206
6 Charge and Energy Transfer Processes
ΔE
∗
=
(ΔE r + λ)
2
4λ
.
(6.73)
Here ΔE r is the reaction energy, i.e., the energy difference between products and
reactants. The λ parameter is the “reorganization energy,” i.e., the surplus of energy
the products X
+
+Y would have if the ligands or solvent molecules did not readapt
to the new charge distribution. In the model it is computed as the energy of X
+
+Y
at the equilibrium values of R X and R Y for X+Y
+ , or vice versa. We shall call Δ X
and Δ Y the differences in the equilibrium values of R X and R Y between reactants
and products. Then
λ =
K
2
Δ
2
X + Δ
2
Y
(6.74)
where K is the unified force constant of both coordinates. We see that the vertical
excitation energy of X+Y
+ to the charge transfer state X
+
+Y is T v = λ + ΔE r
(remember that ΔE r < 0 in Figs. 6.8 and 6.9). Vice versa, the excitation energy of
X
+
+Y to the state X+Y
+ is λ − ΔE r , so one can obtain the reorganization energy
if both charge transfer bands can be identified in the spectra. The activation energy
can also be written
ΔE
∗
=
T
2
v
4(T v − ΔE r )
.
(6.75)
It should be noted that, with a proper statistical treatment of all the coordinates
(or states) of the chemical environment of the redox pair (ligands, solvent, etc.),
all the energies that appear in Eqs. (6.73)–(6.75) would be replaced by free energies, as discussed in Sect. 4.4. In conclusion, the thermal reaction rate will therefore
contain an Arrhenius factor exp(−ΔE
∗
/K B T ) that can be related to the reaction
energy and to a spectral quantity. However, once the transition state is reached,
the X+Y
+ state must be converted adiabatically to X
+
+ Y; i.e., a transition must
occur between the two diabatic states. Marcus evaluated the transition probability through the Landau–Zener rule, the key parameter being the matrix element
Ψ X,S 0 Ψ Y,D 0
ˆ
H el
Ψ X,D 0 Ψ Y,S 0
. When this interaction is very small, the probability
that an electron jumps from X to Y is low and most molecules will keep going uphill
on the initial diabatic PES, until they turn and go back to the reactants minimum.
On the contrary, when the interaction is strong, the probability of staying on the
ground-state adiabatic PES is high and a pure transition state theory treatment is
adequate.
The optical mechanism corresponds to the blue arrows in Fig. 6.8, right panel, and
to the red pathway in Fig. 6.9. In this case, the system is directly excited to the charge
transfer excited state, i.e., to the other diabatic state. A steep slope, corresponding
to the rearrangement of the chemical environment, brings the system to the crossing
seam and beyond. Now a weak interaction makes probable to continue on the same
diabatic PES and to reach the products minimum, so the quantum yield will be high.
However, if the interaction between X and Y is really small, also the charge transfer
band in the absorption spectrum will be weak.
6 Charge and Energy Transfer Processes
ΔE
∗
=
(ΔE r + λ)
2
4λ
.
(6.73)
Here ΔE r is the reaction energy, i.e., the energy difference between products and
reactants. The λ parameter is the “reorganization energy,” i.e., the surplus of energy
the products X
+
+Y would have if the ligands or solvent molecules did not readapt
to the new charge distribution. In the model it is computed as the energy of X
+
+Y
at the equilibrium values of R X and R Y for X+Y
+ , or vice versa. We shall call Δ X
and Δ Y the differences in the equilibrium values of R X and R Y between reactants
and products. Then
λ =
K
2
Δ
2
X + Δ
2
Y
(6.74)
where K is the unified force constant of both coordinates. We see that the vertical
excitation energy of X+Y
+ to the charge transfer state X
+
+Y is T v = λ + ΔE r
(remember that ΔE r < 0 in Figs. 6.8 and 6.9). Vice versa, the excitation energy of
X
+
+Y to the state X+Y
+ is λ − ΔE r , so one can obtain the reorganization energy
if both charge transfer bands can be identified in the spectra. The activation energy
can also be written
ΔE
∗
=
T
2
v
4(T v − ΔE r )
.
(6.75)
It should be noted that, with a proper statistical treatment of all the coordinates
(or states) of the chemical environment of the redox pair (ligands, solvent, etc.),
all the energies that appear in Eqs. (6.73)–(6.75) would be replaced by free energies, as discussed in Sect. 4.4. In conclusion, the thermal reaction rate will therefore
contain an Arrhenius factor exp(−ΔE
∗
/K B T ) that can be related to the reaction
energy and to a spectral quantity. However, once the transition state is reached,
the X+Y
+ state must be converted adiabatically to X
+
+ Y; i.e., a transition must
occur between the two diabatic states. Marcus evaluated the transition probability through the Landau–Zener rule, the key parameter being the matrix element
Ψ X,S 0 Ψ Y,D 0
ˆ
H el
Ψ X,D 0 Ψ Y,S 0
. When this interaction is very small, the probability
that an electron jumps from X to Y is low and most molecules will keep going uphill
on the initial diabatic PES, until they turn and go back to the reactants minimum.
On the contrary, when the interaction is strong, the probability of staying on the
ground-state adiabatic PES is high and a pure transition state theory treatment is
adequate.
The optical mechanism corresponds to the blue arrows in Fig. 6.8, right panel, and
to the red pathway in Fig. 6.9. In this case, the system is directly excited to the charge
transfer excited state, i.e., to the other diabatic state. A steep slope, corresponding
to the rearrangement of the chemical environment, brings the system to the crossing
seam and beyond. Now a weak interaction makes probable to continue on the same
diabatic PES and to reach the products minimum, so the quantum yield will be high.
However, if the interaction between X and Y is really small, also the charge transfer
band in the absorption spectrum will be weak.