Occupied-Orbital Fast Multipole Method for Efficient Exact Exchange Evaluation

We present an efficient algorithm for computing the exact exchange contributions in the Hartree–Fock and hybrid density functional theory models on the basis of the fast multipole method (FMM). Our algorithm is based on the observation that FMM with hierarchical boxes can be efficiently used in the exchange matrix construction, when at least one of the indices of the exchange matrix is constrained to be an occupied orbital. Timing benchmarks are presented for alkane chains (C₄₀₀H₈₀₂ and C₁₅₀H₃₀₂), a graphene sheet (C₁₅₀H₃₀), a water cluster [(H₂O)₁₀₀], and a protein Crambin (C₂₀₂H₃₁₇O₆₄N₅₅S₆). The computational cost of the far-field exchange evaluation for Crambin is roughly 3% that of a self-consistent field iteration when the multipoles up to rank 2 are used.