We present results of theoretical calculations based mainly on semiempirical (INDO) quantum chemical electronic structure calculations, and in one case (Section 2.44.4.3), on large-scale MD simulations.17The ‘‘spectroscopic’’ Hamiltonian (INDO/s) parametrized by Zerner and co-workers for systems including both organic and transition M-based moieties,80is very useful for treating electronic properties of TMCs, including evaluation of Hif.4,5,40 Alternatively, the method is not parametrized for calculating features of potential energy surfaces, so that necessary bond lengths and force constants must be obtained from independent sources. Given our particular focus onHif, it is notable that in several cases where comparison has been made, INDO/s results forHif
are found typically to be within25% ofab initioSCF results81(see also Newton, 200069).
2.44.4.1 Role of Metal/Ligand Mixing
While ET between TMCs may often be viewed nominally as metal-to-metal charge transfer (MMCT), the actual contact between the TMCs (whether in a bimolecular process or in a binuclear complex) is often established primarily by the Ls.16,22,40,65,82,83
Hence the effectiveness of the overall MM electronic coupling (manifested inHif) depends crucially on the extent to which the transferring charge is partially delocalized onto the Ls, both in the initial ( i) and final ( f) states at the TS. This will be true whether the Ls are in direct contact (as in encounter complexes in bimolecular ET between TMCs with single coordination shells), or indirectly, as mediated by tethers. The specific examples considered inSections 2.44.4.1.1and2.44.4.1.2involve bimolecular ET processes.
The influence of ML mixing on Hif can be cast in terms of superexchange theory (Section 2.44.3.1.1), as illustrated inFigure 2, where the collective B is represented by a pair of Ls (n=2) in contact. The relative importance of ET and HT transfer is then controlled by the ML hopping integrals (T, in Equation (15)) and, respectively, the energy gaps for metal-to-ligand charge transfer (MLCT) and ligand-to-metal charge transfer (LMCT) (the L–L coupling element t(see Equation (15)) is also needed to ‘‘complete the circuit’’).16
For a binuclear TMC system, one may employ a simpler se model based on a single B site (n=1 in Equation (15)), yielding the following superposition of ET and HT pathways, developed specifically for ET involving low-spin Ru2þand Ru3þsites.4,5,71,74
HifẳTMLTM0L
2EML þ TLM0TLM
ELM ð20ị Electronic Coupling Elements and Electron Transfer Theory 579
where ML (M0L) and LM (LM0) refer, respectively, to MLCT and LMCT processes, andEMLand ELMare effective MLCT and LMCT energy gaps.40,71Equation (20)has been used with reason- able success71,74to correlate optical data for MLCT and LMCT (right-hand side) withHifestimates based on optical data for MMCT (left-hand side), also known as the intervalence transition (IT).
A measure of the extent of ML mixing is reflected in GMH effective D/A separation distances (rDA) inferred from experimental optical data (Equations (18) and(19)), as illustrated inTable 1 (see also Brunschwig et al. 19986). The rDA values are appreciably less than the nominal values based on molecular structure, and the spectroscopic (adiabatic) value can be significantly smaller than the diabatic value.4,5
2.44.4.1.1 Electronically saturated ligands
Water and ammonia are familiar examples of electronically saturated Ls which lacklow-lying orbitals and, hence, are expected to serve as electron donors, thus leading to dominance of the HT se pathway (via LMCT) in establishing the magnitude ofHif.40Electron exchange in some model aquo and ammine Fe2þ/3þ, Ru2þ/3þ, and Co2þ/3þcomplexes with a common linear MLLM
geometry yielded proportionality between Hif and the square of the ML covalency parameter, thus underscoring the role of L-mediated overall M–M coupling.82Alternatively,Hifmagnitudes for complexes of a given ML6type can depend strongly on the relative orientation of the TMCs in the bimolecular TS, displaying variations of nearly an order of magnitude.40 For the relatively
‘‘intimate’’ face-to-face (F–F) approach geometry, intermolecular LM (and perhaps even MM) as well as LL contacts may be significant. Complexity of contacts precludes any simple variation of Hifwith rDA(of the type expected for homologous DBA systems (Equation (14)) if comparisons are made among systems with different orientations.
2.44.4.1.2 Metallocene-based electron transfer systems
In exchange between ferrocene/ferrocenium (Fc/Fcþ) or the cobalt analog (Cc/Ccþ, where Cc denotes cobaltocene), the unsaturated cyclopentadienyl (Cp) L can serve as an electron donor (LMCT), but also an A (via a somewhat higher energy MLCT process).16,80 Thus the relative importance of ET and HT pathways for Hif is not immediately obvious. Group theory and analysis of INDO/s results for Hif suggest that ET and HT are, respectively, the dominant se pathways for the ferrocene and cobaltocene exchange. Using quasi-cylindrical symmetry, we find that the transferring charge for Fc and Cc resides in orbitals, respectively, of (3dx2y2or 3dxy) and (3dxzor 3dyz) symmetry (where z is the quasi-cylindrical axis), while the highest occupied and lowest unoccupied orbitals of Cp are, respectively, of and symmetry. Hence the LMCT
Table 1 Effective separation of D/A sites (rDA(A˚)).
