6 Triplet-Energy (or Electron) Transfer between Molecules

Ch015.qxd



12/22/2006



15.6



10:39 AM



Page 411



Triplet-Energy (or Electron) Transfer between Molecules



411



Figure 15.14 Correspondence between the transfer of energy from an electronically excited donor

to an acceptor in its ground state (upper diagrams) and the corresponding emission and absorption

spectra. Only the coupled transitions identified in the upper diagrams conserve energy and may

occur. (a) Transfer between identical molecules. (b) Exothermic transfer that maximises the spectral

overlap. (c) Highly exothermic transfer, beyond the optimal spectral overlap.



better overlap between the donor emission and the acceptor absorption spectra. This overlap goes through a maximum, shown in Figure 15.14b, and then decreases for even more

exothermic reactions (Figure 15.14c). This latter situation is similar to that of the energygap law.

The simplest way to calculate the transfer rate from the initial state s to the final state k

is to use the classical expression for the transfer rate at a given energy E such that E Ͼ ⌬E‡,

where ⌬E‡ is the energy of the crossing point between the two states. Such an expression

is the product between the probability of conversion at the crossing point (Pks) and the frequency of passage over the crossing point

w ( E ) = 2 Pks



(15.76)



There is a factor of 2 in this expression because the energetic condition E Ͼ ⌬E‡ requires

that the turning point of the vibration exceeds the distortion to attain the crossing point,

|xv| Ͼ |x‡|, such that the conversion between the two states is attempted twice during each

vibrational motion.

The classical probability of transition probability was first formulated by Landau and

Zener [23,24], and was presented in eq. 5.53. Using the notation of this chapter

Pks =



2



e

Vks



2



1

r ss − sk



(15.77)



where r ϭ dx/dt is the velocity of passage over the crossing point at x‡, and ss and sk the slopes

of the potential energy curves describing the initial and final states at that point, dV(x‡)/dx,



Ch015.qxd



12/22/2006



10:39 AM



Page 412



412



15. Transitions between Electronic States



respectively. For the specific case of electronic states represented by unidimensional harmonic oscillators with the same force constant f, the difference between the slopes is

ss − sk =



d ⎛ 1 2⎞ d ⎡1

2

⎤

fx ⎟ − ⎢ f ( d − x ) − E 0 ⎥ = fd

⎝

⎠ dx ⎣ 2

dx ⎜ 2

⎦



(15.78)



The velocity of the vibrational motion of this oscillator is defined as the length travelled

per unit time. Given the classical amplitude of the oscillation with an energy E



2E



xv =



(15.79)



2



the velocity of passage over x‡ is



( x v )2 − ( x ą )



2



r=



(15.80)



Rearranging the above expressions leads to the expression for the rate of transfer at an

energy E Ͼ ⌬E‡

w(E ) =



2



1



2



Vks



( )



2

2

f ⎡( x v ) − x ‡ ⎤

⎢

⎥

⎣

⎦



fd 2



(15.81)



The actual classical transfer rate at a given temperature T requires averaging over a

Boltzmann distribution of energies



∫

w=



ϱ

E



w ( E ) e − E kBT d E



∫



ϱ

0



(15.82)



e − E kBT dE



This procedure yields a compact result because



∫



ϱ

0



e − E kBT dE = kBT



(15.83)



and eq. (15.79) can be used to express the other integral involved as



∫ (E −

ϱ



E‡



E‡



)



−1 2



e − E kBT dE =



k BT e −



E ‡ kBT



(15.84)



The resulting expression

w=



2



Vks



2



⎛

E‡ ⎞

exp ⎜ −

⎠

⎝ k BT ⎟

2 fd 2 kBT

1



(15.85)



can be further modified using the quadratic relation between the energy barriers and the

reaction energies, discussed in Chapter 7, eq. (7.6),



(



E0 −



)



2



=4



E‡



(15.86)



