I. Introduction

The goal of artificial photosynthesis is to mimic the green plants and other photosynthetic organisms that use sunlight to make high-energy chemicals.^1-8 This is a challenging goal because success requires integration of multiple chemical functions in a stable chemical architecture. As a result, artificial photosynthesis has developed more slowly than other approaches to solar energy conversion, namely, solid-state solar cells, interfacial TiO₂ photovoltaic cells, and organic thin film devices.^9-23

The molecular complexity of the natural photosynthetic apparatus is both an object lesson that illustrates the difficulties involved and an inspiration to research in this area.^24-32 With successful application, artificial photosynthesis would shortcut the billions of years of evolution that led to working photosynthetic membranes. Using chemical design and the principles of chemistry and physics, its goals are simpler and far less demanding.

In natural photosynthesis in the higher green plants, a complex reaction scheme uses solar energy to convert H₂O into O₂ and reducing equivalents, which appear as NADPH. In photosystem I, the reducing equivalents in NADPH are used to reduce CO₂ to carbohydrates, eq 1, or in bacteria, used directly as a reductive energy source.^25-32 In artificial photosynthesis, the goal is to harness the energy of the sun to drive high-energy small-molecule reactions such as water splitting, eq 2, or CO₂ reduction, eq 3.

The high-energy chemicals that form in these reactions can be recombined to extract the stored chemical energy. An ultimate goal is water splitting with the photochemically produced hydrogen and oxygen recombined in high-efficiency fuel cells for electricity production (Scheme 1).

Scheme 1. Photochemical Water Splitting Coupled to a Hydrogen/Oxygen Fuel Cell

Other small-molecule reactions are potential targets, and applications may exist in "green chemistry" for making high-value-added chemicals. For example, a high-efficiency photoelectrochemical synthesis (PES) cell has been reported for the production of Br₂ and H₂O₂.³³

The energy-storage reactions in eqs 1-3 are all oxidation-reduction (redox) reactions and can be divided into half-reactions. Each half-reaction involves a multiple electron change. This and the high-energy content of intervening 1e^- intermediates such as CO₂^- or OH ensure that both the photochemical fuel-forming reactions and their subsequent recombination are slow under ambient conditions. Both half-reactions must be catalyzed if they are to occur at appreciable rates at or near room temperature.

Resolution into half-reactions provides the basis for a "modular" approach to artificial photosynthesis. As in natural photosynthesis, separate half-reactions can be addressed separately and combined at a later stage into a single device. Complexity is unavoidable because of multifunctional requirements (light absorption, energy transfer, electron transfer, redox catalysis). There is a need to arrange and integrate functional groups and to provide an overall structural hierarchy.

A system analysis leads to the concept of the "integrated modular assembly" as described in an Accounts of Chemical Research article in 1989¹ and more recently in an article in Coordination Chemistry Reviews.³⁴ The current account is a "do it yourself" guide to constructing working assemblies for artificial photosynthesis.

Two approaches have dominated research in this area. One has been based on porphyrins and metalloporphyrins as chromophores and their incorporation into molecular assemblies.^2,3,6 A second is based on the metal-to-ligand charge-transfer (MLCT) excited states of metal polypyridyl complexes. Both approaches utilize the same underlying physical principles, and both utilize systematic molecular assembly strategies.

II. Integrated Approach to Artificial Photosynthesis Based on Polypyridyl Complexes. Functional Elements

II.A. Spatially Integrated Molecular Assemblies. In the integrated modular approach, reaction "modules" are linked together to construct reactive molecular assemblies. Energy conversion is based on light absorption and electron transfer. There is an analogy with natural photosynthesis where both CO₂ reduction in photosytem I and water oxidation in photosystem II are triggered by light absorption and excited-state electron transfer in the reaction centers of photosynthetic membranes.^24,30,35-41

Figure 1 illustrates what may be the minimum number of elements and processes required in a molecular assembly for artificial photosynthesis:

Figure 1 Diagram illustrating the essential elements in an assembly for artificial photosynthesis and the sequence of events that occurs after light is absorbed. The abbreviations are C = chromophore (light absorber), D = electron transfer donor, A = electron-transfer acceptor, catred = catalyst for chemical reduction, and catox = catalyst for chemical oxidation. The reaction illustrated is the photochemical splitting of water into H2 and O2. In the absence of a light-harvesting array, multilayer structures are required to achieve sufficient light absorption.

(1) Light absorption, either at a single "reaction center" chromophore (C) or by excitation of an antenna array, followed by energy-transfer sensitization of C*. Single-molecule light absorption is low, requiring the use of antenna arrays or multilayer structures for efficient light harvesting. The absorbance (Abs) of a surface layer of chromophore of coverage (in mol/cm²) and molar extinction coefficient (in M^-1 cm^-1) is given by Abs = 10³. Even with = 10 000, Abs = 10^-3 for a monolayer coverage of 10^-10 mol/cm². Absorption of 90% of the light requires Abs = 1.

(2) Electron-transfer quenching, of a donor-chromophore-acceptor (D-C-A) array either oxidatively, D-C*-A D-C⁺-A^-, or reductively, D-C*-A D⁺-C^--A. The driving force for either is a favorable free-energy change with G < 0.

(3) Redox separation by electron transfer, D-C⁺-A^- D⁺-C-A^- or D⁺-C^--A D⁺-C-A^-, also driven by G < 0. The combination of processes in (2) and (3) uses free-energy gradients to give spatially separated oxidative and reductive redox equivalents stored as D⁺ and A^-. This is analogous to a p/n junction in a Si semiconductor, which separates photochemically produced electron-hole pairs.^42-47

(4) Electron-transfer activation of catalysts, from A^- to a catalyst for reduction, cat_red, and to D⁺ from a second catalyst, cat_ox, for oxidation. The potentials of the D^+/0 and A^0/- couples dictate E' for the individual fuel-forming half-reactions and the overall free-energy change, G. G cannot exceed the free-energy content of the excited state above the ground state, G_ET, with G (eV) = -F[E(D^+/0) - E(A^0/-)] G_ET (eV). F is the Faraday constant, 96 485 C/mol of electrons or 1 eV/V in SI units.

(5) Multiple electron transfer and repetition of the light absorption-electron-transfer sequence to give the required number of reductive or oxidative equivalents at cat_ox and cat_red to carry out the half-reactions. For CO₂ reduction to HCOOH, n = 2, and for water oxidation to O₂, n = 4.

(6) Reaction of the activated catalysts, with H₂O, H⁺, CO₂, etc., to give the final energy conversion products returning the catalysts to the cat_ox and cat_red states, ending a catalytic cycle.

II.B. Photoelectrochemistry. Photoelectrochemical Synthesis (PES) Cells. Another approach, utilized in PES cells, is to carry out the half-reactions at the separate electrodes of an electrochemical cell. The potential required to drive the reaction is provided by light rather than by a battery or some other applied energy source.

This simplifies integration. The half-reactions are connected by electron transfer through an external circuit with ion flow between cell compartments to maintain charge neutrality. It does require stable interfacial links to attached molecules and molecular assemblies. These links must also support facile electron transfer to and from electrode or semiconductor surfaces.

In a PES cell, the electrode(s) can function in two ways.

(i) Electrode as an Electroactive Interface. In one approach, the electrode acts only as a conductive interface for transferring electrons between catalyst assemblies at separate electrodes. This is schematically illustrated in Figure 2. In Figure 2, light absorption and electron transfer occur initially to give (surface 1)-A^--C-D-(cat_ox)⁺. This is followed by interfacial and interelectrode electron transfer to give (surface 2)-(cat_red)^-. Repetition would build up multiple redox equivalents and initiate the half-reactions.

Figure 2 Schematic diagram for an integrated, modular PES cell for water splitting illustrating photoinduced electron transfer from catalyst catox to catalyst catred.

As diagrammed in Figure 2, the PES cell for water splitting would also produce a photopotential and photocurrent driven by the potential difference between the surface A^0/- and cat_red^0/- couples. More complex surface structures such as (surface 1)-A-C-C-C-C-C-cat_ox could help meet the requirement for multiple molecular light absorption but only if efficient, random-walk C* C energy transfer occurs to the quencher where electron transfer occurs.^34,48

Appropriate, transparent, high-surface-area electrodes are known for such applications, for example, nanoparticle thin films of doped SnO₂.⁴⁹ There is also an extensive photochemistry of surface-adsorbed, monolayer D-C-A molecular assemblies on Au surfaces.⁵⁰

(ii) Electrode as an Active Modular Component. The electrode itself can be an active element. An example, pioneered by Grätzel and co-workers, is photoinjection into TiO₂ by the excited states of adsorbed chromophores including polypyridyl complexes of Ru^II such as cis-Ru(4,4'-(COOH)₂bpy)₂(NCS)₂.^9,10,51,52 The carboxylic acid groups bind by the formation of surface ester bonds and by H bonding.

