Pigment-pigment interaction

Bacteriochlorophylls in LH complexes do not only interact with their surrounding protein matrix, but also with each other. Actually, Frenkel-excitons develop by virtue of intermolecular interactions, where electrostatic (Coulomb) interactions are by far the dominant ones [61]. The Coulomb interaction V_nm between two

2.7 Frenkel-excitons

chromophores n and m can be written as:

V_nm = 1 4π₀

Xq_nq_m

r_nm , (2.11)

where is the dielectric constant of the medium, 0 the electric permittivity of free space and r_nm the relative distance between the electronic or nuclear charges q_n and q_m of the chromophores. If the pigments are not permanently charged and their mutual distance is large compared to their size, the interaction energy, Vnm, between them, can be approximated by the dipole-dipole term in the multipole expansion of the Coulomb interaction (higher order terms fall off more rapidly with the mutual distance rnm between the chromophores) [14, 59, 61]:

V_nm= µ² 4π₀

| {z }

V0

· 1 r³_nm · 1

µ²

~

µ_n·~µ_m−3(~µ_n·~r_nm) (~µ_m·~r_nm) r²_nm

| {z }

κnm

. (2.12)

Here, ~µ_nand ~µ_m represent the strength and the orientation of the transition-dipole moments of the molecules n and m, respectively. V₀ defines the coupling strength and, assuming that the interacting chromophores are identical (|~µn|=|~µm|=µ), it is,|~µ_n| · |~µ_m|=µ_n·µ_m =µ². κ_nm is an orientational factor which covers the mutual orientation of the transition-dipole moments by means of normalized vectors. Thus, the dipole-dipole interaction can be divided into a distance dependent part, 1/r_nm³ , and an orientational part, described by κ_nm.

Figure 2.11: Schematic illustrations of the dipole-dipole interaction. (a) Two dipoles ~µ_n and ~µ_m and their connecting vector ~r_nm drawn in-plane. α, β and γ indicate the mutual angles between the transition-dipoles and their connecting vector and between each other.

(b) Characteristic mutual orientations of the transition-dipole moments and the corresponding values for the orientational parameter κ_nm (see text).

With the angles α, β and γ, defined in Figure 2.11a, it is possible to write down

2 Light-harvesting complexes

κ_nm in the following way:

κ_nm = 1

With this, it is easy to understand that κnm can adopt the following characteristic values: κ_nm = −2 (2) for a collinear head-to-tail (head-to-head) arrangement of the transition-dipoles, and κ_nm = 0 for mutually perpendicular transition-dipoles (Figure 2.11b).

For the B850 aggregate in LH2 complexes and the B880 assembly in LH1 com-plexes, the dipole-dipole approximation might be questioned, as the distance be-tween neighboring pigments is comparable to their dimensions. Therefore, other approaches to calculate the spectral properties of these pigment assemblies were chosen as well, taking the spatial distribution of the transition-dipole moments [62, 63] or the exchange interaction [64, 65] into account. However, in [64] and [65]

it was also shown that the Coulomb coupling between the pigments still dominates over the exchange coupling. Furthermore, it is noteworthy that the dipole-dipole approximation is not only reasonable if the intermolecular distance r_nm is larger than the size of the chromophore, but even when r_nm is larger than the dipole ra-dius, defined as, a=µ_i/e ,whereµ_i denotes the magnitude of the transition-dipole moment of the chromophore and e is the elementary unit of charge [61]. For the numerical simulations performed in this work (see chapter 5), a dipole strength of µ_i = 8.8 D was assumed for the Q_y transition-dipole moment of the BChl a molecules, corresponding to a dipole radius a ≈ 2Å. Comparing this value to the mutual separation between neighboring BChl a molecules in a LH1 pigment ag-gregate [9], r_nm ≈ 10Å, it can be seen that the prerequisite for the dipole-dipole approximation, r_nm a, is not perfectly fulfilled in this case. Nevertheless, it can be expected that the dipole-dipole interaction approach provides at least a qual-itative description of the processes inside a LH1 complex. Further, to avoid this inaccuracy as much as possible, for the numerical simulations of chapter 5, the effective Hamiltonian approach [66] was adopted in part. In this approach fixed values are assumed for the interaction between neighboring pigments (based on a fit of the resulting exciton spectrum to extensive quantum chemistry calculations) and the dipole-dipole interaction is only applied for pigments which are further apart (second, third, ... neighbors). In [66] it was shown that by means of the effective Hamiltonian the results of extensive quantum chemistry calculations on the B850 aggregate of LH2 complexes, can be reasonably approximated. In chap-ter 5, due to the fixed positions of the pigments inside a dimer unit, the value of the intradimer interaction is fixed, whereas the interdimer interaction (between neighboring pigments of adjacent dimer units) and the second-neighbor interaction are calculated with the help of the dipole-dipole formula (2.12). Thus, the afore-mentioned inaccuracy is not completely eliminated for the numerical simulations

2.7 Frenkel-excitons

in chapter 5, but is still present for the interdimer interaction. Here, a compromise had to be made, since on the one hand, due to the basic construction principle of LH1 complexes, where it is assumed that the dimer units inside the LH1 complex experience a certain degree of conformational freedom with respect to the overall aggregate [20], it is not reasonable to assume a fixed value for the interdimer in-teraction. On the other hand, the dipole-dipole approach may not be the most accurate way to calculate the varying values for the interdimer interaction around the aggregate. However, for the rather simple numerical simulations of chapter 5, where other approximations are also made (e.g. not taking the coupling to phonon modes into account), it is an appropriate solution to use the dipole-dipole formula (2.12) for the calculation of the interdimer interaction, at least to gain a general idea of the excited state properties in LH1 complexes, depending on the mutual arrangement of the dimer units.