J. Phys. Chem. 1996, 100, 7411-7417
7411
Calculations of Intramolecular Reorganization Energies for Electron-Transfer Reactions Involving Organic Systems Steen Jakobsen, Kurt V. Mikkelsen,* and Steen U. Pedersen* Department of Chemistry, Aarhus UniVersity, Langelandsgade 140, DK-8000 Aarhus C, Denmark ReceiVed: NoVember 29, 1995; In Final Form: February 5, 1996X
The determination of intramolecular reorganization energies for self-exchange outer-sphere electron-transfer (ET) reactions is presented using a recently published method. From ab initio calculations, the electronic structure and equilibrium geometry, q, for the neutral, qeq(0), and monocharged, qeq(+/-), organic molecules are obtained along with the force constant matrix F. A projected force constant f is determined as f ) (∆qTF∆q)/|∆q|2, where ∆q ) qeq(0) - qeq(+/-). The intramolecular reorganization energy is obtained as λi ) ((f(0)f(+/-))/(f(0) + f(+/-)))|∆q|2. It is shown that, due to the inclusion of the coupling between the various vibrational coordinates, this approach is an improvement compared to conventional methods for calculating the intramolecular reorganization energy. The method is tested on the reductive ET of aromatic hydrocarbons, the oxidative ET of substituted benzenes, and oxidative/reductive ET of benzyl radicals.
Introduction Thirty years ago nearly all organic reactions were explained through the concept of polar mechanisms based on the movements of electron pairs. The SN2 mechanism is an example of the success of the polar concept where the reaction is explained as an attack of the electron pair of the nucleophile on the electrophilic center with subsequent expulsion of the leaving group. In the transition state both the nucleophile and the leaving group are partially bonded to the central carbon atom. In modern organic chemistry one recognizes that electrons are far from always being paired.1 In many aspects it seems more important to describe chemical reactions by the movements of single electrons. In the sixties this was shown by Kornblum and Russell and later by Bunnett in the description of the SRN1 reaction.2-5 In organic electrochemistry it was also soon recognized that electrons could be donated from or accepted by the electrode one by one depending on the potential and the electrode.6 The classical SN2 mechanism is overall a transfer of one electron from the nucleophile to the leaving group. The transfer of the electron can take place when the nucleophile has inferior interaction with the electrophile, in which case the electron transfer reaction (ET) is called outer-sphere. On the other hand, when the nucleophile and the electrophile can interact more strongly in the sense of forming a partial bond between the reactive centers, we are within the area of inner-sphere electron transfer. The interaction between the nucleophile (or the electron donor) and the electrophile (or the electron acceptor) will always stabilize the transition state relative to the transition state for the outer-sphere ET, and consequently the rate for an inner-sphere ET is faster than for an outer-sphere ET, assuming that everything else remains unchanged.7,8 This leads, of course, to the situation that whenever possible the ET reaction between a donor and an acceptor will always be inner-sphere except when the interaction is diminished.9 Sterical hindrance prevents the close interaction between the donor and acceptor and therefore prevents the inner-sphere ET but still allows the outer-sphere ET, which is much less geometrically restricted,8,10-12 in accordance with the experimental findings that the change in * Authors to whom correspondence should be addressed. X Abstract published in AdVance ACS Abstracts, April 1, 1996.
