Orbital-specific hybrid functional

Despite the quantitative victory of molecular orbital (MO) theory, much of our qualitative understanding of electronic structure is still couched in terms of local bonds and lone pairs, that are key conceptual elements of the valence bond (VB) picture. VB theory is essentially the quantum chemical formulation of the Lewis concept of the chemical bond [1,2]. Thus, a chemical bond involves spin-pairing of electrons which occupy valence atomic orbitals or hybrids of adjacent atoms that are bonded in the Lewis structure. In this manner, each term of a VB wave function corresponds to a specific chemical structure, and the isomorphism of the theoretical elements with the chemical elements creates an intimate relationship between the abstract theory and the nature of the... 

Here the crystallographic indices in the subscripts refer to the hybrid molecular orbital directions. Because the Schrbdinger Equation governing electron motion is linear, any combination of wave functions that solve it will also be a solution. In other words, choosing the hybrid orbitals or the atomic orbitals as a starting point for the calculation must yield identical results. The most flexible and general approaeh is not to be restricted to specific hybrid orbitals but rather to consider all possible orbital-by-orbital interactions of the fundamental atomic states. These states apply to a given atom in any environment. Thus, their use is valid for any material in which the atom occurs. As an example of a specific interaction, one can ask how does the Px orbital on one atom interact with the orbital on another atom. 

To introduce some of the basic ideas of molecular orbital theory, let s look again at orbitals. The concept of an orbital derives from the quantum mechanical wave equation, in which the square of the wave function gives the probability of finding an electron within a given region of space. The kinds of orbitals that we ve been concerned with up to this point are called atomic orbitals because they are characteristic of individual atoms. Atomic orbitals on the same atom can combine to form hybrids, and atomic orbitals on different atoms can overlap to form covalent bonds, but the orbitals and the electrons in them remain localized on specific atoms. 

This means that the atomic orbitals in the hybrids must have the same symmetry properties as A) and E. More specifically, it means that one orbital must have the same symmetry as Aj (which is one-dimensional) and two orbitals must have the same symmetry, collectively, as E (which is two-dimensional). This means that we must select one orbital with Ai symmetry and one pair of orbitals that collectively have E symmetry. Looking at the functions listed for each in the right-hand column of the character table, we see that the s orbital (not listed, but understood to be present for the totally symmetric representation) and the orbitals match the Aj symmetry. However, the 3d orbitals, the lowest possible d orbitals, are too high in energy for bonding in BF3 compared with the 2s and 2p. Therefore, the 2s orbital is the contributor, with A) symmetry. 

The chemist s sketches, which are typically drawn to emphasize directionality of the sp hybrid orbitals, and a contour plot of the actual shape, are shown in Figure 6.44. Each of these contours can be rotated about the x-y plane to produce a three-dimensional isosurface whose amplitude is chosen to be a specific fraction of the maximum amplitude of the wave function. These isosurfaces demonstrate that sp hybridization causes the amplitude of the boron atom to be pooched out at three equally spaced locations around the equator of the atom (see Fig. 6.42). The 2p orbital is not involved and remains perpendicular to the plane of the sp hybrids. The standard chemist s sketches of the sp hybrid orbitals and a contour plot that displays the exact shape and directionality of each orbital are shown in Figure 6.44. The isosurfaces shown in Figure 6.43 were generated from these contour plots. 

The hosts outlined above generally utilize their metal centers to provide interactions with anions. The orbital overlap between unfilled heteroelement orbitals and filled anion orbitals yields bonding interactions. In the case of the mercury based receptors, the role of the metal atoms is partly structural, with their sp hybridization linear geometry specifically yielding large macrocyclic cavities of suitable diameters for anion encapsulation. The metal atoms have often been functionally used as an NMR handle on the anion coordination process. 

In this regard, it is pertinent to note an analogy with the main group elements. Specifically, the radial distribution functions of the 2s and 2p are more comparable than the ns/np pairs for higher principal quantum numbers and thus hybridization is a common feature of the 2" row elements but not for those of the higher periods. The origin of the similar size of the 2s and 2p orbitals is that the 2p orbitals, being the first orbitals with p symmetry, are not subject to Pauli repulsion and therefore penetrate the core relatively more effectively than do 3p orbitals. 

---