Nuclear quantum effects slow down the energy transfer in biological light-harvesting complexes

Abstract

We assess how quantum-mechanical effects associated with high-frequency chromophore vibrations influence excitation energy transfer in biological light-harvesting complexes. After defining a classical nuclear limit that is consistent with the quantum-classical equilibrium, we include nuclear quantum effects through a variational polaron transformation of the high-frequency vibrational modes. This approach is validated by comparison with fully quantum-mechanical benchmark calculations and applied to three prototypical light-harvesting complexes. For light-harvesting complex 2 of purple bacteria, the inter-ring transfer is 1.5 times slower in the quantum treatment than in the classical treatment. For the Fenna-Matthews-Olson complex, the transfer rate is the same in both cases, whereas for light-harvesting complex II of spinach, the transfer is 1.7 times slower in the quantum treatment. The effect is most pronounced for systems with large excitonic energy gaps and strong vibronic coupling to high-frequency modes. In all cases, nuclear quantum effects are found to be unimportant for the directionality of energy transfer.

Fig. 1.. Exciton energy transfer in LH2.

(A) LH2 structure, composed of the B850 ring (red, sites 1 to 16) and the B800 ring (blue, sites 17 to 24) embedded in a protein scaffolding (shaded). (B) Density of bath reorganization energies, $\mathrm{\Lambda }\left(\mathrm{\omega }\right)={\sum }_k\mathrm{\hslash }{\mathrm{\omega }}_k{g}_k^2\mathrm{\delta }(\mathrm{\hslash }\mathrm{\omega }-\mathrm{\hslash }{\mathrm{\omega }}_k)$ . This dimensionless quantity is partitioned into a low-frequency part that contains the solvent and discrete modes below $k_{\mathrm{B}}T$ (red), a high-frequency part containing discrete modes between $k_{\mathrm{B}}T$ and $\mathrm{\hslash }{\mathrm{\omega }}_{\text{max}}$ (teal), and an off-resonant part containing discrete modes above $\mathrm{\hslash }{\mathrm{\omega }}_{\text{max}}$ (purple). For visual purposes, each discrete mode has been broadened by γ = 10 cm⁻¹ such that its contribution to $\mathrm{\Lambda }\left(\mathrm{\omega }\right)$ is $\frac{2}{\mathrm{\pi }}{g}_k^2{\mathrm{\omega }}_k^2\frac{\mathrm{\gamma \omega }}{{({\mathrm{\omega }}_k^2-{\mathrm{\omega }}^2)}^2+{\mathrm{\omega }}^2{\mathrm{\gamma }}^2}$ . (C) Time-dependent population of the B850 ring after an initial excitation on site 17 in the B800 ring. Shades around lines indicate two standard errors in the mean. (D) Time-dependent populations of a few representative sites.

Fig. 2.. Energy transfer in FMO.

(A) Site labeling. (B) Spectral density plotted as in Fig. 1. (C) The MASH population dynamics is essentially the same with or without the VPT.

Fig. 3.. Exciton dynamics in the major LHCII complex of spinach.

(A) Structure (49) showing the assignments of Chla and Chlb chlorophylls in red and blue, respectively, and the orientation relative to the stromal and lumenal layers. (B) Spectral density plotted as in Fig. 1. (C) Time-dependent population of the Chla sites (dotted lines are biexponential fits).