Orthogonal hybrid orbitals

All three orbitals are in the same plane, and each one is 120° from each of the other orbitals. The remaining p orbital is orthogonal to (perpendicular to the plane of) the three hybridized orbitals. 

It is usually of interest to express the atomic orbitals as functions of the hybrid orbitals. As the transformation matrix is orthogonal, its inverse is... 

The orbitals containing the bonding electrons are hybrids formed by the addition of the wave functions of the s-, p-, d-, and f- types (the additions are subject to the normalization and orthogonalization conditions). Formation of the hybrid orbitals occurs in selected symmetric directions and causes the hybrids to extend like arms on the otherwise spherical atoms. These arms overlap with similar arms on other atoms. The greater the overlap, the stronger the bonds (Pauling, 1963). 

Let us now choose to locate a second equivalent, though not necessarily orthogonal, hybrid orbital h2 in the x-z plane. The axis of h2 will be chosen to make the angle a with the axis of hi. In order for hi and h2 to be equivalent, the values of the angular parts of the hybrids must be identical in their respective directions. This condition is satisfied for any separation angle a by the following expression for h2 ... 

We now have two normalized, equivalent hybrid orbitals, one directed along the z axis and the other making an angle of a with the first. However, these orbitals are not necessarily orthogonal to each other for arbitrary a. The allowed hybrid angles a are those for which the overlap integral (hi h2) vanishes... 

Dewar and coworkers offered an alternative view.79 In their model the d and d)Z orbitals are hybridized to give two orbitals which are directed toward the adjacent nitrogen atoms (Fig. 16.25). This allows for formation of three-center bonds about each nitrogen. 9 This scheme, sometimes called the island model, results in delocalization over selected three-atom segments of the ring, but nodes are present at each phosphorus atom since the two hybrid orbitals of phosphorus are orthogonal to each other. Evidence has been offered in support of both models, but neither theory has been confirmed to the exclusion of the other. A third viewpoint holds that rf orbital participation is relatively unimportant in the bonding in these molecules. 1... 

B. Fdrster-Coulson-Moffitt Orbitals and Non-orthogonal Valence Bond Hybrid Orbitals... 

TABLE 1. Hybridization and interorbital angles in cyclopropane using orthogonal hybrid orbitals... 

TABLE 2. Hybridization and overlap values obtained with non-orthogonal hybrid orbitals within a valence bond calculation of cyclopropane"... 

As can be seen from Table 2, the amount of s-character of the CH hybrid orbital of 1 is indeed increased relative to that of the CH hybrid orbitals of ethene, ethane and methane while at the same time the s-character of the CC hybrid orbitals is decreased. This seems to confirm predictions based on model calculations with orthogonal hybrid orbitals. However, closer inspection of the data in Table 2 reveals that s-character of the CH hybrid... 

Notice that the four hybrid orbitals all are orthogonal to each other. ... 

Let us compare the MO and VB descriptions of HX molecules. The simplest VB treatment would consider simply the overlap between the singly-occupied AOs ls(H) and 2p2(F), leading to the formation of a o bond. The filled 2s(F) orbital together with the filled 2px v(F) orbitals would constitute the three lone pairs the valence state corresponds to the configuration s2p2p2p. If we admit that 2s(F) should be considered, the VB procedure would be to construct a pair of orthogonal s-p hybrid orbitals ... 

---