494
J. Phys. Chem. B 2006, 110, 494-500
Temperature- and Pressure-Dependence of the Outer-Sphere Reorganization Free Energy for Electron Transfer Reactions: A Continuum Approach Swati R. Manjari and Hyung J. Kim* Department of Chemistry, Carnegie Mellon UniVersity, Pittsburgh, PennsylVania 15213-2683 ReceiVed: July 20, 2005; In Final Form: October 10, 2005
The outer-sphere reorganization free energy for electron-transfer reactions in polar solvents and its variations with temperature and pressure are studied in the dielectric continuum framework by extending the recent fluctuating cavity description [J. Chem. Phys. 2005, 123, 014504]. The diabatic free energies are obtained as a function of three variables, i.e., radii of two spherical cavities for the donor and acceptor moieties of an electron-transfer complex and a solvent coordinate that gauges an arbitrary configuration of solvent orientational polarization. Equilibrium cavities relevant to the reactant and product states are determined via the variational principle. This incorporates cavity size readjustment accompanying electron transfer and related electrostrictive effects. Another important consequence of the variational determination of equilibrium cavities is that their size depends on thermodynamic conditions. The application of the theoretical formulation presented here to electron self-exchange shows that in contrast to the prediction of the standard Marcus theory, the solvent reorganization free energy decreases with temperature. This is in excellent accord with a recent experiment on a mixed valence dinuclear iron complex in acetonitrile [J. Phys. Chem. A 1999, 103, 7888]. It is also found that electrostriction makes a significant contribution to outer-sphere reorganization. Model calculations for the dinuclear iron complex system show that about 25-30% of the total solvent reorganization free energy arises from cavity size changes, while solvent repolarization is responsible for the rest.
1. Introduction The rearrangement of solvent dipoles plays an important role in a wide range of different reactions involving charge shift in solution. Examples include electron/proton transfers and SN1/ SN2 reactions. Since the seminal work by Marcus,1 there have been extensive efforts2 to understand its influence on free energetics and dynamics of electron transfer (ET) processes often represented as
Dq1,R + Aq2,R f Dq1,P + Aq2,P; e ) q1,P - q1,R ) q2,R - q2,P (1) where D and A are the donor and acceptor moieties of an ET complex, respectively, q1 and q2 are their electric charges, e is elementary charge that is transferred, and R and P denote the reactant and product states. The prediction of the original Marcus theory for the free energy cost associated with the solvent dipole rearrangement accompanying ET in eq 1 is1
{
λM(a1, a2, r) ) e2
}(
)
1 1 1 1 1 + - , ∞ 0 2a1 2a2 r
(2)
where D and A, separated by a distance r, are assumed to be placed in spherical cavities of radius a1 and a2 immersed in a solvent characterized by optical and static dielectric constants ∞ and 0. Despite its approximate nature mainly due to its underlying continuum solvent description, λM has provided a very useful and powerful tool to analyze and interpret ET free energetics in solution.1,2 Hereafter, we refer to λM in eq 2 as the Marcus reorganization free energy. Recently, the Marcus * To whom all correspondence should be addressed.
