Orbital Symmetry

Chapter29

Molecular Orbitals. Orbital Symmetry

Molecular orbital theory

•  The structure of molecules is best understood through quantum mechanics. Exact quantum mechanical calculations are enormously complicated, and so various methods of approximation have been worked out to simplify the mathematics. The method that is often the most useful for the organic chemist is based on the concept of molecular orbitals: orbitals that are centered, not about individual nuclei, but about all the nuclei in the molecule.

• What are the various molecular orbitals of a molecule like? What is their order of stability? How are electrons distributed among them? These are things we must know if we are to understand the relative stability of molecules: why certain molecules are aromatic, for example. These are things we must know if we are to understand the course of many chemical reactions: their stereochemistry, for example, and how easy or difficult they are to bring about; indeed, whether or not they will occur.

• We cannot learn here how to make quantum mechanical calculations, but we can see what the results of some of these calculations are, and learn a little about how to use.

• In this chapter, then, we shall learn what is meant by the phase of an orbital, and what bonding and antibonding orbitals are. We shall see, in a non-mathematical way, what lies behind the Hiickel An + 2 rule for aromaticity. And finally, we shall take a brief look at a recent and absolutely fundamental development in chemical theory: the application of the concept of orbital symmetry to the understanding of organic reactions. 

Wave equations. Phase

• In our first description of atomic and molecular structure, we said that electrons show properties not only of particles but also of waves. We must exam little more closely the wave character of electrons, and see how this is involved in chemical bonding. First, let us look at some properties of waves in general. Let us consider the standing waves (or stationary waves) generated by the vibration of a string secured at both ends: the wave generated by, say, the plucking of a guitar string (Fig. 29.1), As we proceed horizontally along the string from

• left to right, we find that the vertical displacement the amplitude of the wave increases in one direction, passes through a maximum, decreases to zero, and then increases in the opposite dhcction. The places where the amplitude is zero are called nodes. In Fig. 29.1 they He in a plane the nodal plane- -perpendicular to the plane of the paper. Displacement upward and displacement downward correspond to opposite phases of the wave. To distinguish between phases, we arbitrarily assign algebraic signs to the amplitude: plus for, say, displacement upward, and minus for displacement downward. If we were to superimpose similar waves on one another exactly out oj phase that is, with the crests of one lined up with the troughs of the other they would cancel each other: that is lo say, the sum of their amplitudes, f and , would zero.

• The differential equation that describes the wave is a wave equation. Solution of this equation gives the amplitude, <, as a function, /(Y), of the distance, .v, along the wave. Such a function is a wave function. Now, electron v\avc^ are described by a wave equation of the same general form as that for string waves. The wave functions that are acceptable solutions to this equation again give the amplitude, <^, this time as a function, not of a single coordinate, but of the three coordinates necessary to describe motion in three dimensions. It is these electron wave functions that we call orbitals.

• Any wave equation has a set of solutions an infinity of them, actually each corresponding to a different energy level. The quantum thus comes naturally out ot the mathematics.

Molecular orbitals. LCAO method

• As chemists, we picture molecules as collections of atoms held together by bonds. We consider the bonds to arise from the overlap of an atomic orbital of one atom with an atomic orbital of another atom. A new orbital is formed, which is occupied by a pair of electrons of opposite spin. Each electron is attracted by both positive nuclei, and the increase in electrostatic attraction gives the bond its strength, that is, stabilizes the molecule relative to the isolated atoms. 

• This highly successful qualitative model parallels the most convenient quantum mechanical approach to molecular orbitals: the method of linear combination of atomic orbitals (LCAO). We have assumed that the shapes and dispositions of bond orbitals are related, in a simple way to the shapes and dispositions of atomic orbitals. The LCAO method makes the same assumption mathematically: to calculate an approximate molecular orbital, 0, one uses a linear combination (that is, a combination through addition or subtraction) of atomic.

• $ is the molecular orbital (f>A is atomic orbital A B is atomic orbital.

• The rationale for this assumption is simple: when the electron is near atom A, resembles ^A ; when the electron is near atom B, resembles ^B . Now this combination is effective that is, the molecular orbital is appreciably more stable than the atomic orbitals only if the atomic orbitals A and B : (a) overlap to a considerable extent; (b) are of comparable energy; and (c) have the same symmetry about the bond axis. These requirements can be justified mathematically. Qualitatively, we can say this: if there is not considerable overlap, the energy of is equal to either that of A or that of ^B ; if the energy of ^A and < B are quite different, the energy of is essentially that of the more stable atomic orbital. In either case, there is no significant stabilization, and no bond formation.