In a Grätzel cell, which utilizes a mesoporous, high-surface-area TiO₂ film, excitation and quenching occur by electron transfer to the conduction band of the semiconductor, TiO₂-Ru^II + h TiO₂-Ru^{II* -}TiO₂-Ru^III. Surface Ru^III is subsequently reduced by I^-, 2TiO₂-Ru^III, 3I^- 2TiO₂-Ru^II, I₃^-. The cell is completed by I₃^- reduction at an inert cathode, 2e^-(cathode) + I₃^- cathode + 3I^-. This sequence provides the chemical basis for a photovoltaic device with a maximum photopotential arising from the potential difference between the Fermi level in the conduction band and E' for the I₃^-/I^- couple.

Photoinjection and surface reduction of Ru^III are complete on the nanosecond time scale. Back electron transfer from the bulk of the semiconductor to adsorbed Ru^III is far slower. This and the small transit volumes for conduction-band electrons to the underlying electrode result in efficient collection of the photoproduced electrons.

Possible application of this approach to artificial photosynthesis and the fuel-forming reaction in eq 3 is illustrated in Figure 3, which highlights the electrode as the initial electron acceptor rather than a molecular unit in a molecular assembly.

Figure 3 Schematic diagram illustrating the photoelectrochemical reaction between CO and H2O to give HCOOH and O2 initiated by excitation and photoinjection from an adsorbed chromophore to the conduction band of TiO2.

The modular approach in Figure 3 is general including possible applications in "green chemistry". Reactions such as olefin epoxidation by water, eq 4, or dehydrogenation of alcohols, eq 5, could be accessible by varying the catalyst linked to the phtotoanode.

Such cells would produce a photocurrent at a maximum photopotential equal to the potential difference between the Fermi level of the TiO₂ conduction band and E' for the cat_red^-/0 couple. Photochemical dehydrogenation of 2-propanol in eq 5 has been demonstrated for [(4,4'-(CO₂H)₂bpy)(4,4'-Me₂bpy)Ru_a^II(dpp)Ru_b^II(tpy)(OH₂)]⁴⁺ adsorbed on TiO₂, although with low efficiency (section 7.3, ref 53).

III. Modules. Light Absorption and Excited States

III.A. Introduction. (i) Excited-State Redox Potentials. Quenching and flash photolysis measurements were used in the early 1970s to demonstrate excited-state electron transfer. Early work focused on MLCT excited states of [Ru(bpy)₃]²⁺ (bpy is 2,2'-bipyridine) and its derivatives and the -* excited states of porphyrins.^54-56

[Ru(bpy)₃]²⁺ absorbs light in the visible region with _max = 452 nm in CH₃CN ( = 13 000 M^-1 cm^-1). MLCT excitation, ¹(d⁶) ¹(d⁵*), is followed by rapid ( < 1 ps) relaxation to the lowest triplet state, [Ru(bpy)₃]²⁺*, ³(d⁵*). The lowest "triplet" is split by low symmetry and spin-orbit coupling into three low-lying states separated by ~30 cm^-1. Because of spin-orbit coupling in the d⁵ core, low-lying triplet excited states are mixed with higher-lying singlet excited states. The lifetime of [Ru(bpy)₃]²⁺* is ~1 s, depending on the medium and temperature.^57-59

[Ru(bpy)₃]²⁺* undergoes facile electron transfer. The impact of excited-state formation on redox potentials is illustrated in Scheme 2. These data show that the oxidizing and reducing abilities of [Ru(bpy)₃]²⁺ are enhanced in [Ru(bpy)₃]²⁺* by G_ES = 2.1 eV, the free-energy content of the excited state above the ground state. Ground-state (GS^+/0/GS^0/-) and excited-state redox potentials (ES^+/0/ES^0/-) are related by E(ES^+/0) = E(GS^+/0) - G_ES/F and E(ES^0/-) = E(GS^0/-) + G_ES/F.^54-56

Scheme 2. Excited- and Ground-State Redox Potentials (vs SCE in CH3CN) for [Ru(bpy)3]2+ in CH3CN at 25 Ca a The potentials are related by E(ES+/0) = Eo(GS+/0) - GES/F and E(ES0/-) = E(GS0/-) + GES/F with F the Faraday (1 eV/V).54-56

At pH = 8, [Ru(bpy)₃]²⁺* is thermodynamically capable of oxidizing water (E(pH = 8) = -0.76 V) and reducing CO₂ (E = -0.67 V). The full excited-state energy can be utilized by initial oxidative quenching

(ii) Molecular Assemblies for Directed (Vectorial) Excited-State Electron Transfer. Excited-state electron transfer was demonstrated by flash photolysis experiments in solutions containing [Ru(bpy)₃]²⁺ in the presence of electron-transfer donors, such as 10-methylphenothiazine (10-MePTZ), or acceptors, such as methylviologen (MV²⁺) (Scheme 3).^56,60-64

	Figure 4 Early donor-chromophore-acceptor assembly.74,75
	Scheme 3. Reaction Scheme Illustrating Energy Conversion Based on Excited-State Electron-Transfer Quenching

The reaction in Scheme 3 provided an early conceptual basis for artificial photosynthesis. Visible light was used to drive an endoergic chemical reaction, G = +1.3 eV, in which neither of the reactants absorb visible light.⁶⁴

The next step was to combine these functions in D-C-A assemblies in which oxidative and reductive equivalents are created and separated by free-energy gradients.^{1-3,34,58,59,65-73} An early example based on polypyridyl complexes is shown in Figure 4. Excitation of the Ru(bpy) MLCT chromophore and electron transfer led to (PTZ⁺)Ru^II(MV⁺), with the transiently stored redox equivalents as PTZ⁺ and MV⁺ spatially separated in a single molecule.^74,75

Application of transient laser techniques, first on the nanosecond and then picosecond time scales, with absorption monitoring provided the key evidence for photochemical electron transfer in these assemblies. By using these techniques, it was possible to observe the loss of characteristic absorption features for the excited state and characteristic growth in absorbances for PTZ⁺ at 480 nm and MV⁺ at 605 nm. These measurements also allowed the rate constants for the following MV⁺ PTZ⁺ back-electron-transfer reaction to be measured.

The D-C-A assembly in the photosystem II reaction center from a recent 3.5-Å-resolution crystal structure is shown in Figure 5.²⁴ Excitation of chlorophyll P₆₈₀, Chl_D1 in Figure 5, is followed by electron transfer through pheophytin Pheo_D1 to the quinone acceptor, Q_A. Subsequent reduction of P₆₈₀⁺ occurs by electron transfer from tyrosine Tyr_Z, which is H-bonded to histidine-190 (His-190).

Figure 5 Molecular structure of the reaction center of photosystem II illustrating the TyrZ-ChlD1-PheoD1-QA donor-chromophore-acceptor (D-C-A-A') array. Reprinted with permission from ref 24. Copyright 2004 American Association for the Advancement of Science.

III.B. Light Absorption and Energy-Transfer Sensitization. (i) Absorption. Light absorption or antenna and energy-transfer sensitization initiate the electron-transfer sequences in Figures 1 and 2. The peak intensity in the solar spectrum appears in the visible just below 500 nm, and the spectrum extends well into the near-IR. In a single-junction photovoltaic cell, an ideal sensitizer would convert sunlight to electricity below a threshold wavelength of 920 nm.¹⁰ The energy at this wavelength, 10 870 cm^-1 (1.35 eV), approaches a minimum threshold for the energy required for a useful fuel-forming reaction.

Single-molecule absorption is low, and spectral envelopes are typically relatively narrow.^76-79 In the spectrum of [Ru(bpy)₃]²⁺ in Figure 6, the bands at 452 and ~320 nm arise from ¹(d⁶) ¹(d⁵*¹) MLCT transitions to the first and second *-acceptor levels on bpy. The intense band at 292 nm arises from a bpy-based * transition. Because there are three bpy ligands and three d orbitals, the detailed electronic structure is complex, giving rise to a series of closely spaced, overlapping MLCT bands.^78,80-89

Figure 6 Absorption spectra of [Ru((4,4'-COOEt)2bpy)(dpb)(dtc)}+ (- - -), [Ru(4,4'-COOEt)2bpy)((4,4'-Me)2bpy)(dtc)]+ (···), and [Ru(bpy)3]2+ (-) in CH3CN at room temperature.

