J. Phys. Chem. B 1998, 102, 677-686
677
Activation Energy of Electron Transfer between a Metal Electrode and Reagents of Nonspherical Form and Complicated Charge Distribution. Cr(EDTA) Complexes† Renat R. Nazmutdinov,*,‡,§ Galina A. Tsirlina,| Yurij I. Kharkats,⊥ Oleg A. Petrii,| and Michael Probst‡ Institute of General and Inorganic Chemistry, UniVersity of Innsbruck, Innrain 52a, A-6020 Innsbruck, Austria, Moscow State UniVersity, Vorob’eVy Gory, Moscow GSP-3, 119899, Russia, and Institute of Electrochemistry, Russian Academy of Science, Leninskii Prospekt 31, Moscow 117071, Russia ReceiVed: July 1, 1997; In Final Form: September 3, 1997X
An approach is suggested in order to investigate the mechanism of interfacial electron-transfer processes with reagents of nonspherical form and complicated charge distribution. Chelate chromium(III) ethylenediaminetetraacetate (EDTA) complexes are regarded as a good model system. A series of SCF quantum chemical calculations at the ZINDO/1 level were performed for the systems Cr(EDTA)(H2O)n- (n ) 0, 1, 2, 3, and 4). A quinquedentate complex Cr(EDTA)H2O- was found to be energetically more favorable compared to Cr(EDTA)-. The interaction of Cr(EDTA)- with a cadmium electrode (modeled as a 19-atomic planar cluster) was investigated as well. The analysis of both the potential energy surfaces and the partial charge transfer allows the explanation of the absence of “specific” interaction between Cr(EDTA)- complexes and mercurylike metals observed experimentally. A way to estimate the inner-sphere contribution (Ein) to reorganization energy is proposed. The inner-sphere asymmetry (nonequality of Ein values for reduction and oxidation) was found, which plays a significant role in the theoretical analysis of the activation energy. To obtain estimates of the solvent reorganization energy, the shape of reagents was approximated by ellipsoids. The detailed atomic charge distribution in oxidized and reduced complexes was employed to provide a “microscopic” description of electrical double layer effects. Additional ab initio SCF calculations with several basis sets of different quality were performed for the analysis of the charge distributions. The activation energies for several orientations of Cr(EDTA)- and Cr(EDTA)H2O- relative to the metal surface were calculated for two values of the electrode charge densities. It is concluded that a sufficient interpretation of relevant experimental data is not possible without calculations of the reaction pre-exponent.
I. Introduction The charge transfer across electrochemical interfaces is of considerable interest for theory and fundamental for understanding the mechanism of a manifold of electrode processes. Recently, important steps have been made from general theoretical concepts toward a study of some electron-transfer reactions occurring near an electrode/solution interface. Schmickler developed the theory of adiabatic electrochemical reactions using the Anderson-Newns model of the chemisorption at a metal surface.1 This approach was employed to describe both the iodide ion discharge and the redox Fe3+/Fe2+ and Zn2+/ Zn+ processes.2,3 A series of pioneering molecular dynamics (MD) simulations were performed in order to estimate the activation energy (∆Ea) of the iodide ion discharge4,5 and of the reduction of Fe3+ proceeding at a Pt(100)/water interface.6-9 There the primary attention was paid to the calculation of solvent reorganization energy from the “first principles”. The harmonic approximation for polar media (linear response theory) was demonstrated to be valid near a metal/water boundary, which †
Dedicated to L. I. Krishtalik on the occasion of his 70th birthday. * Corresponding author. University of Innsbruck. § Permanent address: Kazan State Technological University, K. Marx Str. 68, 420015 Kazan, Republic Tatarstan, Russia. | Moscow State University. ⊥ Russian Academy of Science. X Abstract published in AdVance ACS Abstracts, November 1, 1997. ‡
coincides with the similar conclusion made earlier for the bulk systems.10 It was assumed in ref 8 that the Franck-Condon barrier for the iodide ion discharge plays a minor role and the reaction was treated as an ion-transfer process. The estimations of ∆Ea made in these works are in reasonable agreement with experiment, whereas a calculated value of the symmetry coefficient is about 0.1.5 Computer simulations in this area are restricted to classical MD schemes so far. Although ab initio MD (Car-Parrinello type) methods were already employed for the modeling of electrochemical interfaces,11 we do not know of any attempt to use this new powerful tool for investigations of the kinetics of electrode processes. The mechanism of interfacial reactions of electron transfer was also studied using quantum chemical SCF MO LCAO methods at semiempirical level. The activation energy of electrode reactions CH3X + e ) CH3 + X- (X notes Cl, Br, and I) was estimated by German et al.12 on the basis of the PM3 method. The cluster model of an electrode surface is a new branch of theoretical interfacial electrochemistry13 and was also employed in the studies of kinetic problems. An attempt to provide a microscopic description of the catalytic reactions with indium aqua- and hydroxocomplexes was made in ref 14. In this work the indium surface was modeled by a cluster and all quantum chemical calculations were performed at the CNDO/2 level. The power of computer simulations and quantum chemical methods available nowadays is sufficient for a thorough study of the mechanism of various electrode processes on the basis
S1089-5647(97)02130-5 CCC: $15.00 © 1998 American Chemical Society Published on Web 01/22/1998
678 J. Phys. Chem. B, Vol. 102, No. 4, 1998
Nazmutdinov et al.
