Coupling strengths between chromophores from real-time TDDFT

used in the field of natural light harvesting systems [Hu+02;CGK06;SR06;JCM14].

The coupling strengths between the chromophores are often modeled by, e.g., Förster’s dipole coupling or more involved approaches [För48;LKH99;CGK06;Hu+02]. In the following, I use real-time TDDFT with TDLDA to determine the coupling strength between two resonant sodium dimers and between theQy transitions of two resonant BChls. The sodium dimers are those already introduced in the last section with tran-sition dipoles along the z-direction and separated along the x-axis. For the two BChls I used the structure from [ONS10] rotated by45^◦ around the z-axis and subsequently by−45^◦ around the x-axis to orient the bacteriochlorin ring approximately parallel to the xy-plane. I duplicated this structure and separated the two resulting BChls along the z-axis to align them face-to-face.

From real-time TDDFT I computed the coupling strengths V^Dav mainly in the frequency domain as described above. I compare the TDDFT results to the strengths from a pure Coulomb-like coupling between the chromophores. The full Coulomb coupling matrix element can be expressed through the transition densities ρAA^∗ and ρ_DD^∗ and reads [För48;SFK09;JCM14]

V^Coul= e² 4πε0

Z Z ρ_AA^∗(r)ρ_DD^∗(r⁰)

|r−r⁰| d³rd³r⁰. (5.25) V^Coulis often approximated through a Förster-type coupling between the transition dipoles [För48;SFK09;HKK10]

V^dd =κe²|r_AA^∗||r_DD^∗|

4π0R³ . (5.26)

κ = eDD^∗ ·eAA^∗−3(eDD^∗·eR)(eAA^∗ ·eR). eAA^∗ and eDD^∗ are the unit vectors in the direction of the respective transition dipoles of donor and acceptor. e_Ris the unit

0.05

(a) Two sodium dimers parallel to each other.

0.001

Figure 5.8: Coupling strengths from transition densities, transition dipoles, and Davy-dov splitting in frequency and time domains for two sodium dimers and two BChls against the inter-chromophore distance.

vector in the direction of the relative vector between the two chromophores, i.e., ex

for the sodium system and e_z for the BChla system. In the following, I calculated the transition dipoles from the transition densities through

r0j = Z

rρ0j(r)d³r . (5.27)

Sodium dimers The different coupling strengths for the sodium system are displayed in figure 5.8a against the inter-chromophore distance R on a logarithmic scale. Per construction, the orientation factor for this system isκ= 1. As expected, the dipole-dipole strength shows theR⁻³behaviour from equation (5.26). The coupling strengths from dipole-dipole and transition density coupling approach each other for large dis-tances per construction through equation (5.27). I tested this explicitly for disdis-tances up to 1000 a₀ (not shown) to ensure the validity of the computation of V^Coul, which is described in appendix E.2.3 along with other numerical details.

For distances of less than 20 a0 the dipole-dipole coupling begins to differ signif-icantly from the full Coulomb coupling. Figure 5.9 shows the relative error of the dipole-dipole coupling compared to the full Coulomb coupling. The error behaves as a power law⁴⁵ ∝R⁻² and deceeds the 10%threshold at distances R≥20 a0.

The Davydov strengths in figure 5.8a fit perfectly to those of the pure Coulomb coupling forR >16 a₀. I tested this explicitly up to R= 50 a₀. ForR= 16 a₀,V^Coul differs from V^Dav by ≈ 3% and by ≈ 10% for R = 14 a0. The sodium dimers can still be described through two coupled chromophores for R <16 a₀ but the coupling strengths cannot be calculated as a pure Coulomb coupling between the unperturbed transition densities of the single chromophores.

Finally, forR <10 a₀ the spectrum cannot be interpreted in the sense of the two-level donor-acceptor model and a Davydov evaluation is not possible. The deviations from the donor-acceptor model already begin at R = 11 a0. They become visible

45The next higher multipole order neglected by the dipole-dipole coupling decays ∝ R⁻⁵ which explains, compared to the dipole-dipole strength∝R⁻³, the∝R⁻² behaviour of the error.

5%

10%

50%

100%

14 16 18 20 30 40 50 60

(Vdd-VCoul)/VCoul

Distance R [a₀] Na₂ - Na₂ BChl - BChl

Figure 5.9: Error of the dipole-dipole coupling approximation compared to the full Coulomb coupling for different distances.

through the unequal heights of the two spectral lines and other excitations appearing at similar energies. The corresponding data points are enclosed in brackets in figure 5.8a.

In addition, I evaluated the coupling strength between the sodium dimers in the time domain by fitting the beat frequency to the envelope ofδN_A (V^Beat) and fitting the slope (V^Slope) as shown in figure 5.6 (a1) and (a2). As expected, the beat-frequency evaluation in the time domain according to [HKK10] leads to the same results as the Davydov evaluation in the frequency domain. To get absolute values for the coupling strengths from the slope evaluation, I fixed the missing proportionality constant to the coupling strength from the beat-frequency evaluation at 50 a0, i.e., V^Slope(R = 50 a₀) =^! V^Beat(R = 50 a₀). The values of V^Slope shown in figure 5.8a are relative to this fixpoint and coincide well with the Davydov and beat-frequency evaluation. Still, for smaller distances the evaluation of the slope is less accurate since the beat becomes faster and the fitting range narrower.

Bacteriochlorophylls The coupling strength between the BChls for distances be-tween 14 a0 and 60 a0 from dipole-dipole and full Coulomb coupling are shown in figure 5.8b. The orientation factor between theQ_y transition dipoles isκ= 0.975842 for this setup. The coupling strengths behave qualitatively similar to the ones of the sodium system but are smaller. The relative error of the dipole approximation com-pared to the full Coulomb coupling is also shown in Figure 5.9. Since the BChls are much larger than the sodium dimers, the error of the dipole approximation is much larger at the same distance. Still, it shows the sameR⁻² behaviour. The error of the dipole coupling reaches the10%level at a distance of R= 50 a₀⁴⁶.

In addition to the discussed data, I computed a single coupling strength from the Davydov splitting approach for the same setup with an inter-BChl distance of R = 8 ˚A ≈ 15.12 a₀. The data point is marked in figure 5.8b. The predicted cou-pling strength of V^Dav = 0.02570 eV exceeds the one from pure Coulomb coupling V^Coul = 0.02469 eV by about 4 %. Hence, the Coulomb coupling as described by the unperturbed transition densities of the single BChls is still valid for this setup at

46Note that the calculation is done in vacuum.

R ≈15 a₀. For smaller distances, one can again expect deviations betweenV^Dav and V^Coul. Numerical details can again be found in appendix E.2.3.