theory has been extended to nondipolar but quadrupolar solvents, where reorientation of solvent quadrupoles plays a central role.3 There has been considerable interest in the temperature dependence of ET kinetics and reorganization free energy to gain detailed insight into reaction pathways, controlling factors and electronic coupling.4-14 In highly polar solvents, it was found that outer-sphere reorganization free energy λ tends to decrease as the temperature T increases.10-14 Despite its widespread success in describing ET free energetics, the direct application of eq 2 fails to capture this λ behavior with T.11,13,15 To see this, we differentiate λM with respect to T at constant pressure p
( )
∂λM ) ∂T p
- e2
{ ( ) ( ) }( 1 ∂∞ 1 ∂0 - 2 2 ∂T p ∞ 0 ∂T
p
)
1 1 1 + - , (3) 2a1 2a2 r
where (...)X means that the variable X is held fixed for the operation in parentheses. Because the cavities are assumed to be fixed in conventional continnum theories, viz., ∂a1/∂T ) ∂a2/ ∂T ) 0, they do not contribute to T-variations of λM. Even though both the (∂∞/∂T)p and (∂0/∂T)p terms are negative, the contribution from the former dominates because 0-2 , ∞-2. As a consequence, eq 3 usually yields positive (∂λM/∂T)p for highly polar solvents, i.e., λM increases with T in the conventional ET description. This is at variance with experimental findings.10-14 By contrast, analyses12,15 based on a molecular solvent description16 lead to (∂λ/∂T)p < 0. Since the solvent density and structure around the ET complex decrease with T,
10.1021/jp0536145 CCC: $33.50 © 2006 American Chemical Society Published on Web 12/09/2005
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J. Phys. Chem. B, Vol. 110, No. 1, 2006 495
the overall strength of the solute-solvent interactions becomes reduced. This results in lowering of λ with increasing T. This exposes the importance of the alterations of the solvent distribution near the ET complex with temperature, which are not properly reflected in eq 3 based on the continuum description. Recently, we have constructed a fluctuating cavity description17 by extending an earlier theory by Kim,18 where cavity radii are allowed to undergo thermal fluctuations and to vary with external conditions in a continuum solvent. As such, this description takes into account electrostriction19-23 and related dynamic effects23-25 (e.g., cavity relaxation dynamics) on solution-phase reactions and solvation processes in a consistent way within its formulation. Its application to a variety of nonpolar and polar solvents shows that cavity size at equilibrium increases with T along an isobar and decreases with p along an isotherm.17 It is interesting to note that in a couple of earlier attempts,26,27 similar cavity behaviors were posited to reproduce certain experimental results (i.e., dependence of the water ionization constant and solvent reorganization free energy on thermodynamic conditions) using a continuum solvent description. In this article, we apply the theoretical formulation of ref 17 to electron-transfer reactions without introducing any ad hoc assumptions on cavity behaviors and analyze variations of the outer-sphere solvent reorganization free energy with T and p. The outline of this paper is as follows. In section 2, we briefly review the fluctuating cavity decription of ref 17 and extend it to electron-transfer reactions in the two-sphere cavity description of Marcus. To be specific, the diabatic free energies for the reactant and product states are obtained as a function of the cavity radii and a collective coordinate s that gauges an arbitrary solvent orientational polarization configuration. The resulting theory is applied to a symmetric ET system and the T and p dependence of its λ is analyzed there. Model calculations for a mixed-valence dinuclear iron complex studied in ref 12 are presented in section 3. Section 4 concludes. 2. Theoretical Formulation Here we give a brief reprise of the fluctuating cavity formulation of ref 17 and apply it to describe electron-transfer reactions in polar solvents. 2.1. Solvation Free Energy. The Gibbs free energy change ∆G when an ionic solute of charge q is transferred from a fixed position in the gas-phase to a fixed position inside a spherical cavity in a solvent is17
