The versatility of the Cholesky decomposition in electronic structure theory

Abstract

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

Automatic Generation of Accurate and Cost-Efficient Auxiliary Basis Sets

“…Given that the increase in the size of the OBS is accompanied by an increase in the maximum angular momentum of the corresponding ABS ( eq 9 ), a likely explanation for the decreasing error with increasing OBS is that the one-center approximation that is the cornerstone of the DF/RI method 12 is becoming more accurate: energetically important two-center products of OBS functions are captured to better and better accuracy, when the ABS becomes larger and larger. Increasing the cardinal number of the OBS adds higher polarization shells in the autogenerated ABS and also adds more radial functions to the lower angular momenta in the ABS.…”

“…In the DF/RI approach, the two-electron integrals of the OBS are approximated with the help of an auxiliary basis set (ABS) as 6 where A and B are functions in the ABS, and ( A | B ) −1 denotes the A , B element of the inverse of the two-index Coulomb overlap matrix ( A | B ). Traditionally, ABSs are manually optimized for ground-state energy calculations for each OBS for each level of theory, 11 , 12 but some automatic algorithms for generating an ABS from the given OBS have also been suggested; 13 − 18 see ref ( 12 ) for a recent review.…”

“… 24 , 25 Choosing the basis functions by a pivoted Cholesky decomposition of the (Coulomb) overlap matrix—which is a so-called Gram matrix—is analogous to using an optimal Gram–Schmidt orthogonalization. 12 …”

Revisiting Gauge-Independent Kinetic Energy Densities in Meta-GGAs and Local Hybrid Calculations of Magnetizabilities

“…Although one needs to be careful about mixing auxiliary basis sets for different families, we tested the accuracy of this RI-J approximation with a few of the functionals considered in this work, and the resulting magnetizabilities with the RI-J approximation coincided with values obtained without it to the reported number of digits, which is consistent with results recently computed for NMR shieldings. , We note that accurate auxiliary basis sets with controllable accuracy for RI calculations can nowadays be easily generated automatically , and recommend such autogenerated auxiliary basis sets to be used when tailored basis sets are not available. We refer the reader to ref for a recent review of further automatic auxiliary basis generation techniques.…”

“… 105 , 106 We note that accurate auxiliary basis sets with controllable accuracy for RI calculations can nowadays be easily generated automatically 107 , 108 and recommend such autogenerated auxiliary basis sets to be used when tailored basis sets are not available. We refer the reader to ref ( 109 ) for a recent review of further automatic auxiliary basis generation techniques.…”