Simplifications in the generation and transformation of two‐electron integrals in molecular calculations

Abstract

AbSbCtSApproximate numerical linear dependence among the columns of the two-electron integral matrix and the positiveness of the Coulomb operator are exploited in order to reduce the number of integrals that need to be calculated when a numerical accuracy is given by the machine in use or by the choice of the investigator. Numerical results presented indicate that the method leads to an algorithm for carrying out the two-electron integral four-index transformation which in practice can be achieved in a small fraction (-4-$) of the time required to generate the integrals by one of the fastest available Gaussian integral programs, Almlof's MOLECULE. This effectively removes one of the major bottlenecks of computational quantum chemistry.La dtpendance lintaire numtrique approximative entre les colonnes de la matrice formke des inttgrales B deux tlectrons et la positivitt de I'opirateur de Coulomb ont t t e exploittes pour rtduire le nombre d'inttgrales a calculer pour une prtcision numtrique donnte par I'ordinateur ou par le choix du chercheur. On prtsente des rtsultats numkriques, qui indiquent que notre mtthode mtne a un algorithme pour la transformation a quatre indices des intkgrales a deux tlectrons, qui ne requiert qu'une petite fraction (-$-$) du temps nkcessaire pour gkntrer les mbmes intkgrales par un des programmes pour des orbitales gaussiennes les plus rapides, le MOLECULE de Almlof. Ceci enlttve d'une facon efficace une des difficultts principales des calculs de la chimie quantique.Die approximative numerische lineare Abhangigkeit zwischen den Spalten der von Zweielektronenintegralen gebildeten Matrix und die Positivitat des Coulomboperators werden ausgenutzt urn die Anzahl der Integrale zu reduzieren, die fur eine von der Rechenmaschine oder dem Forscher gegebene Genauigkeit gebraucht werden. Die vorgelegenen numerischen Ergebnisse zeigen, dass die Methode zu einem Algorithmus fur die Vierindextransformation der Zweielektronenintegrale fuhrt, der in der Praxis ausgefuhrt werden kann in einem Bruchteil (-4-4) der &it, die mit einem der schnellsten gausschen Integralprogramme, dem Almlof'schen MOLECULE, fur die Erzeugung der Integrale gebraucht wird. Dadurch wird eine der wichtigsten Schwierigkeiten in quantenchemischen Berechnungen beseitigt.

Gaussian basis sets for molecular applications

“…Beebe and Linderberg demonstrated that CD of two-electron integrals can also be thought of as the determination of a linearly dependent set of atomic orbital product functions, [196] which are commonly referred to as the Cholesky basis, and it is very important to note that using a Cholesky basis as an ABS makes CD and DF formally equivalent. Early CD implementations were significantly more expensive than DF as they still required four-center two-electron integrals to be evaluated, and the Cholesky basis had a strong dependence on molecular geometry.…”

Atomic Cholesky decompositions: A route to unbiased auxiliary basis sets for density fitting approximation with tunable accuracy and efficiency

“…On the other hand, if we consider an isolated atom, the CD of its ERI matrix 36 provides an upper bound for the value of ⌬ min , Eq. ͑3͒,…”

“…35 With this spirit and with the additional aim at rectifying the drawbacks of the standard auxiliary basis sets, three of the present authors proposed 63 to merge the DF approximation with the Cholesky decomposition ͑CD͒ approximation of Beebe and Linderberg. 36 The latter had been around for about 30 years as the forgotten twin brother of the DF method. Although CD presented some advantages over the DF approximation, strict error control in particular, it did not become popular.…”

Polarizability and optical rotation calculated from the approximate coupled cluster singles and doubles CC2 linear response theory using Cholesky decompositions

“…1,2 The algorithm is designed to avoid storage, in core as well as on disk, of four-index intermediates along the lines deviced by Hättig and co-workers [10][11][12][13][14] for the RI-CC2 method. Through examples involving as many as 2.8 billion coupled cluster doubles amplitudes, we have demonstrated the large-scale applicability of the implementation.…”