2 is placed in r2. The vectors r1 and r2 define the position of the charges in the lattice. 3. Results A. Effective Ion-Ion Interaction. There is one detail in the model that has not been discussed before but that has to be clarified before starting to analyze the effective ion-ion interaction data. A polarizability and a dipole are assigned to each lattice point. However, when an ion is placed in one lattice point, there is an ambiguity with respect to what should be done with the polarizability and dipole moment of that point. Data that illustrates this situation is presented in Figure 2. Three curves are shown in the Figure. They are all obtained with a lattice-site polarizability of 0.06 and a lattice dipole moment of 0.015. Together, these values result in a dielectric permittivity of 4.90. The curves shown in Figure 2 show the eff values calculated according to eq 8. Charges that are so small (0.001) that no effect of dielectric saturation should be present are used. Two of the curves, the upper and the lower ones, are calculated when the dipole moment of the lattice sites with ions have been set to 0.0. The upper curve corresponds to the interaction between a positive and a negative ion, and the lower curve corresponds to a situation when the two ions have similar charges. The curve in the middle is also obtained when the lattice sites with ions have a dipole moment. In this case, the same curve is obtained for both combinations of ions (+ + or + -). Note that in both cases the absence of a dipole moment on the lattice sites with ions will result in an extra repulsive interaction energy term between the ions. For the situation with the same charges on the ions, this extra term will lower the calculated eff, but for ions with opposite charges, the opposite will be true. Figure 2 also shows that the effect is present only for the shortest distances (less than 3 l.u.). From this Figure, it can also be seen that there are rather large oscillations in the calculated eff values. They are due to the orientation of the interionic axis relative to the main axis of the lattice. High values

Figure 3. (a) Effective permittivity calculated according to eq 8 for a lattice-site polarizability of 0.06 l.u. and a lattice-site dipole moment of 0.015 l.u. The charges that were used are 0.001, 0.4, 0.8, and 1.0. Both ions have the same sign on their charges. The ordering of the curves is such that higher effective permittivities are obtained for the smaller charges, except for distances where the dips in the curves occurs (4, 6, and 8 l.u.). Here, the ordering is the opposite. The bulk permittivity is 4.90. (b) Effective permittivity calculated according to eq 8 for a lattice-site polarizability of 0.06 l.u. and a lattice-site dipole moment of 0.015 l.u. The charges that were used are 0.001, 0.4, 0.8, and 1.0. The ions have different signs on their charges. The lowest curve corresponds to the charge 0.001. This curve is followed by the curves corresponding to the charges 1.0, 0.8, and 0.4. The ordering is different for a few geometries where the difference between the curves is so small that it cannot be seen on the graph. The bulk permittivity is 4.90.

of the eff are observed when no lattice points are located along the interionic axis, and low values are observed when as many lattice points as possible are located along this axis. This has consequences for the way that ions will approach each other in solution. Ions with the same charge will prefer orientations of their solvation shells that are such that no solvent molecules are located on the interionic axis, whereas ions with opposite charges prefer solvent molecules to be located along this axis. This issue will be further discussed in the last section. We will now continue this analysis by presenting effective dielectric constants, defined as in eq 8, for the medium discussed above. To make the situation as realistic as possible, we will assume that for the lattice sites where there is an ion the dipoles have been removed but the polarizabilities remain. In Figure 3, the effective dielectric permittivities calculated according to eq 8 are presented as a function of the interionic distance for different values for the ionic charges. In Figure 3a, the + + situation is shown, and in Figure 3b, the + - combination is presented. The four curves in Figure 3a and b corresponds to


Figure 4. Schematic illustrating the three regions used to discuss the effect of dielectric saturation on the effective ion-ion interaction.