of modern theories of the charge transfer in polar media,15 and most of the approaches developed in references cited above can in principle be extended to other electrochemical systems and, in particular, also to large complex ions of nonspherical form and complicated charge distribution. This leads, however, to additional problems and requires models that are not completely elaborated up to now. The inner-sphere reorganization, for example, is expected to contribute significantly to the total activation energy for such systems. Another problem to be solved is the interaction of nonspherical charge distribution with the field of the electric double layer (EDL). One of the challenging problems of electrochemical kinetics remains understanding the dependence of rate constants of redox reactions on the nature of an electrode. Recent attempts to handle this problem16,17 are based on the analysis of experimental data using theoretical expressions for the pre-exponent (transmission coefficient)18,19 and assume a constant value of the tunnelling distance for all reagent species. Redox processes with the participation of chromium(III) ethylene-diaminetetraacetate Cr(EDTA)- can be suggested as a good model system to investigate the problems considered. Cr(EDTA)- chelate complexes have a relatively large size and obviously nonspherical form and are soluble in many polar solvents. Electron-transfer reactions with such complexes were already studied experimentally in ref 20 (bulk water solutions) and in refs 21-23 (a mercury electrode/water interface). Very recently Hecht and Fawcett have reported the results on experimental studies of the kinetics of redox reactions with Cr(EDTA)- complexes at a mercury electrode for a series of nonaqueous solvents.24,25 These reactions were found to proceed without successive chemical stages. According to the capacity measurements reported in ref 26, Cr(EDTA)- species do not reveal any specific adsorption from water solutions at a mercury electrode and negatively charged mercury-like metals. Moreover, a drastic difference in the rate of Cr(EDTA)- reduction when passing from Cd to Hg electrode was also observed,27 and this interesting effect deserves a serious interpretation at microscopic level. The resulting (measured) rate constant of the electron-transfer reaction 〈κ〉 is often written in the integral form,15
〈κ〉 )
∫κ(z) P(z) dz
(1)
where κ(z) is the “partial” rate constant being a function of the metal-reactant distance z; P(z) is a distribution function calculated from statistical mechanics. This is a good approximation for molecules of symmetric form (halogenid ions in water solutions and some metal aqua-complexes). For complex nonspherical reactants with nonuniform charge distributions eq (1) should be rewritten as
〈κ〉 )
∫κ(z,Ω) P(z,Ω) dz dΩ
(2)
This form accounts for dependencies of partial kinetic parameters on both the metal-reagent distance and angular degree of freedom Ω. The orientation of large nonspherical molecules should be apparently sensible both to the electrode material and to the surface charge, and one can expect an interplay between different orientations contributing to the resulting 〈κ〉 value. The orientations can differ in partial transmission coefficients (orbital overlapping factors) and in the activation energies. The concept of the orientation effects in electrochemical kinetics was briefly discussed long ago28 but to the best of our knowledge, it was never considered quantitatively for real electrochemical systems.
This paper is organized as follows. First the quantum chemical method employed is briefly described. In the next section we present the results concerning the structure of Cr(EDTA)- complex ion in aqueous solution. In parts IV and V the calculations of the inner- and outer-sphere components of the reorganization energy are discussed. The results of quantum chemical calculations based on the cluster model for a cadmium surface and attempts to take into account the electric field of the electrical double layer (EDL) are summarized as well. The values of total activation energy of the Cr(EDTA)reduction on mercury and cadmium electrodes are collected in part VI. A detailed analysis of the wave functions as well as the calculations of resonance integrals and transmission coefficients will be reported separately together with related results.29 II. Method We have performed a series of SCF MO LCAO quantum chemical calculations at semiempirical INDO/1 (intermediate neglect of differential overlap) level. Up to now different INDO/1 schemes have been successfully employed to get information about electronic structure of transition metal complexes with large ligands.30 The version of INDO/1 used in this work was elaborated and tested originally by Zerner and co-workers.31-33 We used the ZINDO/1 implementation in the HyperChem code (version 4.5).34 Constant Slater d orbital exponents for transition metals (ξd), for example are employed in this program package instead of the distance dependent ones (ξd a + b/R) proposed in original work.31 A double-ξ representation for the radial part of d orbital R3d(r) of chromium atom was used to calculate overlap integrals,
R3d(r) ) C1r2 exp(-ξd1r) + C2r2 exp(-ξd2r)
(3)
The σ-σ and π-π overlap weighting factors held a value of 1 in our calculations. As almost all molecules considered have an open electronic shell, the unrestricted version of the Hartree-Fock scheme (UHF) was employed. Compared with the less sophisticated CNDO/2 method most of the exchange interactions are included into the INDO Hamiltonian. This provides a reliable description of the electronic states with high spin multiplicity, which is first of all important for transition metal complexes. Geometry optimization was done by the Polak-Ribiere conjugate gradient algorithm method. To test some results, we also performed several single-point ab initio SCF calculations with four basis sets of different quality using the Gaussian 94 program.35 For the systems of interest we have found experimental data that in principle can be compared with predictions of quantum chemical calculations. Water solutions of Cr(EDTA)- complexes have been studied by several spectroscopic techniques.36,39-45 The IR absorption data37 had been reported for crystalline salts containing Me(EDTA) fragments for some transition metals Me. A chromium(III) complex showed a shift of about 280 cm-1 in the asymmetrical vibration of the COO group. Our calculations on acetate acid (CH3COOH), Cr(EDTA)-, and Cr(EDTA)H2O- complexes predict a value of about 100 cm-1 that agrees only qualitatively with experiment. Another comparison with experimental data is possible using the results of X-ray studies of the crystal structure of aqua ethylenediaminetriacetatoacetic acid chromium(III).38 A Cr(HEDTA)H2O neutral complex was computed with the full optimization of molecular geometry for two different spin configurations (Figure 1). The quartet state was found to be the lowest lying one, about 8 kcal mol-1 lower than the douplet
Activation Energy of Electron Transfer
J. Phys. Chem. B, Vol. 102, No. 4, 1998 679 TABLE 2: Reaction Heat (∆H) Calculated for Several Model Processes (kcal mol-1) (1) (2) (3) (4) (5) (6)
process
-∆H/kcal mol-1
5H2O ) (H2O)5 Cr(EDTA)- + H2O ) Cr(EDTA)H2OCr(EDTA)- + 3H2O ) Cr(EDTA)(H2O)3Cr(EDTA)- + 3H2O ) Cr(EDTA)(H2O)3- (*) Cr(EDTA)- + 2H2O ) Cr(EDTA)(H2O)2Cr(EDTA)- + 4H2O ) Cr(EDTA)(H2O)4-
95.4 59.2 163.7 151.4 53.4 139.2
III. Reagents in Bulk Solution
Figure 1. Molecular structure of Cr(H2O)HEDTA (Table 1).
TABLE 1: Selected Bond Lengths (Å) and Angles (deg) Calculated for a Cr(H2O)HEDTA Molecule (Figure 1). Experimental Data38 Are in Parentheses Cr-O8 Cr-O6 Cr-N1 C9-C10 N1-C1 C6-O8
2.175 (1.935) 2.260 (2.002) 2.392 (2.141) 1.482 (1.503) 1.440 (1.497) 1.345 (1.305)
C4-C3 N1-Cr-N2 N1-Cr-O6 O6-Cr-O7 O9-Cr-N2
1.494 (1.515) 72.08 (84.78) 103.26 (99.31) 98.57 (89.51) 83.55 (80.90)
state. As can be seen from Table 1, the bond lengths of the HEDTA fragment and its angles are described rather well, whereas the Cr- N and Cr-O bond lengths are somewhat overestimated. It has to be kept in mind, however, that a crystalline substance was investigated in ref 38, whereas we dealt with an isolated molecule. Taking this circumstance into account, we can maintain that agreement with experiment is quite good. This conclusion does not look surprising, because the nonspectroscopic versions of the INDO method describes correctly first of all the molecular geometry of transition metal complexes.32
Figure 2. Several forms of Cr(EDTA)(H2O)n- complexes: (d) Cr(EDTA)(H2O)3- (*).
