J. Phys. Chem. B 2008, 112, 11079–11086
11079
Ab Initio Calculations on the Intramolecular Electron Transfer Rates of a Bis(hydrazine) Radical Cation Weiwei Zhang,† Wenjuan Zhu,† WanZhen Liang,*,† Yi Zhao,*,‡ and Stephen F. Nelsen*,§ Department of Chemical Physics, UniVersity of Science and Technology of China, Hefei, 230026, P. R. China; State Key Laboratory of Physical Chemistry of Solid Surfaces and Department of Chemistry, College of Chemistry and Chemical Engineering, Xiamen UniVersity, 361005, P. R. China; and Department of Chemistry, UniVersity of Wisconsin, 1101 UniVersity AVenue, Madison, Wisconsin 53706-1396 ReceiVed: April 25, 2008; ReVised Manuscript ReceiVed: June 28, 2008
Electron transfer (ET) rates of a charge localized (Class II) intervalence radical cation of a bis(hydrazine) are investigated theoretically. First, the intramolecular ET parameters, i.e., reorganization energy, electronic coupling, and effective frequency, are calculated using several ab initio approaches. And then, the extended Sumi-Marcus theory is employed to predict ET rates by using the parameters obtained. The results reveal that the rates of three isomers of [22/hex/22]+, oo+[22/hex/22]+, io+[22/hex/22]+, and oi+[22/hex/22]+, are agreement with the experiment quite well while the rate of isomer ii+[22/hex/22]+ is about 1000 times larger than those of the others. The validity of different ab initio approaches for this system is discussed. 1. Introduction Charge-localized intervalence compounds (Robin-Day Class II systems)1,2 may be characterized as having two charge-bearing units attached to a bridge, with which they have an electronic interaction, and which differ in charge. These compounds can be symbolized as M-B-M+ or M-B-M-,3-5 and some of their parameters for ET have been obtained from their optical spectra6 on the basis of the classical Marcus-Hush theory.7,8 For the dimeric system [22/hex/22]+, 13C NMR experiments demonstrated that the conformations present the bicyclic in different double nitrogen inversion isomers, shown in Figure 1. The radical cations of interest here will have one bicyclic ring neutral and the other present as the radical cation, with rather different geometries, flatter nitrogens in the radical cation, as indicated for the “monomers”, [22/22]0 and [22/22]+, also shown in Figure 1. There are two different cations from oi[22/hex/22], because either the “inner” or “outer” bicyclic hydrazine could be oxidized. Although these compounds have been studied experimentally by the Nelsen group9-11 and semiempirical calculations have been published by the Nelsen and Hupp groups,12,13 few discussions of ab initio calculations have been made because it is nontrivial to obtain very good structures and some other ET parameters using HF and general DFT methods. In this study, ab initio calculations are performed to confirm that they can also achieve good results. The rates of ET reactions can be estimated according to classical Marcus theory14,15 or many other semiclassical approaches by using standard free energy difference (G0), reorganization energy (λ), and electronic coupling (Hab). Thus, accurately calculating these parameters becomes the key step to get the rates of ET reactions. Robin and Day assigned three classes of intervalence compounds according to the electronic coupling strengths.1 Class * To whom correspondence should be addressed. E-mail: yizhao@ xmu.edu.cn,
[email protected],
[email protected]. † University of Science and Technology of China. ‡ Xiamen University. § University of Wisconsin.
Figure 1. Structures of three neutral [22/hex/22] isomers, and the monohydrazine compounds [22/22]0 and [22/22]+.
I refers to compounds that have no electronic interaction between donor and acceptor states. Class II refers to compounds that have a small enough electronic interaction to make charge localize on one charge-bearing unit. Class III compounds have such a large interaction that charge is delocalized over the whole molecule. On the basis of state symmetry, for Class II and Class III compounds, the donor and acceptor states should have the same local symmetries if the electronic coupling has a nonzero value. In such a case, ET is symmetry-allowed process.16 From a molecular orbital point of view, the symmetry of the orbitals relating to ET in diabatic representation must be the same for both Class II and Class III to make electron transfer. However, the symmetry of the orbitals at the delocalized transition state (TS) for both Class II and III in the adiabatic representation becomes different because of the electronic coupling. For Class II, the two-state model is widely used to calculate the intramolecular electronic coupling from both theoretical studies17-19 and optical spectrum estimations.6,20,21 One of the theoretical approaches to estimate the electronic coupling is based on Koopmans’ theorem (KT),22-24 i.e., using the energy gap of the molecular orbitals of proper symmetry for an allowed transition at the delocalized TS.25-30 It has recently been pointed out that there is a problem doing this. An allowed transition for a symmetrical molecule must involve orbitals of different symmetry; the transition dipole moment disappears if the symmetry of the states involved is the same. However, the symmetry of the states involved in the charge transfer absorption for a Class II (localized) system that Hush theory addresses is the same; it is a Mulliken-type charge transfer absorption. The
10.1021/jp8036507 CCC: $40.75 2008 American Chemical Society Published on Web 08/07/2008