charges 0.001, 0.4, 0.8, and 1.0. If we use a length scale of 1 l.u. ) 6 au (close to the size of a water molecule), then these charges correspond to 0.00245, 0.98, 1.96, and 2.45. They can thus be seen as representative pf a system with no saturation, a 1:1, a 2:2, and a 3:2 electrolyte. If parts a and b of Figure 3 are compared, it is seen that the effect of saturation is larger for the + + system and that only minor effects of saturation can be seen for the + - system. The ordering of the curves is also different for the + + and the + - systems. For the + + system, the lowest curve (where the largest deviation from ideal behavior is seen) corresponds to the highest charges, and the effect is monotonically decreasing with the ionic charge. For the + - system, the lowest curve is obtained for the charge 0.001; then comes the system with the highest charges (1.0) followed by the 0.8 and 0.4 systems at the shorter distances. At longer distances, the ordering of the curves is such that the lowest effective dielectric permittivities correspond to the smallest charges, and higher values are obtained when the charges are increased. We noticed above that when the lattice sites with ions were augmented with a polarizability and a dipole then the + + and the + - curves were the same for small charges, which is a consequence of the fact that the polarization due to different charges is additive under these conditions. We also noticed that when the dipole of the charged sites was removed an extra repulsive force between the charges appeared, which resulted in a smaller effective dielectric permittivity for the + + system and a higher one for the + - system. However, for larger charges, a more complex pattern emerged. A straightforward way to understand the results presented above both for the + + and the + - situation is to divide the system into three regions, as is indicated in Figure 4. Regions 1 and 2 are areas around each of the two charges where the field from the charges is so strong that the dielectric response from the region that is experienced by the other charge is smaller than the response from a corresponding piece of bulk matter. The lack of dielectric response from these regions will result in an effective repulsion between the ions and a lowering of the effective dielectric permittivity when the charges are large and of the same sign. When the charges are large and have opposite signs, the repulsion results in an increased effective dielectric permittivity. The outer region will solvate the two charges. Saturation effects in this region will result in a poorer solvation of two ions with the same charge when they approach each other and a lowering of the effective permittivity for charges with the same sign. When the ions have opposite signs, their solvation from this


Figure 5. Effective permittivity calculated according to eq 8 for a lattice-site polarizability of 0.06 l.u. and lattice-site dipole moments of 0.010, 0.015 and, 0.019 l.u. The charge that was used is 0.001. The upper and lower curves in each group correspond to the + - and the + + situation, respectively, with no dipole assigned to the sites with charges.

outer region will be improved as the ions approach each other. This behavior corresponds to an extra effective attractive force between the ions, and a decreased effective permittivity will thus be observed from this region for charges with opposite signs. As a result, we can see that there are two contributions that have the same consequences for charges with the same sign and that result in a lowering of the effective dielectric permittivity. For ions with opposite signs on their charges, the two effects have different signs, and the complex pattern discussed above emerges. So far we have studied how the magnitude of the charge influences the dielectric response of a medium with relatively low bulk permittivity. We have seen that there are effects of dielectric saturation and that these effects disappear when the ions are separated by 3-4 boxes. It is natural to pose the question, how will the permittivity change when the properties of the lattice particles are changed? In Figure 5, we present the effective permittivities obtained when the dipole moment of the lattice cells is changed and the polarizabilities of the lattice sites are kept fixed at 0.006 au3 using the charge 0.001. Three different values of the dipole moment are used, and two curves corresponding to the + + and + - systems are presented for each dipole moment. As could be expected from the analysis above, the + - curve in general corresponds to higher values than does the + + curve. The dielectric bulk permittivity values according to the CM equations are 2.93, 4.90, and 10.26. It is also clearly seen that the oscillatory behavior of the effective permittivity increases drastically with increasing dipole moment. In Figure 6a and b, the variation in the calculated effective permittivities as a function of the charges of the interacting ions is presented for the system with a bulk permittivity of 10.26, which was discussed above. The trends are the same as that previously shown in Figures 3a and b. If Figures 3a and 6a are compared, it is also seen that both the absolute and relative influence of the charges on the calculated dielectric permittivities are larger for larger dipole moments and short distances, indicating an increased importance of dielectric saturation at higher dielectric permittivities. In this section, we have seen that there is a relatively large influence on the effective ion-ion interaction for highly charged ions from dielectric saturation effects and that this influence increases the repulsion between similarly charged ions compared


Figure 6. (a) Effective permittivity calculated according to eq 8 for a lattice-site polarizability of 0.06 l.u. and a lattice-site dipole moment of 0.019 l.u. The charges that were used are 0.001, 0.4, 0.8, and 1.0. Both ions have the same sign on their charges. The ordering of the curves is such that higher effective permittivities are obtained for the smaller charges, except for distances where the dips in the curves occurs (4, 6, and 8 l.u.). Here, the ordering is the opposite. The bulk permittivity is 4.90. (b) Effective permittivity calculated according to eq 8 for a lattice-site polarizability of 0.06 l.u. and a lattice-site dipole moment of 0.019 l.u. The charges that were used are 0.001, 0.4, 0.8, and 1.0. The ions have different signs on their charges. The lowest curve corresponds to the charge 0.001. This curve is followed by the curves corresponding to the charges 1.0, 0.8, and 0.4. The ordering is different for a few geometries where the difference between the curves is so small that it cannot be seen on the graph. The bulk permittivity is 4.90.

