Thermally-induced or “activated” ET can take place in both intramolecular and intermolecular contexts. In the intermolecular (bimolecular) case, the donor, D, and acceptor, A, reactant species first diffuse together and upon encounter form what is known as the “precursor complex” (associated pair) as shown in Figure 1.1. In the classical picture, the precursor complex reorganizes its nuclear coordinates through a thermal activation barrier, ΔGth, to form a transition state configuration relevant to the
ET event. This transition state is located at, or near the intersection point of the corresponding reactant and product potential energy surfaces.
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In the schematic illustration of the ET processes shown in Figure 1.1, a2 and a3
represent the average coordination sphere radii, assumed to be proportional to metal- ligand bond lengths, of the two reactant complexes involved in the ET reaction. M and M+ represent the reactants respective oxidation states, and a* represents the radii
of the thermally activated reactants in which a2 and a3 have “compressed” or
“expanded” respectively, such that the nuclear coordinates are equal at the intermediate, transition state geometry (independent of electronic state). In this representation, the reactants start in an already formed associated pair or “symmetric binuclear encounter complex” (labeled REACTANT). From this starting point, ET may occur through one of two pathways: thermal ET (lower pathway, represented by kth) or optical ET (upper pathway, represented by hν, vide infra).
Figure 1.1 An illustration of optical (represented by “hν”, upper pathway) and
thermal (represented by “kth”, lower pathway) ET processes relevant to a symmetric
binuclear complex (or bimolecular encounter complex if the wavy “bridge” is omitted).15
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At the transition state geometry, the electron is then able to be transferred on a rapid timescale (defined by a tunneling frequency, el, vide infra), such that the
nuclear coordinates and momenta are unchanged during the electronic transition as required by the Franck-Condon principle. The Franck-Condon principle comes from the fact that the time scales of electronic density fluctuations (< 10-15 sec), and
presumably any transitions between allowed electronic wavefunctions, ψel, are much faster than nuclear motion which occurs on the vibration/libration timescale of 10-13 - 10-11 sec. Therefore, it can be assumed that the nuclei remain “frozen” with respect to their positions and momenta during an electronic transition.11, 15 The Franck-Condon principle is also related to the Born-Oppenheimer approximation by which the wavefunction for some molecular system is divided into two parts; the separated electronic,ψel, and vibrational (or nuclear, χnu) wavefunctions. It is the probability density overlap between nuclear wavefuntions of two different electronic states which
directly yields the quantity known as the Franck-Condon “factor”.16, 17 These are
most often discussed with respect to spectroscopic (“vertical”) transitions between ground and excited electronic states, but they may be applied to thermally-induced barrier crossings as well. In our case, the nuclear coordinates, which remain frozen during the thermally-activated transition, would include both metal-ligand and all other skeletal bond lengths, as well as solvent shell configurations which are electrostatically coupled to the location of the probability density centroid of the “exchanging” electron (corresponding to the difference at the transition state between the system being on either the reactant’s or product’s electronic surface).
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Once the electron is transferred within the encounter/precursor complex and some degree of vibrational relaxation has begun, the resulting successor complex is now considered to be “locked” in the products electronic configuration for at least some number of vibrations which can then lead to full relaxation in the thermally- equilibrated product state. This means that the bond lengths of the two complexes, as well as the “solvent shell” around the products now re-adjust to the new electrostatic field corresponding to the products electronic surface. The products then separate and diffuse apart. In the bimolecular thermal ET case, it should be noted that the displacement associated with nuclear relaxation from the activated precursor/successor intermediate nuclear configuration (transition state geometry) to the ET product state is about half of the total nuclear displacement corresponding to going from fully equilibrated reactants to equilibrated products (vide infra).
1.2.2 Optical ET
The two reactants, donor and acceptor, can exist together in close proximity in an electrostatically- disfavored “like-charged” encounter complex, a favored, unlike- charged “mixed-valence” ion pair, or in a covalently-bound bridged binuclear complex as shown in Figure 1.1 (where the wavy line represents the bridge). In some cases, the bimolecular, non-bridged encounter (or “precursor”) complex is a stable species, such as the known class of ion pairs of the composition (NH3)5RuIIIL/MII(CN)6 where M = FeII, RuII, or OsII. These strongly-charged (+3)
and (-4) acceptor and donor ions are now electrostatically held in an overall (-1)
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M~M+ as in Figure 1.1 or by a general associated donor-acceptor pair [D,A], can
then result in sudden ET from donor to acceptor, and this vertical or “optical” ET transition necessarily occurs without any change in nuclear coordinates as required by
the Franck-Condon principle (vide supra). In covalently linked D-A mixed-valence
dimers as in Figure 1.1, the bridging group holds the two metal centers at a fixed distance and typically modulates the quantum interaction between D and A (or “M”
and “M+”). Now both optical and thermal ET may occur just as in the case of an
associated bimolecular ion pair, but now there is no associative step to form the precursor complex and spectroscopic study of the optical ET is facilitated.15
The “charge-transfer state” or “intervalence-transfer excited state” arrived at upon vertical transition caused by photon absorption is necessarily created in a vibrational excited state of the new electronic surface since nuclear positions are slow to adjust to the new charge distribution. In the case of symmetrical mixed-valence dimer systems where there is no driving force or “0-0” band energy, this Franck- Condon state (or initially-populated vibrational excited state) is at an energy λ above
the ground state.15 The energy λ is known in the ET literature as the “nuclear
reorganizational energy” (vide infra).18 The excited state thus formed can then relax
to form the thermally-equilibrated products “redox isomer” of the former reactants ground state, now described by the product’s electronic distribution [D+, A-] and with nuclear coordinates corresponding to, and in equilibrium with, the new electronic wavefunction.
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