Ch015.qxd



12/22/2006



15.6



10:39 AM



Page 413



Triplet-Energy (or Electron) Transfer between Molecules



413



and associating the reorganisation energy with the displacement between the curves representing the electronic states (Figure 15.7),

=



1 2

fd

2



(15.87)



The final expression for the classical rate for triplet–triplet (or electron) transfer,

w=



2



1



e 2

ks



V



4



⎡

exp ⎢ −

⎢

k BT

⎣



(



)



E0 −



2



4 k BT



⎤

⎥

⎥

⎦



(15.88)



must be regarded with the caution that follows from the numerous approximations

involved in its derivation. First, the problem of energy transfer between molecules is multidimensional, whereas in deriving the above expression only one effective frequency was

e

used. Second, the Landau–Zener probability is only valid when Vks  1/2 r2, that is, for

Ϫ1

an electronic coupling of 5 kJ mol the reorganisation energy must exceed 20 kJ molϪ1,

which involves a difficult balance between having enough adiabaticity and at the same

time a transfer rate that is competitive with the radiative lifetime. Third, the quadratic relation between the energy barriers and reaction energies must be obeyed. Finally, the classical model assumes that the electronic states cross at a given energy, used to define the

activation energy ⌬E‡, but in fact there are two molecules (and four electronic states) in

the reactants, and as many in the products. The crossing point is not rigorously defined and

is replaced by the reorganisation energy defined in eq. (15.87), where it is assumed that all

the electronic states have the same frequency . However, the value of is not calculated

from the energy of the crossing point, but from the displacement of the electronic states in

each species (Figure 15.7).

A more fundamental approach to the problem of energy transfer can be made by writing the classical probability distribution for the donor in the excited state that takes the

form at R

⎛ f ⎞

( R) = ⎜

⎝ 2 k BT ⎟

⎠



12



⎛ f ( R − R )2 ⎞

e

exp ⎜ −

⎟

⎜

2 k BT ⎟

⎝

⎠



(15.89)



This has a clear analogy with the wavefunction of the harmonic oscillator at its ground

vibrational state, eq. (15.59). The energy transfer involves a vertical transition between the

two energy curves of the donor. Given (R), the distribution of energies associated with

the donor, ι( Ј), which represents the energies required to transfer the triplet energy from

the donor, is

ι ( Ј) =



1

2



2

D



⎡ ( Ј + ED −

exp ⎢ −

2 D2

⎢

⎣



D



)



2



⎤

⎥

⎥

⎦



(15.90)



where ED is the triplet energy of the donor, D2 ϭ kBT/f the square of the standard deviation of the respective Gaussian distribution and D the difference between the energy of the



Ch015.qxd



12/22/2006



10:39 AM



Page 414



414



15. Transitions between Electronic States



triplet state in its equilibrium geometry and the energy it would have in that state at the

equilibrium configuration of the ground state eq. (15.87). The centre of the Gaussian distribution is placed at the ground-state equilibrium configuration of the donor, with an

energy EDϪ D that is lower than its triplet energy (Figure 15.15) [25]. This displacement

is related to the difference between the nuclear configuration of the initial triplet state and

the final ground state of the donor. This energy difference, D, is the reorganisation energy

of the donor. A similar spectrum can be formulated for the ground-state acceptor,



( )=



1

2



2

A



⎡ ( + EA +

exp ⎢ −

2 A2

⎢

⎣



A



)



2



⎤

⎥

⎥

⎦



(15.91)



where EA is the triplet energy of the acceptor and A its reorganisation energy.

Mathematically, the possibility that the energy taken from the donor is between Ј and

Ј ϩ d Ј, is ( Ј)d Ј and the possibility that the energy given to the acceptor is between

and ϩ d , is ( )d . To conserve energy in the process, the transfer from a particular

energy Ј in the donor distribution ι( Ј) must be paired with the transfer in the acceptor distribution ( ) of the same energy. In Figure 15.15 this corresponds to a horizontal process.