Up to now the structure of Cr(EDTA)- complexes in aqueous solutions remained a matter for discussions.39-45 The main problem seems to be the possible penetration of a water molecule into the inner coordination sphere, which can lead to the formation of a stable quinquedentate Cr(EDTA)H2Ocomplex. The latter structure was accepted in works.40-43 However, according to Wheeler and Legg’s studies by deuteron NMR spectroscopy45 EDTA4- forms a sexidentate complex with Cr(III) between pH 3.5 and 6.5. The same conclusion was reached in ref 44 on the basis of Raman spectroscopy investigations. We believe that a quantum chemical study of certain model structures can bring this controversial situation somewhat to light. The calculations of several molecules [Cr(EDTA)(H2O)n]z (n ) 0, 1, 2, 3, and 4) were performed both for oxidized (z ) -1) and for reduced (z ) -2) forms for different values of the spin multiplicity. The electronic states with the highest spin number (ns) were obtained to be energetically more favorable for both forms. In the case of a sexidentate Cr(EDTA)- (Figure 2a) the lowest energy was found for a quartet state (ns ) 4), whereas for Cr(EDTA)2- a triplet configuration (ns ) 3) has the lowest energy. The difference in the total energy was found to be 56 kcal mol-1 for Cr(EDTA)- (between quartet and douplet states) and 131 kcal mol-1 for Cr(EDTA)2- (between triplet and singlet states). Judging from the charge distributions, all complexes under investigation reveal a strongly covalent character of the CrEDTA binding that supports conclusions made earlier in work37 on the basis of IR measurements. Estimates of ∆H of several model reactions are collected in Table 2. Here we intrude, however, into the realm of speculations, where one should be careful and avoid far-reaching quantitative conclusions. Since our studies were aimed at modeling of reactions in water solutions, we need a measure of the water-water interaction energy (∆Hw). Process 1 in Table 2 can be suggested in order to provide relevant information. In
(a) Cr(EDTA)-; (b) Cr(EDTA)(H2O)-; (c) Cr(EDTA)(H2O)3- ; and
680 J. Phys. Chem. B, Vol. 102, No. 4, 1998 an open pentamer (H2O)5 one water molecule forms four H-bonds with nearest neighbors, and therefore, a quantity of ∆H(1)/4 (-24.9 kcal mol-1) may be treated as a reasonable approximation for ∆Hw. Although INDO fails to describe H-bonds46 and the result obtained is strongly overestimated, this value can be used, nevertheless, for some qualitative predictions. The calculations performed for a Cr(EDTA)H2O- molecule point to the formation of a complex with the quinqedentate coordination of Cr(III) (Figure 2b). It can be seen from Table 2 that ∆H(2) < ∆Hw, and therefore, process 2 should be feasible. In such a structure a water molecule is bonded directly to the chromium atom, which results in a significant stretching of one of the Cr-COO bonds. Moreover, the noticeable deformation of the water molecule (H1OH2) (R(O-H1) ) 1.260 Å and R(OH2) ) 1.013 Å) may be regarded as precursor of the reaction,
Nazmutdinov et al. TABLE 3: Selected Bond Lengths (Å) and Bond Angles (deg) Calculated for Cr(EDTA)z and Cr(EDTA)H2Oz Complexes (z ) -1 and -2). O1 Refers to the Nearest Oxygen Atoms Lying at the Same Plane as the Nitrogen Atoms (“Equatorial” Coordination), O2 Marks the Two Nearest “Axially” Coordinated Oxygen Atoms, and Ow Is the Oxygen Atom of a Water Molecule Cr(EDTA)- Cr(EDTA)2- Cr(EDTA)H2O- Cr(EDTA)H2O2Cr-O1 Cr-O2 Cr-O1a Cr-Ow Cr-N O1-Cr-O1 O2-Cr-O2 O1-Cr-O1a N-Cr-N a
Cr(EDTA)H2O- S Cr(EDTA)OH2- + H+ A similar equilibrium process in water solutions was considered earlier in ref 42. After three water molecules were brought into play, we found a minimum of the total energy corresponding to a striking structure (Figure 2c), where one water molecule is “wedged” into the chelate fragment COO-Cr-COO, forming a chemical bond directly with the Cr atom. Thus, we observed herein a complex with seven-coordinated Cr(III). Since seven-coordination for chromium is rare,47 we assumed that this molecule might correspond to an assumed transition state of the SN1 reaction II in Table 2. To prove this hypothesis, a “broken” transition complex Cr(EDTA)(H2O)3- (*) (Figure 2d) was computed as well. It can be seen from Table 2 that the difference between ∆H values for processes 3 and 4 amounts to 12.3 kcal mol-1. As this value is less than ∆Hw, the transition structure involving a seven-coordinated Cr atom may lead to the formation of Cr(EDTA)H2O-. A EDTA molecule is known to have both hydrophobic (-CH3-CH3-) and hydrophilic (-COO-) fragments. To study the interaction of different EDTA fragments with water molecules, we modeled processes 5 and 6 (Table 2). In a Cr(EDTA)(H2O)2- complex two water molecules contact with the hydrophobic edge of EDTA, whereas in Cr(EDTA)(H2O)4they are attached to four COO groups. It is easy to see from Table 2, that ∆H(6)/4 < ∆Hw < ∆H(5)/2. Thus, our calculations allows the qualitative description of the hydrophilic and hydrophobic nature of EDTA-water interaction. Although the formation of a quinqendentate Cr(EDTA)H2Ocomplex was found to be more favorable compared with a sexidentate Cr(EDTA)-, the simultaneous existence of both Cr(EDTA)H2O- and Cr(EDTA)- seems possible bearing in mind that complexes of Cr(III) in water solutions are very inert to substitution.43 IV. Inner-Sphere Reorganization Energy It can be seen from Table 3 that electron transfer results in a noticeable change in the molecular geometry of both Cr(EDTA)and Cr(EDTA)H2O- complexes. Weaker chemical binding between the chromium atom and ligands in the reduced form leads to the stretching of the Cr-O and Cr-N bonds, as well as to the distortion of the bond angles. Other geometrical parameters of the molecules are changed very slightly, and, therefore, this part of inner-sphere coordinates does not affect the activation barrier. We attempted to estimate the inner-sphere contribution to the total reorganization energy of the Cr(EDTA)-
2.115 2.157
2.258 2.233
2.253 126.36 156.66
2.423 143.12 148.40
81.07
75.41
2.112 2.127 3.588 2.089 2.248
2.276 2.247 3.694 2.218 2.379
156.64 146.60 81.64
149.44 157.92 77.39
The oxygen atom of EDTA substituted by a water molecule.