S0022-3654(95)03525-8 CCC: $12.00
activation entropy is much larger for inner-sphere ET than for outer-sphere ET.10 Physical organic chemists are familiar with linear free energy relations like the Bronsted or Hammett relations. The quadratic free energy relation proposed by Marcus for outer-sphere ET represents still a fairly simple but powerful relation that is perfect for rationalizing experimental results.13 For the outer-sphere ET reaction
A•- + B h A + B•-
(1)
Marcus13 put forward an equation that describes how the activation energy ∆G# (and hence the rate constant) is connected to the driving force ∆G° for the outer-sphere ET reaction 1
∆G# )
(λ + ∆G°)2 4λ
(2)
λ, the reorganization energy, is four times the activation energy at zero driving force and expresses the energy required to reorganize the molecules and solvent in order to achieve the appropriate transition state configuration for an isoenergetic electron transfer. The Marcus equation usually contains a work term due to Coulombic interaction between the reactants, but in all cases where one of the reacting species has no net charge, this contribution vanishes. According to the Marcus equation only two parameters, the change in free energy, ∆G°, and the reorganization energy, λ, are needed for expressing the activation energy, ∆G#. The ∆G° can be expressed as the difference in redox potentials between the electron donor and acceptor, eq 3, and the latter can often be measured by electrochemical methods. Redox potentials for transient species can either be
∆G° ) -nF(E°B - E°A)
(3)
measured by fast voltammetric techniques or by indirect methods or be calculated from thermochemical cycles from gas-phase data. The reorganization energy for the outer-sphere ET reaction 1 can be obtained from the Marcus cross relation
λ)
(λA + λB) 2
© 1996 American Chemical Society
(4)
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where λA and λB are the reorganization energies associated with the self-exchange ET reactions 5 and 6, respectively.
A•- + A h A + A•-
(5)
B•- + B h B + B•-
(6)
The reorganization energy can be split up in two parts14
λ ) λi + λo
(7)
where λi is the intramolecular (inner-sphere) contribution to the total reorganization energy and λo is the solvent (outer-sphere) contribution. The Born model for the solvation of charged species in a dielectric continuum gives for the calculation of the solvent reorganization energy
λo )
(
)(
)
1 1 1 1 1 + - (∆q)2 op st rA r •- R A
(8)
Here, the influence from the solvent is taken into account via st and op, the static and optical dielectric constants, respectively. rA and rA•- are the cavity radii of the reactants, R is the distance between the two reacting species, and ∆q is the charge transferred (for a description of the underlying assumptions, see e.g., refs 13a-d). Several authors15-19 have modified eq 8, yet the basic physics remains the same. The intramolecular reorganization energy is the energy required to reorganize the intramolecular degrees of freedom, qi, and assuming the harmonic approximation for the displacement of these coordinates, δqi, λi can be calculated from
λi ) 1/2∑kHjkδqjδqk
(9)
jk
where kH is the Hooke’s law force constant which is approximated by an averaged force constant for the coordinate. This averaged force constant is given20,21 by
f r fp kH ) 2 (fr + fp)
(10)
where fr and fp are the force constants for reactants and products, respectively. If the general modes qj in eq 9 are replaced by normal modes, Qj and eq 10 is inserted, the intramolecular reorganization energy becomes
fr fp (δQj)2 λi ) ∑ (f + f ) j r p
(11)
As shown by Chandler et al.22,23 and Song et al.,24,25 the quantum modes of the intramolecular vibrations do not change the expression of the classical intramolecular reorganization energy. The quantum effect of these modes leads to modifications of the electron transfer coupling element.24,25 Therefore, we do not introduce any quantum corrections to the classical intramolecular reorganization energy. The reorganization energy can be measured for the self-exchange ET reaction when all species involved are stable. The rate constants for the selfexchange ET reaction are obtained from ESR line-broadening experiments. Reorganization energies for organic ET selfexchange reactions have been measured for the reduction of aromatic compounds26 and oxidation of highly stabilized vinylic systems or aromatic amines.27
