On the use of spatial symmetry in atomic‐integral calculations: An efficient permutational approach

Abstract

AbstractsThe minimal number of independent nonzero atomic integrals that occur over arbitrarily oriented basis orbitals of the form % ( r ) . Yl,(n) is theoretically derived. The corresponding method can be easily applied to any point group, including the molecular continuous groups C,, and Dmh. O n the basis of this (theoretical) lower bound, the efficiency of the permutational approach in generating sets of independent integrals is discussed. It is proved that lobe orbitals are always more efficient than the familiar Cartesian Gaussians, in the sense that GLOS provide the shortest integral lists. Moreover, it appears that the new axial GLOS often lead to a number of integrals, which is the theoretical lower bound previously defined. With AGLOS, the numbers of two-electron integrals to be computed, stored, and processed are divided by factors 2.9 (NH,), 4.2 (C5Hs), and 3.6 (C&) with reference to the corresponding CGTOS calculations. Remembering that in the permutational approach, atomic integrals are directly computed without any four-indice transformation, it appears that its utilization in connection with AGLOS provides one of the most powerful tools for treating symmetrical species.Le nornbre minimal d'inttgrales atomiques indtpendantes et non nulles qui apparaissent entre des orbitales d e base d e la forme % ( I ) . Y,,@) et ayant des orientations arbitraires, est dtrivt thtoriquement. La rnithode correspondante peut h e aistment appliqute a n'importe quel groupe ponctuel, y compris les groupes moltculaires continus C,, et Dmh. A partir de cette borne inferieure thtorique, I'efficacitb d e la mtthode permutationnelle de gintration des ensembles d'inttgrales independantes, est discutte. O n dtmontre que les orbitales "lobes" sont toujours plus efficaces que les Gaussiennes Carttsiennes, dans la mesure ou les premieres fournissent les listes d'inttgrales les plus courtes. D e plus, il apparait que les nouvelles fonctions "axiales" (AGLO) conduisent souvent a un nombre d'inttgrales qui est la borne thtorique infirieure difinie plus haut. Avec les AGLOS, les nombres d'inttgrales bitlectroniques qui doivent &tre calcultes, conservtes et utilisies par la suite, sont divists par des facteurs de 2.9 (NH3). 4.2 (C5Hs), et 3.6 (C6H6) par rapport aux calculs correspondants effectuts avec des gaussiennes cartisiennes. Si I'on se souvient que dans I'approche permutationnelle les inttgrales atomiques sont calcultes directement, sans aucune transformation quatre indices, il apparait que I'utilisation de cette mtthode avec des fonctions de base de type AGLO, constitue un des outils les plus puissants permettant d e traiter les rnoltcules symttriques.

Axial gaussian‐lobe orbitals: Optimization of highly angularly dependent functions

“…This is precisely the case in the Gaussian-lobe formalism and we have defined the lobe realization of AAOS [l], that we called axial Gaussian-lobe orbital (AGLO). The main practical advantages of such functions are (ii) The molecular symmetry (if even) can be introduced in a powerful manner [3] via a permutational method, yielding to two-electron integral lists of minimal length.…”

“…(ii) The molecular symmetry (if even) can be introduced in a powerful manner [3] via a permutational method, yielding to two-electron integral lists of minimal length.…”