Mathematically, the probability of transition with Ј between Ј and Јϩd Ј and between

and ϩ d , is ( Ј)d Ј ( )d , with Ј ϭ . The density of states in eq. (15.36) is this probability per unit of energy. Thus, dividing the probability of transition by one of the energy

intervals, and integrating over the other one, gives the total density of states

J =∫



+ϱ

−ϱ



( ) ( )d



=

2



⎡ ( + ED −

∫− ϱ exp ⎢− 2 2D

⎢

⎣

+ϱ



1

2

D



2

A



D



)2 − (



+ EA +

2



2

A



A



)2 ⎤ d

⎥

⎥

⎦



(15.92)



Figure 15.15 Probability distribution of energies required for triplet energy transfer from a donor to

an acceptor. The conservation of energy requires that the transfer is a horizontal process in this diagram, and the nuclear coordinates must pre-organise accordingly.



Ch015.qxd



12/22/2006



15.6



10:39 AM



Page 415



Triplet-Energy (or Electron) Transfer between Molecules



415



If we make the substitution

a=−

d=



− EA



A

D



2

D



d



(15.93)



− ED



the exponential term can be rewritten

⎧ [ − a ]2 [ − d ]2 ⎫

⎡

⎪

⎪

exp ⎨−

−

⎬ = exp ⎢

2 2

2 2 ⎪

A

D

⎣

⎪

⎩

⎭



2

A



a+

2

A



2

D



−



2

A



+

2

A



2



2

D

2

D



2



⎤

⎛

⎥ exp ⎜ −

⎦

⎝



2

A



a2 +

2 2

A



2

D

2

D



d2 ⎞

⎟

⎠

(15.94)



to indicate the presence of the standard integral



∫



+ϱ

−ϱ



⎡ c2 ⎤

c2 exp ⎢ 1 ⎥

⎣ 4c 2 ⎦



e c1 x − c2 x dx =

2



(15.95)



and express the overlap integral in a more compact form

J =∫



+ϱ

−ϱ



ι(



) ( )d



1



=



4



2



⎧ ⎡ E 0 − ⎤2 ⎫

⎪

⎦ ⎪

exp ⎨− ⎣

⎬

2

4

⎪

⎪

⎩

⎭



(15.96)



where 2 ϭ ( D2 ϩ A2)/2, ϭ D ϩ A, and the reaction energy is the difference between

the triplet energies of the donor and the acceptor, ⌬E0 ϭ EDϪEA.

Introducing this Franck–Condon weighted density of states in the Golden Rule, gives

w=



2



e 2

ks



V



1

4



2



⎡

exp ⎢ −

⎢

⎣



(



E0 −

4



2



)



2



⎤

⎥

⎥

⎦



(15.97)



This expression for the Golden Rule is identical to the classical expression, eq. (15.88),

when 2 ϭ kBT. However, it was initially formulated as 2 ϭ kBT/f. This difference can be

solved with the transformation

2



=



2f



⎛

⎞

coth ⎜

2 k BT ⎟

⎝

⎠



(15.98)



/2) and kBT/f at low temperatures

which approaches kBT at high temperatures (kBT

(kBT

/2). In view of this connection between a classical behaviour at high temperatures and the temperature independence at low temperatures, eq. (15.97) is a semi-classical

rate for triplet–triplet energy (or electron) transfer [26]. This formulation takes the reorganisation energy of the donor as the difference in its energy when in the triplet state, but

having the same geometry as the ground state. This is the Stokes shift between the maxima of the emission and absorption envelopes of the donor, without specific consideration

of the vibronic bands. The same applies to the acceptor. Hence, this reorganisation energy

includes the vibrational as well as solvent contributions to the Stokes shift.



Ch015.qxd



12/22/2006



10:39 AM



Page 416



416



15. Transitions between Electronic States



The semi-classical rate is a significant improvement in the definition of the reaction

coordinate for triplet energy transfer. It relates the reorganisation energy of each reactant to

the displacement of the corresponding initial and final states along the reaction coordinate,

without having to assume the presence of a crossing between the reactive states. It is also

less restrictive on the size of the electronic coupling. However, the connection between the

high and low temperatures was introduced in an ad hoc fashion, and does not recognise that

vibronic transitions change the overlap substantially. Further improvement in the treatment

of the quantum effects can be achieved with a quantum-mechanical formulation.