to the value of the repulsion calculated from a simple dielectric model. We have also found that the attraction between ions with opposite charges becomes stronger than what could be expected from a dielectric approach, but in general it is weaker than what could be expected from a lattice without saturation effects. The main reason for this result is, however, not associated with dielectric saturation but instead is linked to the short-range packing in the primitive cubic lattice studied. B. Reorganization Energies. In this paragraph, we will focus on the influence of dielectric saturation on the reorganization process associated with electron transfer. To get a complete picture of the physics behind the standard calculation of the outer-sphere reorganization energies associated with electrontransfer processes, the reader is referred to the original work by Marcus11-13 or Hush.14-15 Here, only a simplified version of the theory will be given. To start, we will assume that we have two ions A and B with charges x and y that are located at a distance s. Each ion is assumed to be surrounded with a first layer of solvent molecules. Here, we will assume that the radii

Karlstro¨m (r) of these solvated ions are the same. We will further assume that the dielectric response associated with the motion of the electronic motion is fast enough to be in equilibrium with the electron-transfer process but that the rotation of the molecules constituting the medium is a slow process compared to the electron-transfer process. The transfer of an electron from one site to another is equivalent to the creation of a dipole moment. Initially, we will therefore study the reorganization energy associated with the inversion of a dipole. The same type of lattice as before is chosen, and as before, a polarizability and a dipole are associated with each site. Technically, the calculations are carried out in the following way. 1. Choose the magnitude of the dipole moment and orient it, for example, along the x axis. 2. Solve the polarization equations for the dipoles and polarizabilities and obtain the solvation energy for the dipoles as well as the polarization of the dipoles and the polarizabilities. 3. Invert the orientation of the central dipole, freeze the polarization of the dipoles corresponding to the medium molecules, and solve the polarization equations for the polarizabilities in the field from the dipoles. 4. Calculate the energy difference between the energy obtained for the equilibrium system under step 2 and the nonequilibrium system under step 3. This energy difference corresponds to 4 times the outer-sphere contribution to a reorganization energy associated with the reorientation of the dipole, defined in a similar way as the outer-sphere contribution for an electron-transfer process by Marcus. 5. Calculate the ratio between this reorganization energy and the solvation energy of the dipole in the lattice. Formally, this reorganization energy associated with the rotation of a dipole can be calculated in the dielectric approximation, which is equivalent to the Marcus model, from

Ereo )

2µ2

6( - op)

r (2op + 1)(2 + 1) 3

(9)

The corresponding solvation energy for a dipole in a medium can be calculated from

µ2 ( - 1) Esolv ) - 3 r (2 + 1)

(10)

To eliminate the dependence of the somewhat arbitrary quantity r, it is convenient to evaluate the ratio between Ereo and Esolv. In this way, we obtain

Q)

12( - op) Ereo )Esolv (2op + 1)( - 1)

(11)

The purpose of this section is to investigate the influence of dielectric saturation on the outer-sphere contribution to reorganization energies. A similar approach has been used by Churg et al. to evaluate the reorganization energy for cytochrome C.16 The physics in the cytochrome system is, however, such that the effects that will be discussed here can hardly be observed. Before calculating the saturation contribution to the reorganization energies, we must investigate how large deviations from ideal dielectric behavior are to be expected from the granularity of the medium. In Table 7, the value of the quantity Q is presented together with the corresponding quantity calculated as indicated above. Two things have been varied in Table 7: the orientational contribution to the dielectric permittivity and



TABLE 7: Reorganization Energies Calculated for the Reorientation of a Dipole as a Function of the Electronic and Orientational Polarizationa dipole moment

electron polarizability

tot

opt

central dipole

Marcus model *4

Geo(l)

reorganization energies Geo(2) Geo(3) Geo(4)

Geo(5)