From molecular geometrya
From two-state analysis of spectral datab
ETsystem Diabatic Adiabatic
(NH3)5(Ru2þL)c
MLCTd
Lẳpz LẳpzHỵ Lẳbpy LẳbpyHỵ 8>
<
>:
3.5 2.2 1.0
3.5 2.1 <0.1
5.6 3.4 2.9
5.6 4.3 3.6
(NH3)5(Ru2þL Ru3þ) NH3)5c ITeẳ Lẳpz
Lẳbpy
6.8 1.4 <0.1
11.3 5.2 5.1r
a Based on separation of M and L midpoint (for MLCT cases) orrMM0(for IT cases), where M and M0denote Ru atom sites. (Table 2 of Newton, 2001;42003,5reproduced by permission of Wiley-VCH). b Based on spectral data analyzed in terms of the two-state GMH model (Equation (17)). In the analysis a value off=1.3 was used to relate the local to the applied external electric field in the Starkmeasurements
E*locẳf E*ext
. c pzpyrazine.6,93,94
pathway (with lower energy gap) does not couple to the transferring electron on Fc, leaving the higher energy ET pathway (via MLCT) as the symmetry-allowed route forHif.16 Table 2shows that calculatedHifvalues based on the lowest energy encounter complexes (separated by3 k cal mol1 from other structures)16bracket experimental values.84The results for Cc imply the adiabatic ET regime (el 1), whereas Fc appears to be near the nonadiabatic/adiabatic boundary. Further good agreement between calculation and experiment has been obtained for intramolecular ET between Fc and Fcþsites linked by an acetylenic (A) or an ethylenic (E) group in DBA structures denoted, respectively, as (FcAFc)þand (FcEFc)þ.85–87
2.44.4.2 Alternative Initial and Final States
Implementation of the TSA becomes ambiguous in cases of near degeneracy (e.g., Newtonet al., 1991;16Broo and Larsson, 1992;20Cacelli and Ferretti, 1998;28Elliottet al., 1998;50Zerneret al., 1980;80Newton, 2003;85 Shin et al., 199688) (common in open shell TMCs, where, for example, perturbation of idealized octahedral symmetry may lead to splitting oft2gandegstates) or other situations where the appropriate choice of iand fis not obvious.15,17,89,90The choice in general involves a tradeoff including thermal access (in the case of initial states), and the state-dependence of activation energy (Gy) and coupling (Hif).15,17
2.44.4.2.1 Near-degeneracy
In cases of near-degeneracy in D or A manifolds, one component state may couple effectively to the B (hopping integral T in Equation (15)), while the other couples weakly due to the nodal structure. An example is provided inTable 3for the case of Fc and Fcþlinked by unsaturated Bs of the type ‘‘APA’’ and ‘‘VPV,’’ where A, E, and P denote, respectively, an acetyenic, an ethylenic, and ap-phenylenic moiety.85In the isolated ferrocene molecule the 3dx2y2and 3dxyorbitals (and hole states) are degenerate, but due to different orientation of nodal planes in the tethered DBA systems, 3dx2y2(which overlaps with a Cp MO having a finite contribution at the carbon atom linked to the B) lies above the 3dxy orbital (which has a node at the linked Cp carbon atom).
Thus, the hole ground states (3dx2–y2) are the ones which yield good overall D/A coupling.
Nevertheless, the weakly coupled 3dxy hole states are calculated to lie only 0.05–0.06 eV higher and thus are thermally accessible at room temperature.
Michl and co-workers have discussed the potential role of (C4R4)Co(C5H5) complexes in conductive two-dimensional assemblies, in which the cyclobutadiene rings may be linked via suitable substituents R.85 Denoting the complex as CbCoCp, where R=H for the present discussion, we note that it may be viewed as basically isoelectronic (as far as the relevant Co and L MOs are concerned) with ferrocene (CpFeCp).5,85 In Table 4we compare the calculated low-lying hole states of CbCoCp and CpFeCp.85 The primary feature of interest is the relative order of the nondegenerate 3dx2y2 and 3dxy hole states in (CbCoCp)þ. While the overall sym- metry of the CbCoCp structure can be at most Cs, the local C4v symmetry of the CbCo moiety plays an important role, with the antibonding Cb pi MO (b1) mixing with the Co 3dx2y2orbital and pushing it below the 3dxyorbital (b2), whose nodal planes contain the Cb carbon atoms and thus in the localC4vsymmetry do not mix with any of the Cb pi MOs. This result has implications for electronic coupling when CbCoCp units are linked, since as discussed above, the 3dxyorbitals yield much less favorable coupling than the 3dx2y2orbitals.