11080 J. Phys. Chem. B, Vol. 112, No. 35, 2008 observed transition for a Class III mixed valence transition state is therefore not closely related to that for a Class II compound, and there is no real reason to expect that half the transition energy for the Class III system would exactly correspond to the electronic coupling for the Class II systems. Nelsen et al.5 pointed out that the simplest model that could lead to electronic couplings for Class III mixed valence compounds that have bridges involves two electronic couplings and three diabatic energies and that the transition energy for what has been called the mixed valence transition for such systems involves all five quantities. For the present systems, the bridge has C-C σ bonds; and the electronic coupling is not so strong that the localized state can be maintained. We have used the neighboring orbital model to analyze the present systems. It is found that the energy gaps among orbitals are very different between Class II and Class III systems. For the systems of Class III, e.g., radical anions of 1,4-dinitrobenzene and 1,4-dinitrobiphenyl, the energy gaps between two neighboring orbitals are very similar and the biggest difference is about twice of the energy gaps. For the present isomers, e.g., oo+[22/hex/22]+, the neutral in cation geometry (NCG) method reveals that the 95th MO is HOMO, and other three neighboring orbitals are 94th, 92th, and 90th MOs, respectively. The energy gap between 95th and 94th MOs is 2400 cm -1. However, the energy gap between 94th and 92th MOs is 28 000 cm -1, 10 times larger than that between HOMO and HOMO-1. It manifests that the MOs used to calculate the electronic couplings have big energy gaps with other MOs. Thus, we think that the orbitals of bridge have small enough couplings with the charge-bearing units, and we can reduce the multiplestate model (four-state models) to the two-state model safely. However, it is not clear whether KT correctly predicts the electronic coupling as mentioned above. In this work, we will adopt various two-state approaches to clarify the validity of the energy split approaches. Besides the conventional KT approach, other approaches include directly calculating the electronic coupling from two diabatic states (two-state model, (TM)),31-33 energy gap between two adiabatic states by using the spin-flip (SF) strategy34,35 and the generalized Mulliken-Hush (GMH) method.54 These approaches have been demonstrated to work well in the relative weak electronic coupling. However, few comparisons have been discussed for the Class II compounds that have as large an Hab/λ as the case considered here. In the calculations of the inner reorganization energy, two methods are adopted. One is the four-point technique introduced by Nelsen.36 It can directly and easily predict the reorganization energy. The other is based on the summarization of the reorganization energy of each mode, and this approach can clarify which mode plays an important role in ET and obtain the effective frequency for ET rate calculations. Another goal of the present work is to investigate the effect of isomers on ET rates. Experimentally, it is not easy to explicitly distinguish the individual contributions coming from isomers because they are usually mixed together. [22/hex/22]+ is a typical example. Although Nelsen’s group has measured its ET rates in various solvents,9 the isomeric effect has not yet been clarified. We will answer such a question via calculating ET rates by using the calculated ET parameters. However, it is nontrivial to estimate the ET rates for the present system because of the solvent effect and strong electronic coupling, which make the well-known Marcus formula in the weak electronic coupling region invalid (see, for instance ref 37). To discard the perturbation limit, we use the extended Sumi-Marcus approach38 to estimate the ET rates, where the “sink” function is
Zhang et al. obtained from the scattering theory.39 Thus, it can cover from weak-to-strong electronic coupling regions, as well as incorporating the dynamical solvent effect. The structure of this article is as follows: in section 2, we briefly review the computational methods for the inner reorganization energy and electronic coupling. Section 3 presents our results from ab initio calculations. Section 4 shows the calculations of ET rates and the concluding remarks are given in section 5. 2. Ab Initio Computational Methods 2.1. Reorganization Energy. The reorganization energy in ET consists of inner and outer parts, λv and λs, respectively. λv corresponds to the change of the intramolecular geometries of the donor and acceptor states while λs comes from the solvent response. Because of computational limitations, one cannot quantum mechanically calculate the system and solvent molecules together. In the present work, we focus on λv only. An easy way to calculate λv is the four-point technique proposed by Nelsen.36 To implement it, either the molecule can be split into the neutral and cationic parts which are calculated separately or the whole bis(hydrazine) molecule can be calculated, as was done using the semiempirical quantum chemical method AM1 previously.12 However, there is a problem with using the whole bis(hydrazine) molecule with ab initio methods. Hartree-Fock calculations get hydrazine radical cations to be too flattened and make λv too large, and the general density functional theory overemphasizes the importance of electron delocalization, causing differences in geometry between the oxidized and neutral ends of bis(hydrazine) to be too small, which in turn causes λv to be calculated to be far too small.40 In this work, DFT is employed but using a suitable density functional. λv is obtained by the energy differences between the relaxed and vertical neutral and cationic compounds at the same geometries. Concretely speaking, if we use n and c to represent the optimized geometries of the neutral and cationic components and the charge is labeled as the superscript, the reorganization energy is then given by36,40,41
λv ) [E(c0) - E(n0)] + [E(n+) - E(c+)]
(1)
where E represents the energy. In eq 1, it is assumed that the precursor pair is n0, c+, and the Franck-Condon excited-state is c0, n+, respectively. Alternatively, assuming that one knows the characteristic vibrational energies and displacements of each mode of the donor and acceptor states of the system [22/hex/22]+, the reorganization energy can be calculated as
λV )
1 2
∑ ωi2∆Qi2
(2)
i
where ∆Qi represents the normal-mode coordinate shift between the donor and acceptor states, and ωi represents the frequency for the i-th mode. In the numerical simulation, the frequency of each mode can be obtained at the optimized geometry from many ab initio software packages. Thus, the key step left is to obtain ∆Qi for each mode. The matrix transformation techniques can be applied for such calculations.42-46 2.2. Methods for the Calculation of Electronic Coupling. According to the classical theory, ET occurs at the seam surface (crossing region) of the donor and acceptor potential energy surfaces. In the adiabatic representation, the seam surface corresponds to the delocalized TS geometry. The electronic coupling in this geometry thus controls the ET rates.
