Comparison of Three Efficient Approximate Exact-Exchange Algorithms: The Chain-of-Spheres Algorithm, Pair-Atomic Resolution-of-the-Identity Method, and Auxiliary Density Matrix Method

We compare the performance of three approximate methods for speeding up evaluation of the exchange contribution in Hartree–Fock and hybrid Kohn–Sham calculations: the chain-of-spheres algorithm (COSX; Neese, F. Chem. Phys. 2008, 356, 98–109), the pair-atomic resolution-of-identity method (PARI-K; Merlot, P. J. Comput. Chem. 2013, 34, 1486–1496), and the auxiliary density matrix method (ADMM; Guidon, M. J. Chem. Theory Comput. 2010, 6, 2348–2364). Both the efficiency relative to that of a conventional linear-scaling algorithm and the accuracy of total, atomization, and orbital energies are compared for a subset containing 25 of the 200 molecules in the Rx200 set using double-, triple-, and quadruple-ζ basis sets. The accuracy of relative energies is further compared for small alkane conformers (ACONF test set) and Diels–Alder reactions (DARC test set). Overall, we find that the COSX method provides good accuracy for orbital energies as well as total and relative energies, and the method delivers a satisfactory speedup. The PARI-K and in particular ADMM algorithms require further development and optimization to fully exploit their indisputable potential.