Orbitals lobes

Figure 1.18 A molecular orbital description of the C=C tt bond in ethylene. The lower-energy, tt bonding MO results from a combination of p orbital lobes with the same algebraic sign and is filled. The higher-energy, -tt antibonding MO results from a combination of p orbital lobes with the opposite algebraic signs and is unfilled.

Molecular modeling helps students understand physical and chemical properties by providing a way to visualize the three-dimensional arrangement of atoms. This model set uses polyhedra to represent atoms, and plastic connectors to represent bonds (scaled to correct bond length). Plastic plates representing orbital lobes are included for indicating lone pairs of electrons, radicals, and multiple bonds—a feature unique to this set. 

Morris and Waring<49) suggested that to effect the necessary overlap to form (64), the 7r-orbital lobes at C-4, C-6 and C-l and C-5 must be twisted from planarity. This could explain the finding that only heavily substituted dienones lead to the bicydo[3.1.0]hex-3-en-2-one photolysis. 

As there are 6 ir electrons to accommodate—two per orbital—the HOMO will be ip3 (11). To form the C—C a bond on cyclisation, the orbital lobes on the terminal carbon atoms of the conjugated system (C2 and C7—the C atoms carrying the Me substituents) must each rotate through 90° if mutual overlap is to occur (p/sp2— sp3 re-hybridisation must also occur). This necessary rotation could be either (a) both in the same direction—conrotatory (12), or (b) each in opposite directions—disrotatory (13) ... 

Conrotatory movement results in the apposition of orbital lobes with opposite phase—an anti-bonding situation, while disrotatory movement results in the apposition of orbital lobes with the same phase—a bonding situation, leading to formation of the cyclohex-adiene (7) in which the two Me groups are cis. 

The difference in stereochemical outcome of these reactions is determined, therefore, by the relative phase of the lobes—at the terminal carbon atoms—of the MOs of these (and other similar) mre systems by the symmetry of their orbitals, that is. As we have seen, the orbital lobes, at the two terminal carbon atoms, have the same phase in the HOMO (tf 3) of the triene (67re), and in the HOMO after irradiation (i/i3) of the diene (4ire) while these orbital lobes have opposite phases in the HOMO (i/r2) of the diene and in the HOMO after irradiation (ifi4) of the triene. Two such terminal lobes with the same phase require disrotatory movement before bond-making/bond-breaking can occur, while two terminal lobes with... 

A Ti bond has a nodal plane passing through the two bonded nuclei and between the 7z molecular orbital lobes. 

The antibonding tt MO results when orbital lobes of opposite sign overlap between adjacent carbon atoms => there is a node between each pair of carbon atoms. 

The essential idea is that orbital lobes of the same sign can lead to favorable overlap (the overlap integral has a value >0). This can occur between orbitals of different types in several ways. Figure 3.6 shows a few of the types of orbital overlap that lead to bonding. As we shall see in later chapters, some of these types are quite important. 

Fig. 2. Contour plots of (Hike bonding orbitals of Zeise s anion. The contour values increase in absolute magnitude with increasing absolute values of the contour labels. The sign of the labels gives the sign of the orbital lobes. The set of contour values plotted is the same for each of the three orbitals. The interior nodes at the various atoms are not shown for clarity of presentation (a) the 5a, orbital, (b) the 6at orbital, and (c) the 7ot orbital showing significant interaction between the ethylene 7r-orbital and the Pt dx, yi orbital. [Reproduced from Rosch et at. (193), by permission of the American Chemical Society.]...

The c/a ratio is greater than for V02, which implies that the n band (i.e. that with d-orbital lobes in the basal plane) is more occupied than in V02 (Goodenough 1971, p. 352). But we think that if it were not ferromagnetic, the n band, in contradistinction to V02, would be wholly above the Fermi energy. The Hubbard correlation term U, however, produces localized moments for the 3d2 states, as explained in Chapter 3, and these, if oriented ferromagnetically, would just fill the tjj band. The filled band (for spin-up electrons) will now overlap the n band, allowing ferromagnetic interaction of Zener or RKK Y type between the d2 moments, as described in Chapter 3. The T2 term in the resistivity could be explained as in Chapter 2, Section 6. 

C2u character table because when x2 and y2 are of the same symmetry, any linear combination of the two will also have that symmetry. Note that although both the and d - orbitals transform as at in this point group, they are not degenerate because they do not transform together, ft would be a worthwhile exercise to confirm that the s, p, and d orbitals do have the symmetry properties indicated in a Cu molecule. Keep in mind, in attempting such an exercise, that the signs of orbital lobes are important... 

Another way to computationally treat unpaired electrons is to employ restricted open-shell HF (ROHF) theory. Here, we encounter another pit-fall. It is an artifact called symmetry breaking (97). Whereas ROHF wave functions are pure spin states, the ROHF wave function may not retain the symmetry of the molecule. Suppose a molecule has C2V symmetry. The wave function should have the same symmetry, e.g., the orbital lobes on either side of the symmetry plane should be identical. However, with symmetry breaking, the two sides are not equal. The unsymmetrical ROHF wave function may even give lower energy than a physically correct (symmetrical)... 

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