ET Rates of a Bis(hydrazine) Radical Cation
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To determinate the TS geometry, a linear reaction coordinate R may be introduced to describe the change of the i-th internal coordinate Qi in the ET process,31,47,48
Qi ) RQia + (1 - R)Qid
(3)
Here Qi refers to the i-th internal coordinate(bond length, bond angle, or dihedral) and there are 3N-6 independent internal coordinates for an N-atom system. Qia and Qid refer to the optimized geometries of the acceptor and donor states, respectively. Taking the symmetric reaction as an example, TS should locate at the geometry with R ) 0.5. Obviously, R ) 0 and R ) 1 refer to the optimized nuclear configurations of the donor and acceptor states, respectively. Although the selection of the different 3N-6 independent internal coordinates predicts a different geometry at a given R, our experience shows that it has a trivial influence on the calculation of the electronic coupling. For this saturated-bridged system, we will show that the electronic coupling estimated from TS geometry is the same as that obtained from the ground-state geometry. There are two strategies to calculate the electronic coupling, one of which works starting from the diabatic representation and the other from the adiabatic representation. In the diabatic representation, two diabatic states are introduced to refer to the donor- and acceptor-localized states. The electronic coupling is straightforwardly related to the overlap of the diabatic wave functions of the two states with the Hamiltonian of the system.31 The TM approach belongs to this category. In this method, the electronic coupling can be obtained by applying the variational principle to the two diabatic states Ψa and Ψb. By defining sab ) 〈Ψa|Ψb〉 and hab ) 〈Ψa|H|Ψb〉, the electronic coupling can be expressed as
Hab ) (E1 - E2)/2 )
hab - sab(haa + hbb)/2 1 - sab2
(4)
While in the adiabatic representation the electronic coupling is given by half of the energy gap of two adiabatic eigenvalues at TS geometry under Born-Oppenheimer approximation. In the well-known KT approach,22,40 these adiabatic eigenvalues are approximated by the energies of the molecular orbitals (MOs). In practice, the electronic coupling is estimated by the energy gap between (usually, and in this case) the highest occupied molecule orbital (HOMO) and HOMO-1, of the neutral compound using the optimized structure of the radical cation.49 For radical anions, the corresponding Koopmans-based method is (usually) to calculate the energy gap between the lowest unoccupied molecule orbital (LUMO) and LUMO+1 of the neutral compound.4 The important feature is that the occupancy of the orbital compared must be the same; whether there are two, one, or no electrons in an MO has a very large effect on the energy calculated for it. In principle, the energy gap of two adiabatic potentials at TS geometry can be straightforwardly obtained from ab initio calculations. However, it is still a challenge to accurately calculate it. For instance, if one uses the configuration interaction singles (CIS) approach to calculate the excited-state energy, it fails in most cases to represent the adiabatic energy gap because the energy of the first CIS state is lower than that of the ground state obtained from the Hartree-Fock (HF) approach.34 The reason is that the electron states are nearly degenerate in electron-transfer systems and CIS does not properly include nondynamical correlations. Traditionally, the useful treatment is using the multiconfigurational self-consistent field (MCSCF) or the complete active space self-consistent field (CASSCF).
Unfortunately, these methods need a high computational cost and complex manipulations. So it is nontrivial to apply them to large systems. Recently, Krylov et al. have introduced an alternative solution, the SF approach, to the nondynamical correlation problem.50-53 In traditional single-reference excitedstate model, the excitation from the ground state to excited state is spin-conserving. In the SF approach, however, it takes a highspin configuration as a reference state, and the desired lowspin configurations are generated from this reference state as SF excitation.34 The approach has been applied to calculate the electronic coupling,34,35 and is used here. The above methods are usually limited to calculate the electronic couplings at TS geometries. An alternative approach is GMH. It is usually used to calculate the electronic couplings at the optimized geometries. In GMH for a two-state model, the electronic coupling is given by54-57
Hab )
µab∆Eab
√∆µab2 + 4µab2
(5)