reduction (Ein) using a well-known approach based on the harmonic approximation1,48
∑k m*k
Ein ≈
(ωki ωkf )2 (ωki
+
ωkf )2
(Rkf - Rki )2
(4)
Rkf and Rki are the metal-ligand atom bond lengths for the reduced and oxidized state, respectively; m*k is the “effective” oscillator mass; and wki and wkf are the frequencies of normal vibrations. The summation in eq 4 is taken over all “classical” vibrations (pω e κT). Recently similar quantum chemical calculations at the INDO level were performed by Khan and Zhou49,50 in order to estimate the inner-sphere reorganization energy for certain aqua- and ammine-complexes of transition metals. It is convenient in our case to represent Ein as a sum of two main components, Eeqin and Eaxin (contributions from the “equatorial” and “axial” nearest coordinated atoms (Table 3), respectively),
{
E eq in ≈ 2m*
eq 2 (ωeq i ωf )
eq 2 (ωeq i + ωf )
Cr-O(1) 2 ((RCr-O(1) ) + i - Rf
(RCr-N - RCr-N )2) i f and
{
E ax in ≈ 2m*
ax 2 (ωax i ωf )
ax 2 (ωax i + ωf )
}
(RCr-O(2) - RCr-O(2) )2 i f
}
(5)
(6)
Since the vibrations occur between the ligand and the metal atom, we can define m* as mCrmEDTA/(mCr + mEDTA); therefore m* ≈ 44. The frequencies obtained from additional quantum chemical calculations were 440 cm-1 (405 cm-1) for eq -1 (356 cm-1) for wax (wax), respectively. weq i (wf ) and 413 cm i f These results are in line with the frequencies known for other complexes of transition metals.48 Equations 5 and 6 lead to a noticeable contribution from the Eeqin part (16.46 kcal-1), whereas the Eaxin value is significantly less (0.804 kcal mol-1). A negligibly small value of Ein was assumed earlier in a study22 on the Cr(EDTA)- reduction at a mercury electrode, whereas according to our calculations the resulting Ein amounts to 17.26 kcal mol-1. The quantum modes (hω . kT) do not contribute to the activation energy, but can affect the reaction preexponent
Activation Energy of Electron Transfer
J. Phys. Chem. B, Vol. 102, No. 4, 1998 681
TABLE 4: Inner-Shell Part of the Total Reorganization Energy of Oxidation and Reduction Processes Calculated by Eqs 7, 8 for Several Forms of Cr(EDTA) Complexes (kcal mol-1) redox pair (M2-/M-)
reduction
oxidation
M ) Cr(EDTA) M ) Cr(EDTA)H2O M ) Cr(EDTA)(H2O)4
31.20 28.23 31.20
13.50 13.17 12.56
(tunneling factor).15 However, the change of corresponding geometrical parameters of the complexes (e.g., the lengths of the covalent N-C, C-O, and C-H bonds) was found to be very small, and therefore, this part of inner-sphere plays a minor role in the electron transfer. Equations 4, 5, and 6 are approximate, because an “average” frequency is used for simplicity. Moreover, the parabolic approximation for the outer-sphere coordinates is assumed to be valid also in the vicinity of the activation barrier. We suggest another way to calculate Ein from Marcus theory. Let us assume a difference between the inner-sphere reorganization energies for reduction (R) -
Cr(EDTA) + e ) Cr(EDTA)
2-
(
∆Ea ) Wi + θ(1 - θ) Es +
(
∆I + VE Bin
)
1 - 2θ + (1 - υ)θ2 (1 - (1 - V)θ)2
)
υE Bin + θ∆I (10) 1 - θ + υθ + (1 - 2θ)Es ) 0 (11)
where υ ) A Ein/E Bin and the parameter θ refers to the transfer coefficient. It is easy to see that for the case B Ein ) A Ein an expression for θ takes the well-known form
∆I 1 θ) + 2 2(Es + Ein)
(12)
and eq 10 coincides with eq 9.
and oxidation (O)
V. Reagents in the Interfacial Layer
Cr(EDTA)2- ) Cr(EDTA)- + e Then we can define Ein for process R in the following way,
B Ein ) Etot〈Cr(EDTA)2-〉* - Etot〈Cr(EDTA)2-〉
(7)
where Etot〈Cr(EDTA)2-〉* is the total energy of a Cr(EDTA)2complex with the molecular geometry optimized for Cr(EDTA)(* means nonequilibrium state) and Etot〈Cr(EDTA)2-〉 is the total energy of Cr(EDTA)2- (equilibrium state). The inner-sphere reorganization energy of process O is calculated in a similar fashion,
A Ein ) Etot〈Cr(EDTA)-〉* - Etot〈Cr(EDTA)-〉
(8)
where Etot〈Cr(EDTA)-〉 is the total energy of a Cr(EDTA)complex with the molecular geometry optimized for Cr(EDTA)2and Etot〈Cr(EDTA)-〉 is the total energy of Cr(EDTA)-. We found a significant difference (asymmetry) between B Ein and A Ein that can be regarded as the most important result arising from such an analysis (Table 4). As can be seen from Table 4, water molecules interacting with Cr(EDTA)2-/- complexes in different ways slightly change the results obtained for the nonhydrated complexes. It is also interesting to note that the results of calculations using eqs 5 and 6 (17.26 kcal mol-1) are closer to the estimate obtained by eq 8 (13.5 kcal mol-1). We can, therefore, assume the strong violation of harmonic behaviour of the reaction term describing the reduced state near the Franck-Condon barrier. This asymmetry phenomenon and its possible manifestations in kinetics are considered by us separately.51 According to our preliminary estimates, it is less pronounced for spherical nonchelate complexes. When B Ein ≈ A Ein ) Ein, the well-known Marcus equation is usually applied to calculate the activation energy of the electrontransfer reactions (∆Ea),
∆Ea ) Wi +
where Es is the solvent reorganization energy, Ein is the innersphere contribution to the total reorganization energy, and ∆I is the reaction heat (∆I, ) -Fη + Wf - Wi, η denotes the electrode overpotential). Wi and Wf are energies needed to transfer a reagent in oxidized (i) and reduced (f) states, respectively, from the solution bulk to the reaction layer (work of approach). It is important to note that in the “asymmetric” case (E Bin . A Ein, or vice versa) we come to a generalization of eq 9 in form of the system of equations,51
(Es + Ein + ∆I)2 4(Es + Ein)
(9)
Two components of activation energy (Es and W) depend on the reagent localization in the near-electrode layer. The specific features of these components for nonspherical reagents are considered below. The first problem to be discussed is the nature of reagent-metal interaction. Interaction with a Metal Surface. It follows from the analysis of capacity measurements,26 that EDTA containing complexes of Cr(III) do not reveal any specific adsorption from water solutions neither on a mercury nor on a cadmium electrode. It was also established that the Cr(EDTA)- reduction proceeds at small negative charges of a cadmium electrode, whereas the similar process at mercury can occur only at more negative electrode charge densities.27 In an effort to provide some understanding of these experimentally observed features, we performed quantum chemical model calculations of the adsorption of a single Cr(EDTA)- ion at a cadmium surface. A Cd(1000) face was modeled by a planar 19-atomic cluster (Figure 3a). The positions of the metal atoms in the cluster were held fixed with a