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First-principles investigation of organic photovoltaic materials C$_{60}$, C$_{70}$, [C$_{60}$]PCBM, and bis-[C$_{60}$]PCBM using a many-body $G_0W_0$-Lanczos approach

We present a first-principles investigation of the excited-state properties of electron acceptors in organic photovoltaics including C$_{60}$, C$_{70}$, [6,6]-phenyl-C$_{61}$-butyric-acid-methyl-ester ([C$_{60}$]PCBM), and bis-[C$_{60}$]PCBM using many-body perturbation theory within the Hedin's $G_0W_0$ approximation and an efficient Lanczos approach. Calculated vertical ionization potentials (VIP) and vertical electron affinities (VEA) of C$_{60}$ and C$_{70}$ agree very well with experimental values measured in gas phase. The density of states of all three molecules is also compared to photoemission and inverse photoemission spectra measured on thin-films, exhibiting a close agreement - a rigid energy-gap renormalization owing to intermolecular interactions in the thin-films. In addition, it is shown that the low-lying unoccupied states of [C$_{60}$]PCBM are all derived from the highest-occupied molecular orbitals and the lowest-unoccupied molecular orbitals of fullerene C$_{60}$. The functional side group in [C$_{60}$]PCBM introduces a slight electron transfer to the fullerene cage, resulting in small decreases of both VIP and VEA. This small change of VEA provides a solid justification for the increase of open-circuit voltage when replacing fullerene C$_{60}$ with [C$_{60}$]PCBM as the electron acceptor in bulk heterojunction polymer solar cells.