where µab is the transition moment, ∆Eab is the energy gap, and ∆µab is the dipole moment difference between the initial adiabatic state and the final one. In the calculation, ∆µab is estimated by 2µ1,58 where µ1 is the dipole moment of the donor state. 3. Ab Initio Computational Results 3.1. Geometries. In the present calculations, the Q-Chem software package59 is adopted to get the optimized geometries and the electronic couplings by using KT and SF methods. The electronic coupling from TM is calculated using the NW-Chem program,60 and GMH with DFT and the semiempirical AM1 are manipulated by the Gaussian03 suite.61 The exchangecorrelation functional 50-50 (50% Hartree-Fock + 8% Slater + 42% Becke for exchange and 19% VWN + 81% LYP for correlation)52 is used to obtain the optimized geometries of the system. The 6-31 g(d) basis set is adopted in all ab initio calculations. In order to demonstrate the accuracy of ab initio calculations for the present system, we take the geometry of an isomer of [22/hex/22]+, oo+[22/hex/22]+, as an example to compare it with experiment and other calculations. Table 1 lists the optimized structural parameters of the oo+[22/hex/22]+ with different theoretical methods, as well as the X-ray crystallographic parameters of its monomers, [22/22]0 and [22/22]+ for the purpose of comparison. For the monomers, the bond lengths and angles which are obtained from ab initio approaches, such as HF, DFT with 50-50 functional and DFT with B3LYP are much better than those from AM1. For instance, AM1 predicts the N-N bond length is about 0.08 Å shorter compared with X-ray data while the maximum error from ab initio approaches is about 0.03 Å. However, these small differences may induce a large reorganization energy difference. Comparing the structure of oo+[22/hex/22]+ with its monomer, we find that the bond lengths and angles are very similar to each other, and again the N-N bond length from AM1 is much shorter than that from HF and DFT with the 50-50 functional. However, DFT with the hybrid B3LYP functional is incapable of obtaining the electronic localized state, as previously noted for aromatic bridged examples.40 One thus expects that DFT with 50-50 functional and HF should be good approaches for the present system, at least for the structure optimization. For other isomers, they have the same characteristics as that of oo+[22/hex/22]+ that we do not list here. Further
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TABLE 1: Comparison of X-ray and Optimized Structural Parameters of oo+[22/hex/22]+ and Its Monomersa methods X-rayb AM1c HFc 50-50c B3LYPc AM1 HF 50-50
NN(+)/NN(0)d
CN(+)/CN(0)
∠CNN(+)/∠CNN(0)
∠CNC(+)/∠ CNC(0)
av(+)/av(0)e
1.325/1.492 1.322/1.406 1.313/1.461 1.327/1.468 1.349/1.493 1.321/1.406 1.312/1.457 1.327/1.464
1.472/1.474 1.489/1.486 1.474/1.459 1.464/1.456 1.476/1.470 1.482/1.483 1.467/1.454 1.457/1.451
114.4/108.7 114.2/111.9 114.5/111.2 114.2/111.0 114.1/110.7 114.6/112.3 114.8/111.5 114.5/111.3
126.0/118.6 125.9/116.8 129.8/121.1 128.8/120.6 129.8/121.0 124.8/114.9 128.6/119.2 127.5/118.9
118.3/112.8 118.1/113.6 119.6/114.5 119.1/114.2 119.3/114.2 118.0/113.2 119.4/114.0 118.8/113.8
a Units for bond lengths and angles are angstroms and degrees. b From ref 73. c [22/22]+ and [22/22]0 which are monomers of oo+[22/hex/22]+. d + reflects the structure of the oxidized hydrazine. e The average of the bond angles of the heavy atom (two ∠CNNs and ∠CNC).
TABLE 2: Comparison of the Structural Parameters of Three Reactions of [22/hex/22]+ at the Transition States Obtained with Different Definitionsa methods +
+
c
ii f i i (line ) ii+ f i+i (C2V d) oo+ f o+o (line) oo+ f o+o (C2V) oi+ f o+ie (linef)
r(N1-N2)
r(C1-N1b)
r(C2-N1)
∠C1N1N2
∠C2N1N2
∠C1N1C2
1.394 1.390 1.395 1.390 1.390/1.400
1.440 1.434 1.443 1.431 1.441/1.440
1.463 1.461 1.463 1.461 1.463/1.465
113.15 113.39 113.13 113.41 113.3/113.0
113.62 112.70 113.90 112.72 112.7/112.4
123.26 123.32 122.81 123.15 124.8/123.7
a Units for bond lengths and angles are angstroms and degrees. b The atom numbers are shown in Figure 1. c The geometries evaluated from the linear reaction coordinate. d Both units attached to the bridge have the same geometries via C2V symmetry. e oi+f o+i and io+ f i+o have the same transition state. f Both sides are different due to R * 0.5 for asymmetric ET reactions.
comparison with their inner reorganization energies of the system will show that DFT with 50-50 functional is better than HF. Table 2 lists the detailed TS geometries of three reactions from DFT with the 50-50 functional. The linear reaction coordinate and constrained symmetry (C2V) are used to locate the TS for symmetric ET reactions. For asymmetric ET reactions, the linear reaction coordinate is only used. It is found that bond parameters of these three TS geometries are very close to each other in spite of their different conformations, and the geometries obtained both from the linear reaction coordinate and the constrained symmetry are also very close for symmetric ET, which are consistent with the discussion of the barrier. However, the electronic couplings are significantly different as shown in the following. Table 3 shows the relative energy of the four isomers and the barrier heights for all possible ET. It is found that the order of the relative energies is E(oi+) < E(oo+) < E(ii+) < E(io+). But the energy difference is not so large. Thus, all isomers may have contributions to ET. However, their barriers have big differences. One may estimate the rates with the transition state theory. For example, the ratio of the rates for ii+ f i+o and ii+ f i+I is about exp[-∆Gii+fi+o/kbT]/exp[-∆Gii+fi+i/kbT] ) 1.46 × 10-10 at T ) 300 K and a small different barrier may lead to a large ratio of rates (for instance, the ratio of the rates is 30 for ∆G ) 2 kcal/mol). With similar estimations, one can find that only four reactions
oi+[22/hex/22]+ f o+i[22/hex/22]+ +
+
+
+
+
+
oo [22/hex/22] f o o[22/hex/22] +
+
(6) (7)
ii [22/hex/22] f i i[22/hex/22]
(8)
io+[22/hex/22]+ f i+o[22/hex/22]+
(9)
and
dominate ET. Thus, in the following we only consider above four reactions. 3.2. Reorganization Energy. First, we consider the fourpoint approach for the estimation of the reorganization energy.