•   When we speak of the symmetry of orbitals, we are referring to the relative phases of lobes, and their disposition in space. To see what is meant by requirement (c), that the overlapping orbitals have the same symmetry, let us look at one example: hydrogen fluoride. This molecule can be pictured as resulting from overlap of the s orbital of hydrogen with a p orbital of fluorine. In Fig. 29. 3a, we use the 2px orbital, where the x coordinate is taken as the H F axis. The shaded s orbital overlaps the shaded lobe of the p orbital, and a bond forms. If, however, we were to use the 2pz (or 2py) orbital as in Fig. 29.3A, overlap of both lobes plus and minus would occur and cancel each other. That is, the positive overlap integral would be exactly canceled by the negative overlap integral; the net effect would be no overlap, and no bond formation. The dependence of overlap on phase is fundamental to chemical bonding.

Bonding and antibonding orbitals

• Quantum mechanics shows that linear combination of two functions gives, not one, but two combinations and hence two molecular orbitals: a bonding orbital, more stable than the component atomic orbitals; and an antibonding orbital, less stable than the component.

•  We can see, in a general way, why there must be two combinations. There can be as many as two electrons in each component atomic orbital, making a total of four electrons; two molecular orbitals are required to accommodate them. Figure 29.4 shows schematically the shapes of the molecular orbitals, bonding and antibonding, that result from overlap of various kinds of atomic orbitals. We recognize the bonding orbitals, or and TT, although until now we have not shown the two lobes of a TT orbital as being of opposite phase. An antibonding orbital, we see, has a nodal plane perpendicular to the bond axis, and cutting between the atomic nuclei. The antibonding sigma orbital, or*, thus consists of two lobes of opposite phase. The antibonding pi orbital, TT*, consists of four.

• In a bonding orbital, electrons are concentrated in the region between the nuclei, where they can be attracted by both nuclei. The increase in electrostatic attraction lowers the energy of the system. In an antibonding orbital, by contrast, electrons are not concentrated between the nuclei; electron charge is zero in the nodal plane. Electrons Spend most of their time farther from a nucleus than in the separated atoms. There is a decrease in electrostatic attraction, and an increase in repulsion between the nuclei. The energy of the system is higher than that of the separated atoms. Where electrons in a bonding orbital tend to hold the atoms together, electrons in an antibonding orbital tend to force the atoms.

• It may at first seem strange that electrons in certain orbitals can actually weaken the bonding. Should not any electrostatic attraction, even if less than optimum, be better than none? We must remember that it is the bond dissociation energy we are concerned with. We are not comparing the electrosiatic attraction in an antibonding orbital with no electrostatic attraction; we are comparing it with the stronger electrostatic attraction in the separated atoms.

• There are, in addition, orbitals of a third kind, non-bonding orbitals. As the name indicates, electrons in these orbitals unshared pairs, for example neither strengthen nor weaken the bonding between atoms.

 Electronic configurations of some molecules

• Let us look at the electronic configurations of some familiar molecules. The shapes and relative stabilities of the various molecular orbitals are calculated by quantum mechanics, and we shall simply use the results of these calculations. We picture the nuclei in place, with the molecular orbitals mapped out about them, and we feed electrons into the orbitals. In doing this we follow the same rules that we followed in arriving at the electronic configurations of atoms. There can be only two electrons and of opposite spin in each orbital, with orbitals of lower energy being filled up first. If there are orbitals of equal energy, each gets an electron before any one of them gets a pair of electrons. We shall limit our attention to orbitals containing TT electrons, since these electrons will be the ones of chief interest.

• For the -n electrons of ethylene (Fig. 29.5), there are two molecular orbitals since there are two linear combinations of the two component p orbitals. The broken line in the figure indicates the non-bonding energy level; below it lies the bonding orbital, TT, and above it lies the antibonding orbital, TT*.

•  Normally, a molecule exists in the state of lowest energy, the ground state. But, as we have seen (Sec. 13.5), absorption of light of the right frequency (in the ultraviolet region) raises a molecule to an excited state, a state of higher energy. In the ground state of ethylene, we see, both rr electrons are in the n orbital; this configuration is specified as 7r 2 , where the superscript tells the number of electrons in that orbital. In the excited state one electron is in the TT orbital and the other still of opposite spin is in the TT* orbital; this configuration, TTTT*, is naturally the less stable since only one electron helps to hold the atoms together, while the other tends to force them.