∆G )
4π 3 a p + 4πγ∞a2 - 16πγ∞δ a + 3 4πn
∞ x2 ULJ(x) dx ∫a+δ
( ) 1-
2 1 q , (4) 0 2a
where n is the number density of the solvent, γ∞ is the surface tension of a planar liquid-vapor interface of the solvent, ULJ is the short-range solute-solvent interaction, a is the cavity radius, and δ is a microscopic analogue of the Tolman length, which denotes the displacement of the equimolar surface associated with vanishing superficial density of matter, measured from the surface of tension.28,29 The first term on the righthand side of eq 4 is the reversible work for creating the volume of the cavity in the solvent, and the next two terms describe a similar free energy cost associated with its surface area. They were obtained by integrating the pressure and surface tension
over the cavity volume and surface area changes, respectively. We note that the term involving γ∞δ arises from the curvature dependence of the surface tension. The ULJ term is the contribution of the short-range solute-solvent interactions to solvation free energy, while the last term in eq 4 describes the solvation stabilization of the solute charge through electrostatic interactions with solvent polarizations. While the reader is referred to ref 17 for details of eq 4, several points are worth mentioning here for clarity: First, in our formulation, all surfaces are Gibbs’ dividing surfaces,30 and the surface of tension30 defines the cavity surface. Therefore, changes in the curvature of the cavity surface do not contribute explicitly to ∆G; i.e., differentials of its curvature tensor are absent in variations of ∆G. Second, δ given by17
4πn
∞ 2 r dr ∫0∞r2 g*(r) dr ) 4πn∫a+δ
(5)
incorporates effects related to solvent density inhomogeneity near the cavity surface, where g*(r) is a radial distribution of the solvent around the solute under an arbitrary solvation condition. As such, the theory described here goes beyond its earlier version in ref 18. For simplicity, it is assumed that δ is independent of the curvature of the cavity. Third, the cavity radius a is arbitrary, i.e., out of equilibrium, whereas the solvent polarizations are in equilibrium with the solute charge distribution for given a in eq 4. Thus, ∆G describes, in general, a nonequilibrium solvation free energy in the presence of a spherical cavity of arbitrary size. The equilibrium cavity can be obtained by invoking the variational principle ∂∆G/∂a ) 0,18 which yields
4πa2p + 8π γ∞ a - 16π γ∞ δ - 4πn(a + δ)2 ULJ(a + δ) +
( )
2 1 q 1) 0. (6) 0 2a2
The solution of eq 6 defines the radius aeq of the cavity at full equilibrium.17 Substituting aeq into eq 4, one obtains equilibrium solvation free energy. Finally, both the short-range and longrange solute-solvent interactions as well as reversible work associated with the formation of the cavity (“cavitation free energy”) are accounted for in eq 4 in the continuum solvent framework. 2.2. Cavity Size Variations. From the cavity equilibrium condition eq 6, it is easy to see that aeq varies with both the solute charge q and thermodynamic variables, T and p. We first consider alterations of aeq with q. Suppose the cavity size undergoes a change aeqf aeq + ∆aeq when we vary the solute charge q f q - ∆q at constant T and p. To first order in variations, ∆aeq and ∆q are related by
∆aeq ≈ -
( ) ∂aeq ∂q
T,p
∆q
(7)
We differentiate eq 6 with respect to a and obtain
( )
K ha
∂aeq ∂q
T,p
( )
)- 1-
1 q , 0 a 2 eq
(8)
where K h a is the effective force constant for the cavity size fluctuations31 evaluated at a ) aeq
496 J. Phys. Chem. B, Vol. 110, No. 1, 2006
K ha ≡
Manjari and Kim
( ) ∂2∆G ∂a2
(9)
T,p
4 GR(a1, a2, s) ) ER° + πp(a13 + a23) + 4πγ∞(a12 + a22) 3 16πγ∞δ(a1 + a2) + 4πn[
) 8πp a + 8π γ∞ - 4πn(a + δ)[2ULJ(a + δ) +
( )
2
1 q . 0 a3
(a + δ)U′LJ(a + δ)] - 1 -
{ }[ { }[
∫a +δr2 U(2)LJ (r) dr] - 21 ∞ 2
( )
( )
(10)
() ( ) [ () ] () 2
q ∂aeq ∂0 ∂δ ) 16πγ∞ + 2 2 ∂T p ∂T p 2 a ∂T p 0 eq 4π n (aeq + δ)2
2 ∂δ - Rp ULJ (aeq + δ) + aeq + δ ∂T p ∂δ U′ (a + δ), (11) ∂T p LJ eq
4π n (aeq + δ)2