The intense, broad MLCT absorption band for [Ru(bpy)₃]²⁺ at _max = 452 nm in CH₃CN has a bandwidth at half-maximum, _1/2, of ~3600 cm^-1. The spectral width of the visible and onset of the near-IR from 400 to 900 nm is 13 900 cm^-1. To span this entire spectral region with high absorptivity requires either multiple absorptions in a single chromophore or an antenna apparatus containing chromophores that absorb in different spectral regions.

An example of the former strategy for MLCT absorbers is illustrated in Figure 6. In this strategy, multiple acceptor polypyridyl ligands with different *-acceptor levels were used to broaden light absorption over an extended spectral range. The dithiocarbamate anion ligand (dtc^-) was added to shift the MLCT bands to lower energy. This moves higher energy d ₂* MLCT bands from the UV into the high-energy visible, further broadening the visible spectrum. The combined effect makes [Ru((4,4'-COOEt)₂bpy)(dpb)(dtc)]⁺ a "black absorber", which absorbs light broadly throughout the visible into the near-IR.^80,90,91

Band Shapes and Intensities. For a Gaussian-shaped band in the classical limit, the energy maximum, E_abs, and bandwidth at half-height, _1/2, are given in eqs 6 and 7 in the limit that h = h' with no frequency changes for the coupled vibrations and solvent.

is the sum of the reorganization energies for the solvent, _o, and intramolecular vibrations treated classically, _i. Assuming the solvent to be a dielectric continuum, with the ground-to-excited-state absorption approximated as creating a dipole in a sphere, gives the expression for _o in eq 8. In this expression, a is the radius of a sphere enclosing the molecule, D_s and D_op are the static and optical dielectric constants of the solvent, and _g and _e are the point dipole vectors of the ground and excited states, respectively.⁹²

The solvent is a major contributor to and the bandwidth. Its contribution increases as the dipole moment change increases, the molecular radius decreases, and the solvent polarity increases.

The solvent dependence of E_abs also includes the solvent-dependent part of G, w(D_s), which varies with the static dielectric constant of the solvent, as shown in eq 9.

[Ru(bpy)₃]²⁺ is of D₃ symmetry in the ground state, and _g = 0. The difference in E_abs between solvents, E_abs, is given in eq 10. Experimentally, E_abs has been found to vary with D_op, as predicted by eq 10 from E_abs = 21 200 cm^-1 in nitrobenzene to 22 200 cm^-1 in H₂O. These results were in agreement with charge transfer to a single ligand, [Ru^II(bpy)₃]²⁺ + h [Ru^III(bpy^-)(bpy)₂]²⁺*, rather than to an orbital delocalized over all three, [Ru^II(bpy)₃]²⁺ + h [Ru^III(bpy^1/3-)₃]²⁺.⁹³

For low-frequency vibrations, which can be treated classically as harmonic oscillators, _i is given by the sum in eq 11. It is over all coupled vibrations j, those for which there is a distortion between the ground and excited states with Q_e 0. Q_e is the distortion, the change in equilibrium displacement of the normal mode between states, f_j is the force constant, S_j is the electron-vibrational coupling constant, or Huang-Rhys factor, and (=h) is the vibrational quantum spacing. According to eq 11, _i increases as S and increase. S is related to Q_e, the angular frequency ( = 2), and the reduced mass, M, as shown in eq 11a.

Typically, MLCT absorption spectra at room temperature consist of a series of relatively featureless, broad absorption bands with the underlying vibronic components obscured; note Figure 6. This is due to overlapping bands and bandwidths comparable to or larger than the vibrational spacings. Vibronic structure can be observed at low temperatures as bandwidths narrow (eq 7).^78,81-87,89 They are included in the band-shape equations as a sum over vibronic components. This is discussed for emission band shapes in section III.C.(i).^76,77,79

For [Ru(bpy)₃]²⁺, there are 7-8 (bpy) ring-stretching vibrations from 1200 to 1600 cm^-1 coupled to MLCT absorption. They are sufficiently close in energy that they can be treated as a single averaged mode at ~1300 cm^-1 even at 77 K.

Figure 7 Derivatized PS, antenna-reaction center polymer illustrating sensitized electron transfer following MLCT excitation at a RuII antenna site.95

With the exception of transition-metal, lanthanide, and actinide complexes, most molecules are spin singlets in the ground state (S₀, S = 0), and absorption bands of high absorptivity appear in the spectra from transitions from S₀ to singlet excited states, S₁, S₂, ..., S_n. Transitions between states of different spin multiplicity, e.g., S T, with T a triplet state, are weak or not observed. Spin-orbit coupling mixes these states, imparting allowedness to the transitions.

This is an important effect for second- and third-row transition-metal complexes. In the visible absorption spectrum of [Ru(bpy)₃]²⁺, a S₀ T absorption is observed on the low-energy side of the S₀ S₁ absorption at 452 nm, which is lower in absorptivity by a factor of ~10 (Figure 6). These low-energy absorptions are far more pronounced for [Os(bpy)₃]²⁺ because of the larger spin-orbit coupling constant, (Os^III) ~ 3000 cm^-1 compared to ~1000 cm^-1 for Ru(III).^{57-59,78,82,84-89} Spin-orbit coupling mixes states of different spin multiplicities.

(ii) Antenna Effect. Energy Transfer. Absorbing a significant fraction of the solar irradiance (moles of incident photons per unit time) requires multilayers of individual chromophores or antenna sensitization. Antenna arrays can both broaden the region of high light absorptivity to higher energy and increase the effective molecular absorptivity. Following excitation of the antenna, the excited-state energy is transferred to a low-energy "reaction center" where, as shown schematically in Figure 1, electron transfer occurs. Energy in excess of G_ES for the reaction center is dissipated as heat. As for electron transfer, energy transfer is driven by favorable free-energy gradients.

Elaborate antenna arrays are utilized in natural photosynthesis. In photosystem II, there are six chlorophyll a's and one or two -carotenes. In photosystem I, there is an integral antenna system consisting of ~90 Chl molecules and 22 carotenoids in a precisely laid out geometry.^24,26 Application of ultrafast spectroscopies and a recent theoretical treatment by Fleming and co-workers have given detailed insight into antenna dynamics and the electronic and vibrational couplings that lead to rapid, efficient energy transfer to the reaction center.^35-37,95

Antenna arrays have been constructed from chemically linked porphyrins, organic chromophores, and dendrimers.^2,94,96-98 A derivatized polystyrene (PS) example is shown in Figure 7,⁹⁷ in which excitation of a Ru(bpy) "antenna" site is followed by facile energy transfer to a (PTZ)Ru^II(MV²⁺) "reaction center" mimic. Energy transfer causes chromophore sensitization and electron transfer, -Ru^II*(PTZ)Ru^II(MV²⁺)- -Ru^II(PTZ)Ru^II*(MV²⁺)- -Ru^II(PTZ⁺)Ru^II(MV⁺), to give a 1.15-eV redox-separated state based on PTZ⁺ and MV⁺.⁹⁷

Successful light harvesting and application of the antenna effect rely on efficient light collection, energy transfer, and sensitized electron transfer. All must be rapid compared to the lifetime of the antenna molecular excited state(s).

The principles of energy transfer are well understood.^99-103 Application of time-dependent perturbation theory gives the expression in eq 12 for the energy-transfer rate constant, k_en. In eq 12, H is the operator that mixes the electronic wave function for the energy-transfer acceptor (_el') and donor (_el). The associated spin wave functions are _s' and _s. The total vibrational wave functions for the two states including solvent are _vib' and _vib.

There are two mechanisms for energy transfer. In the Förster mechanism, energy transfer is induced by an electrostatic dipole interaction between the electrons in the initial and final states. V_en² varies as 1/R⁶ with R the internuclear separation distance between donor and acceptor.