For less stable molecules with half-lives smaller than 10 s, the ESR line-broadening experiment cannot be used. However, measurements of ET rates at electrodes by electrochemical techniques provide reorganization energies, but there seem to be large discrepancies between reorganization energies determined from self-exchange ET measurements and heterogeneous ET measurements.26,28 It is likely that the cause for this difference is the special environment close to the electrode, the double layer, in the heterogeneous ET measurements. For many transient species the only alternative left is to obtain the reorganization energies from calculations. Sometimes the electron donor and acceptor are both stable on the time scale of the ET, but more often bonds in the acceptor and donor are broken concertedly with the ET. This is always the situation when the electron is transferred directly to the orbital having pronounced electron density on the leaving group. The acceptor with the extra electron will in this case have an extremely short lifetime, in which case the ET is said to be dissociative.29,30 The outer-sphere dissociative ET has been described by Morse curves, and according to this treatment a modified Marcus relation is derived. However, the modified reorganization energy includes the dissociation energy for the cleaved bond, and the redox potential for the acceptor is now redefined as E°RX/R•,X-, reflecting that RX•- has an extremely short lifetime, the order of a vibrational period. The latter potential cannot be measured directly. Many different angles have been used to test for ET characteristics in reactions generally believed to follow polar mechanisms. In the investigation of ET in polar substitution reactions, researchers have used kinetic tests,8,31 stereochemical probes, and activation parameters to uncover possible outersphere ET reactions. In the kinetic tests the substitution rate for an attack of a nucleophile on alkyl halides is compared to the rate of an outer-sphere ET reaction where the donor has the same redox potential and the same self-exchange reorganization energy as the nucleophile. When the rates are similar, the nucleophile behaves as an outer-sphere electron donor, and the substitution reaction involves an outer-sphere ET reaction as the rate-determining step. Normal nucleophilic reactions are inner-sphere ET reactions with substantial stabilization of the transition state by partial bonding between the nucleophile and electrophile, and therefore the rate will be many orders of magnitude faster than the rate expected for an analogue outersphere ET. Aromatic radical anions have been used as outersphere electron donors, and due to their stability, redox potentials and reorganization energies are easily measured by cyclic voltammetry and by ESR line-broadening experiments, respectively. Recently, it was claimed that the self-exchange ET reaction between aromatic compounds also contained some inner-sphere contribution, a question that needs to be addressed further.32 The redox potentials for nucleophiles can in rare cases be measured directly by cyclic voltammetry, but in most cases indirect measurements or thermochemical cycles have to be applied. Concerning the self-exchange reorganization energy for Nuc•/Nuc-, one encounters a plentitude of problems. The usual ESR line-broadening technique cannot be applied due to the instability of Nuc•. In a few cases indirect methods can be applied, but most often one has to rely on quantum chemical calculations. Knowledge about potentials of radicals is important, i.e., for distinguishing between ion and radical chemistry, in determination of the acidity for weak acids, and for understanding the possibility of nucleophiles acting as electron donors. Reduction potentials for radicals can seldomly be measured by normal
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electrochemical techniques. However, recently two methods to measure reduction potentials for transient radicals were published. One of them is based on sensitive phase-modulated voltammetry where radicals are formed photochemically.32 The method requires the radical to be somewhat stable in order to obtain the required minimum steady-state concentration for detection. The other method is indirect and is based on the competition of alkyl radicals to couple with or be reduced by aromatic radical anions.33 If the redox potential of the aromatic radical anion is negative with respect to the reduction potential of the alkyl radical, then the latter will be reduced to an alkyl anion; otherwise, alkylated aromatic hydrocarbons are formed. Linear sweep voltammetry can be used to distinguish between coupling and reduction of the alkyl radicals.34 Crude estimations of the reorganization energies involved in the ET reduction of the alkyl radical can be made from the functional form showing the dependency of the competition on the redox potential of the aromatic radical anion. However, transformation of the kinetic dependent reduction potentials to thermodynamic redox potentials requires knowledge about the self-exchange reorganization energies for radicals/anions, and it is difficult to imagine that these reorganization energies will become experimentally available in near future; therefore, quantum chemical calculations seem to be the only possibility. In this paper we utilize a previously published35 and straightforward method to calculate intramolecular reorganization energies λi from ab initio calculations. The molecules that we have examined are aromatic compounds. Some of them, naphthalene, anthracene, and nitrobenzene, are commonly used as electron donors (in their reduced form), and experiments have shown