The versatility of the Cholesky decomposition in electronic structure theory

• Published in: Wiley interdisciplinary reviews. Computational molecular science Vol. 14; no. 1
• Main Authors: Pedersen, Thomas Bondo, Lehtola, Susi, Fdez. Galván, Ignacio, Lindh, Roland
• Format: Journal Article
• Language: English
• Published: Hoboken, USA Wiley Periodicals, Inc 01.01.2024
• Subjects: auxiliary basis sets; Cholesky decomposition; reduced scaling; two‐electron integrals
• ISSN: 1759-0876; 1759-0884
• DOI: 10.1002/wcms.1692

The resolution‐of‐the‐identity (RI) or density fitting (DF) approximation for the electron repulsion integrals (ERIs) has become a standard component of accelerated and reduced‐scaling implementations of first‐principles Gaussian‐type orbital electronic‐structure methods. The Cholesky decomposition (CD) of the ERIs has also become increasingly deployed across quantum chemistry packages in the last decade, even though its early applications were mostly limited to high‐accuracy methods such as coupled‐cluster theory and multiconfigurational approaches. Starting with a summary of the basic theory underpinning both the CD and RI/DF approximations, thus underlining the extremely close relation of the CD and RI/DF techniques, we provide a brief and largely chronological review of the evolution of the CD approach from its birth in 1977 to its current state. In addition to being a purely numerical procedure for handling ERIs, thus providing robust and computationally efficient approximations to the exact ERIs that have been found increasingly useful on modern computer platforms, CD also offers highly accurate approaches for generating auxiliary basis sets for the RI/DF approximation on the fly due to the deep mathematical connection between the two approaches. In this review, we aim to provide a concise reference of the main techniques employed in various CD approaches in electronic structure theory, to exemplify the connection between the CD and RI/DF approaches, and to clarify the state of the art to guide new implementations of CD approaches across electronic structure programs. This article is categorized under: Electronic Structure Theory > Ab Initio Electronic Structure Methods Software > Quantum Chemistry Quantum Computing > Theory Development A graphical presentation of the truncated Cholesky decomposition. Only the the light‐shaded part of the matrix is approximated.