TABLE 3: Barriers of ET Reactions with the Transition States Obtained from Linear Reaction Coordinate (Units for Energy, kcal/mol) geometry
i+o
o+ o
i+ i
o+ i
Ea
oi+
27.6 20.1 19.4 10.5
21.0 11.4 25.4 20.3
20.0 25.9 5.9 16.4
10.5 20.3 16.4 27.6
0.00 0.18 0.56 0.72
oo+ ii+ io+
a The relative energy of the four isomers obtained from DFT with the 50-50 functional.
TABLE 4: Reorganization Energies and Effective Frequencies for [22/hex22]+ Conformational Isomers conformation ii+
+
fi i oo+f o+o oi+f o+i (io+ f i+o)
λV (four-pointa)
λV (modeb)
ωeffc
33.3 39.0 39.5d
31.4 40.6 46.2
1081 1078 121
a In kcal/mol, calculated from eq 1. b In kcal/mol, calculated from eq 2. c In cm-1, calculated from eq 10. d The average reorganization energy from the geometries of oi+[22/hex22]+ and io+[22/hex22]+.
In this approach, [22/hex/22]+ and [22/hex/22]0 are optimized, and eq 1 is used to calculate the reorganization energy. λv obtained from different ab initio approaches for reaction 7 are 34.3 kcal/mol (AM1), 45.9 kcal/mol (HF), and 39.0 kcal/mol (DFT with 50-50 functional). Nelsen et al. have experimentally measured λv of the mixed isomers of [22/hex/22]+ from the dependence of its charge-transfer band maximum upon solvent.6 The measurement value of λv is 40.9 kcal/mol, which is very close to the result obtained from DFT with 50-50 functional. Table 4 lists λv calculated by DFT with 50-50 functional for other isomers. To compare with the four-point approach, we also use the alternative approach-the sum of the reorganization energy of each mode described in section 2. Using DFT with 50-50 functional, this approach gives 40.6 kcal/mol for the reaction 7, 31.4 kcal/mol for the reaction 8, 46.2 kcal/mol for the reactions 9 and 6. Both approaches predict similar results.
ET Rates of a Bis(hydrazine) Radical Cation
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Figure 3. Side-on view of the three isomers of neutral [22/hex/22], with loops representing the lone pair orbitals added. Figure 2. Reorganization energy components (λi) as a function of frequency (ωi).
However, the latter approach can obtain detailed information of individual modes. In Figure 2, we plot the λv dependence of the mode frequency. In spite of the different λv for them, the behaviors of the mode-dependent λv are very similar, i.e., only several modes have large reorganization energies. These modes are possible to perform the localized state in the strong electronic coupling. One can thus expect that they have the important effects to ET. Furthermore, the intramolecular effective frequency ωeff that is used to calculate the rates for ET can be calculated from Figure 2 by using eq 1062
ωeff )
iωi2λi λv
Figure 4. Structure of poly-NB(n).
(10)
The calculated effective frequencies and reorganization energies are listed in Table 4 for these four reactions. It is worth mentioning that the values of reorganization energies with the two methods are close for symmetric ET, but the difference is found for asymmetric ET for both reorganization energy and effective frequency. This difference may be explained by the stronger interaction between the neutral and cationic parts in the asymmetric reaction than that in the asymmetric one. The four-point model work quite accurately if the neutral and cation parts are far separated. However, the present systems show strong coupling between them. For instance, the Mulliken charge populations on the cationic parts are 0.74, 0.71, and 0.70 for oo+[22/hex/22]+, ii+[22/hex/22]+, and oi+[22/hex/22]+, respectively. Comparing with the average frequency method, the differences indeed become larger when the charge populations are smaller. Thus, we expect that the possible error in the four point approach may come from the neglect of this coupling. To investigate the interaction deeply, we have found that several modes with frequencies 420, 500, and 650 cm-1 correspond to the coupling motions between neutral and cation parts of the system. These modes have larger reorganization energies for the asymmetric reaction compared with the symmetric reactions (see Figure 2). Thus, we expect that the interaction between neutral and cation parts for the asymmetric reaction is stronger than others. From Figure 2, it is also found that high frequency mode (2930 cm-1) has larger reorganization energy for the asymmetric reaction compared with others. It predicts a higher effective frequency for the asymmetric reaction. On the basis of the above analysis, we expect that the average frequency method may be more accuracy than the four-point approach. In the rate calculations, we use both reorganization energies and effective frequencies obtained from the effective frequency method. 3.3. Electronic Coupling. As pointed out long ago by Hoffmann,63 a “3V” arrangement of the lone pair orbitals and
Figure 5. Molecular orbitals and their combinations of oo+[22/hex/ 22]+ at the transition state.