that the total reorganization energies are different. The present paper will address these differences from the calculations of the intramolecular reorganization energies. Measuring oxidation potentials for radicals with the competition method requires that the differences in the self-exchange reorganization energies for the acceptors are small compared to the absolute value for the reorganization energy of the radical/cation; thus, screening of possible electron acceptors is required prior to experimental applications in order to assure that all acceptors have the same self-exchange reorganization energy. Besides the classical method for calculating intramolecular reorganization energies, the literature mainly offers three different approaches concerning the theoretical determination of λi. The approach by Bu et al.36,37 has focused on diatomic molecules and on di- and trihydrides, in their neutral, reduced, and oxidized form. In the case of diatomic molecules,36 the potential energy curve for the molecule is assumed to be represented by a third-order polynomial rather than by the frequently used harmonic approximation, and by the use of spectroscopic data, intramolecular reorganization energies are determined. The approach is slightly different when going to the di- and trihydrides.37 Here, the reorganization energy is defined as
reorganization energy. Furthermore, the ab initio calculation provides Bu with various constants, such as re (the equilibrium distance), f ) d2E/dr2, g ) d3E/dr3, and by employing different analytical expressions for the potential energy, another four values for the reorganization energy ensue. This model has so far only been used for small molecules, e.g., diatomic molecules and di- and trihydrides, in which symmetry could be implied. Extension of the model to calculate intramolecular reorganization energies for larger organic molecules, like benzene or naphthalene, appears to be very difficult, since the model requires knowledge of the analytical expression for the potential energy surface of the molecule, expressions that usually are not available. Grampp38 et al. used a bond-order/bond-length relation to estimate λi for some aromatic and heteroatom-containing aromatic compounds (Q/Q +/• ). The bond lengths are extracted from X-ray crystallographic data, and then a new empirical bond-order/bond-length relation gives the force constants for the vibrational coordinates of Q/Q +/• ; hence, λi is obtained from eq 11, though a correction for nuclear tunneling39 has been included. The problem of finding the bond lengths for unstable molecules, in connection with the approximate nature of this approach, makes it unrecommended for practical use. Larsson et al.40 recently suggested another model for calculating intramolecular reorganization energies. The idea is as follows: The geometry of the neutral molecule A is optimized using an ab initio quantum chemical code which yields a total energy E1. Next, by adding an electron to A while freezing the geometry, the energy, E2, is calculated (this energy difference is equal to the vertical ionization potential of A). Then the geometry of A•- is optimized to give the energy E3. The difference (E2 - E3) is assigned to the reorganization energy λ1. Subsequently, an electron is removed from A•-, and the energy E4 is calculated without changing the geometry. The resulting energy gain is equal to the vertical electron affinity of A•-. The energy difference (E4 - E1) is then set equal to the reorganization energy λ2. Finally, λi is found from
λi ) RE ) 4(∆Er + ∆Ep)
The total energy of a molecule can be represented by a Taylor series of the nuclear degrees of freedom, that is, the vibrational degrees of freedom q ) {qi} around a given reference configuration
(12)
where ∆Er ) Er# - Ereq (Er# is the energy of the reactants at the transition state, and Ereq is the energy of the reactants at equilibrium geometry). A similar expression holds for ∆Ep. Taking, for example, trihydrides, AH3, the three bonds A-H are considered to be equal as well as the three angles ∠HAH; thus, the potential energy surface is a function of two variables. For different values of these variables, single-point ab initio calculations are carried out for the two reacting species, and by locating the transition state, the use of eq 12 gives the
λi ) λ1 + λ2
(13)
Important to note is that the model uses small differences in large total energies computed for neutral and radical anion species for calculating the intramolecular reorganization energy, and inherently, such a procedure is inaccurate. All the above-mentioned approaches utilize properties of isolated molecules leading to estimations of the intramolecular reorganization energies which do not take into account the fact that the donor and acceptor compounds undergo their necessary geometry changes in the presence of each other. The present work utilizes an approach where the interactions between the donor and acceptor are included through a projection/partitioning of the force constant matrix. Theoretical Section