In Dexter transfer, an electrostatic perturbation mixes the electronic wave functions and V_en² varies as exp[-(R/L)]. L is the sum of the average Bohr radii for the donor and acceptor. The 1/R⁶ dependence for Förster transfer, compared to the exponential dependence for Dexter transfer, favors Förster transfer at long distances. Experimentally, V_en values of 1-25 cm^-1 have been reported for Dexter transfer.^48,104-106

In both Förster and Dexter transfer, the electronic wave functions include _s and _s' . The operator inducing energy transfer does not include spin, and in the absence of spin-orbit coupling, energy transfer can only occur between states of the same spin multiplicity. For example, <_s'_s> = 1 for S S' transfer and <_s'_s> = 0 for S T transfer. Spin-orbit coupling mixes the spin character of the states, making spin changes possible but still greatly decreasing the magnitude of V.

The vibrational overlap term in eq 13, F_calc, can be evaluated from the emission spectrum of the donor and absorption spectrum of the acceptor by use of the spectral overlap integrals in eq 13. In eq 13, f_d(E) is the normalized emission spectrum of the donor and F_a(E) is the normalized absorption spectrum of the acceptor. The integration is over the energy ranges of the absorption and emission spectra.¹⁰⁷

As for nonradiative decay and electron transfer, in sections III.C.(ii) and IV, it is also possible to express F_calc and k_en in terms of specific contributions from coupled vibrations and the solvent at the donor and acceptor and the free-energy change, G. These parameters can be obtained by analyzing emission band shapes or Raman excitation profiles.^83,108

Antenna energy transfer has been demonstrated and treated quantitatively in a series of derivatized PS polymers and, as shown in Figure 7, used to sensitize electron transfer at an artificial reaction center.^{34,48,94,97,109} In the PS polymers, the molecular volumes of the complexes are large relative to the repeat unit of the polymer (note Figure 10 below), which ensures an extended, rodlike structure and one-dimensional, intrastrand energy migration.

The energy-transfer experiments were conducted on mixed polymers containing both Ru^II(bpy) "antenna" sites, which were in the majority, and minority Os^II(bpy) trap sites. The energy of the Os^II acceptor is 1.70 eV, with Ru^II* Os^II energy transfer favored by 0.40 eV. Antenna sensitization of Os^II* following excitation at Ru^II can be highly efficient, >90%, depending on the chemical links from the metal complexes to the polymer backbone.³⁴

Analysis of the emission lifetime data revealed a complex energy-transfer mechanism in the polymers based on (1) Ru^II* Ru^II energy migration by site-to-site hopping (k_migration in Scheme 4), (2) long-range, random-walk energy migration along the polymer strands, and (3) Ru^II* Os^II energy transfer. The antenna effect is efficient because energy migration is rapid (nanosecond time scale) compared to the excited-state lifetime of ~1 s.¹⁰⁹

Scheme 4. Kinetic Pathways Leading to Intrastrand Excited-State Energy Transfera a The two distinct pathways shown are energy migration (RuII* RuII) and energy transfer (RuII* OsII). Randon walk is the sum of migration pathways both toward and away from the OsII trap site.

Cross-surface, two-dimensional antenna sensitization has been demonstrated on ZrO₂ nanoparticles containing coadsorbed [Os^II(bpy)₂(4,4'-(COOH)₂bpy)](PF₆)₂ and [Ru^II(bpy)₂(4,4'-(PO(OH)₂)₂bpy)](Br)₂. The average rate constant for cross-surface energy transfer, <k_en>, was found to vary exponentially with distance, consistent with Dexter transfer with an average, close-contact rate constant of <k_en> = 2.7 × 10⁷ s^-1. Energy transfer occurs by long-range, cross-surface, random-walk energy migration at low surface coverages and by percolation chain transfer at high coverages.¹¹⁰

III.C. Excited States. From the Franck-Condon principle, light absorption is rapid on the time scale for nuclear motions. This means that an excited state formed by excitation is initially in the equilibrium coordinates and solvent configuration of the ground state. Subsequent relaxation to the equilibrium coordinates and configuration of the excited state occurs in tens of femtoseconds for the coupled vibrations and in picoseconds for the solvent.^111-115

As noted above, the [Ru(bpy)₃]²⁺ absorption spectrum is dominated by multiple MLCT and * absorptions that occur from the singlet ground state, (d⁶) S₀, to excited states of increasing energy, S₀ S₁, S₂, ..., S_n.^81,82 Subsequent relaxation occurs on the subpicosecond time scale to and through the lowest singlet state, S₁, to the lowest triplet (³(d⁵*¹) T₁), which is largely responsible for the observed excited-state properties.^111,112

As noted above, T₁ is split by low symmetry and spin-orbit coupling into three closely spaced states separated by 30 cm^-1. They behave kinetically as a single state at room temperature. Transient IR and transient resonance Raman measurements have shown that the excited electron is localized on a single bpy ligand, [Ru^III(bpy^-)(bpy)₂]²⁺*, rather than delocalized over all three, [Ru^III(bpy^1/3-)₃]²⁺*, in the thermally equilibrated excited state.^116,117

The results of transient polarization measurements are consistent with the excited electron hopping among ligands (or rather combined electron-electron hole transfer leading to rotation of the excited-state dipole)³⁴ within a few picoseconds.^118,119 In the lowest triplet state ³(d⁵*¹), the hole in the d⁵ core resides in the d orbital lying along the reduced bpy ligand. Rapid rotation of the excited electron between ligands is important in assemblies such as the one shown in Figure 4 in order to ensure that the excited electron is available at all three ligands regardless of which is initially involved in the excitation.

(i) Radiative Decay. Excited states decay by a combination of radiative (emissive) and nonradiative decay or undergo chemical change. The rate constant for radiative decay, k_r, varies with the transition moment, M, and the inverse cube of the average emission energy, <^-3>^-1, as shown in eq 14. The transition moment is related to the transition dipole, , by M = e. For transitions between states of common spin parentage, e.g., S₁ S₀, is relatively large and k_r rapid because it scales with ². Radiative decay typically occurs on the nanosecond time scale for organic * singlets, consistent with the high absorptivities of the intense S₀ S₁ absorption bands that appear in their spectra.

Radiative decay is far slower for excited-state triplets because of the spin change involved in a T₁ S₀ transition. The latter can occur on the seconds time scale. Radiative lifetimes for transition-metal complex excited states undergoing spin changes can be far shorter because spin-orbit coupling mixes singlet character into excited states of higher spin multiplicity. For example, k_r = 3 × 10⁴ s^-1 for the T₁ S₀ radiative transition in [Os(bpy)₃]²⁺*.^83,120

Radiative decay results in emission. The relationship between absorption and emission is illustrated in Figure 8, assuming that they interrelate the same two states. In the case of [Ru(bpy)₃]²⁺, these are S₀ and T₁. In Figure 8, E_em and E_abs are the energies of the emission and absorption maxima and the reorganization energy as defined previously. Gaussian band shapes are assumed in the classical limit with h_vib = h_vib' for the coupled vibrations and solvent.

Figure 8 Schematic energy-coordinate diagram in the average mode approximation illustrating the relationship between absorption and emission. See the text.

Figure 8 shows that the red shifts observed for emission compared to absorption are a natural consequence of the Franck-Condon principle. The change in the electronic configuration between ground and excited states causes changes both in the intramolecular structure and in the surrounding medium. Light absorption occurs rapidly on the time scale for nuclear motions and gives the excited state initially in the nuclear configurations of the ground state. Similarly, emission from the thermally equilibrated excited state occurs to nonequilibrium nuclear configurations in the ground state.

In the classical limit with h_vib = h_vib', E_em is related to , G, and E_abs as shown in eq 15. In eq 15, the spectroscopic quantities E_em and E_abs are related to the free-energy difference between states, G, rather than to the internal energy difference, E. This has been shown to be the case based on both a classical free-energy surface analysis¹²¹ and a quantum mechanical, harmonic oscillator analysis in the limit of small frequency changes.¹²²

Because emission typically occurs from a single state, analysis of emission spectral profiles is far simpler than for absorption, where there are usually complications from overlapping bands; note Figure 6. At temperatures below ambient, vibronic structure appears in the emission spectrum of [Ru(bpy)₃]²⁺*, arising from 7-8 coupled (bpy) ring stretching vibrations from 1200 to 1600 cm^-1. They appear as a single, unresolved progression at ~1300 cm^-1 at 77 K,^{57,83,117,120} with further resolution into the separate components at lower temperatures.^84-87

Emission Spectral Fitting. MLCT emission spectra have been fit by mode averaging and straightforward line-shape equations^{76,77,83,92,120,122} and the excited states investigated by transient resonance Raman and infrared spectroscopies.^117,123-127 In the average-mode approximation, the coupled (bpy) vibrations are treated as an average harmonic oscillator, with S defined as the sum in eq 16a and h the weighted average in eq 16b. The summations are over the coupled vibrations within a limited range of frequencies, as noted above from ~1200-1600 cm^-1 for the (bpy) vibrations.

