Improving the Efficiency of Electrostatic Embedding Using the Fast Multipole Method

This paper reports the improvement in the efficiency of embedded-cluster model (ECM) calculations in ORCA thanks to the implementation of the fast multipole method. Our implementation is based on state-of-the-art algorithms and revisits certain aspects, such as efficiently and accurately handling the extent of atomic orbital shell pairs. This enables us to decompose near-field and far-field terms in what we believe is a simple and effective manner. The main result of this work is an acceleration of the evaluation of electrostatic potential integrals by at least one order of magnitude, and up to two orders of magnitude, while maintaining excellent accuracy (always better than the chemical accuracy of 1 kcal/mol). Moreover, the implementation is versatile enough to be used with molecular systems through QM/MM approaches. The code has been fully parallelized and is available in ORCA 6.0.