nearest neighbour distance of 2.98 Å taken from the bulk. Quantum chemical calculations were performed for several values of the Cr-Cd distance without the additional optimization of molecular geometry. The adsorption near the potential of zero charge was assumed, which agrees with experimental conditions mentioned above. We investigated two different orientations of the adsorbate (Figure 3b,c). The dependence of the total energy of the adsorption complex and the partial charge transfer as a function of R(Cr-Cd) is shown in Figure 4. The cadmium-Cr(EDTA)- interaction was found to be significant for both orientations and leads to a noticeable charge transfer from the ion to the metal. As can be seen from Figure 4, the specific interaction (or, in other words, chemisorption) occurs only in a certain interval of the Cr-Cd distances (“contact” adsorption region). Beyond a R(Cr-Cd) value of 5 Å (orientation in Figure 3b) or 7 Å (orientation in Figure 3c) the partial charge transfer becomes practically zero. This demonstrates obviously that the “specific” interaction manifested in the electrochemical measurements no longer occurs at such distances. We can conclude, therefore, that a Cr(EDTA)- ion
682 J. Phys. Chem. B, Vol. 102, No. 4, 1998
Nazmutdinov et al. TABLE 5: Geometric Parameters of Cr(EDTA)- and Cr(EDTA)H2O- Complexes Described as Particles of Ellipsoidal Shape (Å)
a
Figure 3. Planar cluster modeling the surface of a cadmium electrode (a) and two different orientations of Cr(EDTA)- relative to the metal surface (b and c).
ellipsoid axis
Cr(EDTA)-
Cr(EDTA)H2O-
a b c reff
5.23 4.53 3.36 4.36
5.61 4.53 3.36 4.49 (4.9)a
Experimental estimate.22
a significant amount of the water desorption energy exceeding the complex-metal interaction energy (∆Ei-Me). We estimated the number of water molecules to be desorbed (nw) using a simple geometrical approach and found nw to be about 9. Since our quantum chemical calculations predicted a value of 36 kcal mol-1 for the desorption energy of a single water molecule adsorbed on-top at the cadmium cluster with the dipole moment parallel to the surface (∆Edes), it is evident from Figure 4 that nw∆Edes > ∆Ei-Me. Thus, at least one layer of water molecules appeared to be between a Cr(EDTA)- complex and a cadmium electrode surface. As cadmium is widely regarded by electrochemists as a mercury-like electrode, a similar picture of the Cr(EDTA)- -mercury interaction is likely. This allows us to deal in practice only with the electrostatic part of and Wf and Wi components of eqs 9-11 (interaction of the reagent with the field of the electric double layer). Outer-Sphere Reorganization Energy. The outer-sphere part of reorganization energy (Es) results from a nonequilibrium solvent response. Since we employed “continuum” models to calculate Es, estimations of the geometrical parameters of reagents were of primary importance in this part of the investigation. Both Cr(EDTA)- and Cr(EDTA)H2O- were described as ellipsoids. Parameters characterizing the ellipsoids (Table 5) were estimated using the molecular modeling program package HyperChem. The relaxation of molecular geometry after the electron transfer was neglected. Finally, Es was calculated on the basis of the model of conducting ellipsoids developed in work52 that seems to work well for large molecules with high polarizability. We considered two different orientations of the ellipsoids: axis c is perpendicular (eq 13) and parallel (eq 14) to an electrode surface,
Es ) and
Es )
(
){
(
)}
(
){
(
)}
a2 + (ceff)2 1 1 1 1 1opt st 2reff 4x 12x2
a2 + (ceff)2 1 1 1 1 1+ opt st 2reff 4x 24x2
(13)
(14)
where reff (effective radius of the ellipsoidal figure) is defined in the following way:
reff )
xa2 - c2 F(θ,φ)
F(∂,φ) is the elliptic integral of the first kind with parameters Figure 4. Total energy of the Cr(EDTA)-‚‚‚Cd19 adsorption complex (a) and the charge of the cadmium cluster (b) as a function of the Crelectrode distance (solid line, orientation Figure 3b; dashed line, orientation Figure 3c).
does not penetrate into the first water monolayer on the electrode surface in real electrolyte solutions. This points admittedly to
x
θ ) arcsin
a2 - c2 and φ ) a2
x
a2 - b2 a2 - c2
ceff ) (b + c)/2 and opt and st are optical and static dielectric constants of the medium (1.8 and 78 for water, respectively). In eqs 13 and 14 x is the ellipsoid center-metal surface distance.
Activation Energy of Electron Transfer
J. Phys. Chem. B, Vol. 102, No. 4, 1998 683
Figure 5. Selected orientations of the Cr(EDTA)- complex relative to the electrode surface.
In our calculations one layer of water molecules between the reagent and electrode was assumed; therefore, x ) reff + 2rw (where rw is the effective radius of a water molecule (1.5 Å)). As shown in Table 5, the values of reff calculated both for Cr(EDTA)- (4.36 Å) and for Cr(EDTA)H2O- (4.49 Å) are in reasonable agreement with an estimate (4.9 Å) obtained from the diffusion coefficient measurements.22 Values of the outer-sphere reorganization energy calculated from eqs 13 and 14 range from 14.27 to 15.08 kcal mol-1, and both forms of the reagents are practically not distinguishable in this sense. We attempted to investigate the influence of spatial dispersion of st on the solvent reorganization energy using equations derived by Liu and Newton (model of interlayer).53 As expected, this effect plays a minor role for our systems. We also found that Es calculated by the simple Marcus formula54 (derived for a conducting sphere near a metal surface) with the value of reff taken from Table 5 is very close to the result obtained using eqs 13 and 14. Interaction with the Field of the Electric Double Layer. We considered in the present work four major orientations for Cr(EDTA)- (Figure 5) and five for Cr(EDTA)H2O- (Figure 6) needed for further calculations. A rigorous self-consistent description of the interaction of charged species with the nonuniform field of the EDL is a very complicated problem just from the viewpoint of classical electrostatics. We suggest a few assumptions to handle that problem in a more simple way. Let us represent the charge distribution in the atomic B1,...,R Bn) by a sum, positions Fcoul (R
Fcoul(R B1,..,R Bn) ≈ qpoint(R B1,...,R Bn) + qhyb(R B1,...,R Bn) (15) where a set of the point atomic charges of qpoint(R B1,...,R Bn) is calculated from Mulliken population analysis. The second term, qhyb(R B1,...,R Bn), arises from the fact that coordinates of the “charge center” of atoms do not coincide in general with their geometrical position (“hybridization” effect). Semiempirical quan-
Figure 6. Selected orientations of the Cr(EDTA)(H2O)- complex relative to the electrode surface.