The spectra are constructed from a series of vibronic lines separated by vibrational spacing, = h, with the angular frequency = 2. Relative heights are a function of the electron-vibrational coupling constant S and vary as S^v/v!. The bandwidth of each vibronic component includes both the solvent reorganization energy, _o (eq 8), and the reorganization energy from low-frequency modes treated classically, _i,L (eq 11a); note eq 17b below.

Analysis of emission spectral profiles gives (1) E_o, the energy of the v = 0 v' = 0 vibronic band, the energy gap, (2) _1/2, the bandwidth at half-maximum for each vibronic band, and (3) S and . These parameters characterize the excited state: (1) S is a measure of Q_e and the extent of excited-state distortion in the acceptor ligand.^122,128 (2) E_o is related to G_ES, as shown in eq 17a. (3) _1/2 gives _o,L (eqs 7 and 17b). (4) The relative rate constants for nonradiative decay can be calculated from E_o, S, and _o,L; see below. From an emission spectral fitting of [Ru(bpy)₃]²⁺*, S = 1.0 with = 1300 cm^-1.^83,129

(ii) Nonradiative Decay. In nonradiative decay, the transition between excited and ground states is induced by "promoting modes", which have the symmetry properties required to mix the electronic wave functions of the initial and final states. The excess energy when nonradiative decay occurs is channeled into "acceptor" modes and the solvent. The acceptor modes are typically high- or medium-frequency vibrations for which Q_e 0.

Nonradiative decay occurs through a series of channels from a discrete set of vibrational levels, v, in the excited state to final levels v' in the ground state. Expressions for k_nr have been derived from time-dependent perturbation theory.^120,130-136 In the limiting case of a single coupled vibration, or average vibration (eq 16), the contribution to k_nr from the v = 0 v' channel, k_nr(v = 0 v'), is given by eq 18, with G = G_ES. This expression assumes that k_BT (k_BT = 200 cm^-1 at 25 C) so that only the v = 0 level in the excited state is appreciably populated and contributes to nonradiative decay. The term _o,L is the sum of the solvent reorganization energy and coupled low-frequency vibrations treated classically (eq 17b).

k_nr(v = 0 v') is the product of three terms:

(1) The preexponential term contains both the barrier crossing frequency, 2V_k²/, and the classical density of states in the coupled solvent and low-frequency vibrational modes, (4RT_o,L)^-1/2. V_k is the vibrationally induced electronic coupling matrix element for the promoting mode(s). V_k is defined in eq 19 for a single mode of reduced mass M_k, quantum spacing _k, and normal coordinate Q_k.

(2) The square of the vibrational overlap integral with <_vib'(v')_vib(v=0)>² = (S^v'/v'!) exp(-S') for a harmonic oscillator. It gives the extent to which the vibrational coordinates in the initial and final vibrational levels are coincident along normal coordinate Q. Vibrational overlap is illustrated in Figure 10 for electron transfer in the inverted region. The transition between _vib and _vib' is referred to as nuclear tunneling in the physics literature. It is a quantum effect arising from the probabilistic uncertainty in spatial coordinates for particles at the quantum level.

(3) The classical barrier crossing, exp{-[(G(v,v') - _o,L)²/4_o,LRT]}, includes contributions from all other coupled vibrations treated classically and the solvent. It gives the fraction of molecules at temperature T at the barrier crossing in each of the coupled low-frequency vibrations and the solvent. It ensures that the v v' channel occurs with energy conservation.

As shown in eq 20, k_nr is the sum over all channels, k_nr = _v'k_nr(v)'. This result can be generalized to include any number of coupled vibrations, levels above v = 0, and '.^{83,92,120,130-136}

In the limits E_o (=G - _o,L) S and k_BT, k_nr is given by the famous "energy gap law" in eq 21.^130-136 This result illustrates the central role of the energy gap in nonradiative decay.¹²⁰

All of the parameters in eq 21 except C_k are available from emission spectral fitting. This has allowed quantitative application of the energy gap law to nonradiative decay for Ru(II), Os(II), and Re(I) MLCT excited states.^{57,83,92,120,127,137}

A more inclusive treatment with S and values included for all coupled vibrations on a mode-by-mode basis acquired by the analysis of Raman excitation profiles has also been applied to MLCT excited states.^83,138-141

(iii) Excited-State Lifetimes and Reactivity. In the absence of chemical reactions, excited states decay by a combination of radiative and nonradiative decay with lifetimes, , related to k_r and k_nr as in eq 22a. The emission quantum yield in eq 22b is defined as _em = k_r/(k_r + k_nr).

Many MLCT excited-state properties are predictable based on this result and those in previous sections. The energy gap is a key parameter. It defines emission and low-energy absorption maxima and excited-state redox potentials. It also helps to determine , with k_r varying as E_o^-3 (eq 14) and k_nr as exp(-E_o) (eq 20).¹⁴²

Variations in the surrounding ligands have been used to vary E_o and systematically to demonstrate the validity of the energy gap law.^{83,92,120,129,137,142} There are other ways to manipulate . As predicted by eq 21, k_nr should decrease and increase if S is decreased. This can be accomplished by enhanced electronic delocalization of the excited electron in the acceptor ligand, which has the effect of decreasing structural changes. Enhanced rigidity has been shown to have the same effect.^91,142-144

Manipulation of the energy gap has led to the synthesis of "designer excited states" in which excited-state properties are manipulated systematically by varying the ligands.^{57,78,80,120,137,142-144} This includes the series [Os(bpy)(L)₄]ⁿ⁺ and [Os(phen)(L)₄]ⁿ⁺. Variations in the nonchromophoric ligands L in this series were used to vary MLCT excited-state lifetimes from 0.041 to 4.3 s and emission quantum yields from 0.003 to 0.44 in CH₃CN at 23 ± 2 C as E_o varied from 13 080 to 19 790 cm^-1.^120,142

In these Os complexes, metal-ligand mixing with electron-donating ligands such as Cl^- decreases E_o by increasing the energy of d. Ligands such as CO have the opposite effect in stabilizing d by d-*(CO) back-bonding.

In the series [Re^I(4,4'-X₂bpy)(CO)₃Cl], variations in X from electron donor (e.g., OMe) to electron acceptor (e.g., OOCEt) were used to vary the energy of the *(4,4'-X₂-bpy) acceptor levels, with E_o varying from 14 050 to 18 100 cm^-1. This provides a second way to vary the energy gap systematically.¹³⁷

MLCT emission energies and lifetimes are solvent-dependent because of the influence of the solvent on E_o and _o (eqs 10 and 15). Emission energies for [Ru(bpy)₃]²⁺* vary from 606 nm in dichloromethane to 630 nm in N,N-dimethylformamide at 25 C, consistent with the dielectric continuum prediction in eq 10.^93,145

MLCT lifetimes are weakly temperature-dependent because of the temperature dependence of E_o, which influences both k_r and k_nr. The temperature dependence arises from entropic differences between the excited and ground states.¹⁴⁶ Additional temperature effects, which arise from thermal population and decay through higher lying excited states, appear; see eq 23 below.^147,148

Reactivity. In the new electronic configuration of an excited state, lowered barriers can open new pathways for reaction including decomposition.

