Lobes, of orbitals

We may also look at this reaction from the opposite direction (ring closing). For this direction the rule is that those lobes of orbitals that overlap (in the HOMO) must be of the same sign. For thermal cyclization of butadienes, this requires conrotatory motion (Fig. 18.3). In the photochemical process the HOMO is the %3 orbital, so that disrotatory motion is required for lobes of the same sign to overlap. 

Orbital lobes of Orbital lobes of like sign pointed opposite sign pointed toward each other toward each other... 

There are many cases in which the prediction of precisely zero overlap can be made, irrespective of the relative sizes of the orbitals or the inter-nuclear distance. These predictions follow from the symmetry properties of the orbitals and are thus independent of scale factors, because there must always be exactly equal areas (volumes) of positive and negative overlap. It should now become obvious why it is important to remember not only the shapes but also the signs of the lobes of orbitals. 

Sigma (o ) orbital (Section 1.13) A molecular orbital formed by end-on overiap of orbitals (or lobes of orbitals) on adjacent atoms. Sigma orbitals may be bonding (orbitals or lobes of the same phase sign overlap) or antibonding (orbitals or lobes of opposite phase sign overlap). 

Conservation of orbital symmetry is a general principle that requires orbitals of the same phase (sign) to match up in a chemical reaction. For example, if terminal orbitals are to combine with one another in a cyclixation reaction as in pattern. A, they must rotate in the same dii ection (conrotatory ovei lap). but if they combine according to pattern H. they must rotate in opposite directions (disrotatory). In each case, rotation takes place so that overlap is between lobes of the it orbitals that are of the same sign. 

Other methods are also used to contrast the regions of an orbital where the signs of the wave function are differ ent Some mark one lobe of a p orbital + and the other — Others shade one lobe and leave the other blank When this level of detail isn t necessary no differentiation IS made between the two lobes... 

FIGURE 2 9 Each half filled sp orbital overlaps with a half filled hydrogen Is or bital along a line between them giving a tetrahedral arrangement of four ct bonds Only the major lobe of each sp orbital is shown Each orbital contains a smaller back lobe which has been omitted for clarity... 

Figure 2.3 The shapes of orbitals for the s electron pair, the three pairs of p electrons with obitals mutally at right angles, and the sp orbitals which have the major lobes pointing towards the apices of a regular tetrahedron.

Woodward and Hoffmann speeulated that the preferred motion was that whieh involved constructive (bonding) overlap between the terminal lobes of the highest-occupied molecular orbital (HOMO). 

Figure 1.3 Representations of s, p, and d orbitals. The s orbitals are spherical, the p orbitals are dumbbell-shaped, and four of the five d orbitals are cloverleafshaped. Different lobes of p orbitals are often drawn for convenience as teardrops, but their true shape is more like that of a doorknob, as indicated.

The asymmetry of sp3 orbitals arises because, as noted previously, the two lobes of a p orbital have different algebraic signs, + and -. Thus, when a p orbital hybridizes with an s orbital, the positive p lobe adds to the s orbital but the negative p lobe subtracts from the s orbital. The resultant hybrid orbital is therefore unsymmetrical about the nucleus and is strongly oriented in one direction. 

Figure 8.2 The structure of a secondary vinylic carbocation. The cationic carbon atom is sp-hybridized and has a vacant p orbital perpendicular to the plane of the tt bond orbitals. Only one R group is attached to the positively charged carbon rather than two, as in a secondary alkyl carbocation. The electrostatic potential map shows that the most positive (blue) regions coincide with lobes of the vacant p orbital and are perpendicular to the most negative (red) regions associated with the ir bond.

What do molecular orbitals and their nodes have to do with pericyclic reactions The answer is, everything. According to a series of rules formulated in the mid-1960s by JR. B. Woodward and Roald Hoffmann, a pericyclic reaction can take place only if the symmetries of the reactant MOs are the same as the symmetries of the product MOs. In other words, the lobes of reactant MOs must be of the correct algebraic sign for bonding to occur in the transition state leading to product. 

For a bond to form, the outermost tt lobes must rotate so that favorable bonding interaction is achieved—a positive lobe with a positive lobe or a negative lobe with a negative lobe. If twro lobes of like sign are on the same side of the molecule, the two orbitals must rotate in opposite directions—one clockwise and one counterclockwise. This kind of motion is referred to as disrotatory. 

Mixed labeling involving both x and a orbitals occurs in certain molecules the 5BU molecular orbitals of ra is-2-butene (III.78) and transoid 1,3-butadiene (III.65) are labeled ch3> ( cc) and 7rcn2> ( cc) because one lobe of the x orbital overlaps well with the adjacent CC bond-orbital to form a delocalized combination. In cisoid acrolein, orbitals 9A and 10A are labeled TCH2y nodal surfaces of the two localized orbitals coincide and allow for a delocalized combination (III.G8). 

Describe the orientation of the lobes of the px, px, and p -orbitals with respect to the reference Cartesian axes. 

The lobes of two d-orbitals on the same atom occupy markedly different regions of space. As a result, electrons in different d-orbitals are relatively far apart and repel one another only weakly. 

In this paper we amplify Powell s discussion, which is in some respects misleading. For example, Powell made the following statement Unlike the familiar four-lobed cubic d orbital, the pyramidal d orbital has only rather inconspicuous lobes of opposite sign. Each orbital is not quite cylindrically symmetrical about its own axis of maximum probability. In fact, the pyramidal d orbital that he discusses in detail is far from cylindrically symmetrical about its own axis of maximum probability, and the other pyramidal d orbital is also far from cylindrically symmetrical. In the equatorial plane about the axis of maximum probability the functions of Powell s first set (which we shall call II) vary from —0.3706 in two opposite directions to —1.7247 in the orthogonal directions. Each of these functions has almost the same value (strength) in the latter directions as in the principal directions, for which its value is 2.0950. The functions of the other set (which we call I) vary in this plane from —0.7247 to —1.4696, their value in the principal direction being 2.1943. 

Each lobe of the d 2 yi orbital interacts predominantly with one point charge. The repulsive effects relate to the electron density within any given orbital so we might describe the interaction in units of lobe repulsion and say that, for the dp. yi orbital, this amounts to 4 = 16 repulsion units (4 squared because electron density oc jF). 

The ground term of the cP configuration is F. That of is also F. Those of and d are " F. We shall discuss these patterns in Section 3.10. For the moment, we only note the common occurrence of F terms and ask how they split in an octahedral crystal field. As for the case of the D term above, which splits like the d orbitals because the angular parts of their electron distributions are related, an F term splits up like a set of / orbital electron densities. A set of real / orbitals is shown in Fig. 3-13. Note how they comprise three subsets. One set of three orbitals has major lobes directed along the cartesian x or y or z axes. Another set comprises three orbitals, each formed by a pair of clover-leaf shapes, concentrated about two of the three cartesian planes. The third set comprises just one member, with lobes directed equally to all eight corners of an inscribing cube. In the free ion, of course, all seven / orbitals are degenerate. In an octahedral crystal field, however, the... 

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