The full quantum-mechanical treatment of the vibrations coupled to the changes of the

electronic state is rather sophisticated. A convenient simplification is to consider only one

vibrational mode for the donor, D, and another one for the acceptor, A. This is a dramatic simplification, but the result can be generalised to multi-mode processes using a

suitable procedure. Another simplification is the classical treatment of the low-frequency

modes, attributed to the medium and to low vibrational frequencies of the reactants and

products. This is a good approximation if

kBT. For example, v Ͻ 100 cmϪ1 correϪ1

sponds to a spacing of Ͻ1 kJ mol between the vibrational levels, which is less than the

thermal energy at room temperature, RT ϭ 2.5 kJ molϪ1. A difference of 200 cmϪ1 between

the simulation of the anthracene absorption and emission was found in Figure 15.11.

For a system with the restrictions mentioned above, the Franck–Condon factor for

triplet energy transfer combines the description of the vibrational levels done by eqs.

(15.68) and (15.69), with the reorganisation energies included in eqs. (15.90) and (15.91).

Integrating as in eq. (15.92), gives

ϱ



1



J vЈ, v =



4



ϱ



∑∑e



2



− SD



e



− SA



v = 0 vЈ = 0



⎡

v

v

SDЈ SA

exp ⎢ −

⎢

vЈ! v !

⎣



(



E 0 − + vЈ

4



D



+v



)



2



A



2



⎤

⎥

⎥

⎦



(15.99)



where

2



=



+

2



2

D



2

A



=(



D



+



A



) k BT =



k BT



(15.100)



as a consequence of the approximation made for the low frequency modes. The overlap

integral JvЈ,v is also known as the Franck–Condon weighted density of states. The corresponding triplet energy transfer rate is

w=



2



e 2

ks



V



∞



1

4



Setting D ϭ ϭ

tions, the result

w=



∞



∑∑e



− SD



A,



Dϭ



ϭ



2



e 2

ks



2



e



− SA



v = 0 vЈ = 0



V



A



1

4



⎡

v

v

SDЈ SA

exp ⎢ −

⎢

vЈ! v !

⎣



(



E 0 − + vЈ

4



D



)



2



+v



A



2



⎤

⎥ (15.101)

⎥

⎦



and S ϭ SD ϩ SA gives, after some algebraic manipula-



∞



∑e

k T

B



v=0



−S



(



⎡ ΔE 0 − + v

Sv

exp ⎢ −

⎢

v!

4 k BT

⎣



)



2



⎤

⎥

⎥

⎦



(15.102)



Ch015.qxd



12/22/2006



15.6



10:39 AM



Page 417



Triplet-Energy (or Electron) Transfer between Molecules



417



This expression for the quantum-mechanical rate for triplet energy (or electron) transfer

remains numerically indistinguishable from eq. (15.101) even when D and A differ by

as much as 10% [27].

The overlap between the vibronic bands of the reactants and products is illustrated in

Figure 15.16 [25]. This representation is inspired by the model developed by Marcus

[28,29] for electron transfer, which will be discussed in detail in Chapter 16. The curve for

the reactants represents the potential energy of the system when the donor is in the triplet

state and the acceptor in the ground state. The product curve represents the system when

the donor is in the ground state and the acceptor in the triplet state. The Franck–Condon

principle requires that the nuclear configuration is the same immediately after the transfer,

that is, the transfer is vertical. Conservation of energy requires that the transfer is horizontal. The only possible way to meet both requirements is at the crossing point. However,

the reaction coordinate should have as many dimensions as there are degrees of freedom

for nuclear motion in the system, and the crossing point is, in fact, a large ensemble of



Figure 15.16 Nuclear motion accompanying triplet energy transfers, emphasising the quantum

nature of the vibrational modes. The vibrational energy levels are shown with the squares of the

vibrational wavefunctions superimposed on them. The reaction coordinate represents a combination

of the positions of all the nuclei disturbed by the electronic transition, as it evolves from the reactants (excited donor and ground-state acceptor) to the products (ground-state donor and excited

acceptor).



