Integral bounds and rigorous screening algorithms for reduced scaling in explicitly correlated, semi-numerical, and non-Hermitian quantum chemistry

The properties of chemical systems can be determined computationally by solving the physical equations that govern them. This typically requires the calculation of a very large number of molecular integrals that take various forms depending on the approximations used. Most of these integrals are negligible and avoiding the calculation of negligible integrals can increase computational efficiency immensely. In this work, novel upper bounds and screening algorithms are developed for this purpose. These bounds and algorithms are applicable to a wide range of quantum chemical theories and can be used in combination with the state-of-the-art integral approximation methods that are at the heart of today’s most efficient numerical implementations. A Schwarz-type bound that captures the decay of four-center two-electron integrals due to the decreased interaction of distant charge distributions is developed and tested in the context of Hartree-Fock and range-separated density functional theory, and on four-center integrals over short-range correlation factors that arise in explicitly correlated theories. An integral partitioning procedure is developed which leads to extremely flexible upper bounds, integral partition bounds (IPBs), for molecular integrals over any number of elec- trons, any number of basis function centers, and various combinations of integral operators. The procedure allows for the inexpensive calculation of rigorous extents for charge distri- butions within these various contexts. The IPBs are completely separable into two-center factors, which capture all the sources of asymptotic decay. This allows for the formulation of scaling-consistent screening algorithms, even for the three- and four-electron integrals that arise, e.g., within explicitly correlated Møller-Plesset perturbation theory (MP2-F12). The IPBs are used to increase the efficiency and reliability of semi-numerical tech- niques for calculation of the exchange matrix in Hartree-Fock and hybrid DFT calcula- tions, where real space numerical quadrature is used to approximate electron repulsion integrals. Similarly, a framework for very efficient MP2-F12 working equations based on optimal combinations of resolution-of-the-indentity, density-fitting, and numerical quadra- ture approximations are given. They reduce the fifth-order formal scaling of the MP2-F12 method to fourth-order, while drastically reducing the cost of the initially most expensive terms. The resulting equations involve various types of sparse integral tensors which can all be treated using IPBs, and asymptotically linear scaling implementations are possible. Furthermore, a Schwarz bound is developed and implemented for screening the novel four-center two-electron integrals that arise in the treatment of resonance states using the method of complex basis functions within non-Hermitian quantum chemistry. This is a crucial step for increasing the efficiency and reach of the method