0.01225 0.01225 0.01225* 0.01817 0.01817

0.05 0.05 0.05 0.05 0.05

3.16 4.59 4.59 7.10 7.10

1.79 1.79 1.79 1.79 1.79

0.001 0.001 0.001 0.001 1.000

1.654 2.035 2.035 2.274 2.274

1.954 2.454 2.552 2.813 0.962

1.859 2.454 2.274 2.813 0.962

1.754 2.149 2.260 2.418 2.164

1.830 2.328 2.316 2.633 1.600

1.737 2.281 2.352 2.587 2.025

a The values presented are obtained using the Claussius-Mossotti equation. The values obtained by the Marcus model have been multiplied by 4 to be comparable. * indicates that the lattice site with the reorienting dipole also has a polarizability similar to the other lattices sites. A radius of 8 l.u. is used for the lattice. The geometries Geo(n), N ) l-5 are defined in Table 8 and in the text.

TABLE 8: Lattice Sites in the Vicinity of the Solute Dipole or the Charges That Are Left without Polarizability and a Dipole Modeled by the Langevin Modela lattice site

assigned value

0, 0, 0 0, 0, 1 0, 1, 1 1, 1, 1 0, 0, 2

Geo(1) Geo(2) Geo(3) Geo(4) Geo(5)

a For a given geometry, the specified lattice site and lattice sites corresponding to lower geometry values do not contain aLangevin dipole or polarizability., which means that, for example, for Geo(3), the lattice sites 0, 0, 0; 0, 0, 1; and 0, 1, 1 and symmetry-related sites are left without aLangevin dipole or polarizability.

the size of the dipole (i.e., how many of the lattice sites are left empty in the vicinity of the dipole). This issue is illustrated in Table 8, where the lattice sites in the central part of the cavity are presented and numbers are assigned to the sites. Note that only symmetry-unique sites are presented in the Table. If a value n is assigned to a lattice site, then for geometries labeled Geo(m), where m is equal to or larger than n, this lattice site will not contain any polarizability or dipole. Table 7 shows that the deviations in Q due to the granular character of the medium are on the order of 5-30% and that the agreement is slightly better for lower dielectric permittivities and much better for larger distances from the reorienting dipole to the nearest effective lattice site. One can also note that the agreement is slightly worse if the lattice sites specified in Table 7 are assumed to have electronic polarizabilities. This assumption causes the lattice model to be in better agreement with the assumption behind the Marcus model. Most of the lattice data presented in Table 7 are calculated with a central dipole of 0.001, a value that is so small that no effect of dielectric saturation is present. For comparison, we have also included data for a central dipole of 1. In this case, a clear deviation from the ideal dielectric behavior is observed for the smallest cavities. Dielectric saturation effects lower the outer-sphere contribution to the reorganization energies. It should, however, be noted that a dipole of 1.0, in the dimensionless units that are used in this work, for a reasonable a value of the lattice constant (3Å) corresponds to a very large dipole of around 35 D. The calculations presented so far have shown that the influence of the granularity of the medium on the calculated Q values is rather small even for the smallest cavities and that one can expect dielectric saturation effects to reduce the outersphere reorganization energies in electron-transfer processes. There is, however, a fundamental difference between the solvation of ions and of dipoles. The field and thus the dielectric saturation effects from an ion are much more long-ranged than are those from a dipole. One may thus expect the influence of dielectric saturation on reorganization energies for highly

charged species to be larger that what has been indicated above. Note that the mechanism for the saturation of the reorganization energies is somewhat different in the ionic case than in the dipolar case. In the ionic case, the major part of the saturating field is not changed because of the reorganization of the charge, and it is to a large extent this permanent part of the field that saturates the dielectric response. To study this phenomenon, the following calculations have been performed. 1. Two charges q1 and q2 have been placed in the lattice at points (1, 0, 0) and (-1, 0, 0), and the polarization equations have been solved for the system, yielding an energy (El) and the polarization at each lattice point. 2. The polarization corresponding to the dipoles is frozen, the charges are changed to q2 and q1, and the polarization equations are solved for the electronic part of the polarization, keeping the orientational part of the polarization frozen. The energy E2 is thus obtained. 3. The energy difference E2 - E1 corresponds to 4 times the reorganization energy of the electron-transfer process as defined by Marcus. The outer-sphere contribution to the process described above can, according to Marcus,l1-13 be calculated from

∆G/0 )

( )(

)

(∆Z)2 1 1 1 1 4 s r opt

(12)