tum chemical methods (particularly, INDO) allow one to extract µhyb from the total dipole moment of a molecule,55
µtot ) µpoint + µhyb
(16)
where µpoint is calculated on the basis of qpoint(R B1,...,R Bn). In our case µhyb includes sp and pd components. It should be noted that unlike µpoint, the “hybrid” part of the total dipole moment is a function of the electronic density matrix and does not depend on a choice of the molecular coordinate system even for charged systems. Then the electrostatic energy of field-reagent interaction (W components in eqs 9-11) can be written as
W)
∫ψ(x) Fcoul(RB1,...,RBn) dΩ ≈ ∑j qpoint(xj) ψ(xj) + µhyb(x)BF(x*)
(17)
where x is a coordinate normal to the metal surface, ψ(x) is the potential distribution within the electrical double layer, µhyb(x) is the x-projection of µhyb, B F(x) is the field strength, and x* is the reagent center (the ellipsoid center in our case). Since in our case the plane of closest approach of the ellipsoid center of Cr(EDTA)- and Cr(EDTA)H2O- ranges from 5 to 9 Å, most atoms of the reagents are positioned in the diffuse part of the EDL. The behaviour of ψ(x) inside the diffuse layer is described usually in terms of the well-known Gouy-Chapman theory,
ψ(x) )
{ {
} { }}
FψOHP 4RT x′ arctan h tanh exp F 4RT λ
(18)
where ψOHP is the potential of outer Helmholts plane (OHP), x′ is the distance between a point charge in the diffuse layer and
684 J. Phys. Chem. B, Vol. 102, No. 4, 1998
Nazmutdinov et al.
TABLE 6: “Hybrid” Part of the Total Dipole Moment Projected on the Major Ellipsoid Axes (µhyb(x)), Diagonal Components of the Tensor of Electronic Polarizability (rLL), and Parameter of the Solvent Reaction Field (f) Calculated for Cr(EDTA)z and Cr(EDTA)H2Oz Complexes (in Parentheses) ellipsoid axes µhyb(x)(z ) -1)/D µhyb(x)(z ) -2)/D RLL/Å3 f
c
a
b
4.49 (3.00) 4.69 (3.45) 18.58 (19.81) 1.42 (1.42)
0.11 (0.56) 0.10 (0.15) 48.16 (23.07) 1.25 (1.23)
0.49 (0.61) 0.53 (-0.12) 20.40 (19.81) 1.29 (1.42)
OHP, λ ) x0RT/2cF2, c is the background electrolyte concentration, and is the bulk value of the dielectric constant of water (78). A few atoms can also fall into the space between inner and outer Helmholtz planes, which is characterized by the linear potential drop and an assumed value of the water dielectric constant in the bulk. Details of the calculation of ψ(x) can be found in ref 26. There are two additional electrostatic effects which also should be included into the analysis. The first effect is polarizability (R) in terms of a linear response to an external electric ind field B F(x) resulting in an induced dipole moment µhyb(x) ; the second one describes the solvent influence on the charge distribution in the reagents. To describe both effects, we employed an approach based on the Onsager reaction field model and considered in detail for particles of ellipsoid shape in Bo¨ttcher’s monograph.56 Some R components obtained in the present work (Table 6) are close to an estimate (≈18 Å3) used by Newton et al.57 to describe the polarizability of ferrocene and cobaltocene complexes. It was found, however, that for EDL fields of 106 V/cm, which are induced by electrode charges up to 20 µC/cm2, the polarizability correction is negligible. The solvent affects the dipole moment, increasing it by a factor f of 1.3-1.4 depending on orientation. solv µeff(x) ) fµind hyb(x)
(19)
solv Here µeff(x) is the resulting dipole moment induced by the solvent reaction field and parameter f is calculated according to ref 56. Both R and f parameters are a function of the geometrical characteristics of reagents (ellipsoid axes), the static dielectric constant of water ( ) 78), and an ellipsoid particle (i). As we computed the electronic properties of complexes, a value of 2 (close to the optical dielectric constant for water) was chosen as a reasonable estimate for i. Since molecular charge distributions play an important role in our estimations of W components, it would be useful to compare the atomic charges in a Cr(EDTA)- molecule calculated by different quantum chemical methods. The complexes were computed also at the unrestricted Hartree-Fock level using the fixed molecular geometry optimized by ZINDO/1 calculations. The chromium orbitals 3s3p3d4s4p were described by a simple (3s,3p,5d,5s,2p). Gaussian basis set (LanL2MB) or a (3s,3p,4d,d′,4s,s′,p,p′) Gaussian basis set of DZ quality (LanL2DZ).58 The effect of the inner-shell electrons was included in the relativistic effective core potential by Hay and Wadt.58 We employed STO-3G59 (included to LanL2MB), D95V (included to LanL2DZ)60 and 3-21G61 basis sets to describe C, N, O, and H atoms. It can be seen from Table 7 that the ZINDO/1 partial charges are qualitatively reasonable and fall into the range of SCF-derived charges. The values of the resulting dipole moment (µtot) calculated in the same coordinate system are also presented in Table 7. This quantity
TABLE 7: Mulliken Atomic Charges (e) of Selected Atoms and the Resulting Dipole Moment (µtot/D) of Cr(EDTA)Calculated by Different Quantum Chemical Methods atom
ZINDO/1
LanL2MB
ab initio LanL2DZ
LanL2DZ/3-21g
Cr O1a O2a N C1 C2
0.489 -0.419 -0.443 -0.070 0.463 -0.007
0.304 -0.144 -0.393 -0.313 0.271 -0.034
1.298 -0.587 -0.663 -0.526 0.407 -0.253
1.461 -0.498 -0.514 -0.737 0.276 -0.259
µtot
11.80
4.42
7.35
9.30
a
Nearest coordinated oxygen atoms.
