Arno Liberles Fairieigh Dickinson University Teaneck. New Jersey 07666
A great many quantum mechanical concepts are included in modern introductorv organic chemistrv courses: vet because the course is frequently kmmathemaiical, the& ideas are discussed qualitatively and, as a result, less precisely. Consequently, the student may end up with an inaccurate mental picture of what is actually being described. Such an initial misconception can he difficult to correct a t a later stage in the student's development, and it seems worthwhile to spend some extra time to ensure a proper initial understanding. One important concept is that of delocalization. When does it take place; what is a suitable measure of delocalization; and what about it imparts stability? Delocalization The idea of delocalization, which is now well-rooted in the jargon of organic chemistry, results from the popularity of the Hiickel approach to molecular-orbital theory with its calculation of delocalization energies as a measure of stahility (1-4). Since the original Hiickel method applied only tom-systems, the student is sometimes led to believe that only these electrons are delocalized. However in molecular orhital theory delocalization is not so restricted, and the a as well as the "nonhonding" or "lone-pair" electrons are also delocalized. All electrons are delocalized in the sense that all molecular orbitals can involve atomic orhitals from every nuclear center. In a planar system such as benzene, there are a- and a-type molecular orhitals, and while hoth types are delocalized, they do not mix. However, if the system does not contain the proper vlane of svmmetw. ". even a-a sevaration is not mathematicallv possible. In such cases, while the molecule may contain a double bond in the classic sense, no molecular orhitals can he considered as just a or m. ~ e p e n d i on n ~the system, a-m mixing may he slight, or there may he nothing resembling a r-molecular orbital. Examples are propene in the 30' methyl-rotated geometry (I),cvclohexene (II), and other alkenes, ketones, etc. not having the necessary plane of reflection.
In the linear combination of atomic orhitals appro1 (LCAO), a molecular orhital JIR takes the form =
Propene in the 30"-geometry (I) is neither a reactant nor a transition state. However, this geometrvdoes illustrate the idea of n-n mixing, and in Table 1 we present $ 1 2 for the system.' The y,z -plnne contains the cnrhon atoms, and the molecular orbital is ddinitrlv n n - t-. v w with lnree coefficients -~~~~~~~ multiplying p,~,and p,z. Yet a-mixing does take place, not only in the hyperconjugative sense, hut also involving the 2s, 2py, and 2pz orhitals on the carbon atoms as well as the 1s orhitals on the olefinic hvdroeens. In cvclohexene the a - m . mixing is much greater, and for systems such as cyclooctatetraene nothine" resemhline a classic a-molecular orhital can he found. The concevt of a-delocalization is not generallv introduced in the early ;tages of an introductory organic colrse because it is not necessary to do so in order to explain the behavior of most organic molecules. Thus the student may he left with the impression that r-electrons are only sometimes delocalized and that o-electrons almost never are. Since this is incorrect, it may well he worthwhile to point out that o and nonbonded electrons are also delocalized over the entire system but that one need not frequently invoke this fact in order to explain the behavior of the molecules. Since all electrons are delocalized, one needs a suitable criterion to determine whether, in fact, such delocalization lowers the energy. Based on the "particle-in-a-box" problem, the expectation value of the kinetic energy operator T would seem to fulfill this requirement. It is not meant to imply that a auantum mechanical calculation is necessarv in order to d i s k delocalization; it simply means that delocalization can be associated with changes in the kinetic energy of a system. If the number of nodes in the wavefunction does not increase, greater delocalization lowers the kinetic energy of the system; the more localization raises it. If the number of nodes wavefunction does increase, then even increased delocalization can he accompanied by an increase in the kinetic energy of the system. Of course, for real systems, one could equally well use the potential energy of the system since these two quantities are related hoth by the virial theorem and by summing, to the total energy. Employment of the virial theorem, however, leads to some interesting conclusions. Reactants, intermediates, products, and transition states are all stationarv states: that is. thev renresent ~ o i n t on s the electronic energy surface where ~ E ? ~ R ' (isRa shuctural parameter) is zero. For such points, the virial theorem relating ~
~
-
~~
-
x aikmi
where the 4; are atomic orbitals generally centered a t a nucleus. Table 1. C1
Delocalization
'The molecular orbitals are the result of single determinantal abinitio SCF calculations using the 3G basis set. See Ref: ( 1 5 ) .
The Coefficientr for $ , , o f Propene in the 3OD6eometry C2