Table 2 Calculated and experimental estimates ofHiffor (CpFeCp)o/þand (CbCoCp)o/þa.
Encounter
Intermolecular contactc Fe()d Co()d
geometryb rMM rCC Calc Exp84 Calc Exp84
Coaxial (D5h) 6.75 3.39 135 35 920 175
T-shaped (Cs) 6.50 3.63 99 70
a Newtonet al. (1991).16 b Lowest-energy structures.16 c M–M and shortest bimolecular carbon–carbon contact distances (A˚).
d Symmetry-type of D and A orbitals, based on pseudo-cylindrical symmetry.
Electronic Coupling Elements and Electron Transfer Theory 581
Other examples of near-degeneracy effects involve ET in low-spin Fe2þ/3þand Ru2þ/3þcouples complexed to aza aromatic Ls. The case of tris bpy Fe2þand Fe3þsites linked by three polymethyl- ene chains ((CH2)m) with approximate C3symmetry (schematically represented by structure1(m) is especially interesting.50While the lst2gstates of the Fe3þsite are split into ground (2A) and excited (2E) states separated by only 100 cm1, the larger spin orbit coupling constant (estimated at 440 cm1) requires the use of spin-orbit adapted states in analyzing both the thermal and optical ET processes, in which a single thermally accessible initial state is significantly coupled to two separate final states, estimated to be split by 3kBT at room temperature.50 If the spin orbit coupling were absent, only the ground state ET process would have appreciable weight.50
2.44.4.2.2 High-spin vs. low-spin states
The mechanism of Co2þ/3þ(NH3)6aqueous exchange requires careful consideration of the roles of hs and ls electronic states.4,5,15,17,89,90For Co2þand Co3þ, these states correspond, respectively, to the t52ge2g(hs)=t62geg(ls) andt52geg(hs)=t62g(ls) electronic configurations. The (zeroth-order) ground states are hs(2þ) and ls(3þ). In the absence of spin-orbit mixing, the ground state process, while
Table 3 Quasi-degenerate hole states in (FcAPA Fc)þand (Fc EPEFc)þ.a
Fe 3d-hole typeb
Hif
(cm1)
Relative energy (eV)c
(FcAPAFc)þ 3dx2y2 177 0.000
3dxy 2 0.058
(FcEPEFc)þ 3dx2y2 140 0.000
3dxy 0.4 0.051
a Reproduced by permission of the American Chemical Society fromACS Symposium Series,2003, 844, 196 (A, P, and E denote, respectively, acetylenic, phenylene, and vinyl moieties). b Dominant character of calculated hole state, where z is the Fc axis and thex-axis is aligned with the single bonds linking the Fc Cp rings to the B. c Based on INDO/s SCF/CI calculations.
Table 4 Comparison of low-lying hole states for ferrocene and (C4H4)Co(C5H5).a
Hole stateb
Relative energy (eV)c
Ferrocene 3dx2y2, 3dxy 0
3dz2 0.48
(C4H4)Co(C5H5)b 3dxy 0
3dz2 0.27
3dx2y2 0.33
a Reproduced by permission of the American Chemical Society fromACS Symposium Series,2003, 844, 196
b The 3d-type hole states are based on a coordinate system where thez-axis is perpendicular to the ring planes and where thex-andy-axis are parallel to the diagonals of the C4 ring. c The ground and low-lying excited states were obtained from separate direct SCF calculations.
