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Assessment of the Ab Initio Bethe-Salpeter Equation Approach for the Low-Lying Excitation Energies of Bacteriochlorophylls and Chlorophylls

Bacteriochlorophyll and Chlorophyll molecules are crucial building blocks of the photosynthetic apparatus in bacteria, algae and plants. Embedded in transmembrane protein complexes, they are responsible for the primary processes of photosynthesis: excitation energy and charge transfer. Here, we use ab initio many body perturbation theory within the $GW$ approximation and Bethe-Salpeter equation (BSE) approach to calculate the electronic structure and optical excitations of Bacteriochlorophylls $a$, $b$, $c$, $d$ and $e$ and Chlorophylls $a$ and $b$. We systematically study the effects of structure, basis set size, partial self-consistency in $GW$, and the underlying exchange-correlation approximation, and compare our calculations with results from time-dependent density functional theory, multireference RASPT2 and experimental literature results. We find that optical excitations calculated with $GW$+BSE are in excellent agreement with experimental data, with an average deviation of less than 100\,meV for the first three bright excitations of the entire family of (Bacterio)chlorophylls. Contrary to state-of-the-art TDDFT with an optimally-tuned range-separated hybrid functional, this accuracy is achieved in a parameter-free approach. Moreover, $GW$+BSE predicts the energy differences between the low-energy excitations correctly, and eliminates spurious charge transfer states that TDDFT with (semi)local approximations is known to produce. Our study provides accurate reference results and highlights the potential of the $GW$+BSE approach for the simulation of larger pigment complexes.