TABLE 8: Works of Approach (W/kcal mol-1) Calculated for Several Orientations of Oxidized (i) and Reduced (f) Forms of Cr(EDTA)- and Cr(EDTA)H2O- Complexes Relative to the Electrode Surface (Figures 5 and 6) and Two Different Values of the Surface Charge (qs). A Concentration of 0.1 M for the Supporting Electrolyte Was Assumed in the Calculations qs ) -2.9 µC cm-2
qs ) -13.5 µC cm-2
Wi
Wf
Wi
Wf
0.81 0.53 0.48 0.62
1.55 1.17 1.06 1.24
2.37 1.31 1.31 1.52
3.64 2.37 2.30 2.56
0.83 0.53 0.48 0.60 0.58
1.57 1.17 1.01 1.18 1.18
2.49 1.31 1.27 1.64 1.59
4.47 2.99 2.56 3.14 3.16
Cr(EDTA)a b c d Cr(EDTA)H2Oa b c d e
may be regarded as a more objective characteristic of the charge distribution. It can be seen from Table 7 that ZINDO/1 predicts a maximal µtot value; however, the corresponding ab initio SCF results also increase noticeably with increasing basis set quality. The results collected in Table 8 demonstrate a drastic difference in the Wi(f) values calculated for orientation a and other orientations. These effects cannot be found by using traditional schemes of the EDL corrections treating the reagent as a simple point charge. The distribution of the atomic point charges was found to play an important role for all orientations. The contribution of spread charge (µhyb) is the most pronounced only for a and b orientations (Figures 5 and 6), where it amounts to 10% of the resulting value. As follows from simple estimates,29 all orientations are competitive even for high values of the electrode charges. This is especially important for the calculation of resulting current density, which depends crucial on the orientation distribution of reagents. This way to calculate the Wi(f) components can be regarded as “microscopic” psi-prime (ψ′) corrections and should be helpful for studies of the mechanism of interfacial reactions of electron transfer with large nonspherical complex ions. Let us define the “microscopic” discreteness coefficient (λ):
λ* ) W cli /Wi
(20)
where W cli refers to ψ′ corrections made in the traditional (classical) manner.27 It has been found that the relating λ* values are close to 1 for d and e orientations, but 1 in the case of orientations b and c. In this approach we neglected the dielectric nature of the ellipsoid resulting in the difference of electric field inside and outside the particle. An exact analytical solution for this
Activation Energy of Electron Transfer
J. Phys. Chem. B, Vol. 102, No. 4, 1998 685
TABLE 9: Activation Energy of Electron Transfer (∆Ea/ kcal mol-1) Calculated for Several Orientations of Cr(EDTA)- and Cr(EDTA)H2O- Complexes Relative to the Electrode Surface (Figures 5 and 6) and Two Different Values of the Surface Charge (qs) Relating to the Same Overvoltage (0.05 V) for Cadmium and Mercury Electrodes qs ) -2.9 µC cm-2 (Cd) qs ) -12.5 µC cm-2 (Hg) Cr(EDTA)a b c d Cr(EDTA)H2Oa b c d e
10.57 10.24 10.10 10.26
12.29 10.72 11.07 11.31
9.84 9.51 9.30 9.45 9.45
12.04 10.80 10.45 10.91 10.90
Acknowledgment. It is a pleasure to thank Prof. A. M. Kuznetsov (Moscow) for helpful discussions. This work was supported in part by O ¨ AAD and Russian Foundation for Fundamental Research, Project No. 96-03-32370a. A support from the Austrian FWF is also gratefully acknowledged. Appendix The diagonal components of the polarizability tensor (RLL) and parameter f were calculated in the following way:56
and
abc
(22)
∫0∞(s +dsR2)R
abc 2
(L ) a, b, and c)
(23)
References and Notes
For the transfer coefficient, θ values of 0.55 ( 0.02 were found (eq 11) for all orientations. This can be compared with values obtained from experimental data on the basis of traditional ψ′ corrections.21-25,27 The value 0.5 ( 0.05 for Hg27 is in good agreement with the theoretical predictions, whereas θ ) 0.63 for Cd27 is slightly higher. Other results were also reported for a mercury electrode (0.5821 and 0.522). Surprisingly low values of the transfer coefficient (0.31-0.38) were found in works24,25 on nonaqueous media. The values of the activation energy calculated for different orientations of the hydrated and nonhydrated complexes are shown in Table 9. The main features are the lower ∆Ea values for b and c orientations determined by the smaller repulsive double-layer contributions. Furthermore, it can be seen that the activation energy for all orientations slightly increases with increasing of the negative electrode charge. A small difference in ∆Ea observed for Cr(EDTA)- and Cr(EDTA)H2O- complexes originates mainly mainly from the inner-sphere contribution. It is also evident from Table 9 that the difference of rate constants for Cd and Hg (about 2 orders of magnitude)27 cannot be interpreted only on the basis of the knowledge of activation energies without additional calculations of the pre-exponential factor.29
3{1 + (i - 1)AL}
+ (i - )AR
where R2 ) (s + a2)(s + b2)(s + c2).
VI. Transfer Coefficient and Activation Energy
i - 1
{ + (1 - )AR}{1 + (i - 1)AR}
where a, b, and c are the major ellipsoid axes, is the static dielectric constant of water, and i is the dielectric constant of an ellipsoid particle. Parameter AL in eqs 21 and 22 depends on the geometrical characteristics of the ellipsoid,
AL )
problem is known only for the case of a constant external field.56 Qualitative analysis of this complicated case, however, predicts the increasing difference in the electrostatic energy W values for orientations considered.