Ch015.qxd



12/22/2006



10:39 AM



Page 418



418



15. Transitions between Electronic States



nuclear configurations. Some of these are represented in Figure 15.14, which gives a more

exact microscopic description of the energy transfer process.

The expression obtained for the quantum-mechanical rate is based on the description of

the initial and final states of the electronic transition as displaced harmonic oscillators.

Additionally, the derivation was based on a single vibrational mode for each reactant. It was

assumed that an average vibrational frequency could describe the modes that promote the

electronic transition and accept the excess energy released in the course of that transition.

However, a more detailed analysis of the absorption and emission bands of the aromatic

hydrocarbons that are most amenable to simulations with displaced harmonic oscillators,

reveals that even these systems have “shoulders” in the main progression bands, which are

probably associated with other vibrational modes. Figure 15.17 shows the overlap between

the fluorescence of naphthalene and the absorption of anthracene modelled with the highfrequency progression band discussed above, but it is evident that a subsidiary mediumfrequency band, probably associated with CCC bending vibrations, is also present.

For these multi-mode processes it is still possible to employ the average-mode approximation [27,30], and take S as the sum of the Sj values for the coupled high- and mediumfrequency vibrations

S = ∑ Sj

j



(15.103)



Figure 15.17 Normalised fluorescence of naphthalene (ES ϭ 31150 cmϪ1) and absorption of

anthracene (ES ϭ 26400 cmϪ1), modelled by one vibrational mode each. Naphthalene: v D ϭ 1420

cmϪ1, D ϭ 150 cmϪ1, SD ϭ 0.82; Anthracene: v D ϭ 1450 cmϪ1, D ϭ 150 cmϪ1, SD ϭ 0.70. Solid

lines: experimental spectra; dotted lines: simulated spectra; dashed line: calculated spectral overlap.



Ch015.qxd



12/22/2006



10:39 AM



Page 419



15.6



Triplet-Energy (or Electron) Transfer between Molecules



and



as the weighted average of the quantum spacing

=



∑S

j



j



∑S



419



j



(15.104)

j



j



The average-mode approximation combines the vibrational progressions needed for a

better simulation of absorption and emission spectra. Unfortunately, for triplet-energy (and

electron) transfers the relevant singlet–triplet absorption and phosphorescence (or chargetransfer bands) are rarely available for such simulations. It should also be noted that bimolecular transfer rates tend to be controlled by diffusion when the overlap is large, and that

for small overlaps minor errors in the simulation of the spectra lead to large errors in JvЈ,v.

Moreover, when the reaction energies exceed 60 kJ molϪ1, the higher frequency modes

such as the CH stretching vibrations become better energy acceptors than the CC modes,

and the information carried by the absorption and fluorescence spectra is of little use.

Finally, highly exothermic reactions result in the population of highly excited vibrational

levels, as illustrated in Figure 15.14, and these are strongly anharmonic. The participation

of additional accepting modes in the reaction coordinate and the increased anharmonicity

of the accepting modes as the reactions become more exothermic make it very unlikely

that the reorganisation energy is independent of the reaction energy.

The observation of large deuterium isotope effects is a good diagnostic for the participation of CH or other bonds to hydrogen in the reaction coordinate for energy or electron transfer. The paradigmatic example is the quenching of O2(1⌬g) is solution. The bimolecular rate

constant of radiationless deactivation of singlet oxygen drops from 4.3ϫ103 MϪ1 secϪ1

in H2O to 2.6ϫ102 MϪ1 secϪ1 in D2O, a KIE of 16.5. The rate is further reduced to

0.75 MϪ1 secϪ1 in perfluorodecalin, a solvent without C᎐H bonds. This spectacular effect

is due to the deactivation by coupled vibronic O2(1⌬g, vЈ ϭ 0 → vЈ ϭ m) and vibrational

X᎐H(v ϭ 0 → v ϭ n) transitions [31]. Charge recombinations between aromatic hydrocarbons and tetracyanoethylene have much smaller KIE, usually not exceeding a factor of 2,

with the particularity that the deuteration of methyl hydrogens in methylated benzene

donors gives a larger KIE than that of phenyl hydrogens [32].