• For 1,3-butadiene, with four component /; orbitals, there are four molecular orbitals for TT electrons (Fig. 29.6). The ground state has the configuration 0i 2^2 2 J that is, there are two electrons in each of the bonding orbitals, $i and ^2 The higher of these, ^2 > resembles two isolated n orbitals, although it is of somewhat lower energy. Orbital 3.

•  The allyl cation has IT electrons only in the bonding orbital. The free radical has one electron in the non-bonding orbital as well, and the anion has two in the non-bonding orbital. The bonding orbital $\ encompasses all three carbons and is more stable than a localized TT orbital involving only two carbons; it is this delocalization that gives allylic particles their special stability. We see the symmetry we have attributed to allylic particles on the basis of the resonance theory; the two ends of each of these molecules are equivalent.

• Finally, let us look at benzene. There are six combinations of the six component p orbitals, and hence six molecular orbitals. Of these, we shall consider only these combinations, which correspond to the three most stable molecular

• orbitals, all bonding orbitals (Kig. 29,8). Each contains a pair of electrons. The lowest orbital, ^\ , encompasses all six carbons. Orbitals ^ 2 an(J 4*3 are of different shape, but equal energy; together they provide as does ^ equal electron density

Aromatic character. The Hickel 4/i + 2 rule

• In Chap. 10 we discussed the structure of aromatic compounds. An aromatic molecule is flat, with cyclic clouds of delocalized/ed n electrons above and below the plane of the molecule. We have just seen, for ben/erne, the molecular orbitals that permit this delocalizes/anion. But delocalize/anion alone is not enough. For that special degree of stability, we call aromaticity\\ the number of n electrons must conform to Hockel's rule: there must be a total of (4/i -f 2) n electrons.

• In Sec. 10.10, we saw evidence of special stability associated with the "magic" numbers of 2, 6, and 10 rr electrons, that is, with systems where n is 0, 1, and 2 respectively. Problem 5 (p. 447) described the nm. spectrum of cyclooctadecanonaene, which contains 18 rr electrons (n is 4). Twelve protons lie outside the ring,

• are deshelled, and absorb downfield; but, because of the particular geometry of the large flat molecule, six protons lie inside the ring, are shielded (see Fig. 13.4, p. 419), and absorb upfield. The spectrum is unusual, but exactly what we would expect if this molecule were aromatic.

• The cyclopentadienyl cation has four electrons. Two of these go into the lower orbital. Of the other two electrons, one goes into each orbital of the lower degenerate pair. The cyclopentadienyl free radical has one more electron, which fills one orbital of the pair. The anion has still another electron, and with this we fill the remaining orbital of the pair. The six rr electrons of the cyclopentadienyl anion are just enough to fill all the bonding orbitals. Fewer than six leaves bonding orbitals unfilled; more than six, and electrons would have to go into antibonding orbitals. Six n electrons gives maximum bonding and hence maximum stability.

• Figure 29.10 shows the molecular orbitals for rings containing five, six, and seven s/> 2-hybridized carbons. We see the same pattern for all of them: a single orbital at the lowest level, and above it a series of degenerate pairs. It takes (4/i 4- 2) TT electrons Lovill a set of these bonding orbitals: 2 electrons for the lowest orbital, and 4 for each of n degenerate pairs. Such an electron configuration has been likened to the rare gas configuration of an atom, with its closed shell. It is the filling of these molecular orbital shells that makes these molecules aromatic.

• In Problem 10.6 (p. 330) we saw that the cyclopropyl cation is unusually stable: 20 kcal/mole more stable even than the allyl cation. In contrast, the 

Orbital symmetry and the chemical reaction

• A chemical reaction involves the crossing of an energy barrier. In crossing this barrier, the reacting molecules seek the easiest path: a low path, to avoid climbing any higher than is necessary; and a broad path, to avoid undue restrictions on the arrangement of atoms. As reaction proceeds, there is a change in bonding among the atoms, from the bonding in the reactants to the bonding in the products. Bonding is a stonily/mg factor, the stronger the bonding, the more stable the system. If a reaction is to follow the easiest path, it must take place in the way that maintains nut \imp homing during the reaction pi-mess. Now. bonding, as we visual/c it, results from overlap of orbitals. Overlap requires that portions of different orbitals occupy the same space, and that they he of the Wm paw.

• This line of reasoning seems perfectly straightforward. Yet the central idea, that the course of reaction can be controlled hy orbital symmetry, was a revolutionary one, and represents one of the really giant steps forward in chemical theory. A number of people took part in the development of this concept: K. Fukui in Japan, H. C. Longuet-Heggems in bogland. But organic chemists became aware of the power of this approach chiefly through a series of papers published in 1965 by R. B. Woodward and Roald Hoffmann working at Harvard University.