four σ bonds of the bridge, such as is present in ii[22/hex/22] (see Figure 3), leads to significantly larger electronic coupling than do other possibilities, because of the overlap properties of the orbitals involved. Paddon-Row, Hush, and co-workers have used photoelectron spectroscopic measurements of the energy separation between symmetric and antisymmetric π-dominated orbitals with the simple two-state model to estimate Hab values for poly-V-aligned norbornenes (poly-NB(n), see Figure 4) as a function of n, although the energy differences proved too small to measure by this method for n > 2 (that is, with eight or more bonds between the alkenyl groups) or if the σ-bond framework were not aligned in the poly-V (all exosubstituted) arrangement.64-66 Before discussing the electronic coupling, we analyze the properties of the MOs at the TS geometry. We should point out that the two delocalized MOs, which are used to estimate the electronic coupling according to Koopmans’ theory, need to satisfy the condition that the electronic cloud should localize on donor and acceptor states, respectively, after summing and subtracting MO wave functions of HOMO and HOMO-1.67,68 As an example, Figure 5, a and b, shows electronic density distributions of the HOMO and HOMO-1, respectively, of the neutral molecule at the geometry of cation with C2V symmetry for oo+[22/hex/22]+. Figure 5, c and d, displays the MO densities after summing and subtracting of the HOMO and HOMO-1 densities. Indeed, they are dominantly localized at the donor and acceptor sides of the system, and only partially delocalized to the σ-bridge.
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TABLE 5: Calculated Electronic Couplings (cm-1) for Four Reactions of [22/hex/22]+ ET reaction ii+ f i+i oo+ f o+o oi+ f o+i io+ f i+o
linea C2Vb line C2V line line
KT(HF)
KT(DFT)
SF(CIS)
SF(TDDFT)
TM(HF)
GMH(TDDFT)c
3402 3731 1097 1317 1865 1865
2963 3182 988 1207 1755 1755
3715 3938 1440 1641 2077 2077
3083 3292 1105 1282 1852 1852
2835 3038 950 1108 1510 1510
2878 1052 1557 1564
a The transition state geometries evaluated from the linear reaction coordinate. b The transition geometries obtained by C2V symmetry. c The optimized geometries obtained from DFT with 50-50 functional at the TD-MPW1K/6-31g(d) level.
Table 5 lists the electronic couplings obtained from various approaches. The geometries at TS are determined by two methods, the linear reaction coordinate with optimized geometries of the donor and acceptor states (line) and constrained symmetry (C2V). For a given reaction, KT with DFT, SF with TDDFT, and TM approaches in both geometries (line and C2V) predict close electronic coupling values, and the maximum error is about 15%. It manifests that the KT approach is indeed valid for the present system. The more detailed comparison displays that SF predicts the largest electronic coupling while TM produces the smallest one. However, KT with HF and SF with CIS predict too large electronic couplings compared with those from DFT. Those differences will be discussed in the following. To investigate the geometry dependence of the electronic coupling, we also calculate the electronic coupling strength at the optimized geometry using GMH method, shown in Table 5. The transition energies used are as follows: 43.67 kcal/mol for reaction 6, 41.73 kcal/mol for reaction 7, 40.12 kcal/mol for reaction 8, and 41.46 kcal/mol for reaction 9. In general, transition energies have the similar values to the reorganization energies for symmetric reactions with weak couplings. However, these energies become larger than reorganization energies if the coupling is strong. For instance, two energy values are very different for the ii+[22/hex/22]+ f i+i[22/hex/22]+ reaction while they are similar for the oo+[22/hex/22]+ f o+o[22/hex/ 22]+ reaction. Compared with other electronic couplings, it is found that GMH predicts nearly the same values obtained from TM at the TS. Thus, we expect that the electronic couplings do not strongly depend on the geometries. A similar conclusion has been obtained by Pati et al. who used TM to estimate the electronic couplings at the optimized and TS geometries on some carborane molecules.69 For different reactions, all the calculations show that the electronic coupling for the reaction 8 is about 3 times larger than that of the reaction 7 in spite of the similar parameters of their geometries. For the reactions 6 and 9, the electronic couplings are the same because their reactions satisfy the detailed balance principle, and they have the same TS geometries and the same MO properties for the determination of the electronic coupling. 4. Electron-Transfer Rates It may be difficult to directly explore which approach used above is the most suitable for the present system without comparing the calculated results from them with experimental data. Since the ET rates have been measured experimentally,9 we can compare them with calculated ET rates from the parameters thus obtained. In order to incorporate the strong electronic coupling and solvent dynamics, we adopt the extended Sumi-Marcus model proposed recently for the calculation of ET rate.38 Here, we outline the main procedures. Introducing a single coordinate