(a) Ligand Loss from dd Excited States. Low-lying dd [metal-centered (MC)] excited states complicate the excited-state properties of ruthenium(II) polypyridyl complexes by contributing to nonradiative decay and causing decomposition. In these d⁵d*¹ excited states, a d* metal-ligand antibonding orbital (e_g in O_h symmetry) is occupied, which leads to metal-ligand bond breaking.^149-162

The dd states are accessed by thermal activation and barrier crossing from MLCT states after they are formed by excitation. As shown by the k₂ term in eq 23, this introduces a temperature dependence into . There is an additional contribution to the temperature dependence arising from thermal population and decay through a fourth MLCT state, the k₁ term in eq 23. Nonradiative decay through this upper MLCT state is rapid because its enhanced singlet character causes greater mixing with the ground state.^147,148

Population of dd states can lead to efficient ligand loss. For the [Ru(bpy)₃]²⁺ analogue cis-[Ru(bpy)₂(py)₂]²⁺ (py is pyridine), MLCT excitation leads to loss of a py ligand with quantum yields (the ratio of chemical events to total molecules excited) as high as 0.3. In nonpolar solvents with added anions such as NCS^-, [Ru(bpy)₃]²⁺ is also photochemically unstable with a bpy ligand being replaced by ion-paired NCS^-, for example.¹⁶³

Ligand-loss photochemistry is inhibited in rigid media^148,164-173 and at low temperatures where thermal population and decay through the dd states is unimportant (eq 23). Ligand-loss photochemistry can also be avoided by stabilizing the lowest MLCT state, as in [Ru((4,4'-COOEt)₂bpy)(dpb)(dtc)]⁺ in Figure 6, so that the energy gap for the k₂ term in eq 23 is large.^{80,91,152,164} It can be avoided in molecular assemblies for artificial photosynthesis by ensuring that the time scale for energy or electron transfer is far shorter than the time scale for thermally activated dd MLCT barrier crossing. To put this into perspective, ₂^-1 = k₂ exp[-(E₂/RT)] from eq 23, with ₂ ~ 0.1-0.3 s for [Ru(bpy)₃]²⁺* at room temperature in solution.

For most Os(II) and Re(I) complexes, dd states are not a complicating feature at least at room temperature. The splitting between the d and d* orbitals (10 Dq) is greater [~30% greater for Os(II)], and dd states are not significantly populated. By contrast, in related complexes of Fe(II), dd states are lowest lying or easily accessible thermally. This greatly limits their usefulness in electron- or energy-transfer applications.

(b) Other Decomposition Pathways. The chromophore [Ru(4,4'-(COOH)₂bpy)₂(NCS)₂] has been one of the dyes of choice in photovoltaic applications involving Grätzel-type cells and adsorption on TiO₂. This is true even though photoinjection results in the Ru(III) form of the adsorbed complex on the surface and it is unstable with respect to intramolecular NCS^- Ru^III electron transfer and release of (NCS)₂.^10,174,175

Even with this instability, this adsorbed dye is stable through 10⁸ redox cycles, which corresponds to 20 years of operation in natural sunlight.¹⁰ In this case, stability is a consequence of the difference in time scales for the decomposition reaction (seconds) and re-reduction of surface-bound Ru(III) by I^- (nanoseconds) [section II.B(ii)].

IV. The Modules. Photochemical Electron Transfer

IV.A. Intramolecular Electron Transfer. The initiating event for the energy conversion reactions in Figures 1 and 2 is intra-assembly electron transfer. In the classical limit, the rate constant k_ET is given by eq 24.^{92,100-103,176-182}

As before, is the sum of intramolecular, _i, and solvent, _o, reorganization energies. In this case, they pertain to electron transfer, e.g., [Ru(bpy)₃]²⁺*, MV²⁺, [Ru(bpy)₃]³⁺, MV⁺, and are different from _i and _o for nonradiative decay of [Ru(bpy)₃]²⁺*, for example. The free-energy dependence of k_ET in eq 24 has been tested for bimolecular electron-transfer quenching of [Ru(bpy)₃]²⁺*. Analysis of these data also gave an estimate for the excited-state redox potentials in Scheme 2.^56,183-186 In the limit of weak electronic coupling between the electron-transfer donor and acceptor, the nonadiabatic limit, the barrier crossing frequency, _ET, is the electron tunneling frequency, _e. It is related to and the electron-transfer matrix element, H_DA, as shown in eq 25. H_DA is the resonance energy arising from orbital mixing between the donor and acceptor orbitals.

Even with H_DA only tens of cm^-1 in magnitude, electron tunneling is rapid on the time scale for nuclear motions. This is the adiabatic limit, in which the barrier crossing is dictated by the frequency or frequencies of the slowest coupled nuclear motion or motions, _n. These are typically collective dipole reorientation modes in the solvent or translation of ion-paired counterions. In this limit, _ET is given by eq 26, with _n the frequency of the slowest mode or an average frequency if there is more than one contributor. In the intermediate dynamic region, _ET is given by the kinetic average in eq 27.

If there is a coupled high-frequency vibration or averaged vibration with k_BT and assuming = ', k_ET is given by eq 28. As before, _o,L includes both _o and low-frequency modes treated classically, _i,L (eq 11) but for electron transfer. The result in eq 28 can be fully generalized to include multiple vibrations, frequency changes, etc.^{100-103,179,187-196}

If the solvent is treated as a dielectric continuum and electron transfer as occurring between spheres of radii a₁ and a₂ at an internuclear separation distance d, _o is given by eq 29. As before, D_s and D_op are the static and optical dielectric constants of the solvent.^{92,103,114,176-178,187,197-201} More sophisticated treatments are available by separating _o into components arising from orientational fluctuations of solvent dipoles and density fluctuations^202-204 and by use of frequency-resolved cavity models.^205-207

Based on eq 29, the magnitude of _o and its contribution to the electron-transfer barrier decrease as (1) the radii of the redox sites (a₁ and a₂) increase, (2) the electron-transfer distance (d) decreases, and (3) the static dielectric constant (D_s) decreases and the optical dielectric constant (D_op) increases. Large reactants, small electron-transfer distances, and nonpolar solvents all increase k_ET.

The coupled vibrations and the extent to which they create a barrier through the (S^v/v!) exp(-S) and _i,L terms depend on the reaction. In reductive electron-transfer quenching of [Ru(bpy)₃]²⁺*, [Ru^III(bpy^-)(bpy)₂]²⁺*, D, [Ru(bpy)₃]⁺, D⁺, an electron is added to a d orbital, d⁵*¹ + e^- d⁶*¹. Structural changes occur in the Ru-N bonding framework, which couples low-frequency (Ru-N) vibrations to electron transfer. In oxidative electron-transfer quenching, [Ru(bpy)₃]²⁺*, A, [Ru(bpy)₃]³⁺, A^-, an electron is lost from a ligand-based *(bpy) level and (bpy) vibrations are coupled much as in nonradiative decay.

Both the classical, eq 24, and quantum results, eq 28, show that the vibrational contributions to the barrier decrease as h and Q_e (and S) decrease. Q_e is minimized for couples such as d⁵/d⁶ [Ru(bpy)₃]^3+/2+ or *⁰/ *¹ MV^2+/+, where electron transfer involves orbitals that are not involved in the -bonding skeleton of the molecule.

IV.B. The Inverted Region. Both the classical and quantum results in eqs 24 and 28 predict that the electron-transfer barrier decreases as -G increases. This is only true until -G = , at which point classical electron transfer becomes barrierless. As illustrated in Figure 9, the barrier then increases as -G is increased further. Figure 9 illustrates the variation in the electron-transfer barrier in the classical limit at fixed according to eq 24 for three cases: (A) in the normal region with -G < , (B) at the barrierless point, where -G = and k_ET = _ET, and (C) in the inverted region, where -G > .

Figure 9 Free energy-coordinate curves illustrating the influence on the classical barrier to electron transfer of increasing -G at fixed reorganization energy, , according to eq 24. The three cases illustrated are for electron transfer in (A) the normal region with -G < , (B) the barrierless point where -G = , and (C) the inverted region with -G > .

The inverted region was initially predicted by Marcus,^201,208 and the decrease in k_ET with -G has been observed experimentally many times.^92,209-226 It provides the basis for understanding such phenomena as chemiluminescence and electrochemiluminescence, in which energetic electron transfer gives excited-state products, e.g., [Ru(bpy)₃]³⁺ + C₂O₄^- [Ru(bpy)₃]²⁺* + 2CO₂.^211,227-230

Figure 10 Schematic energy-coordinate (E-Q) diagram illustrating electron transfer in the inverted region (A) by a vibrational channel below the intersection region and vibrational wire function overlap between the initial and final vibrational levels (see the text) and (B) by classical barrier crossing.

In the normal region, electron transfer occurs by thermal activation and barrier crossing. In the inverted region, excess energy is released in the electron-transfer act. As in nonradiative decay with a coupled high-frequency vibration or vibrations, electron transfer occurs through a series of vibrational channels rather than by a classical barrier crossing. Both quantum and classical barrier crossings are illustrated in Figure 10, with the former illustrating vibrational wave function overlap between the initial and final vibrational wave functions.

