Good or bad vibrations for the Fenna-Matthews-Olson complex?

Electronic and vibronic coherences in the FMO complex using GPU HEOM by Kreisbeck and Kramer
Time-evolution of the coherence for the FMO complex (eigenstates 1 and 5 ) calculated with GPU-HEOM by Kreisbeck and Kramer, J. Phys. Chem Lett. 3, 2828 (2012).

Due to its known structure and relative simplicity, the Fenna-Matthews-Olson complex of green sulfur bacteria provides an interesting test-case for our understanding of excitonic energy transfer in a light-harvesting complex.

The experimental pump-probe spectra (discussed in my previous post catching and tracking light: following the excitations in the Fenna-Matthews-Olson complex) show long-lasting oscillatory components and this finding has been a puzzle for theoretician and led to a refinement of the well-established models. These models show a reasonable agreement with the data and the rate equations explain the relaxation and transfer of excitonic energy to the reaction center.

However, the rate equations are based on estimates for the relaxation and dephasing rates. As Christoph Kreisbeck and I discuss in our article Long-Lived Electronic Coherence in Dissipative Exciton-Dynamics of Light-Harvesting Complexes (arxiv version), an exact calculation with GPU-HEOM following the best data for the Hamiltonian allows one to determine where the simple approach is insufficient and to identify a key-factor supporting electronic coherence:

Wendling spectral density for FMO complex
Important features in the spectral density of the FMO complex related to the persistence of cross-peak oscillations in 2d spectra.

It’s the vibronic spectral density – redrawn (in a different unit convention, multiplied by ω2)  from the article by M. Wendling from the group of Prof. Rienk van Grondelle. We did undertake a major effort to proceed in our calculations as close to the measured shape of the spectral density as the GPU-HEOM method allows one. By comparison of results for different forms of the spectral density, we identify how the different parts of the spectral density lead to distinct signatures in the oscillatory coherences. This is illustrated in the figure on the rhs. To get long lasting oscillations and finally to relax, three ingredients are important

  1. a small slope towards zero frequency, which suppresses the pure dephasing.
  2. a high plateau in the region where the exciton energy differences are well coupled. This leads to relaxation.
  3. the peaked structures induce a “very-long-lasting” oscillatory component, which is shown in the first figure. In our analysis we find that this is a persistent, but rather small (<0.01) modulation.

2d spectra are smart objects

FMO spectrum calculated with GPU-HEOM
FMO spectrum calculated with GPU-HEOM for a 3 peak approximation of the measured spectral density, including disorder averaging but no excited state absorption.

The calculation of 2d echo spectra requires considerable computational resources. Since theoretically calculated 2d spectra are needed to check how well theory and experiment coincide, I conclude with showing a typical spectrum we obtain (including static disorder, but no excited state absorption for this example). One interesting finding is that 2d spectra are able to differentiate between the different spectral densities. For example for a a single-peak Drude-Lorentz spectral density (sometimes chosen for computational convenience), the wrong peaks oscillate and the life-time of cross-peak oscillations is short (and becomes even shorter with longer vibronic memory). But this is for the experts only, see the supporting information of our article.

Are vibrations good or bad? Probably both… The pragmatic answer is that the FMO complex lives in an interesting parameter regime. The exact calculations within the Frenkel exciton model do confirm  the well-known dissipative energy transfer picture. But on the other hand the specific spectral density of the FMO complex supports long-lived coherences (at least if the light source is a laser beam), which require considerable theoretical and experimental efforts to be described and measured. Whether the seen coherence has any biological relevance is an entirely different topic… maybe the green-sulfur bacteria are just enjoying a glimpse into Schrödinger’s world of probabilistic uncertainty.


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