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Quasiparticle Self-Consistent $GW$-Bethe-Salpeter equation calculations for large chromophoric systems

The $GW$-Bethe-Salpeter Equation (BSE) method is promising for calculating the low-lying excited states of molecular systems. So far, it has only been applied to rather small molecules, and in the commonly implemented diagonal approximations to the electronic self-energy it depends on a mean-field starting point. We describe here an implementation of the self-consistent and starting-point independent quasiparticle self-consistent (qs$GW$)-BSE approach which is suitable for calculations on large molecules. We herein show that eigenvalue-only self-consistency leads to an unfaithful description of certain excitonic states for Chlorophyll dimers while the qs$GW$-BSE vertical excitation energies (VEE) are in excellent agreement with spectroscopic experiments for Chlorophyll monomers and dimers measured in the gas phase. On the other hand, VEEs from time-dependent density functional theory calculations tend to disagree with experimental values and using different range-separated hybrid (RSH) kernels changes the VEEs by up to 0.5 eV. We use the new qs$GW$-BSE implementation to calculate the lowest excitation energies of the six chromophores of the photosystem II (PSII) reaction center (RC) with nearly 2000 correlated electrons. Using more than 11000 (6000) basis functions, the calculation could be completed in less than 5 (2) days one a single modern compute node. In agreement with previous TD-DFT calculations using RSH kernels on models that do also not include environment effects, our qs$GW$-BSE calculations only yield states with local character in the low-energy spectrum of the hexameric complex. Earlier work with RSH kernels has demonstrated that the protein environment facilitates the experimentally observed interchromophoric charge transfer. Therefore, future research will need to combine correlation effects beyond TD-DFT with an explicit treatment of environment electrostatics.