On the promotion energy of an atom in a molecule I. Mayer ⇑ Chemical Research Center, Hungarian Academy of Sciences, H-1525 Budapest, P.O.Box 17,Hungary a r t i c l e i n f o Article history: Received 26 July 2010 In final form 3 September 2010 Available online 7 September 2010 Dedicated with affection to my wife, Dr. Márta Révész. a b s t r a c t Using the concept of the effective minimal basis set introduced some time ago, a proper definition is pro- posed for the atomic promotion energy in the molecule, which the atom can be assigned after the orbital deformations are introduced but before any bonding, delocalization and charge transfer effects are taken into account. The first pivoting calculations indicate that these promotion energies can be quite substan- tial and are characteristic for the chemical nature of the atom. Ó 2010 Elsevier B.V. All rights reserved. 1. Introduction Our qualitative understanding of molecular structure strongly relies on the notion of the minimal basis set. We think in terms that the atoms enter the molecules with their 1s; 2s; 2p; etc. orbi- tals (or their hybrids), while we do our calculations by using larger and larger basis sets. This contradiction can be resolved by intro- ducing an effective minimalbasis set of an atom, which can be extracted from the results of a large scale ab initio calculation using an extended basis set. That can be done either in the framework of a ‘Hilbert-space analysis’ performed in terms of the basis orbitals assigned to the different atoms [1,2] or in terms of the ‘three- dimensional’(3 D) analysis [2,3] performed in the physical space. It has always been found for SCF wave functions 1 that such an a posteriori analysis results in as many effective atomic orbitals (AO-s) with non-negligible occupation numbers as is the number of functions in a classical minimal basis set. The delimitation between the strongly occupied and the essentially unoccupied effec- tive AO-s is usually very sharp—the occupation numbers fall by orders of magnitude. 2 Originally [1] the orbitals of the effective minimal basis were found by looking for molecular orbitals for which Mulliken’s net atomic population on the given atom is maximal, or at least sta- tionary (Magnasco–Perico localization criterion [4]), and renormal- izing the intraatomic parts of the orbitals obtained—it was shown that they form an orthonormalized atomic basis [1]. The further analysis[2] revealed that these effective atomic orbitals are in a close connection with McWeeny’s natural hybrid orbitals [5], and the stationary values obtained for the net orbital populations can be considered as occupation numbers—our free program [6] also uses an algorithm based on this analogy. At the same time these orbitals are only in a quite loose connection with the concepts of ‘natural hybrid orbitals’ and ‘natural atomic orbitals’ of Weinhold and coworkers [7–9]. In the 3 D case the effective AO-s can be ob- tained [3] by applying a localization functional representing the di- rect 3 D generalization of Mulliken’s net atomic populations. Although it was obvious that the ‘effective minimal basis’ orbi- tals are essentially those by which the atom participates in the molecular wave function,and thus represent its promoted state, no actual atomic many-electron wave functions built up of these orbitals were constructed till date and, accordingly,no promotion energies were calculated by their use. Now they became of some importance in light of the ‘dilemma’ we have recently encountered [10]. We considered several related energy decomposition schemes falling in two main categories depending on the treatment of the kinetic energy integrals.One of them gives numbers which are ‘on the chemical scale’ and have quite appealing values at the equi- librium molecular geometries, but exhibit a counter-intuitive dis- tance dependence,another results in numbers with too large absolute values (very large positive one-center ‘promotion ener- gies’ and very large negative bonding energy components) but ‘cor- rect’distance behaviour. For that reason we started to consider in detail the individual effects contributing to the change of one-cen- ter energy contributions during the molecule formation—promo- tion, electron accumulation/depletion and the specialeffect of enhanced ionic terms in the Hartree–Fock wave function. The pres- ent Letter is devoted to the first of these—the promoted ‘reference state of an atom in a molecule’ and the respective promotion energy. 0009-2614/$ - see front matter Ó 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.cplett.2010.09.007 ⇑ Fax: +36 1 325 7554. E-mail address: mayer@chemres.hu 1 It was mentioned in [2] that these concepts can be mutatis mutandis applied to correlated wave functions, too, but they are out of our present scope—here we deal with SCF wave functions only. 2 Exceptions are some hypervalent atoms which can exhibit also a tail of slightly populated orbitals of basically d-character [1]. But even in such cases it is always evident which orbitals are the true valence ones and which orbitals are only participating in the ‘back-donation’. Chemical Physics Letters 498 (2010) 366–369 Contents lists available at ScienceDirect Chemical Physics Letters j o u r n a l h omepage: w w w . e l s e v i e r . c o m / l o c a t e / c p l e t t