In eq 12, ∆Z corresponds to the amount of charge that is moved, s is the radius of the reactants, and r is the distance that the charge is moved. To choose a realistic value for ∆Z, we use the fact that the lattice constant typically should correspond to 6 au, which means that a real charge of 1.0 should correspond to a charge of 1/x6 ≈ 0.4 in the lattice model. The distance r in eq 12 should obviously be 2 l.u.to get optimal agreement between eq 12 and the model. We have chosen a value of 1 for s, and as in the dipolar study, we will vary the lattice sites that carry a polarizability. In Table 9, some representative estimates of the reorganization energy, obtained when the transferred charge is varied, are presented. Note that to make the comparison easier all energies have been divided by (∆Z/0.4)2. Two different situations have been studied. In the first case, all lattice points except the ones with charge have a polarizability and a dipole moment, and in the second case, the dipoles have also been deleted from the nearest neighbors to the ions. The latter situation is what is assumed in standard electron-transfer theory where the first solvation shell is assumed to be rigid. One should also note that the estimate made by Marcus corresponds to moving the charge to the midpoint between the two lattice points and that it consequently applies to only 1/4 of the estimates made in this work. The data presented in Table 9 are obtained for a lattice-site polarizability of 0.05 and a dipole moment of


Karlstro¨m

TABLE 9: Normalized Reorganization Energies Calculated for Different Charges and Lattice Points with Dipole Momentsa

TABLE 11: Normalized Reorganization Energies Calculated for Different Charges and Lattice Points with Dipole Momentsa

charge x

charge y

4*Marcus

Geo(l)

Geo(2)

charge x

charge y

4*Marcus

Geo(1)

Geo(2)

0.004 0.012 0.12 0.4 0.8 1.2

0.000 0.008 0.08 0.0 0.4 0.8

0.0334 0.0334 0.0334 0.0334 0.0334 0.0334

0.0822 0.0820 0.0696 0.0496 0.0269 0.0208

0.0242

0.004 0.4 0.8 1.2

0.000 0.0 0.4 0.8

0.0277 0.0277 0.0277 0.0277

0.0888 0.0484 0.0185 0.0115

0.0225 0.0190 0.0113 0.0075

0.0216 0.0144 0.0100

a The outer-sphere reorganization energies according to Marcus are given for comparison. All lattice points have dipole moments except the ones with charges in Geo(1). For Geo(2), the dipoles have also been removed from the lattice points that are nearest neighbors to the ions. A radius of 8 l.u. is used for the lattice. The charge is transferred from 1, 0, 0 to -1, 0, 0. A lattice-site polarizability of 0.05 and a dipole moment of 0.0181658 have been used. The geometries Geo(n), N ) 1-5 are defined in Table 8 and in the text.

TABLE 10: Normalized Reorganization Energies Calculated for Different Charges and Lattice Points with Dipole Momentsa charge x

charge y

4*Marcus

Geo(1)

Geo(2)

0.004 0.4 0.8 1.2

0.000 0.0 0.4 0.8

0.0251 0.0251 0.0251 0.0251

0.0742 0.0432 0.0170 0.0106

0.0190 0.0165 0.0101 0.0068

a The outer-sphere reorganization energies according to Marcus are given for comparison. All lattice points have dipole moments except the ones with charges in Geo(l). For Geo(2), the dipoles have also been removed from the lattice points that are nearest neighbors to the ions. A radius of 8 l.u. is used for the lattice. The charge is transferred from 1, 0, 1 to 0, 1, 0. A lattice-site polarizability of 0.05 and a dipole moment of 0.0181658 have been used.

0.0181659. These values correspond to an optical dielectric permittivity of 1.7948 and a permanent permittivity of 7.0966 according to the Clausius-Mossotti equation. If the data in Table 9 are analyzed, one easily realizes that when the smallest charges are used one hardly sees any effect of dielectric saturation, not even for the Geo(1) system. It is also obvious from Table 9 that large contributions to the reorganization energy can be expected from the first solvation shell of the ions. In fact, the numbers presented in Table 9 suggest that 2/3 of the reorganization energy associated with reorientation of the solvent molecules originates from the first hydration shell of the ions. Note that this contribution is in no way related to the inner-sphere contribution to the reorganization energy in the Marcus formalism. In that formalism, the inner-sphere contribution is due to changes in the ionsolvent molecule distances. The three last choices of charge in Table 9 correspond to charge transfer in systems with charges of 1/0, 2/1, and 3/2 because a charge of 0.4 in the model corresponds to a real charge of 1, as was discussed above. The values in the Table indicate clearly that large effects from dielectric saturation can be expected for the 1/0 system and that more than 50% of the reorganization energy has disappeared for the 3/2 system. It could be of interest to investigate how sensitive the calculated reorganization energies are to the distance between the ions. In particular, one may note that one lattice site is common for the first solvation shell of the two ions for the geometries studied in Table 9. In Table 10, reorganization energies calculated when a charge is transferred from lattice site 1, 0, 1 to lattice site 0, 1, 0 are listed. As has been seen above, the solvation energies for an ion in these two lattice sites