where Rp is the coefficient of thermal expansion of the solvent, i.e., Rp ) V-1(∂V/∂T)p ) - n-1(∂n/∂T)p. The pressure dependence of aeq can be obtained in a similar way. The result is
( )
K ha
∂aeq ∂p
T
) - 4πaeq2 + 16πγ∞
[
4π n (aeq + δ)2
() ∂δ ∂p
-
T
] ()
q2 2
20 aeq
2
( ) ∂0 ∂p
+
T
2 ∂δ + κT ULJ (aeq + δ) + aeq + δ ∂p T
( )
4π n (aeq + δ)2
1 0
1
]
q1,R q2,R q1,R2 q2,R2 + + + a1 a2 0 r
]
1 1 1 1 1 + - s2 (13) ∞ 0 2a1 2a2 r
4 GP(a1, a2, s) ) EP° + πp(a13 + a23) + 4πγ∞(a12 + a22) 3
1 q ∆q 1. 0 a 2 eq
Equation 10 shows that as the magnitude of q diminishes, the equilibrium cavity radius increases. This so-called electrostrictive effect plays an important role in solvation and reaction free energetics involving ionic and dipolar species.19-23 As we will see below, it also makes an important contribution to λ in ET. We turn to cavity size variations with thermodynamic conditions. If we differentiate eq 6 with respect to T at fixed p, we obtain17
K ha
1-
e2
In eq 9, U′LJ(x) is the first derivative of ULJ(x) with respect to x. Substituting eq 8 into eq 7, we obtain
h a-1 ∆aeq ≈ K
∫a∞+δr2 U(1)LJ (r) dr +
∂δ U′ (a + δ), (12) ∂p T LJ eq
where κT is the isothermal compressibility of the solvent, κT ) - V-1(∂V/∂p)T ) n-1(∂n/∂p)T. According to our previous model study for various neat polar and nonpolar solvents, the equilibrium cavity size aeq increases with T and decreases with p.17 This is mainly due to contributions from the short-range solute-solvent interactions, which tend to become more repulsive as T grows and p reduces. It was found that excess solvent density near the cavity surface plays an important role in the variations of the ULJ contributions with thermodynamic conditions. 2.3. Electron-Transfer Reactions. We now apply the solvation theory described above to nonadiabatic ET between a donor and an acceptor in a polar solvent. This is straightforward and we present here only the results for diabatic free energies GR and GP relevant to the reactant and product states in the presence of an arbitrary solvent (orientational) polarization configuration and spherical cavities of arbitrary size:
16πγ∞δ(a1 + a2) + 4πn[
{ }[ { }[
∫a∞+δr2 U(2)LJ (r) dr] - 21 2
∫a∞+δr2 U(1)LJ (r) dr +
e2
1-
1
] ]
2 q2,P2 q1,P q2,P 1 q1,P + + + 0 a1 a2 0 r
1 1 1 1 1 + - (s - 1)2 (14) ∞ 0 2a1 2a2 r
Here ER° is the sum of the individual in-vacuo energies for D and A in the reactant state and EP° is the corresponding quantity for the product state.32 The second through fifth terms on the right-hand side of eq 13 (eq 14) describe the solvation free energies for the individual D and A moieties in the reactant (product) state [cf. eq 4], while the sixth term there represents the electrostatic interaction between the two in solution. These termssthe second through sixth in eqs 13 and 14sdescribe the solvation effect when solvent polarizations are in equilibrium with the charge distributions of the ET complex (even though the cavities are, in general, out of equilibrium) [see section 2.1 above]. Deviations from these equilibrium polarizations are accounted for via the last terms in eqs 13 and 14. There s is a collective solvent coordinate that gauges in an effective way an arbitrary configuration of solvent permanent dipole moments in the continuum approach, i.e., fluctuations of orientational polarization arising from rotations and translations of solvent nuclei. The electronic polarization arising from solvent induced dipole moments, on the other hand, is assumed to be always at equilibrium because its response is extremely fast. When s ) 0, the solvent orientational polarization is in equilibrium with the R state, i.e., Dq1,R + Aq2,R. The s ) 1 case corresponds to polarization equilibrium with the P-state charge distribution. Notice that for given cavity radii, the free energy difference between s ) 0 and s ) 1 on either GR or GP is the Marcus reorganization free energy λM [eq 2]. Equilibrium solvation states corresponding to the ET reactant and product states in the diabatic description are determined by the free energy minimization condition
