U-orbitals

Figure 8-3 Approximation to the U Orbital of Hydrogen by 2 Gaussians. The upper curve is the sum of the lower two curves.

In the notation of Eq. (9-29), 4 i(ri) = 1 If the U orbitals are normalized, then the spinorbitals 1 ja(l), etc. are normalized because a and P are normalized. If we take just the expanded determinant for two electrons without 1 / V2, the normalization constant, and (omitting complex conjugate notation for the moment) integrate over all space... 

The remaining AOs are the four H 1, two C 1, and four C 2p orbitals. All lie in the molecular plane. Only two combinations of the C 2s and H U orbitals meet the molecular symmetry requirements. One of these, nearest-neighbor atoms. No other combination corresponds to the symmetry of the ethylene molecule. 

FIGURE 3.14 Each C H bond in methane is formed by the pairing of an electron in a hydrogen U-orbital and an electron in one of the four sp hybrid orbitals of carbon. Therefore, valence-bond theory predicts four equivalent cr-bonds in a tetrahedral arrangement, which is consistent with experimental results. 

FIGURE 3.30 Two p-orbitals can overlap side by side to give bonding (below) and antibonding (above) u-orbitals. Note that the latter has a nodal plane between the two nuclei. Both orbitals have a nodal plane through the two nuclei and look like p-orbitals when viewed along the internuclear axis. 

Table 3. Total energy as a function of the position of the floating Gaussian (zg) for six strengths of the external electric field (F = 0.0,0.001,0.01,0.1, 0.25, and 0.5 a.u.h (All data in a.u., equilibrium distance = 1.474 a.u., orbital...

The next two peaks are associated to the CN groups. The first one contains the contribution of the bands based on the ag and b u orbitals of TCNQ lying at 3.51 and 3.63 eV from the LUMO, and will be labelled a (jr(ag, b f)), while the second peak contains the contribution of bands based on four orbitals of TCNQ lying at... 

Note that the electrons are listed from left to right in increasing order of the energies of the orbitals they occupy and that different b]u orbitals are distinguished oy using a prime instead of the more cumbersome superscript numbers, which might now be confused with the superscripts indicating the presence of two electrons. 

It will be demonstrated in Section 9.5 that the relative energies of the doubly and triply degenerate sets of d orbitals into which the set of five d orbitals is split in a tetrahedral or octahedral environment are as shown in Figure 9.2. Thus, when there is a single d electron in an ion in an octahedral environment, it will occupy one of the t2g orbitals and the energy required to promote it to an en orbital is A0 for the same ion in a tetrahedral environment, the electron will occupy an e orbital and the energy required to promote it to a u orbital will be A,. 

Fig. 15.12 Calculated contour diagrams for the isoiobal u orbitals of [MnHj]"- (left) and (CH,) (right). The contours are plotted in a plane passing through manganese and three hydrogen atoms and through carbon and one hydrogen. (From Hoffmann, R. Angew. Chem. Itil. Ed. Engl. 1982, 21, 711-724. Reproduced with permission.]...

If it should happen that the ag orbital lies below rt , then the curve from the same species cannot cross. In that case, the steep rise over most of its course of the curve from the flj sa orbital could still be regarded as due to the reasons already described, i.e., to its trying to reach it but an " avoided crossing would have to be drawn, with the result that the curve from the a, orbital which " tended to proceed to ag would be drawn in fact to proceed to the r.u orbital. However, we shall see below that the balance of evidence is not unfavourable to avoided crossing " is necessary. 

Figure 1.3 The linear combination of U orbital functions on hydrogen atoms A and B to yield a new orbital function, ij>MO =

The hydrogenolysis of allylic ether and acetate (235 236, R=alkyl or COCHj) should also take place more easily when the compound can adopt a conformation in which the OR group can become parallel to the u orbital of the double bond (69). The same stereochemical requirement must also be necessary in the hydrogenolysis of a substituent in a benzylic position (70). 

It should be clear that the selection rules apply to concerted processes between spatially accessible parts of a molecule. Antarafacial [1,3], [1,5], or [3,3] reactions involving u-orbital transfers between carbon atoms are not possible. Antarafacial migrations do become feasible when the transition state forms a cycle of at least 7 or 8 atoms Woodward and Hoffmann (1965b) cite the [1,7] example (76). 

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