The versatility of Cholesky decomposition of electron repulsion integrals

Over the past three decades, the resolution-of-the-identity (RI) approximation of electron repulsion integrals (ERIs) over Gaussian-type orbitals has become a stan- dard component of accelerated and reduced-scaling implementations of first-principles electronic-structure methods. For the first two decades, the closely related approach based on Cholesky decomposition (CD) of the ERIs lived a more secluded life, fo- cusing mainly on high-accuracy methods such as coupled-cluster theory and multi- configurational approaches. In the past decade, however, the CD technique has been increasingly deployed across quantum chemistry as a numerically robust, computation- ally efficient, and highly accurate approach to on-the-fly generation of auxiliary basis sets for the RI approximation. Starting with a summary of the basic theory underpin- ning both the CD and RI approximations, we provide a brief and largely chronological review of the evolution of the CD approach from its birth in 1977 as a purely numerical procedure for handling ERIs to its current state as the perhaps most powerful on-the-fly generator of auxiliary basis sets.