∞
1 E(q) ) ∑ {(q - q0)‚∇)nE(q0) n)0n!
(14)
where q is a vector containing the internal coordinates for the molecule. The vector q0 contains the internal coordinates for the molecule at the reference configuration. In the harmonic
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approximation we arrive at
E(q) ) E(q0) + GT∆q + ∆qTH∆q
(15)
where superscript T indicates transposition of the vector. ∆q is the displacement from the reference configuration given by
∆q ) q - q0
(16)
The first and second derivatives are the gradient and Hessian matrices
Gi(q0) ) {(∂E(q)/∂qi)q)q0}
(17)
Hij(q0) ) {(∂2E(q)/∂qi∂qj)q)q0}
(18)
Since our method employs geometry-optimized structures (at the ab initio level of theory), the gradient vanishes identically. From the Hessian we may extract the force constant matrix F as
F ) H(q ) qeq)
TABLE 1: Calculated Equilibrium Carbon-Carbon Bond Distance rc-c in Neutral Benzene and Intramolecular Reorganization Energies for the Benzene/Benzene Radical Anion Couple basis set
rc-c (Å)
λi (10-2 au)a
λi (10-2 au)b
3-21G 3-21G* 3-21+G 3-21+G* 6-31G 6-31G* 6-31+G 6-31+G* 6-311G 6-311G* exptlc
1.3846 1.3846 1.3876 1.3876 1.3883 1.3862 1.3901 1.3881 1.3878 1.3855 1.397
1.020 1.020 0.803 0.803 0.932 0.967 0.766 0.786 0.940 0.962
2.625 2.625 2.002 2.002 2.473 2.698 1.958 2.109 2.403 2.595
a Using projected force constants, eq 21. b Calculated from eq 11, the classical scheme. c Reference 42.
from eq 21. For comparison, λi was determined with the aid of eq 11 in all cases. The calculations were carried out on the Silicon Graphics computer at Aarhus University.
(19) Results and Discussion
For the self-exchange ET reaction 5 a full geometry optimization is carried out for both A and A•-, and at the same time the force constant matrix, F, is obtained. A projected force constant f is then determined by
f)
∆qTF∆q |∆q|2
(20)
where |∆q| is the norm of the vector ∆q. This projected force constant includes the off-diagonal elements in the Hessian, and thus we take account of the coupling between the different vibrational modes, something that traditionally has been omitted. The present calculations show that the interactions between the various modes make an important contribution to the total intramolecular reorganization energy. The expression for λi becomes
λi )
f r fo
|∆q|2
(fr + fo)
(21)
where the indices r and o refer to the reduced and oxidized forms of A, respectively. Worth noticing is the use of force constants projected along the reaction coordinate as seen from eq 20, such that they correspond to generalized reaction coordinates. Computational Procedure The calculations were performed using the Guassian 92 program.41 For all molecules, neutral, reduced, or oxidized, no symmetry constraints were imposed during geometry optimization. Furthermore, the unrestricted Hartree-Fock calculations were performed using the lowest possible symmetry of the wave function employing the 6-31G* basis. In the case of benzene/ benzene radical anion, the optimizations were also performed with different basis sets. From the full geometry optimization of A and A•- (A•+), the equilibrium geometry qeq and the force constant matrix F resulted, and subsequently the force constants were projected according to eq 20 with ∆q ) qeq(A) - qeq(A•-). The intramolecular reorganization energy λi was finally calculated
We have performed an analysis of the effect of changing the basis set in the case of benzene/benzene radical anion, the results of which are presented in Table 1. The calculated C-C bond length in the neutral benzene molecule as a function of the basis set is presented, along with the intramolecular reorganization energy calculated from eq 21 or eq 11. The 3-21G basis set, with or without polarization/diffuse functions, predicts an equilibrium C-C bond length that is too short, and this is only slightly compensated when extending the basis set to the 6-31G level (with or without polarization/diffuse functions added). If we add polarization functions to 3-21G or 3-21+G, it does not affect the value of either the bond length in the neutral benzene molecule or the value of the intramolecular reorganization energy, whereas using, e.g., the basis set 6-31G* results in a smaller C-C bond length (and a larger λi) compared to those with 6-31G. One would expect that adding polarization functions to a given basis set will change the geometry and/or the force constant matrix, at least to some extent. Since this is not the case for 3-21G and 3-21+G, we conclude that these four (3-21G, 3-21G*, 3-21+G, and 3-21+G*) basis sets are too primitive in the case of benzene. Probably the best basis set for describing benzene is 6-31+G, where the calculated C-C bond length agrees reasonably well with the experimental