a

The outer-sphere reorganization energies according to Marcus are given for comparison. All lattice points have dipole moments except the ones with charges in Geo(l). For Geo(2), the dipoles have also been removed from the lattice points that are nearest neighbors to the ions. A radius of 8 l.u. is used for the lattice. The charge is transferred from 1, 0, 1 to 0, 1, 0. A lattice-site polarizability of 0.05 and a dipole moment of 0.0198 have been used.

are not exactly the same, so two reorganizations can be calculated, one where the charge is transferred from one site to the other and one where the transfer is in the opposite direction. The data presented in Table 10 is the average value. (The two estimates deviate less than 1% from this value.) The charge is, in this case, moved x3 l.u. In Table 11, data obtained for the same systems as studied in Table 10 but with a lattice-site dipole moment of 0.0198 are presented. A lattice-site dipole moment of 0.0198 together with a polarizability of 0.05 corresponds to a dielectric permittivity of 10.3371 according to the CM equation. A comparison of the data in Tables 10 and 11 suggests that the saturation effects become more important for higher permittivities. 4. Implications of Dielectric Saturation for Real Systems In this work, some simplified model systems have been studied to gain insight into the importance of dielectric saturation to the solvation of ions in general and to the electron-transfer process in particular. It has been shown that dielectric saturation can be expected to be of some importance to the solvation of mono-valent ions and of large importance to divalent or highercharged ions made from one atom. This conclusion is based on the behavior in dielectric media with relatively low dielectric permittivity values (2-10). There is, however, no reason to believe that the conclusion should not be valid for systems with higher dielectric permittivity. In fact, the calculations presented here indicate that the effect should be somewhat more important in systems with larger dielectric permittivity values. It is well known to all chemists that divalent and trivalent positive ions in aqueous solutions normally undergo hydrolysis reactions in which a proton leaves a water molecule that is next to the multivalent ion, thereby reducing the charge of the ion by 1. This behavior is most likely a manifestation of dielectric saturation effects. If the solvent (water) had been capable of fully solvating the ion, there would be no reason for the proton to leave the water molecule. If we instead focus on the importance of dielectric saturation effects on electron-transfer processes occurring in solution, we see that it is of importance in two ways. First of all, it will affect the probability that reactants come close to each other, and second, it will affect the possibility that the electron jumps once the two reacting species have come close. The first effect, which has been studied before, is of importance when the reactants have the same charge. If so, dielectric saturation effects will increase this contribution to the activation energy significantly. On the other hand, the transfer of an electron will be facilitated because the outer-sphere reorganization energy is reduced by the dielectric saturation by a large amount.

Electron-Transfer Processes in Condensed Matter Probably the most studied electron-transfer process in aqueous solution is the one that occurs in the Fe2+/Fe3+ system. The total experimental activation energy for this process at zero ionic strength is between 14.9 and 12.5 kcal/mol (see ref 17 for further details). Theoretical estimates of different contributions to this activation energy predict that there is an inner-sphere contribution to the activation energy that is linked to changes in the vibrational frequencies of the six water molecules in the first solvation shell of each of the two the Fe ions of 6-8.5 kcal/ mol17,18. Furthermore, theoretical modeling based on the Marcus outer-sphere reorganization energy model estimates such a contribution to be 4-6.5 kcal/mol.10,17-20 This outer-sphere contribution is meant to include the contribution to the reorganization energy from all of the water molecules except from those present in the first solvation shell. Finally, one normally estimates a contribution to the activation energy from bringing the ions together of 5 kcal/mol by using eq 2 and a dielectric permittivity of 80.17 In all, these contributions add up to between 15 and 20 kcal/mol, slightly more that the experimental estimates. One could, however, expect tunneling to be of some importance and to lower the barrier somewhat. On the basis of statistical mechanical simulations, Kuharski et al. have estimated this barrier to be around 20 kcal/mol,21 thus indicating reasonable agreement between theory and experiment. The data calculated in this work are not of the quality that they allow for any exact values of the activation energy for the considered process to be specified. First, the inner-sphere reorganization energy that originates mainly from the changes in the Fe-O distances associated with the electron-transfer process is not considered at all in this work. From the data presented here, it seems clear, however, that the outer-sphere reorganization energies are smaller than what is normally assumed, probably around 2 kcal/mol, and that the work needed to bring the ions together is larger than what is normally assumed. Finally, it is interesting to note that one can expect effects from how two solvating ions approach each other in a solvent. In particular, for the Fe3+-Fe2+ system in water, one may expect them to approach with the C3 axis defined by the solvation shell