• Very often in organic chemistry, theory lags behind experiment; many facts are accumulated, and a theory is proposed to account for them. This is a perfectly respectable process, and extremely valuable. But with orbital symmetry, just the reverse has been true. The theory lay in the mathematics, and what was needed was the spark of genius to see the applicability to chemical reactions. Facts were sparse, and Woodward and Hoffmann made predictions, which have since been borne out by experiment. All this is the more convincing because these predictions were of the kind called *' risky": that is, the events predicted seemed unlikely on any grounds other than the theory being .

Electrocyclic reactions

• Under the influence of heat or light, a conjugated polyene can undergo isomerization to form a cyclic compound with a single bond between the terminal carbons of the original conjugated system; one double bond disappears, and the remaining double bonds shift their positions. For example, 1,3,5-hexatrienes yield 1 ,3-cyclohexadienes: 

• It is the stereochemistry of electrocyclic reactions that is of chief interest to us. To observe this, we must have suitably substituted molecules. Let us consider first the interconversion of 3,4-dimethylcyclobutene and 2,4-hexadiene (Fig. 29.12). The cyclobutene exists as cis and trans isomers. The hexadiene exists in three forms: cis, cis; cistrons\ and trans, trans. As we can see, the cis cyclobutene yields only.

• orbital that will form the bond that closes the ring. Bond formation requires overlap, in this case overlap of lobes on C-l and C-4 of the diene: the front carbons in Fig. 29.14. We see that to bring these lobes into position for overlap, there must be rotation about two bonds, C, C2 and C3 C4 . This rotation can take place in two different ways: there can be conrotatory motion, in which the bonds rotate in the same direction,

 

• In the excited state of hexatriene, </ 2

• is the HOMO, and once again we see a

• reversal of symmetry: here, conrotatory motion is the favored process.

• What we see here is part of a regular pattern (Table 29.1) that emerges from

• the quantum mechanics. As the number of pairs of -n electrons in the poiyene

• increases, the relative symmetry about the terminal carbons in the HOMO alternates regularly. Furthermore, symmetry in the HOMO of the first excited state is

• always opposite to that in the ground state.

Cytoderms reactions

• This is an example of cycloaddition, a reaction in which two unsaturated molecules combine to form a cyclic compound, with IT electrons being used to form two new a bond. The Diels-Alder reaction is a [4 + 2] cycloaddition, since it involves a system of 4 tr electrons and a system of 2 IT electrons. Reaction takes place very easily, often spontaneously, and at most requires moderate application of heat. There are several aspects to the stereochemistry of the Diels-Alder reaction. (a) First, we have taken for granted correctly that the diene must be in conformation (s-cis) that permits the ends of the conjugated system to reach the doubly bonded carbons of the dienophile. (b) Next, with respect to the alkene (dienophile) addition is clear-cut syn (Problem 8, p. 880); this stereospecificity is part.of the evidence that the Diels-Alder.

• On this basis, let us examine the [4 + 2] cycloaddition of 1,3-butadiene and ethylene, the simplest example of the Diels-Alder reaction. The electronic configurations of these compounds and of dienes and alkenes in general have been given in Fig. 29.5 (p. 931) and Fig. 29.6 (p. 932). There are two combinations: overlap of the HOMO of butadiene (fa) with the LUMO of ethylene (TT*); and overlap of the HOMO of ethylene (TT) with the LUMO of butadiene (fa). In either case, as Fig. 29.20 shows, overlap brings together lobes of the same phase. There is a flow of electrons from HOMO to LUMO, and bonding occurs.

Sigma tropic reactions

• A concerted reaction of the type, in which a group migrates with its a bond within a TT framework an Ene or a polyene is called a sigma tropic reaction. The migration is accompanied by a shift in w bonds. For example:

• This does not mean that rearrangement actually involves the separation and reattachment of a free radical. Such a stepwise reaction would not be a concerted one, and hence is not the kind of reaction we are dealing with here. Indeed, a stepwise reaction would be a (high-energy) alternative open to a system if a (concerted) sigma tropic rearrangement were symmetry-forbidden.

• In the transition state, there is overlap between the HOMO of one component and the HOMO of the other. Each HOMO is singly occupied, and together they provide a pair of electrons.

• In 1970, H. Klooster Ziel (of the University of Technology, Eindhoven, The Netherlands) reported a study of the rearrangement of the diastereomeric 6,9- dimethyl 4.4] nona-l,3-dienes (cis-Vll and trans-VU) to the dimethylbicyclo- [4.3.0] nonadienes VIII, IX, and X. These reactions are completely stereospecific.