sp 3 Hybrid orbitals and ionization energies of methane from PNOF5 Jon M. Matxain a,c,⇑ , Mario Piris b,c,d , Jose M. Mercero a , Xabier Lopez a,c , Jesus M. Ugalde a,c a Kimika Fakultatea, Euskal Herriko Unibertsitatea UPV/EHU, P.K. 1072, 20080 Donostia, Euskadi, Spain b Faculty of Chemistry, University of the Basque Country UPV/EHU, P.K. 1072, 20080 Donostia, Euskadi, Spain c Donostia International Physics Center (DIPC), 20018 Donostia, Euskadi, Spain d IKERBASQUE, Basque Foundation for Science, 48011 Bilbao, Euskadi, Spain article info Article history: Received 19 January 2012 In final form 14 February 2012 Available online 23 February 2012 abstract The Piris Natural Orbital Functional, in its 5th version (PNOF5), has been used to obtain the localized bonding picture of methane that agrees well with the Pauling proposition of hybrid orbitals, and to pre- dict the two-peak structure of its vertical valence ionization energy spectrum. The ionization energies have been calculated using the extended Koopmans’ theorem. The calculated PNOF5 values are in good agreement with the corresponding experimental data. It is demonstrated that it is possible to reconcile sp 3 orthogonal hybrid orbitals with the experimental vertical ionization energies observed for the CH 4 within the framework of the natural orbital functional theory. Ó 2012 Elsevier B.V. All rights reserved. Methane is a molecule of T d symmetry, where the central car- bon atom has a tetrahedral conformation, with each hydrogen atom placed at the vortexes of the tetrahedron. All CAH bonds are equivalent, i.e., have equal bond-lengths: 1.087 Å. In addition, this molecule shows two peaks in the vertical valence ionization energy (IE) spectrum, one at 14.4 eV and the other at 23 eV [1]. In most chemical textbooks, the tetrahedral structure of meth- ane is explained by considering that the central atom has sp 3 hybridization. Note that neither of the 2s and 2p orbitals of the car- bon atom are oriented towards the position of hydrogens in the tetrahedron. Therefore, no strong overlap can take place between these carbon orbitals and the hydrogen orbitals, according to the Lewis model. This problem was solved in 1931 by Linus Pauling, who proposed [2] that sp 3 hybrid orbitals built as linear combina- tion of these 2s and 2p orbitals would fulfill this requirement. Fur- thermore, inspired in the empirical Lewis model, Heither and London proposed [3] the Valence Bond (VB) theory, considering that covalent bonds are a result of the strong overlap between atomic orbitals of neighbor atoms. In this vein, each sp 3 hybrid orbital of carbon atom would overlap strongly with the 1s of hydrogen atoms leading therefore to four equivalent CAH bonds [4]. This model accounts nicely for the tetrahedral structure of methane. The simplest one-electron model for ionization is based on the Koopmans’ theorem (KT) [5], which states that if an electron is re- moved from a canonical Hartree–Fock (HF) [6,7] orbital, then the IE is the negative of the corresponding orbital energy. According to KT, removing one electron from either of such equivalent sp 3 hybrid orbitals could lead to only one peak in the valence vertical ionization energy spectrum of methane. Based on this hypothesis, some authors have claimed [8] that the hybrid orbital idea should be removed from the chemistry textbooks, and chemistry should be exclusively explained based on delocalized orbitals. Certainly, the molecular orbital (MO) theory [9,10] combines the unhybridized 2s and 2p orbitals of carbon with the four 1s orbitals of hydrogens to form eight delocalized MOs. According to the symmetry of the molecule, the lowest lying valence orbital has A 1 symmetry, while the next level is threefold degenerate state of T 2 symmetry. These orbitals explain perfectly both peaks of the ionization spectrum according to the KT. Nevertheless, the interpretation of bonding delocalized orbitals is not trivial. Since the HF energy is invariant under unitary transformations of the orbitals, many a posteriori localization schemes have been pro- posed in the literature [11–14] in order to bring closer the MO the- ory to the chemical intuition. However, the resulting orbitals are not longer the canonical orbitals and can be neither associated to KT IEs. It is worth noting that VB theory can predict the two peaks of the IE spectrum by considering different resonant structures for the CH þ 4 , which leads to one A 1 and threefold degenerated T 2 states, accounting for the two observed peaks [4]. There is hence no con- tradiction between the VB theory and the IE spectrum of methane. Nonetheless, non-orthogonal orbitals are used by the VB method. Consequently, some interpretative problems arise owing to the non-orthogonality of the VB structures, which is also the bottle- neck for efficient developments in VB theory. One possible way is to employ orthogonalized atomic orbitals, however, the number of structures that must be admitted rapidly becomes vast [15]. In this vein, the Piris Natural Orbital Functional [16–18], in its 5th version (PNOF5) [19,20], might represent a viable alternative 0009-2614/$ - see front matter Ó 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.cplett.2012.02.041 ⇑ Corresponding author at: Kimika Fakultatea, Euskal Herriko Unibertsitatea UPV/EHU, P.K. 1072, 20080 Donostia, Euskadi, Spain. E-mail address: jonmattin.matxain@ehu.es (J.M. Matxain). Chemical Physics Letters 531 (2012) 272–274 Contents lists available at SciVerse ScienceDirect Chemical Physics Letters journal homepage: www.elsevier.com/locate/cplett