J. Phys. Chem. B, Vol. 106, No. 20, 2002 5311 of one ion to point toward that of the other ion. This result is found by statistical mechanical simulations of the Fe3+-Fe2+ system in water.22,23 Acknowledgment. I am indebted to Professor H. Wennerstro¨m and Dr. Bengt Jo¨nsson for stimulating discussions and to NFR (The Swedish Natural Research Council) for financial support. References and Notes (1) Warshel A.; Chu Z. T. Structure and ReactiVity in Aqueous Solution. Characterization of Chemical and Biological Systems; Cramer, C. J., Truhlar, D. G., Eds.; ACS Symposium Series; American Chemical Society: Washinigton, DC, 1994; p 71. (2) Muller, P. M.; Warshel, A. J. Phys. Chem. 1995, 99, 17516. (3) Wesolowski, T.; Muller, R. P.; Warshel, A. J. Phys. Chem. 1996, 100, 15444. (4) Papazyan, A.; Warshel, A. J. Phys. Chem. B 1997, 101, 11254. (5) Papazyan, A.; Warshel, A. J. Chem. Phys. 1997, 107, 7975. (6) Papazyan, A.; Warshel, A. J. Phys. Chem. B 1998, 102, 5348. (7) Karlstro¨m, G.; Malmqvist P.-Å. In Ultrafast Reaction Dynamics and SolVent Effects, AIP Conference Proceedings 298, Royamont, 1993; Gauduel, Y., Rossky, P. J., Eds.; p 59. (8) Bo¨ttcher, C. J. F. Theory of Electric Polarization; Elsevier Scientific Publishing Company: Amsterdam, 1973; p 163. (9) Warshel, A.; Levitt, M. J. Mol. Biol. 1976, 103, 227. (10) Bo¨ttcher, C. J. F. Theory of Electric Polarization; Elsevier Scientific Publishing Company; Amsterdam, 1973; p 215. (11) Marcus, R. A. Annu. ReV. Phys. Chem. 1964, 15, 155. (12) Marcus, R. A. J. Chem. Phys. 1965, 43, 679. (13) Marcus, R. A.; Sutin, N. Biochim. Biophys. Acta 1985, 811, 265. (14) Hush, N. S. Trans. Faraday Soc. 1961, 57, 557. (15) Hush, N. S. Electrochim. Acta 1968, 13, 1005. (16) Churg, A. K.; Weiss, R. M.; Warshel, A.; Takano, T. J. Phys. Chem. 1983, 87, 1683. (17) Tembe, B. L.; Friedman, H.; Newton, M. D. J. Chem. Phys. 1982, 76, 1490. (18) Brunschweig, B. S.; Logan, J.; Newton, M. D.; Sutin, N. J. Am. Chem. Soc. 1980, 102, 5798. (19) Brunschweig, B. S.; Creutz, C.; Macartney, D. H.; Sham, T.-K.; Sutin, N. Faraday Discuss. Chem. Soc. 1982, 74, 113. (20) Weaver, M. J. Chem. ReV. 1992, 92, 463. (21) Kuharski, R. A.; Bader, J. S.; Chandler, D.; Sprik, M.; Klein, M. L.; Impey, R. J. Chem. Phys. 1988, 89, 3248. (22) Kumar, P. V.; Tembe, B. L. J. Chem. Phys. 1992, 97, 4356. (23) Babu, C. S.; Madhusoodanan, M.; Sridhar, G.; Tembe, B. L. J. Am. Chem. Soc. 1997, 119, 5679.

On the Influence of Dielectric Saturation and Medium Granularity on

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