Universal R12 suited basis sets for atoms from lithium to fluorine STANISLAV KEDZ ˇ UCHy, JOZEF NOGAyz* and PIERRE VALIRON} yInstitute of Inorganic Chemistry, Slovak Academy of Sciences, SK-84536 Bratislava, Slovakia zDepartment of Physical and Theoretical Chemistry, Faculty of Natural Sciences, Comenius University, SK-84215 Bratislava, Slovakia }Laboratoire d’Astrophysique, UMR 5571 CNRS, Universite´ Joseph Fourier, BP 53, F-38041 Grenoble Cedex 9, France (Received 5 August 2004; in final form 28 September 2004) Extended basis sets suitable for explicitly correlated R12 calculations have been constructed for the first row atoms from Li to F. These should have a fairly universal character, being defined as a compromise to describe atomic ground states and both the positive and negative ions. At the same time, the potential to create numerical instabilities is minimized. Energies with the final ‘universal’ basis sets differ at the 10 5 –10 6 E h level from the ‘R12 optimal’ energies for individual systems and provide both very good ionization potentials and electron affinities. 1. Introduction During the last two decades great progress has been achieved in development of high precision ab initio methods that treat the Coulomb singularity explicitly via the wave function ansatz. Various approaches to this problem were some time ago extensively reviewed by Klopper [1] and current developments are summarized in a recent monograph edited by Rychlewski [2]. Among these methods, the so-called R12 theories, based on the idea of Kutzelnigg [3], are special through the fact that one essentially works with one electron basis outset, as used in conventional ab initio configuration interaction type calculations. Terms linear in the inter-electronic coordinate ðr 12 Þ are explicitly introduced in the wave function expansion, yet calculation of difficult many electron integrals can be avoided within the so-called standard approximation [4]. In short, the essence of R12 methods can be regarded as a generalization of the finding of Kutzelnigg [3], who for the two electron helium atom showed that in order to satisfy the electron–electron cusp condition [5] it was sufficient to extend the usual (conventional) wave function expansion by augmenting the reference deter- minant by a single linear r 12 term. More generally, for n electrons with the operator of inter-electronic coordi- nates ^ r ¼ P n k>l r kl Kutzelnigg’s ansatz can be written as jCi¼ c ^ rjFiþ ^ OjFi, ð1Þ where jCi is the desired r 12 -dependent final wave function. ^ O is an arbitrary wave operator that trans- forms the reference jFi to a ‘conventional’, that is, configuration interaction type expansion. The factor c in front of ^ r should theoretically be 0.5, to ensure the proper value of the wave function derivative for r 12 ! 0 as required by the electron–electron cusp condition. If jFi is a Slater determinant, by the action of ^ r on jFi one creates a wave function consisting of n ðn  1Þ=2 determinants in which a particular product of (occupied) orbitals ’ i ðkÞ ’ j ðl Þ has been replaced by r kl ’ i ðkÞ ’ j ðl Þ. Further generalization enables one to introduce an operator ( ^ R) which replaces the products ’ i ðkÞ ’ j ðl Þ by different pairs of occupied orbitals r kl ’ m ðkÞ ’ n ðl Þ, since the Pauli principle is not violated. At the same time ^ R can include different variational parameters for such R12 configurations and also takes care for outprojection of the contributions that overlap with the conventional configuration space. ^ R can be used together with a ‘conventional’ excitation operator in the exponential ansatz for the coupled cluster R12 (CC-R12) wave function [6–8]. The latter was first used a dozen years ago [9], being inspired by the orbital invariant second- order Møller–Plesset R12 approach (MP2-R12) by Klopper [10]. Early R12 theories were not orbital invariant [11, 12]. Various R12 ansatz can slightly differ [6, 10, 13, 14], as well as the approximate evaluation of matrix elements pertinent to the method. Three and four electron integrals that would formally arise can be avoided by means of a standard approximation [4, 6] which assumes a fair saturation of the one electron basis to ensure the * Corresponding author. email: uachnoga@savba.sk Molecular Physics, Vol. 103, No. 6–8, 20 March–20 April 2005, 999–1005 Molecular Physics ISSN 0026–8976 print/ISSN 1362–3028 online # 2005 Taylor & Francis Group Ltd http://www.tandf.co.uk/journals DOI: 10.1080/00268970412331332952