∂GR(a1, a2, s) ∂GR(a1, a2, s) ∂GR(a1, a2, s) ) 0; (15) ) ) ∂a1 ∂a2 ∂s ∂GP(a1, a2, s) ∂GP(a1, a2, s) ∂GP(a1, a2, s) )0 ) ) ∂a1 ∂a2 ∂s
(16)
The solutions a1,R, a2,R, and sR (a1,P, a2,P, and sP) to eq 15 (eq 16) define the cavity radii for D and A and solvent orientational polarization, which are in full equilibrium with the reactant (product) state charge distribution of the ET complex. We note that the size of equilibrium cavities, in general, differs between the R and P states due to electrostriction, i.e., a1,R * a1,P and a2,R * a2,P. This arises from alterations of the charge distributions of D and A during the course of ET. As mentioned above,
Outer-Sphere Reorganization Free Energy
J. Phys. Chem. B, Vol. 110, No. 1, 2006 497
the equilibrium polarizations for the reactant and product states are given by sR ) 0 and sP ) 1, respectively. The outer-sphere reorganization free energy λ is the free energy cost for rearrangement of solvent configurations between those equilibrated with R and with P on the GR(a1, a2, s) or GP(a1, a2, s) surface. In contrast with the standard Marcus theory, not only the solvent orientational polarization gauged by s but also cavity radii a1 and a2 readjust in our formulation. Furthermore, the latter two introduce anharmonicity in GR and GP, so that λ becomes state-dependent in general; i.e., λ usually varies with the diabatic surface upon which the solvent reorganization occurs. We thus introduce λR and λP
∂λS 1 ∂0 1 ∂∞ ) - e2 2 - 2 ∂T ∞ ∂T p 0 ∂T p
p
×
∂a(m+1) 1 eq 1 e2 1 1 + (m) - + (m+1) (m+1) 2 r 2 ∂T p ∞ 0 (aeq 2aeq 2aeq ) 1
1
∂a(m) eq (m) 2 ∂T (aeq ) 1
e2((m + 1)2 - m2) ∂0 ∂T 2 2
+
p
0
1
(m+1) p aeq ∂a(m+1) eq
e2((m + 1)2 - m2) 1 1 12 0 (a(m+1))2 eq
-
∂T
1
a(m) eq
-
∂a(m) eq (m) 2 ∂T (aeq )
λP ) GP(a1,R, a2,R, sR) - GP(a1,P, a2,P, sP),
(17)
which describe reorganization free energy on the reactant- and product-state diabatic surfaces, respectively. We note that statedependent λ similar to eq 17 has been observed and studied previously by several different groups.33-38 In view of eqs 13 and 14, we see easily that cavitation and solute-solvent shortrange interactions in general contribute to λR and λP, in addition to solute-solvent electrostatic interactions. For the special case of electron self-exchange
D+m + A+(m+1) f D+(m+1) + A+m (D ) A)
(18)
λR and λP in eq 17 simplify to (m+1) (m) , a(m) , aeq ); λR ) λP ) λS ) λM(a(m+1) eq eq , r) + λcav(aeq
]
e2((m + 1)2 - m2) 1 1 1 λcav ≡ 1, (19) 2 0 a(m+1) a(m) eq eq where subscript S is to emphasize the symmetric nature of ET, (m+1) a(m) are the radii of the cavities in equilibrium with eq and aeq +m +m D ()A ) and D+(m+1)()A+(m+1)), respectively. It should be noticed that in addition to the usual Marcus term λM arising from orientational polarization, there is an extra contribution λcav to solvent reorganization. As mentioned above, the latter is due to electrostriction arising from the cavity size readjustment (“cavity resizing”) during ET.39 Since the cavity radius increases as the magnitude of charge enclosed diminishes [see section 2.2 above], λcav in eq 19 is always positive. Thus, the cavity resizing tends to increase the solvent reorganization free energy. We also notice that due to the symmetric nature of the reaction, direct contributions from cavitation and ULJ to solvent reorganization vanish for ET in eq 18, viz., terms involving p, γ∞, δ or ULJ are absent in λS. However, they do contribute in an indirect way through equilibrium condition eq 6. Because the equilibrium cavity size varies with thermodynamic conditions in our formulation [eqs 11 and 12], a(m+1) eq and a(m) eq in eq 19 change with T and p. This has an important consequence for variations of λS with thermodynamic conditions, which we consider next. 2.4. T and p Dependence of Outer-Sphere Reorganization Free Energy for Symmetric ET. We differentiate λS in eq 19 with respect to T to deduce
-
p
1
λR ) GR(a1,P, a2,P, sP) - GR(a1,R, a2,R, sR);
{ }[
( ) [ ( ) ( )] [ ] { }[ ( ) ( )] ( )[ ] { }[ ( ) ( )] p
. (20)