Absorption and fluorescence are usually registered using a continuous irradiation of the

sample, that is, in steady-state conditions. It is also possible to measure both absorption

and fluorescence intensities at different wavelengths as a function of time. Ware and coworkers [33] followed the fluorescence spectra after excitation on the nanosecond

timescale, and observed a redshift of the fluorescence maxima as a function of time. Ultrafast techniques have more recently been employed to measure the time-dependence of

absorption and fluorescence spectra, and confirmed the generality of this phenomenon for

charge-transfer transitions in condensed phases, known as the “dynamic Stokes shift”.

The time evolution of the fluorescence spectrum of a suitable probe molecule, following excitation by an ultra-short optical pulse, can be used to monitor the dynamic Stokes

shift response function



S (t ) =



v (t ) − v (ϱ)



v ( 0) − v ( ϱ)



(15.105)



Ch015.qxd



12/22/2006



10:39 AM



Page 420



420



15. Transitions between Electronic States



where v (x) represents the peak of the fluorescence spectrum at times t, zero and infinity.

Expressed in this way, the frequencies need not be referenced to their (usually unknown)

gas-phase values. This function carries information on the polar solvation dynamics. The

fastest processes develop on a 10 fsec timescale and correspond to the intra-molecular

relaxation of high-frequency modes. These processes remain very difficult to follow, such

that on the scale of tens of femtoseconds and extending to the picoseconds, there is a continuous shift of the fluorescence spectrum to the red as solvent relaxation proceeds.

Solvation is non-exponential in time, and cannot be interpreted with only one time constant. This is in conflict with the description of the dielectric constant of polar solvents by

the frequency dependence of the Debye dispersion [34]



( )=



op



+



−

1− i



op



(15.106)

D



where and op(ϭ nD2) are the static and optical (infinite) frequency dielectric constants and

D is the Debye relaxation time. This simple model of a polar solvent as a dielectric continuum predicts a single exponential shift of the florescence spectrum with a time constant

approximately equal to the solvent longitudinal relaxation time L ϭ ( op/ ) D, whereas the

dynamic Stokes shifts reveal several time constants, some of which are considerably shorter

than L. Water is a particularly “fast” solvent, with Ͼ50% of the solvation response happening within the first 50 fsec [35], which corresponds roughly to 660 cmϪ1 in frequency. This

should be compared with D ϭ 8.2 psec, op ϭ 4.21, ϭ 78.3 and L ϭ 440 fsec for water. The

ultra-fast relaxation of water was assigned to the vibrational (rotational) motions of the water

molecule. It has a particular incidence on preventing the re-crossing of the reaction freeenergy barrier, because it shows that water can stabilise the reaction products before they

may re-cross to the reactants and reduce the reaction rate. However, the description of the

full range of S(t) requires additional frequencies. In water, some of these frequencies are

associated with the hindered translation of the hydrogen-bonded network (180 cmϪ1), the

hydrogen-bonding bend (60 cmϪ1), and the diffusive motions (2–10 cmϪ1). In fact, the entire

dielectric dispersion spectrum of ( ) must be considered in the simulation of S(t), including

the low-frequency part, which is well described by the Debye formula, and the high-frequency

part, which contains various contributions from inter- and intra-molecular vibrational modes

of the solvent. Fleming and co-workers employed seven frequencies in the range indicated

above to reproduce the dynamic Stokes shift that they observed in water [36].