value x for describing the solvated-structure fluctuations, the reaction populations P(x,t) at each coordinate value x at time t satisfy the diffusion-reaction equation
∂P(x, t)/ ∂ t ) [L - k(x)]P(x, t)
(11)
Here k(x) is the sink function representing the intramolecular reaction at a given solvent coordinate x. In the original SumiMarcus approach,70 k(x) is calculated from the perturbation theory. In order to incorporate the strong electronic coupling effect, the R-matrix scattering approach was employed instead of the perturbation theory.38,39 The generalized Smoluchowski operator L in eq 11 is given by
L ) D(t){∂2/ ∂ x2 + β ∂ / ∂ x[dVi(x)/dx]}
(12)
where β ) 1/kbT, T is temperature, D is the diffusive coefficient. Then, the ET rate can be defined as
Ket ) lim -
d ln
∫ dx P(x, t)
(13)
dt
tf∞
To transform eq 11 into the Hermitian form, we use the substitution
p(x, t) ) P(x, t)/g(x)
(14)
where g(x) is the square root of equilibrium solution of eq 11 in the absence of the reactions. Then we can obtain
p(x, t) ) exp((H(x) - k(x))t)p(x, 0) where
H(x) ) D(t)
[(
∂2 βD(t) β dV(x) 2 d2V(x) + 2 2 dx ∂x2 dx2
)
(15)
]
(16)
Suppose |n〉 is the eigenvector of (H(x) - k(x)), then
p(x, t) )
∑ exp((H(x) - k(x))t)|n 〉 〈n|p(x, 0) ) n
∑ exp(λnt)|n 〉 Cn (17) n
From the definition of the rate (see eq 13), we can get ket ) λ0, where λ0 is the smallest eigenvalue of (H(x) - k(x)). In the calculation of ET rates, the diffusive coefficient is taken as D ) 0.0193 cm -1 which corresponds to MeCN solvent. The other parameters used are summarized as follows: electronic coupling Hab ) 950 cm-1 (2835 cm-1, 1510 cm-1), inner reorganization energy λv ) 40.6 kcal/mol (31.4 kcal/mol, 46.2 kcal/mol), and effective frequency ωeff ) 1078 cm-1 (1081 cm-1, 1218 cm-1) for the reactions 7, 8, 6 and 9. The solvent reorganization energy (9.0 kcal/mol), which is taken from experimental data, is used for all isomer reactions. By using the above parameters, the calculated ET rates at T ) 298 K are as follows: 1.27 × 108 s-1 (reaction 7), 1.71 × 1011 s-1 (reaction 8), 2.16 × 108 s-1 (reaction 6), and 7.28 × 108 s-1 (reaction
ET Rates of a Bis(hydrazine) Radical Cation 9). Surprisingly, the rate for reaction 8 is much larger than the others. It can be easily qualitatively explained from the larger Hab and lower λv of it, which makes a smaller reactive energy barrier and the larger rate. One thus expects that the rates are quite dependent on the conformation presents. Compared with the experimental data (1.3 × 108 s-1), the rate of reaction 8 is about 1000 times larger, and the rates from other three isomers are consistent with the experiment well (the thermal average rate of the three ET processes is 2.14 × 108 s-1). It manifests that the ESR measurement may correspond to these three isomers of [22/hex/22]+ components. Because of the good agreement of the ET rate for those three isomers of [22/hex/22]+ theoretically and experimentally, we believe that the structure parameters listed for the prediction of the ET rates should be reasonably accurate. It thus becomes possible for us to clarify the accuracies of the approaches used in the structure calculations. Among the approaches used for the calculation of the electronic couplings, rather similar results are yielded despite the electronic coupling obtained from TM approach is used to predict ET rates. However, SF with CIS and KT with HF give larger electronic couplings than that from TM. The possible reason for SF is that the correlation may not be fully account for based on the CIS,35 such as, it often gets a large transition energy. It is not certain how much difference introduced by this correlation. While to KT with HF, the error may come from the replacement of DFT MOs by HF ones. However, it is known that the general literature of DFT states that only the energy of HOMO has physical significance. This energy equals to the first ionization potential and other orbital energies have no real physical qualities.71,72 But recent investigations4,49 show that Koopmans’ theorem calculations with DFT give far better transition energies for type A transitions than do HF calculations and it correctly predicts the optical transition energies for type B transitions. Together with the present results, we are convinced that KT with DFT can give better results than KT with HF for some radical ions, such as hydrocarbon and bis(hydrazine) radical cations. It is also noted that the electron clouds are not fully located at donor and acceptor states in the diabatic representation as shown in Figure 5c,d, which may also affect the accuracy of KT. In all, SF with TDDFT, KT with DFT, TM, and GMH are reasonable approaches for the calculations of the electronic couplings, and they yield rather similar results. The detailed comparisons reveal that the approaches which predict the electronic coupling from small to large values are TM, GMH, KT with DFT, and SF with TDDFT, respectively. The maximum error is about 15% in the present calculations. 