formally spin-allowed, would be extremely slow, since it involves a ‘‘three-electron’’ process.15 Spin-orbit mixing of the excited configurations with their ground state counterparts yields a ‘‘one- electron’’ process (transfer of an electron betweenegorbitals at the two sites), but at the cost of a very smallel value (estimated to be103for an apex-to-apex encounter complex, and 104 when orientational averaging is included15). Alternatively, a thermally excited purely ls mechan- ism has the advantage of a reduced reorganization energy and also adiabatic behavior (el 1), which would at least partially offset the energy penalty for thermal hs to ls excitation. In fact, detailed calculations based on bothab initioand INDO/s methods, indicate that the net tradeoff unequivocally favors the ground-state (spin-orbit-enhanced) route.15 The small el values cited above correspond to an effectiveHifin which a large spatial matrix element linkingegorbitals on the two sites (600 cm1 for the apex-to-apex geometry) is scaled by a spin-orbit attenuation factor so=0.014.4,5,15
In comparison with the above picture for Co2þ/3þexchange, a contrasting situation is depicted inFigure 3 for reduction of Co3þ by a reduced bpy L at the Ru2þsite in a tethered binuclear complex14 (see Figure 4), dealt with in more detail in Section 2.44.4.3. Here, the preference for the ls mechanism is dominated by the more favorableelfor the ls final state ( f). The attenuation factor sois expected (on the basis of the approach given in Newton, 199115) to be appreciably larger than the value of 0.014 cited above for Co2þ/3þ exchange, since the spin-orbit mixing controllingHifin the former case is first-order, in contrast to the second-order mixing in the latter case.4,5 The activation free energies (Gy) for the hs and ls processes are estimated to be compar- able, due to compensating effects involving inner-shell reorganization energy and driving force
Co3+ reduction by (Ru2+)(bpy–) Is: Co3+(1A1g) hs: Co3+(1A1g)
Co2+(2Eg) Co2+(4T1g)
(Ru2+)(bpy–)(Co3+)
(Ru2+)(bpy)(Co2+ hs)
(Ru2+)(bpy)(Co2+ Is) (ψi)
(ψ′)
(ψf)
η f
Figure 3 Schematic depiction of energy profiles for reduction of an (OCo3þ(NH3)5) A site by a tethered (bpy)2Ru2þ(bpy) D site17 (see detailed structure in Figure 4) (figure 7 of Derr and Elliott, 1999,51
reproduced by permission of Elsevier Science).
Electronic Coupling Elements and Electron Transfer Theory 583
(Go), each of which is larger for the hs process.4,5 Thus the hs ground final state ( f0) is expected to be formed not directly, but rather via intersystem crossing from the initially formed ls excited state ( f). For related behavior see Yoshimuraet al. (1997)89and Song et al. (1993).90
2.44.4.3 The Role of Activation Parameters in Determination ofHifValues
As a final computational example, we illustrate the importance of understanding activation parameters (both entropic and enthalpic) in comparing calculated Hif values with experiment.
The DBA system is displayed inFigure 4, consisting of a derivatized [(bpy)2Ru2þ(bpy)] D group, a (pro)4B (where proproline and the tetrapeptide is in a polyproline II helical conformation), and a [OCo3þ(NH3)5] A group. The nominal redox process corresponds schematically to Equation (21)(seeFigure 3),
Ru2ỵðbpyịðCo3ỵị !Ru2ỵðbpyịðCo2ỵị ð21ị where the excess electron localized on the bpy linked (by a carbonyl group) to the (pro)4 B reduces the Co3þ site. This system is of interest for detailed theoretical study because it is a member of a homologous series for which a large body of kinetic data exists.92
The details of the theoretical computations (based on MD simulation and quantum chemical calculations) are given in Ungaret al. (1999).17The rate constant (kET) was expressed in terms of the results of the computer simulations,17using a nonadiabatic TST model (Equation (10)). Since the experimental results were analyzed in terms of a phenomenological Arrhenius model,92 agreement between experiment (left-hand side) and theory (right-hand side) would yield the following two equations. For the weakly temperature-dependent prefactor we have:
kBT=h
ð ịexpðSy0=kBị ẳ2Hif2=h
4kBT
ð ị1=2expSy=kB
ð22ị and for the Boltzmann factor:
exp Hyexp
=kBT
h i
ẳexp Hycalc
=kBT
h i
ð23ị In Equation (22), Sy is the ‘‘true’’ activation entropy (see Equation (6)). By contrast, Sy0 is a
‘‘pseudo’’-entropy of activation, which includes the contribution from the nonadiabatic prefactor (seeEquations (9)and(10)) in addition to the quantitySydefined inEquation (6). Thus in order to infer the value of Hif from the experimental data (Sy0), one must know the value of Sy(not directly available from the data of Ogawa, 199392).
Figure 4 Diagram of the ET system discussed in Section 2.44.4.3. The peptide B contains four proline residues, the D is a derivative of (bpy)2Ru2þ(bpy) (with the transferring electron in the initial state largely localized on the bpy L linked to the B), and the A is OCo3þ(NH3)5. (reproduced by permission of the
American Chemical Society fromJ. Phys. Chem. B1999, 103, 7367).
The primary results are summarized in Table 5. The experimental Sy0 value together with the calculatedSyvalue andEquation (22)yields an estimate ofHif(0.1 – 2.0 cm1) close to the calculated range (0.5 – 5.0 cm1). Given the estimated uncertainties,17the experimental and calculated activation energies (Hy) are seen to be in reasonable accord as well. Note that failure to distinguish between the Sy0andSyvalues inTable 5amounts to an error of a factor of20 in the inferredHifmagnitude, and also offers an example in which even ifSwere available from experiment, it would not be a useful guide to estimatingSy.4,5,17