Multicenter orbitals

The relationship between alternative separable solutions of the Coulomb problem in momentum space is exploited in order to obtain hydrogenic orbitals which are of interest for Sturmian expansions of use in atomic and molecular structure calculations and for the description of atoms in fields. In view of their usefulness in problems where a direction in space is privileged, as when atoms are in an electric or magnetic field, we refer to these sets as to the Stark and Zeeman bases, as an alternative to the usual spherical basis, set. Fock s projection onto the surface of a sphere in the four dimensional hyperspace allows us to establish the connections of the momentum space wave functions with hyperspherical harmonics. Its generalization to higher spaces permits to build up multielectronic and multicenter orbitals. 

The localized multicenter orbital holds a certain intermediate ground, and is particularly useful when there are more valence orbitals then electrons in a molecule or transition state. First widely used in the boron hydrides and car-boranes, these three-center and multicenter orbitals provide a coherent and consistent description of much of the structure and chemistry of the upper left side of the periodic table, and of the interactions of metallic ions with other atoms or molecules. 

Each boron or carbon atom has sp hybridization. One of the hybridized orbitals of each atom forms a bond with the terminal hydrogen atom, and the remaining three orbitals (one sp and two p orbitals) give multicenter orbitals of the cluster grouping. The molecular orbital theory shows that borane polyhedra having n vertices have such symmetry that 3n atomic orbitals of the cluster possessing n skeletal atoms form +1 bonding molecular orbitals. ... 

Because of multicenter orbital overlap, the two metal oxygen bonds in II are stronger than in I. On the other hand, I has two very weak four-electron antibonds (due to the overlap of tt with and d ) while II has one very strong four-... 

The boron framework is dissected into polyhedra which are connected to each other by localized bonds, occasionally with the use of multicentered orbitals. 

ADF uses a STO basis set along with STO fit functions to improve the efficiency of calculating multicenter integrals. It uses a fragment orbital approach. This is, in essence, a set of localized orbitals that have been symmetry-adapted. This approach is designed to make it possible to analyze molecular properties in terms of functional groups. Frozen core calculations can also be performed. 

The various MO calculations use different basis sets and have different ways of calculating multicenter coulomb and exchange integrals. The current trend in MO is to expand as a linear combination of atomic orbitals (LCAO). The atomic orbitals are represented by Slater functions with expansion in gaussian functions, taking advantage of the additive rule. When the calculation is performed in this... 

A polymeric structure is exhibited by "beryllium dimethyl," which is actually [Be(CH3)2] (see the structure of (BeCl2) shown earlier), and LiCH3 exists as a tetramer, (LiCH3)4. The structure of the tet-ramer involves a tetrahedron of Li atoms with a methyl group residing above each face of the tetrahedron. An orbital on the CH3 group forms multicentered bonds to four Li atoms. There are numerous compounds for which the electron-deficient nature of the molecules leads to aggregation. 

These same principles can be extended to systems, such as H3+, with delocalized multicenter er-bonding derived solely from the radial s-orbital combinations. The descriptive term, in-plane aromaticity, [44] includes such cases (see Chapter 3.2.1 for further discussion). 

When some boron atoms in non-classical boranes are exchanged by isolobal C+ units, the multicenter bonding MOs look qualitatively the same, but the contribution of carbon hybrid orbitals is larger than those from boron atoms [compare Figures 3.2-3(b) and (c)]. This polarization is due to the higher electronegativity of carbon versus boron atoms. 

The proper way of dealing with periodic systems, like crystals, is to periodicize the orbital representation of the system. Thanks to a periodic exponential prefactor, an atomic orbital becomes a periodic multicenter entity and the Roothaan equations for the molecular orbital procedure are solved over this periodic basis. Apart from an exponential rise in mathematical complexity and in computing times, the conceptual basis of the method is not difficult to grasp [43]. Software for performing such calculations is quite easily available to academic scientists (see, e.g., CASTEP at www.castep.org CRYSTAL at www.crystal.unito.it WIEN2k at www.wien2k.at). 

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