The subsequent processes take place from the lowest vibrational level of the first singlet

state and have already been discussed. One additional point to take into consideration is

that internal conversion leaves the ground state with an excess of heat, which may correspond to temperatures in excess of 1000 K. Its cooling occurs within 1–10 psec, with the

shorter times associated with polar molecules in polar environments [37].

Another experimental method to probe the Franck–Condon excited state is resonance

Raman spectroscopy. Recall that Raman scattering is an inelastic process entailing a transition to a virtual or real molecular state (or sum of states), and a nominally instantaneous

return to the ground state, but in a higher or lower vibrational state. In resonance Raman

spectroscopy the incident radiation nearly coincides with the frequency of the electronic

transition of the sample. Only a few vibrational modes contribute to the scattering of the



Ch015.qxd



12/22/2006



15.7



10:39 AM



Page 421



Electronic Coupling



421



radiation, which simplifies the Raman spectrum and enhances its intensity. The modes that

show resonance Raman activity are the same as those that are Franck–Condon active in the

electronic transition, and their degree of enhancement is related to their displacement upon

formation of the excited state. Consequently, the analysis of vibrational Raman lines

obtained in resonance with charge-transfer transitions can, in principle, reveal the complete

set of mode-specific reorganisation energies associated with the charge separation or recombination accompanying the electronic transition [38,39]. This technique has been applied to

many charge transfers, in particular, in transition-metal complexes and in organic

donor–acceptor complexes. A widely used transition-metal complex that has been studied

using resonance Raman spectroscopy is the Ru(bpy)32ϩ complex in water. Fourteen totally

symmetric vibrational modes were found to be active in the metal-to-ligand charge transfer,

with frequencies ranging from 283 to 1608 cmϪ1. The total reorganisation energy was divided

among these modes, including metal–ligand stretching and bending modes, and ligand

(bipyridine) localised modes. The hexamethylbenzene–tetracyanoethylene (HMB–TCNE)

complex is a typical example of organic donor–acceptor complexes studied by resonance

Raman spectroscopy. There are some differences in the reported results among the several

groups that have studied this system, but more than 10 vibrational modes are associated with

the electronic transition [32,40,41]. The high-frequency modes seem to be the most populated,

but the very low frequency donor–acceptor mode at 165 cmϪ1 is also strongly populated. The

contribution of the solvent and of each vibrational mode to the total reorganisation energy

remains unsettled. In CCl4, where the contribution of the solvent to the reorganisation energy

should be very small, values between 45 and 56 kJ molϪ1 were proposed [32,40]. These values of do not include the contributions from the CH bonds, that are necessary to explain the

KIE mentioned above, but could not be assessed from the resonance Raman spectra.

The picture emerging from ultra-fast techniques and resonance Raman spectroscopy is that

a large spectrum of solvent frequencies, extending to 1000 cmϪ1 in water, and a large number of vibrational frequencies, down to 165 cmϪ1 in donor–acceptor complexes, contribute to

the energy barrier of electron and triplet-energy transfers in solution. A detailed description

of these transfers requires the quantum treatment of a very large number of vibrational modes

and a wide dielectric dispersion spectrum for the solvent. Jortner [42] presented a multi-mode

formalism and compared its results for the HMB–TCNE complex using nine modes, with the

average-mode approximation and with a model using a high single mode with the same reorganisation energy as the multi-mode . The average-mode approximation underestimates the

Franck–Condon factor when the reaction exothermicity exceeds 85 kJ molϪ1. In contrast, a

high-frequency single mode combined with the correct reorganisation energy gives a good

account of the Franck–Condon factors. The lesson to be learnt is that even an incomplete

model that is good enough to reproduce the relevant absorption and emission spectra is also

adequate to calculate the Franck–Condon factors. The calculation of the electron or tripletenergy transfer rates additionally requires an estimate of the electronic coupling.

15.7



ELECTRONIC COUPLING



The factor (4 2 )Ϫ1/2 included in eq. (15.101) is associated with the coupling between the

electronic and the nuclear motions. Consequently, it cannot be exclusively assigned to the