In the limits E_o S, = ', and k_BT, the general electron-transfer result in eq 28 takes the energy gap law form in eq 30. It is different from the expression for k_nr in eq 20 because it contains H_DA rather than V_k and the reorganization energy, _o,L, is for electron transfer. The energy gap E_o is related to G by E_o = G - _o,L.^{120,130,188,231}

This is a remarkable and important result both for natural and artificial photosynthesis. It predicts that following electron-transfer quenching, e.g., D-C*-A D-C⁺-A^-, the rate constant for back electron transfer in the inverted region, D-C⁺-A^- D-C-A, becomes slower as the energy stored increases. Forward electron transfer, D-C⁺-A^- D⁺-C-A^-, is favored at the expense of wasteful back electron transfer, D-C⁺-A^- D-C-A, which dissipates stored energy as heat.

There are important differences between electron transfer in the normal and inverted regions: (1) In the inverted region, the initial energy-coordinate curve is imbedded in the final curve (Figure 10) and emission can compete with electron transfer.^131,232-239 (2) In the normal region, _o contributes to the barrier. In the inverted region, with a coupled high-frequency vibration or vibrations, k_ET increases as _o increases and the solvent's role in energy dissipation is enhanced.²⁴⁰ (3) In the normal region, electron transfer involves a reordering of the electronic distribution within a single state. In the inverted region, a transition occurs between different, weakly coupled states.

Although they are similar, inverted electron transfer and nonradiative decay are fundamentally different processes. In nonradiative decay, donor-acceptor electronic coupling is strong and the operator in H_DA is included in the Hamiltonian defining the two states. They are orthogonal to zero order and mixed by promoting modes [section III.C(ii)]. In inverted electron transfer, electronic coupling mixes the states but is relatively weak, with H_DA < .

High-energy electron-transfer products formed after excited-state quenching, e.g., [Ru(bpy)₃]²⁺*, MV²⁺ [Ru(bpy)₃]³⁺, MV⁺, for which -G > , are called "charge-separated states" or "redox-separated states" (RSSs) to distinguish them from true excited states. In these states, nonradiative decay occurs by electron transfer.

IV.C. Delocalization. Intra-assembly electron transfer in the symbolic molecular assemblies in Figures 1 and 2 occurs through covalent bonds that link the excited state and electron-transfer donors and acceptors. This opens the possibility of significant electronic coupling and delocalization of the transferring electron by orbital mixing. Delocalization decreases the electron-transfer barrier by decreasing structural differences decreasing both Q_e and _i. It also decreases the effective electron-transfer distance, which decreases _o.^241-246 In the limit of strong electronic coupling, the excess electron is delocalized over both redox sites and there is no structural barrier to electron transfer. Based on a semiclassical analysis by Hush, the localized to delocalized transition occurs when 2H_DA . This transition can be induced by changes in the ligands and metal so as to enhance electronic coupling. For example, in [(NH₃)₅Ru^III(4,4'-bipyridine)Ru^II(NH₃)₅]⁵⁺, electronic coupling is weak, 100-200 cm^-1, 2H_DA , and there is a barrier to electron transfer. In [(NH₃)₅Os^II.5 (pz)Os^II.5(NH₃)₅]⁵⁺ (pz is pyrazine), greater d(Os) mixing with the bridging ligand enhances electronic coupling across the bridge and 2H_DA > .²⁴¹ The effect of electronic coupling on the energy of activation, E_a, in the classical limit is given by eq 32.^{176,198,241,247,248}

As H_DA approaches in magnitude, the Born-Oppenheimer separation of electron and nuclear coordinates, which is assumed in the derivation of eqs 24 and 28, is no longer valid. In their theoretical treatment of this problem, Piepho, Krausz, and Schatz (PKS) define electronic wave functions that are independent of nuclear coordinates and then solve the Schrödinger equation, with the nuclear kinetic energy operator included to give mixed electron-vibrational (vibronic) wave functions, energies, and barriers to electron transfer.²⁴⁹ The wave functions are expanded in the complete orthonormal set of harmonic oscillator wave functions to give a general solution for the final mixed-coordinate wave functions. The key elements of the PKS model also appear in earlier work including an analysis of electron-phonon coupling in solids by application of linear response theory.^250-253

IV.E. Kinetic Analysis. The energy conversion scheme in Figure 1 relies on a series of sequential reactions that are kinetically coupled and in competition with others that lead to energy dissipation. The efficiency of formation of the final Redox Separated State (RSS), D⁺-C-A^-, assuming -G > , depends on the relative magnitudes of the rate constants for a series of competing reactions.

Formation of the RSS. In Scheme 5 is summarized the series of competing reactions that occur upon excitation of a D-C-A array following direct or sensitized excitation. The example used is based on the oligoproline array shown in Figure 12.^34,254,255

Scheme 5. Kinetic Scheme Illustrating the Competing Processes That Occur Following MLCT Excitation of the Anq-RuII(b2m)2+-PTA D-C-A Assembly Shown in Figure 12a,254-255 a The abbreviations Anq and PTZ are used for the Anq and PTZ derivatives shown in Figure 12.

Electron-transfer quenching either reductively, k_c in Scheme 5, or oxidatively, k_f, is in competition with excited-state decay, ^-1. The quenching efficiency is given by ₁ = (k_c + k_f)/(k_c + k_f + ^-1). On the basis of the classical result in eq 24, k_c and k_f are dependent on G and reach maximum values when -G = and the reactions are barrierless.

To put this into perspective, the lifetime of [Ru(bpy)₃]²⁺* is ~1 s, depending somewhat on the medium. To achieve ₁ 0.99 for electron-transfer quenching requires that k_c + k_f 10⁸ s^-1. With k_ET = 10¹³ exp{-[(G + )²/4RT]} and = 1 eV, this further requires that G = -0.28 eV (-6.5 kcal/mol). The maximum rate constant, with k_c + k_f = 10¹³ s^-1, would be reached at G = -1 eV.

The desire to minimize the electron-transfer barrier by manipulating G is counterbalanced by the need to maximize the energy available for energy conversion. In the example above, maximizing the quenching rate constant comes at the expense of 1 eV of stored free energy, which could be applied to the fuel-forming reaction.

For the Anq-Ru^II(bpy)-PTZ assembly, reductive quenching occurs in the first step to give Anq-Ru^II(bpy^-)-PTZ⁺. Subsequent bpy^- Anq electron transfer, k_e in Scheme 5, to give Anq^--Ru^II(bpy)-PTZ⁺ is in competition with (bpy^-) PTZ⁺ back electron transfer to give the ground state k_d. The latter occurs in the inverted region, with k_d decreasing as the G content of Anq-Ru^II(bpy^-)-PTZ⁺ increases. The efficiency of the second electron-transfer step is ₂ = k_e/(k_e + k_d). The total efficiency for the formation of the RSS in this case, _RSS, is _RSS = ₁₂.

For the Anq-Ru^II(bpy)-PTZ oligoproline assembly, the efficiency of the formation of Anq^--Ru^II(bpy)-PTZ⁺ varies from 33 to 86% and the stored redox energy from 1.46 to 1.71 eV depending on the solvent.²⁵⁴

Catalyst Activation by Electron Transfer. In Figure 1, the redox equivalents transiently stored in -D⁺-C-A^-- undergo further electron transfer, from A^- to cat_red (k_red) and from cat_ox to D⁺ (k_ox). Electron-transfer activation of the catalysts is in competition with back electron transfer between A^- and D⁺ (k_bet), and the efficiency of catalyst activation, _act, is given by _act = (k_red + k_ox)/ (k_red + k_ox + k_bet). Back electron transfer can occur by long-range electron transfer, -D⁺-C-A^-- -D-C-A-, or stepwise by reversal of the electron-transfer chains, e.g., -D⁺-C-A^-- -D-C⁺-A^-- -D-C-A-.

In a simplified analysis, the efficiency of 1e^- activation of the catalysts, cat_ox-D⁺-C-A^--cat_red (cat_ox)⁺-D-C-A-(cat_red)^-, _ET,1, is given by _ET,1 = _RSS(k_red + k_ox)/(k_red + k_ox + k_bet) = _RSSact, with _act = (k_red + k_ox)/(k_red + k_ox + k_bet).

Assuming that catalyst activation is a 2e^- process, there is a final competition between a second light-induced electron transfer to give (cat_red)^2- and (cat_ox)²⁺, the catalyzed reactions, k_cat,red and k_cat,ox, and (cat_red)^- (cat_ox)⁺ back electron transfer. Even with well-separated catalysts, back-electron-transfer "leakage" through the assembly can have a deleterious effect on the fuel-forming reaction.