RLL )
f)
(21)
(1) Schmickler, W. Interfacial Electrochemistry; Oxford University Press: Oxford, 1996. (2) Schmickler, W. Chem. Phys. Lett. 1995, 237, 152. (3) Schmickler, W. Electrochim. Acta 1996, 41, 2329. (4) Xia, X.; Berkowitz, M. L. Chem. Phys. Lett. 1994, 227, 561. (5) Rose, D. A.; Benjamin, I. J. Chem. Phys. 1994, 100, 3545. (6) Smith, B. B.; Halley, J. W. J. Chem. Phys. 1994, 101, 10915. (7) Straus, J. B.; Calhoum, A.; Voth, G. A. J. Chem. Phys. 1995, 102, 529. (8) Pechina, O.; Schmickler, W.; Spohr, E. J. Electroanal. Chem. 1995, 394, 29. (9) Benjamin, I. Chem. ReV. 1996, 96, 1449. (10) Warshel, A.; Parson, W. W. Annu. ReV. Phys. Chem. 1991, 42, 279. (11) Halley, J. W. Electrochim. Acta 1996, 41, 2229. (12) German, E. D.; Kuznetsov, A. M.; Tikhomirov, V. A. J. Phys. Chem. 1995, 99, 9095. (13) Nazmutdinov, R. R.; Shapnik, M. S. Electrochim. Acta. 1996, 41, 2253. (14) Nazmutdinov, R. R.; Shapnik, M. A.; Malyucheva, O. I. Elektrokhimia 1991, 27, 991. (15) Kuznetsov, A. M. Charge Transfer in Physics, Chemistry and Biology. Physical Mechanisms of Elementary Processes and Introduction to the Theory; Gordon and Breach Science Publishers: Berkshire, 1995. (16) Koene, L.; Sluyters-Rehbach, M.; Sluyters, J. H. J. Electroanal. Chem. 1996, 402, 57. (17) van Venroij, T. G. J.; Sluyters-Rehbach, M.; Sluyters, J. H. J. Electroanal. Chem. 1996, 419, 61. (18) Khan, S. U. M.; Wright, P.; Bockris, J. O’M. Elektrokhimia 1977, 13, 914. (19) Kornyshev, A. A.; Kuznetsov, A. M. Ulstrup, J. J. Phys. Chem. 1994, 98, 3832. (20) Wilkins, R. G.; Yelin, R. E. Inorg. Chem. 1968, 7, 2667. (21) Yamada, A.; Tanaka, N. Bull. Chem. Soc. Jpn. 1969, 42, 1600. (22) Zhang, Z.; Yang, H.; Bard, A. J. J. Am. Chem. Soc. 1987, 109, 1917. (23) Meier, R.; Boddin, M.; Hecht, H.; Schultz, F. A. In Molecular Electrochemistry of Inorganic, Bioorganic and Organometallic Compounds; Pombeiro, A. J. L., McCleverty, J. A., Eds.; NATO ASI Series 385; 1993; pp 489-494. (24) Hecht, M.; Fawcett, W. R. J. Phys. Chem. 1995, 99, 1311. (25) Hecht, M.; Fawcett, W. R. J. Electroanal. Chem. 1995, 396, 473. (26) Khrushcheva, M. L.; Kuz’minova, A. V.; Tsirlina, G. A.; Petrii, O. A.Russ. J. Electrochem. 1996, 32, 865. (27) Khruchsheva, M. L.; Tsirlina, G. A.; Petrii, O. A. Russ. J. Electrochem. 1998, 34, N4. (28) Vlcˇek, A. A. Electrochim. Acta 1968, 13, 1063. (29) Nazmutdinov, R. R.; Tsirlina, G. A.; Kharkats, Yu. I.; Petrii, O. A.; Probst, M. in preparation. (30) Linert, W.; Konecny, M.; Renz, F. J. Chem. Soc., Dalton Trans. 1994, 1523. (31) Bacon, A. D.; Zerner, M. C. Theor. Chim. Acta 1979, 53, 21. (32) Anderson, W. P.; Edwards, W. D.; Zerner, M. C. Inorg. Chem. 1986, 25, 2728. (33) Zerner, M. C. In ReViews in Computational Chemistry II; Lipkowitz, K. B., Boyd, D. B., Eds.; VCH Publishers, Inc.: New York, 1991; pp 313365. (34) HyperChem, Computational Chemistry; Hypercube, Inc.: Waterloo, Ontario, 1994. (35) Foresman, J.B; Frisch, A. Exploring Chemistry with Electronic Structure Methods; Gaussian, Inc.: Pittsburgh, PA, 1996. (36) R.E.Hamm, J. Am. Chem. Soc. 1953, 75, 5670.
686 J. Phys. Chem. B, Vol. 102, No. 4, 1998 (37) Sawyer, D. T.; McKinnie, M. J. J. Am. Chem. Soc. 1960, 82, 4191. (38) Gerdom, L. E.; Baenziger, N. A.; Goff, H. M. Inorg. Chem. 1981, 20, 1606. (39) Ogino, H.; Nagata, T.; Ogino, K. Inorg. Chem. 1989, 28, 3656. (40) Schwarzenbach, G.; Biedermann, W. HelV. Chim. Acta. 1948, 31, 459. (41) Pecsok, R. L.; Shields, L. D.; Schaeffer, W. P. Inorg. Chem. 1964, 3, 114. (42) Thorneley, R. N.; Sykes, A. G.; Gans, P. J. Chem. Soc. (A) 1971, 1495. (43) Sulfab, Y.; Taylor, R. S.; Sykes, A. G. Inorg. Chem. 1976, 15, 2388. (44) Kanamori, K.; Kawai, K. Inorg. Chem. 1986, 25, 3711. (45) Wheeler, W. D.; Legg, J. I. Inorg. Chem. 1984, 23, 3798. (46) Levine, I. N. Quantum Chemistry, 4th ed. Prentice Hall, Inc.: New Jersey, 1991. (47) Drew, M. G. B. In Progress in Inorganic Chemistry; Lippard, S. J., Ed.; John Wiley & Sons: New York, 1977; Vol. 23, pp 67-207. (48) German, E. D.; Kuznetsov, A. M. Elektrokhimia 1990, 26, 931. (49) Khan, S. U. M.; Zhou, Z. Yu. J. Chem. Phys. 1990, 93, 8808.
Nazmutdinov et al. (50) Khan, S. U. M.; Zhou, Z. Yu. J. Chem. Soc., Faraday Trans. 1991, 87, 535. (51) Tsirlina, G. A.; Kharkats, Yu. I.; Nazmutdinov, R. R.; Petriii, O. A. J. Electroanal. Chem. in press. (52) Kharkats, Yu. I. Elektrokhimia 1976, 12, 1284. (53) Liu, Y.-P.; Newton, M. D. J. Phys. Chem. 1994, 98, 7162. (54) Marcus, R. A. J. Chem. Phys. 1965, 43, 679. (55) Minkin, V. I.; Simkin, B. Ya.; Minyaev, P. Theory of Molecular Structure; Visshaya Shkola: Moscow, 1979. (56) Bo¨ttcher, C. J. F. Theory of Electric Polarization; Elsevier Scientific Publishing Co.: Amsterdam, 1973; Vol. 1. (57) Newton, M. D.; Ohta, K.; Zhong, E. J. Phys. Chem. 1991, 95, 2317. (58) Hay, P. J.; Wadt, W. R. J. Chem. Phys. 1985, 82, 270. (59) Here, W. J.; Stewart, R. F. Pople, J. A. J. Chem. Phys. 1969, 51, 2657. (60) Dunning, T. H.; Hay, P. J. In Modern Theoretical Theoretical Chemistry; Schaefer, H. F., III, Ed.; Plenum Press: New York, 1976; pp 1-28. (61) Binkley, J. S.; Pople, J. A.; Here, W. J. J. Am. Chem. Soc. 1980, 102, 939.