value, the difference being only 0.69 pm. With respect to the intramolecular reorganization energies, the value found with the 6-31+G basis, 7.66 × 10-3 au (4.86 kcal mol-1), represents a lower, limiting value, to which the other values converge, when improving the basis set (as judged from the deviation from the equilibrium C-C distance). If we compare our method, which takes advantage of projected force constants, with the classical treatment which uses the diagonal elements of the force constant matrix only, we find that λi is a factor of 2 larger (on average) when obtained from the classical scheme (eq 11). This discrepancy is related to the fact that the classical treatment does not include the nondiagonal elements of the force constant matrix, whereas this study does. It clearly shows that the nondiagonal element in the force constant matrix are very important in calculating λi. We have studied seven substituted benzenes undergoing oxidation and the benzyl radical, undergoing reduction and oxidation, respectively. The results are summarized in Table
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TABLE 2: Intramolecular Reorganization Energies of Substituted Benzenes/Corresponding Oxidized Form (Entries 1-8), Reduced Form (Entry 9), Using 6-31G* as Basis substituent •+
CH3/CH3 NH2/NH2•+ OH/OH•+ F/F•+ PH2/PH2•+ SH/SH•+ Cl/Cl•+ CH2•/CH2+ CH2•/CH2-
λi (10-2 au)a
λi (10-2 au)b
qtot (10-2 au2)c
〈f〉 (au)c
0.833 0.964 1.284 1.142 0.445 0.975 1.032 0.459 0.411
1.323 1.850 2.059 1.639 1.173 1.349 1.210 1.151 3.974
3.385 3.717 5.522 4.587 2.591 5.369 3.980 1.762 2.304
0.49 0.52 0.46 0.57 0.34 0.36 0.45 0.53 0.36
a Using projected force constants, eq 21. b Calculated from eq 11, the classical scheme. c qtot is defined in the text. 〈f〉 is the projected force constant, taken as an average for the neutral and the oxidized (reduced) molecule.
A
B
2. We have defined a parameter qtot, describing the total charge in nuclear coordinates, as
qtot ) ∑(∆qi)2
(22)
i
i.e., how much the molecule changes in geometry upon oxidation or reduction, respectively. It can be seen in Table 2 that λi is in general proportional to qtot, and this is expected from the definition of λi. However, for the benzylic radical, X ) CH2•, this is not so. qtot is smaller at oxidation than at reduction, whereas the reorganization energy is larger at oxidation than at reduction. The reason for this reversed order in λi is found in the averaged projected force constants. For this molecule, 〈f〉 is much larger (0.53 au) at oxidation than at reduction (0.36 au), and this is in particular due to the inclusion of the nondiagonal elements in the force constant matrix. The example illustrates that even though qtot is the main factor in determining the value of the intramolecular reorganization energy, caution should be exercised when the projected force constants are significantly different. We have analyzed the variation in the bond between the benzene ring and the substituent in terms of length and charge distribution for the molecules, in both their neutral and positively charged forms, participating in the oxidation process. The results are displayed in Figure 1. For the neutral molecules the variation in the charge on the substituent can be understood from the Pauling electronegativity scale:
P < S, C < Cl, N < O < F f more electronegative
(23)
Hence, the more electronegative the atom in the substituent, the more the charge is localized on the substituent. The bond length in the neutral molecule follows the size of the central atom in the substituent; i.e. with the largest atom, P, we observe the largest bond length, and going to smaller and smaller atoms (S > Cl > C > N > O > F), the bond length decreases accordingly. Upon oxidation, another feature has to be considered: the polarizability of electrons at the central atom in the substituent, i.e., electrons at P and S and so on, to be donated into the benzene ring. Since d electrons are held less strongly to an atom than are p electrons, we expect PH2 to be more electron donating than NH2, SH more than OH, and Cl more than F, and increasing the core charge in the central atom, e.g., going from PH2 to SH, results in more strongly held electrons. These two effects are shown in Scheme 1. The position of the methyl group is arbitrary; however, CH3 is known to be slightly electron donating due to hyperconjugation.43 The present
Figure 1. Charge distribution and bond distances for the substituted benzenes. (A) X is, from the top, CH3, NH2, OH, or F, respectively. (B) X is, from the top, PH2, SH, or Cl, respectively. qx is the charge on the central atom in the substituent (with H atoms summed into it). rc-x is distance between the central atom in the substituent and the adjacent carbon atom.