More complex catalytic activation schemes, such as the 4e^- oxidation of water, add additional competitive steps that can contribute deleteriously to the overall efficiency.

Based on an analysis by Sergei Lymar of Brookhaven National Laboratory, the minimum rate requirement for efficient activation and reactivity of the catalysts can be estimated based on the AM 1.5 solar irradiance spectrum, with AM 1.5 the accepted spectral standard for sunlight at an angle of incidence of 45. For the solar spectrum at < 600 nm, the incident irradiance is ~30 mW cm^-2 or approximately 1.3 × 10^-7 einsteins cm^-2 s^-1 (assuming 500-nm photons). With a 100% redox separation efficiency and a catalyst coverage density of 1 nm^-2, a system turnover rate, including the catalyst reactions, of ~10² s^-1 is needed to ensure that solar irradiance is rate limiting. This means that each step in the excitation/electron transfer/catalyst activation/fuel generating cycle must occur with < 10 ms. As a useful reference, the individual activation steps in the photochemical oxidation of H₂O by photosystem II occur on the microsecond to submillisecond time scale with up to 50 molecules of O₂ produced per second; see below.

V. Electron-Transfer Activation of Catalysis

V.A. Catalyst Activation. Proton-Coupled Electron Transfer (PCET). Gain or loss of multiple electrons is required to activate the fuel-forming catalysts and avoid high-energy 1e^- intermediates such as CO₂^- or OH. Their impact energetically can be seen by comparing E' values in Scheme 6. The 2e^- entry for CO₂ to CO₂^2- illustrates the deleterious effect of an intermediate that is at high energy in proton content.

Scheme 6

The need for multielectron change necessitates complex pathways and the transfer of more than one electron to or from the catalysts. Charge compensation is required to avoid large increases in stepwise redox potentials in order to achieve redox potential leveling. This is especially important for biological redox couples in nonpolar membrane environments, where charge compensation by the surrounding dielectric is low compared to water, for example. Charge compensation can occur by ion pairing, extrusion, or addition of ions from chemical bonds and, most importantly, by PCET.

The influence of PCET can be seen by the decrease of over 1 V for the 2e^- reduction of CO₂ to HCO₂H compared to CO₂^2- in Scheme 6. PCET is a natural consequence of the effect of electron content on acid-base properties. For example, oxidation of [Fe(H₂O)₆]²⁺ to [Fe(H₂O)₆]³⁺ decreases pK_a,1 from 9.5 to 2.2²⁵⁶ and oxidation of tyrosine (TyrOH) to its cation radical, TyrOH⁺, decreases the pK_a from +10 to -2.^257-260 Both of these couples are pH-dependent over a wide pH range, with the TyrOH⁺/TyrOH couple decreasing by 0.59 mV/pH unit from below pH = 0 to 0.94 V at pH = 7.

The electrostatic effect of PCET can dramatically decrease sequential redox potentials, making the buildup of multiple reductive or oxidative equivalents by electron transfer feasible over a relatively small potential range. As an illustration, E' between the Ru^IV/III/cis-[Ru(bpy)₂Cl₂]^2+/+ and Ru^III/II/cis-[Ru(bpy)₂Cl₂]²⁺ couples in CH₃CN is 1.7 V. It is only 0.11 V between the [cis-Ru^IV(bpy)₂(py)(O)]²⁺/[cis-Ru^III(bpy)₂(py)(OH)]²⁺ and [cis-Ru^III(bpy)₂(py)(OH)]²⁺/[cis-Ru^II(bpy)₂(py)(H₂O)]²⁺ couples in water.^261-263

There is a happy coincidence in this. Both protons and electrons are involved in the fuel-forming half-reactions in eqs 1-3, e.g., 2H₂O O₂ + 4e^- + 4H⁺, and activation of key redox steps also requires PCET, e.g.,

V.B. Electron-Proton Transfer (EPT). In catalyst activation by PCET, the gain or loss of protons can be rate-limiting. An example is the oxidation of cis-[Ru^III(bpy)₂(py)(OH)]²⁺ to catalytically active cis-[Ru^IV(bpy)₂(py)(O)]²⁺, in which loss of a proton is rate-limiting, followed by rapid electron transfer.²⁶⁵ This reaction could occur by electron transfer, followed by proton transfer, but with an energy penalty because E' > 1.6 V (vs NHE) for the cis-[Ru^IV(bpy)₂(py)(OH)]³⁺/cis-[Ru^III(bpy)₂(py)(OH)]²⁺ couple. There is also an energy penalty for the pathway involving proton transfer to give cis-[Ru^III(bpy)₂(py)(O)]⁺ followed by electron transfer. For Ru^III-OH²⁺, pK_a > 13, and loss of a proton at pH = 7 is endoergic by G > +0.42 eV (>+9.7 kcal/mol).²⁶⁵

As shown in the E' diagram at pH = 7 in Scheme 7, the energy-efficient pathway for interconverting cis-[Ru^III(bpy)₂(py)(OH)]²⁺ and cis-[Ru^IV(bpy)₂(py)(O)]²⁺ is by coupled loss of electrons and protons (EPT) to give cis-[Ru^IV(bpy)₂(py)(O)]²⁺ directly. E' for this half-reaction at pH = 7 is only 0.78 V.

Scheme 7. PCET Thermodynamics for the Couples cis-[RuIV(bpy)2(py)(O)]2+/cis-[RuIII(bpy)2(py)(OH)]2+ and cis-[RuIII(bpy)2(py)(OH)]2+/cis-[RuII(bpy)2(py)(H2O)]2+ Illustrating the Energy Advantage of a Coupled EPT Pathway

Although advantageous energetically, the mechanistic details for a EPT pathway are necessarily more complex than either electron or proton transfer because both electrons and protons are transferred simultaneously. This means, for example, that both the electron donor and acceptor must have available orbitals for undergoing coupled EPT.²⁶⁴

EPT is the pathway used in the comproportionation reaction between cis-[Ru^IV(bpy)₂(py)(O)]²⁺ and cis-[Ru^II(bpy)₂(py)(H₂O)]²⁺ shown in eq 33. A tip-off that EPT is operative is the fact that the reaction occurs with k(H₂O)/k(D₂O) = 16. The EPT pathway is illustrated in eq 34. It features electron transfer from a d(Ru^II) orbital to a d(Ru^IV) orbital and proton transfer from _O-H to a lone pair on the oxo group.

PCET and EPT play major roles in biology, in respiration and photosynthesis, for example. As noted in section III.A and Figure 5, in photosystem II D C* A electron transfer from tyrosine Y_Z to Q_A, initiated by oxidative quenching of P₆₈₀* by an adjacent phenophytin, triggers water oxidation. It has been suggested that electron transfer from Y_Z is coupled to proton transfer to H-bonded histidine, His-190 (Scheme 8).^266-271

Scheme 8. Comparative Energetics of Oxidation of Tyrosine YZ by Electron Transfer and by MS-EPT

Because electron and proton transfer involve different electron and proton acceptors, this pathway is called multiple site EPT, or MS-EPT. From the comparative energetics in Scheme 8, MS-EPT offers an energy advantage over electron transfer of 9.5 kcal/mol in the oxidation of Y_Z because it avoids the high-energy radical cation intermediate, TyrOH⁺.

The theory of EPT has been developed by Cukier and Hammes-Schiffer and their co-workers based on somewhat different assumptions.^272-287 Both are based on electron-transfer theory and include proton transfer as a coupled high- or medium-frequency mode, (E-H). In the treatment by Hammes-Schiffer et al., strong electronic coupling between the intial and final proton-transfer states is assumed, giving rise to new adiabatic proton-coupled states. Electronic coupling between the resulting proton adiabatic states is treated subsequently as in electron-transfer theory; see section IV.

Application of the time-dependent perturbation theory with application of the Golden Rule to the transition between the final proton- and electron-coupled states I and II leads to the expression for coupled EPT in eq 35a for D-H- - -A D- - -H-A EPT.²⁷⁴

As in eq 28 for electron transfer, the summations are over a series of vibrational channels, in this case from the initial, coupled D-H- -A vibrational levels to the final coupled D- - -H-A levels . In contrast to eq 28, levels above = 0 are included in the summations. P_I is the Boltzmann population in vibrational level