where (∂aeq/∂T)p is given by eq 11. It is easy to recognize that the first term on the right-hand side of eq 20 is the result of the standard Marcus theory, i.e., eq 3. The second term also arises from λM because cavity radii vary with T in our formulation. We emphasize that this term is absent in the conventional continuum theory since cavities are assumed to be fixed regardless of thermodynamic conditions. Because (∂aeq/∂T)p > 0 [section 2.2 above],17 the first and second terms on the righthand side of eq 20 have opposite signs in highly polar solvents. This alerts that account of cavity size changes with T will be critical to accurate predictions of (∂λS/∂T)p [cf. section 1]. The cavity resizing free energy λcav defined in eq 19 is responsible for the remaining terms in eq 20, which gauge the alteration of the electrostrictive effect with T. The p-derivative of λS can be obtained in a similar fashion. The result is
( )] () [ [ ] { }[ ( ) ( )] ( )[ { }[ ( ) ] ( )] ∂λS ∂p
( )
1 ∂∞ ∞2 ∂p
) - e2
T
1
+ (m+1)
2aeq
1
(m)
2aeq
∂a(m) eq (m) 2 ∂p (aeq ) 1
1
a(m) eq
-
1 ∂0 02 ∂p
T
×
∂a(m+1) eq 1 e2 1 1 1 r 2 ∞ 0 (a(m+1))2 ∂p eq +
T
T
-
e2((m + 1)2 - m2) ∂0 ∂p 2 2 0
1
-
(m+1) T aeq ∂a(m+1) eq
e2((m + 1)2 - m2) 1 1 12 0 (a(m+1))2 eq
∂p
∂a(m) eq 2 ∂p (a(m) ) eq
+
T
1
T
-
T
. (21)
3. Model Calculations for Acetonitrile In this section, we apply the theoretical formulation presented above, in particular, eq 20, to investigate the T dependence of λS for a mixed-valence dinuclear iron complex [Fe(420)3Fe]5+ (420 ) 1,4-bis-[4-(4′-methyl-2,2′-bipyridyl)]ethane) in acetonitrile and make contact with a recent measurement by Derr and Elliott via intervalence charge-transfer transitions.12 We also consider briefly the p dependence of λS using eq 21. 3.1. Models and Methods. Following ref 12, we assume that [Fe(420)3Fe]5+ consists of a donor and an acceptor moiety (D ) A), each placed in a spherical cavity in solution. To take into account the complex structure of the solute where its bulky ligands surrounding metal ions may prevent solvent molecules from reaching all the way to Fe, we consider off-centered
498 J. Phys. Chem. B, Vol. 110, No. 1, 2006
Manjari and Kim
TABLE 1: Cavity Radii and Their First Derivatives with Respect to T and p model
x0 (Å)
σLJ (Å)
LJ/kB (K)
a(2) eq (Å)
a(3) eq (Å)
(∂a(2) eq /∂T)p (10-3 Å/K)
(∂a(3) eq /∂T)p (10-3 Å/K)
(∂a(2) eq /∂p)T (10-4 Å/bar)
(∂a(3) eq /∂p)T (10-4 Å/bar)
I II III IV V VI
0.0 0.0 1.0 1.5 2.0 3.0
5.4 5.4 4.0 3.4 2.6 1.2
500 700 700 700 700 700
5.300 5.408 5.101 5.059 4.822 4.572
5.051 5.178 4.930 4.917 4.712 4.522
2.467 2.478 2.560 2.588 2.629 2.680
2.560 2.554 2.618 2.635 2.665 2.695
-1.727 -1.730 -1.585 -1.550 -1.491 -1.435
-1.459 -1.475 -1.431 -1.426 -1.411 -1.407
TABLE 2: Solvent Reorganization Free Energy and Its Variations with T and p λS (cm-1)
(∂λS/∂T)p (cm-1 K-1)
(∂λS/∂p)T (10-1 cm-1 bar-1)
model
r ) 7.6 Å
r)∞
λcav (cm )
r ) 7.6 Å
r)∞
r ) 7.6 Å
r)∞
I II III IV V VI
6220 5650 5870 5620 5910 5810
13 860 13 300 13 520 13 270 13 560 13 460
2630 2310 1920 1620 1370 670
-7.90 -7.15 -7.27 -6.91 -7.05 -6.51
-4.71 -3.96 -4.08 -3.72 -3.85 -3.32
-1.50 -1.33 -1.23 -1.18 -1.24 -1.42
-9.65 -9.48 -9.38 -9.33 -9.39 -9.57
-1
Lennard-Jones interactions40 between the solvent and the D and A moieties
ULJ(x) ) 4LJ
[( ) ( ) ] σLJ x - x0
12
-
σLJ x - x0
6
(22)
Here x is the distance between the solvent and Fe centers of D and A and x0 gauges the separation between the Fe centers and their ligands. Thus, the major contribution of off-centered ULJ arises from the short-range interactions between the solute ligands and solvent. Six different models were considered (models I-VI). Their values for x0, σLJ, and LJ are compiled in Table 1. Models I and II differ only in LJ, i.e., LJ/kB ) 500 K for the former and 700 K for the latter. For models II-VI, we varied x0 from 0 to 3 Å with LJ/kB fixed at 700 K and adjusted their σLJ values, so that they yield a good agreement with the experimental result for the solvent reorganization free energy λS ≈ 5790 cm-1 for [Fe(420)3Fe]5+ in acetonitrile at 25°.12 The X-ray crystallography result41 for the Fe-to-Fe distance r ) 7.6 Å was employed as the D-to-A separation in the calculations. To evaluate (∂λS/∂T)p via eq 20, we first solved eq 6 to determine a(m+1) and a(m) eq eq , and calculated their T-derivatives with eq 