Model Hamiltonian

With the assumption that only Q_y excitation of the BChl c molecule is of interest, we will adopt the framework of a single excitation Frenkel exciton Hamiltonian, which is here given in the site basis:³⁴

Here, and B_n are the creation and annihilation operators describing an excitation on the nth BChl c molecule. N is the total number of molecules. The first term represents the transition energy of the monomer evaluated as a sum of the energy of the Q_y transition of BChl c in vacuum ω₀ and the energy shift that arises due to interaction with the local environment, Δω_n(t). The second term gives the excitonic couplings J_mn between the different BChl c molecules. For ω₀ we chose the value of 15390 cm^–1, which corresponds to the energy of the Q_y transition of the BChl c monomer in methanol.³⁵ A schematic description of the Bchl c that is surrounded with other chromophores within the aggregate is shown in Figure ​Figure11b). The energy shifts are calculated as the Coulomb intermolecular interaction:³⁶

Here, n labels the specific BChl c molecule. m runs over all M atoms in this molecule, and l runs over L atoms on the surrounding molecules. and are partial charges of the atoms of the molecule of interest that reflect its excited and ground state properties. are the ground state charges on the atoms of the molecules in the local surroundings. The distance between the atoms is given by R_ml(t). In this way, we include the difference in the effects that electrostatic potentials of the local environment have on the ground, compared to the excited states, of the molecules in the aggregate. The partial charges were determined from quantum chemistry calculations as described in the section Quantum Chemical Parametrization. A 20 Å cutoff excluding molecules with a Mg–Mg distance with the central molecules larger than this was used for the electrostatic interactions. This is identical to the cutoff used in previous calculations of the same type on other systems.^24,37 The dependence on the local structure and the dynamic fluctuations of this term lead to what is referred to as diagonal disorder in the Hamiltonian.

We describe the excitonic coupling in eq 1 using the point dipole approximation similarly as in the previous chlorosome study:²⁵

Here, is the transition-dipole moment vector for molecule n, and ) is the distance vector between the centers of the two molecules, which we define with the position of the magnesium atoms. No explicit dielectric screening was included following the protocol of ref (25). We used a transition-dipole moment of 5.48 D in accordance with findings for BChl c monomers in methanol solution,³⁵ which may implicitly account for some of the dielectric screening in the couplings. The direction of the transition-dipole moment was defined to be along the vector between the nitrogen atoms conventionally named N_A and N_C as shown in Figure ​Figure11. This dipole is identical to that used in previous studies on static structures.^12,13,38 The dispersion of molecular orientations and distances within the aggregates will lead to a dynamic distribution in the excitonic couplings, also denoted as off-diagonal disorder.^24,39 For characterization of this disorder we will use the coupling strength which is the signed sum over all pairwise interactions involving a specific BChl c molecule:

S_n(t) gives a single fluctuating quantity per molecule which makes analysis simpler. Dispersion in the exciton couplings will influence the width and the position of the excitonic band.^24,40 We note that the conditions for the validity of the point dipole coupling approximation is possibly not fully met as the chromophores are closely packed. Applying a more accurate extended dipole²⁴ or TrESP⁴¹ coupling approach may improve the current description. Inclusion of such a coupling scheme for the description of excitonic dynamics is computationally demanding, especially for large systems such as chlorosomes and presents a challenge for future studies. As the present coupling model shows good agreement with experiments we do not expect improving the coupling model to affect the conclusions of this paper.