calculations show that the dominant effect is the presence of polarizable d electrons, since the substituents PH2, SH, and Cl all are more positively charged than substituents from the first row in the periodic table. The bond lengths in the cation radicals have been calculated also, but important for the determination of the intramolecular reorganization energy are the changes in bond lengths. However, it is not possible to correlate these alterations in bond distances with the calculated intramolecular reorganization energies (see Table 2). This is merely a manifestation of the principle that the reorganization process is not a local phenomenon but concerns the whole molecular structure. SCHEME 1
The classical treatment overestimates λi; the reason has already been given in the discussion on benzene. Compared to the other molecules in Table 2, the benzyl radical has a smaller λi, both at reduction and oxidation. The optimized structures of the benzyl radical, anion, and cation are all planar; thus, the geometry alterations are modest during ET, and consequently λi is small. In connection with the determination of the potential for the benzyl radical, a preliminary investigation44 has estimated the total self-exchange reorganization energy for the benzyl radical/benzyl anion to be in the vicinity of 10 kcal mol-1, and the calculations in this paper support this observation. (λi for the benzyl radical/benzyl anion is 2.6 kcal mol-1 (see Table 2), and compared to naphthalene (see Table 4), this is about 4 kcal mol-1 smaller. It is expected that the solvent reorganization energy for the
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TABLE 3: Comparison of Calculated Intramolecular Reorganization Energies for the Series Benzene, Naphthalene, Anthracene Upon Reduction (kcal mol-1) present study 6-31G 6-31G* 6-311G* MP2/6-31G 6-31G 6-31G* MP2/6-31G 6-31G 6-31G* MP2/6-31G a
Benzene 5.92 6.14 6.11 Naphthalene 6.06 6.70 Anthracene 6.92 6.31
Larssona et al. 12.73 12.22 7.65 11.88 13.18 4.65 10.61 11.98 3.03
Reference 40.
TABLE 4: Comparison of Reorganization Energies (kcal mol-1)
a
compound
λia
λexptl
anthracene naphthalene nitrobenzene
6.31 6.70 15.80
9.7 12.0 19.3
Present work, eq 22.