11. We employed the methods developed in our previous study on cavity behavior with thermodynamic conditions in neat solvents.17 Briefly, we used experimental information for γ∞ and Rp,42 0 and (∂0/∂T)p,43 and ∞ and (∂∞/∂T)p,42 and eq 22 for ULJ. For the determination of δ and its first derivatives, a mild extension17 of Tolman’s quasi-thermodynamic approach28 to account for oscillations of intrinsic local pressure near the cavity surface was used. The pressure dependence of λS was determined in a similar way. Experimental data for κT42 and (∂0/∂p)T44 were used. Because we were not able to find experimental information on (∂∞/∂p)T, we estimated it via ∞-1(∂∞/∂p)T ≈ κT assuming that ∞ ∝ n. 3.2. Results and Discussion. The numerical results for equilibrium cavity radii for D and A and their first derivatives with respect to T and p are summarized in Table 1. Irrespective (3) of the model descriptions, a(2) eq and aeq increase with T and decrease with p. Because the reactant- and product-state cavities are determined variationally via the equilibration condition eq 6, which depends on the thermodynamic conditions of the system, their radii vary with T and p. We already mentioned that a trend similar to this was found for various pure solvents in a broad polarity range in our previous study.17 As noted
above, the cavity behaviors found here (and also in ref 17) are precisely what was required for dielectric continuum descriptions in earlier studies to reproduce experimental results on variations of water ionization constant26 and solvent reorganization free energy27 with thermodynamic conditions. We make a few remarks for clarity here: First, despite a significant difference in LJ between models I and II, they are characterized by similar cavity radii and first derivatives. This indicates that the results of the model calculations are not that sensitive to the choice of the LJ value.45 Second, as the x0 value is varied across models II-VI with LJ fixed, σLJ changes rather substantially. As x0 increases, i.e., as ULJ becomes more offcenter, σLJ decreases. An estimation based on the recipe proposed in ref 46 yields σLJ ) 4.11 Å for pure acetonitrile in the united atom representation. This suggests that the σLJ value for ULJ gauging mainly the interaction between the solute ligand and CH3CN should not be much smaller than, say 4.0 Å. In this sense, model VI with σLJ ) 1.2 Å may be somewhat unrealistic for [Fe(420)3Fe]5+ in CH3CN. Model II (and also I) with x0 ) 0, on the other hand, does not reflect properly the extended ligand structure of the solute. We thus feel that models III-V provide the best description for [Fe(420)3Fe]5+ among the six different parameter sets considered here. Nevertheless, it should be stressed that all models yield similar results for the (3) T and p derivatives of aeq. Third, a(2) eq > aeq because of electrostriction (eq 10); their differences are 0.17, 0.14, and 0.11 Å for models III-V, respectively. This means that according to our theory, the cavity radius of the donor (acceptor) moiety decreases (increases) by about 0.1-0.2 Å as ET progresses from the R to P states. At the transition state, the donor and acceptor would be characterized by the same cavity size because of symmetry. The results for λS and its first derivatives are presented in Table 2. Because of the adjustments we made for σLJ as mentioned above, models II-VI show a good agreement with the measurement (viz., λS ∼ 5790 cm-1)12 when r ) 7.6 Å is employed in the calculations. For comparison, the results for infinitely separated D and A are also given there. One of the most pronounced features is that λcav defined in eq 19 makes a significant contribution to overall outer-sphere reorganization regardless of the model descriptions. For instance, λcav ) 1920 cm-1 and 1620 cm-1 for models III and IV, respectively, while their λM is about 4000 cm-1 at r ) 7.6 Å. As for model V, we obtained λcav ) 1370 cm-1 and λM ) 4540 cm-1. This means that in terms of free energy, about 25-30% of total solvent
Outer-Sphere Reorganization Free Energy reorganization occurs via cavity resizing despite small changes (