5. Concluding Remarks The electron transfer processes of [22/hex/22]+ have been investigated in detail together with ab initio calculations and the extended Sumi-Marcus approach. Two important parameters, electronic coupling and reorganization energy as well as an effective frequency, have been calculated with several different methods. The obtained electron transfer rates of oo+[22/ hex/22]+, oi+[22/hex/22]+, and io+[22/hex/22]+ agree well with the experiment. Furthermore, the results reveal that electrontransfer rates are heavily dependent on the isomer geometries. For instance, the rate in ii+[22/hex/22]+ f i+i[22/hex/22]+ is about 1000 times larger than that in oo+[22/hex/22]+ f o+o[22/ hex/22]+. Comparing with different ab initio approaches, we have confirmed that the DFT with 50-50 functional is a reasonable approach for the present system to predict its
J. Phys. Chem. B, Vol. 112, No. 35, 2008 11085 structure. For the reorganization energy, both four-point approach and summarizing the reorganization energy of each mode work well. In the calculation of the electronic coupling, TM is still a good choice compared with SF and KT for the present system with strong electronic coupling despite SF and KT with DFT can predict reasonable electronic couplings. Acknowledgment. This work was supported by the National Nature Science Foundation of China (No. 20773115), the National Key Basic Research Foundation Program of China (No. 2007CB815204, No. 2004CB719901), and NSF CHE-0647719 (S.F.N.). The authors also acknowledge a generous allocation of the supercomputer time from the Virtual Laboratory for Computer Chemistry of CNIC and the Supercomputer Center of CNIC, Chinese Academy of Sciences. References and Notes (1) Robin, M. B.; Day, P. AdV. Inorg. Radiochem. 1967, 10, 247. (2) Rosokha, S. V.; Kochi, J. K. New J. Chem. 2002, 26, 851. (3) Nelsen, S. F.; Konradsson, A. E.; Weaver, M. N.; Telo, J. P. J. Am. Chem. Soc. 2003, 125, 12493. (4) Nelsen, S.F.; Weaver, M. N.; Zink, J. I.; Telo, J. P. J. Am. Chem. Soc. 2005, 127, 10611. (5) Nelsen, S. F.; Weaver, M. N.; Lockard, J. V.; Zink, J. I. Chem. Phys. 2006, 324, 195. (6) Nelsen, S. F.; Triebner, D. A., II; Ismagilov, R. F.; Teki, Y. J. Am. Chem. Soc 2001, 123, 5684. (7) Hush, N. S. Electrochim. Acta 1968, 13, 1005. (8) Newton, M. D. Chem. ReV. 1991, 91, 767. (9) Nelsen, S. F.; Adamus, J.; Wolff, J. J. J. Am. Chem. Soc. 1994, 116, 1589. (10) Nelsen, S. F.; Chang, H.; Wolff, J. J.; Adamus, J. J. Am. Chem. Soc. 1993, 115, 12276. (11) Nelsen, S. F.; Ramm, M. T.; Wolff, J. J.; Powell, D. R. J. Am. Chem. Soc. 1993, 119, 6863. (12) Nelsen, S. F. J. Am. Chem. Soc. 1996, 118, 2047. (13) Johnson, R. C.; Hupp, J. T. J. Am. Chem. Soc. 2001, 123, 2053. (14) Marcus, R. A. J. Chem. Phys. 1956, 24, 966. (15) Marcus, R. A.; Sutin, N. Biochim. Biophys. Acta 1985, 811, 265. (16) Jones, G. A.; Paddon-Row, M. N.; Carpenter, B. K.; Piotrowiak, P. J. Phys. Chem. A 2002, 106, 5011. (17) Farazdel, A.; Dupuis, M. J. Comput. Chem. 1991, 12, 276. (18) Coropceanu, V.; Boldyrev, S. I.; Risko, C.; Bredas, J. L. Chem. Phys. 2006, 326, 107. (19) Gao, X.; Tang, S.; Zhou, W. Chem. Phys. Lett. 2007, 445, 297. (20) Nelsen, S. F. AdV. Phys. Org. Chem. 2006, 41, 183. (21) Hush, N. S. Prog. Inorg. Chem. 1967, 8, 391. (22) Koopmans, T. Physica 1934, 1, 104. (23) Paddon-row, M. N.; Wong, S. S. Chem. Phys. Lett. 1990, 167, 432. (24) Jordan, K. D.; Paddon-row, M. N. Chem. ReV. 1992, 92, 395. (25) Creutz, C. Prog. Inorg. Chem. 1983, 30, 1. (26) Broo, A.; Larsson, S. Chem. Phys. 1990, 148, 103. (27) Braga, M.; Larsson, S. Int. J. Quantum Chem. 1992, 44, 839. (28) Braga, M.; Larsson, S. Chem. Phys. Lett. 1992, 200, 573. (29) Braga, M.; Larsson, S. J. Phys. Chem. 1992, 96, 9218. (30) Rodriguez-Monge, L.; Larsson, S. J. Phys. Chem. 1996, 100, 6298. (31) Farazdel, A.; Dupuis, M.; Clementi, E.; Aviram, A. J. Am. Chem. Soc. 1990, 112, 4206. (32) Rodriguez-monge, L.; Larsson, S. Int. J. Quantum Chem. 1997, 61, 847. (33) Li, X. Y.; Tang, X. S.; He, F. C. Chem. Phys. 1999, 248, 137. (34) You, Z. Q.; Shao, Y.; Hsu, C. P. Chem. Phys. Lett. 2004, 390, 116. (35) Yang, C. H.; Hsu, C. P. J. Chem. Phys. 2006, 124, 244507. (36) Nelsen, S. F.; Blackstock, S. C.; Kim, Y. J. Am. Chem. Soc. 1987, 109, 677. (37) Barzykin, A.V.; Frantsuzov, P. A.; Seki, K.; Tachiya, M. AdV. Chem. Phys. 2002, 123, 511. (38) Zhu, W. J.; Zhao, Y. J. Chem. Phys. 2007, 126, 184105. (39) Zhao, Y.; Mil’nikov, G. Chem. Phys. Lett. 2005, 413, 362. (40) Blomgren, F.; Larsson, S.; Nelsen, S. F. J. Comput. Chem. 2001, 22, 655. (41) Li, X. Y. J. Comput. Chem. 2001, 22, 565. (42) Lee, E.; Medvedev, E. S.; Stuchebrukhov, A. A. J. Chem. Phys. 2000, 112, 9015. (43) Hwang, H.; Rossky, P. J. J. Phys. Chem. B 2004, 108, 6723. (44) Mebel, A. M.; Chen, Y.-T.; Lin, S.-H. Chem. Phys. Lett. 1996, 258, 53.
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