benzyl radical is about the same as that for naphthalene, perhaps slightly larger due to the smaller size; hence, the total reorganization energy becomes 9-10 kcal mol-1.) In Table 3 the Hartree-Fock calculated intramolecular reorganization energies for benzene, naphthalene, and anthracene are compared with similar energies calculated by Larsson et al. A significant discrepancy is observed; the presently calculated reorganization energies are about 50% smaller than the ones obtained by Larsson et al. These differences are related to their approach of using isolated molecular properties for calculating the intramolecular reorganization energies. However, when using MP2 calculations, Larsson et al. found that the values for λi drop by a factor of 2-3, and we expect that our method, employing MP2-calculated structures and force constants, will further decrease the reorganization energies. Total reorganization energies, λ, for anthracene,45 naphthalene,26b and nitrobenzene46 in N,N-dimethylformamide, DMF, have been measured in ESR line-broadening experiments. In Table 4 the intramolecular reorganization energies, λi, are shown, calculated from eq 21, along with the experimentally determined total reorganization energies. Note that the total reorganization energy includes solvent reorganization energy. The same values of λi are found for anthracene and naphthalene, whereas the value for nitrobenzene is markedly higher. Under the assumption that the solvent reorganization energy is, more or less, the same for the three molecules (in view of the nearequal size of the three compounds, this is a valid approximation), the present calculations can explain the differences in the experimentally determined λ. Calculation of the solvent reorganization energy in DMF for, e.g., naphthalene results in λ0 ) 16.0 kcal mol-1 (using rA ) rA•- ) 4.8 Å; R ) 2rA), i.e., 4 kcal mol-1 higher than the experimental value for the total reorganization energy. This calls for a more precise method for determining the solvent contribution to the reorganization energy, preferentially one that takes the dynamics of the electron transfer into account. Conclusion The present treatment has clearly demonstrated that the classical scheme for calculating intramolecular reorganization
energies for organic molecules is too primitive. The reason for this failure is understandable: From eq 7, which describes the classical way of calculating λi, it is seen that only the diagonal elements in the force constant matrix contribute to λi. Thereby, one neglects the nondiagonal elements that represent the coupling between two different vibrational modes, which are of importance for large organic molecules. Equivalently, this can be stated as “what happens in one part of the molecule during the actual electron transfer does not influence on the rest of the molecule”. This work shows that this is certainly not true. By including the nondiagonal elements in the description of the reorganization process, properly taken into account in a projected fashion, the value of the intramolecular reorganization enery drops dramatically. It is also illustrated in the differences between Larsson’s and the present results on pure aromatic hydrocarbons. For those investigated systems where experimental values for the total reorganization energy are available, the present calculations on the intramolecular part are able to explain the differences in the observed values, at least on a qualitative basis. The values of λi obtained by Larsson et al. are, at the HF level, higher than the total reorganization energies, and only by inclusion of correlation at the MP2 level, results more consistent with experimental values for the total reorganization energy ensue. The solvent reorganization energy for these systems is, however, overestimated; hence, a more accurate scheme for calculating this term is needed. The oxidation processes of several substituted benzenes have been studied, and although the absolute values for λi are small, significant differences are encountered. Preliminary results on the calculation of intramolecular reorganization energies for oxidations of alkyl radicals show that these are much smaller than the corresponding values for the reduction of the same radicals. The solvent contribution to the total self-exchange reorganization energy is expected to be the same for a given radical, whether it is reduced or oxidized. With this in mind, experimentalists must be careful in choosing electron acceptors, which should ideally have the same total reorganization energy. It is hoped that calculations might be used in choosing acceptors with similar λi. As pointed out by Larsson et al., including correlation at the Møller-Plesset second order perturbation level has a visible effect on the value for the intramolecular reorganization energy, and it is expected that the values found in this paper will be smaller if correlation is included. This aspect is currently being investigated. References and Notes (1) Eberson, L. Electron Transfer in Organic Chemistry; SpringerVerlag: Heidelberg, 1987. (2) Kornblum, N.; Michel, R. E.; Kerber, R. C. J. Am. Chem. Soc. 1966, 88, 5660. (3) Kornblum, N. Angew. Chem. 1975, 87, 797. (4) Russell, G. A.; Danen, W. C. J. Am. Chem. Soc. 1966, 88, 5663. (5) Bunnett, J. F. Acc. Chem. Res. 1978, 11, 413. (6) Peover, M. E.; White, B. S. J. Electroanal. Chem. 1967, 13, 93. (7) Lund, T.; Lund, H. Acta Chem. Scand. 1986, B40, 470. (8) Lund, H.; Daasbjerg, K.; Lund, T.; Pedersen, S. U. Acc. Chem. Res. 1995, 28, 313. (9) Save´ant, J.-M. J. Am. Chem. Soc. 1992, 114, 10595. (10) Daasbjerg, K.; Pedersen, S. U.; Lund, H. Acta Chem. Scand. 1989, 43, 876. (11) Daasbjerg, K.; Pedersen, S. U.; Lund, H. Acta Chem. Scand. 1991, 45, 424.
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