Resolution of Identity and Chain of Spheres
Hartree-Fock theory plays a central role in Quantum Chemistry. It serves on the one hand as a basis for electron correlation approaches while on the other it is also essential for density functional theory. The evaluation of the HF exchange term is a rather time consuming operation for larger molecules and as a result for a wide range of chemically or biologically interesting systems. This motivates the development of various eﬃcient approaches to exchange evaluation. The Chain of Spheres exchange (COSX) algorithm, one of the most eﬃcient methods available for this purpose today, has been developed in the institute since 2009. The algorithm has favorable scaling and contraction properties which makes it especially suitable for larger systems. In combination with the resolution of identity for the Coulomb term (RI-J) it enables large scale HF and DFT calculations. To realize the full potential of any approximation to the SCF energy, it is inevitable to obtain the corresponding gradients and the Hessian. These can then be used in geometry optimizations and to calculate various properties. Gradients are determined in an approximate manner for the COSX algorithm, while for Hessian calculation its main advantage is in the CP-SCF step. The COSX method can also be used beneﬁcially in correlation methods. In these cases the SCF density is replaced by an eﬀective density in which the amplitudes of various excited determinants are contracted with molecular orbital coeﬃcients. Thus, the evaluation of the so called singles Fock term in the LPNO-CCSD method, and the external exchange term in the SCS-MP3, CCSD and EOM-CCSD methods can both be signiﬁcantly accelerated without signiﬁcant loss of accuracy. In the former case the contraction involves the singles, in the latter case the doubles amplitudes, but in both cases the approximated terms are form the bottleneck of the corresponding method.