Charge and Energy Transfer Processes

Abstract

In this chapter we shall present the peculiar features of charge and excitation energy transfer processes (CT and ET) that are of basic importance in photosynthesis, photovoltaics, and other areas of biochemistry and technology. The migration of charge or excitation energy between distinct chromophores implies a dramatic change in the electronic wavefunction, so the general nonadiabatic theory we have already discussed also applies to these processes. However, some peculiar features distinguish charge and energy transfer from other nonadiabatic processes. If the two chromophores are placed in two molecules free to move in gas or liquid phase, the transition can only take place during a collision or encounter, so the kinetics of bimolecular processes plays an essential role. However, just because their interaction is a basic requirement for the process to occur, in structured biological or artificial photosystems the single units are fixed at suitable relative positions and orientations. In typical situations, such arrangements also determine easily discernable spectral features. Whenever the interaction between the involved chromophores is not too large, the initial and final electronic states of the CT or ET process constitute a physically sound diabatic representation, which allows to analyze theoretically the main features of the dynamics.

Problems

6.1

In a study of the quenching of excited anthraquinone (C$_{14}$H$_8$O$_2$) by electron transfer to amines in gas phase [41], the bimolecular rate constant with pyridine (C$_5$H$_5$N) was found to be $2.3\cdot 10^{-19}$ s$^{-1}$ molc$^{-1}$ m$^3$, at $T = 433$ K. Compute the cross section and compare it with the geometrical cross section, obtained by considering two rigid spheres with volumes equal to the molecular volumes. To evaluate the volumes, use the density of solid anthraquinone (1.308 g/cm$^3$) and of liquid pyridine (0.982 g/cm$^3$).

Table 6.4 Wavelengths (nm) of $S_1$ and $T_1$ 0–0 bands

6.2

Compute the average time between two gas-phase collisions and between two encounters in solution, for a given molecule. Make the following “standard” assumptions: hard spheres with radii = 4 Å and molecular masses = 100 a.m.u., $T=300$ K, $P = 1$ atm, concentration of the other solute 1 mol/L, viscosity of the solvent $10^{-3}$ kg$\cdot$m$^{-1}$s$^{-1}$.

Fig. 6.10

Transition dipole moments for three chromophores symmetrically placed at the vertices of an equilateral triangle

6.3

The wavelengths of the 0–0 bands of the $S_0-S_1$ and $S_0-T_1$ transitions are listed in Table 6.4 for five polynuclear aromatic hydrocarbons. Which donor–acceptor pairs satisfy the energy requisites to exhibit FRET? Same question for triplet sensitization. Which compounds are likely to undergo singlet fission? And which triplet–triplet annihilation?

6.4

Prove Eq. (6.62). Can we write a similar relationship for the oscillator strengths?

6.5

Consider an array on n identical chromophores put at the n vertices of a regular polygon, and the n corresponding diabatic states $\eta _i$ of energy $E_{ex}$ in which the excitation is localized in chromophore i (excitons). Assume their transition dipoles are all parallel and the coupling between the excitons is of Förster type. What is the form of the bright state? We shall apply the approximation that the only important couplings are those between first neighbors, with strength V. How good is this approximation, depending on n? Is the bright state an eigenstate within the subspace spanned by the excitons? What is its energy?

6.6

Three identical chromophores A, B, and C are placed at the vertices of an equilateral triangle. Their absorption transition dipoles have a component $\mu _r$ which is radial with respect to the center of the triangle and a component $\mu _p$ perpendicular to the ABC plane, as shown in Fig. 6.10. Note that this could be a model for transitions centered in the equatorial ligands of a trigonal pyramid complex. The distance between the chromophores is R. Show that the excitation transfer dynamics among the four ligands, such as can be observed after a radiation pulse, has a period

$$\begin{aligned} T = \frac{8\pi \hbar R^3}{12\mu _p^2+21\mu _r^2}\; . \end{aligned}$$

Calculate the period with $R=15$ bohr, $\mu _p=0.5$ a.u., and $\mu _r=1$ a.u.

6.7

Same as the previous problem, but with a square arrangement. Four identical chromophores A, B, C, and D are placed at the vertices of a square. Their absorption transition dipoles have a component $\mu _r$ which is radial with respect to the center of the square and a component $\mu _p$ perpendicular to the ABCD plane. The side of the square has length R. Show that the excitation transfer dynamics among the four ligands, such as can be observed after a radiation pulse, has a period

$$ T = \frac{4\pi \hbar R^3}{\left( 4+\sqrt{2}\right) \mu _p^2+2\left( 3+\sqrt{2}\right) \mu _r^2}\; . $$

Calculate the period with $R=15$ bohr, $\mu _p=0.5$ a.u., and $\mu _r=1$ a.u.

6.8

Prove Marcus’ relationship, Eq. (6.73).