A Rationale for the Shape of Simple Huckel Orbitals The utility of one-electron molecular orhitals in discussing reactions of unsaturated molecules makes it desirable to introduce simple Hiickel LCAO orbitals in the elementary course in organic chemistry. The free-electron theory gives an easily visualized mnemonic for the nodal structure of sueh orhitals and even for the magnitude of their atomic coefficients.2 While one can rationalize the atomic orhital signs and the parallel between orhital energy and number of nodes in terms of the number of bonding and antibonding interactions, it might a t first seem more difficult to give the inquisitive student a physical picture of why the atomic coefficients should have particular absolute magnitudes. The temptation is to invoke solution of the secular determinant as a d e w ex machino. Not only are the mathematical details of this procedure obscure to many beginning students, but it is difficult for most people to trace an interpretation through sueh an involved mathematical transformation. From the beginnings the standard variational approach to this problem has involved partial differentiation of energy with respect to each atomic orhital coefficient and construction of the secular determinant, which is then diagonalized after simplification by the Huckel approximations of zero overlap (Stj = 61,) and of standard Coulomb and resonance integrals (Hji = a,Hij = 0 for neighboring atoms, Hjj = O o t h e ~ i s e ) . ~ If these same approximations are applied before the variational treatment, a simple rationale for the atomic orbital coefficients emerges in the ease of homonuelear systems. If we write4 Cc?Hi.
+ CCcccjH,j
the final term in the denominator vanishes by the zero overlap approximation, and the first term in the denominator is a constant (unity) because of normalization. For homonuclear systems the second term in the numerator is a constant ( 0 ) because of normalization, and only the term involving the resonance integrals remains as a variable. The variational treatment then requires only that the (negative) sum
C
c,c#
uiphborr
he minimized (or stationary) with resped to changes in the coefficients, subject of course to normalization. This condition has the clear interpretation that the sum of bond orders in the orhital he a m a ~ i m u m . ~ For example the lowest molecular orhital of the pi-system of 1.3-hutadiene should have no nodes. The most naive guess for a normalized molecular orhital would assign a coefficient of +'h to each of the four atomic orbitals, hut this choice does not maximize the sum of hond orders. Since the central orhitals participate in two bonds while the terminal orbitals participate in only one, a more favorable arrangement would have the coefficients of the central atoms increase a t the expense of the terminal atom coefficients. c, = c, c* = c, 2 , c 9 c,z
+
Simultaneous solution of the normalization equation and the bond order optimization equation
+
2,2 2c,l
=1
+ c,1)/&,
=0 gives c, = 0.3717 and e l = 0.6015, the coefficients of the lowest HBckel molecular orhital. Other solutions of these equations yield the atomic orbital coefficients of higher molecular orhitals. Given the appropriate nodal pattern, the atomic orbital coefficients of honding orhitals maximize the sum of hond orders, and those of anti-bonding orhitals minimize this sum. This method can he used to determine coefficients in other systems, although it is often more cumbersome than diagonalization of the secular determinant. While it of course possess all the frailties of the simple Huckel treatment,6 its explanation of orbital shape in terms of maximum honding is straightforward and physically reasonable.
M2c,c,
1 Alfred P.
Sloan Research Fellow 1971-3; supported in part by a Camille and Henry Dreyfus Teacher-Scholar Grant. %Seefor example Orchin, M., and Jaffb, H. H., "Symmetry, Orbitals, and Spectra (S.O.S.)," Wiley-Inteneience, New York. 1971. DO. 55 ff. Hiickel, E., Z. Physik, 70,204 (1931). Reference 12). Chapter 10. W. B. Ellison has pointed out the analogy with Kaufman's overlap population criterion for molecular stability. See for example Manne, R.,Intern. J. Quantum Chem., 2,69 (1968). 6 Dewar, M. J. S., "The Molecular Orbital Theory of Organic Chemistry," McGraw-Hill, New York, 1969, sec. 3.7. Yale University J. Michael McBridel New Haven, Connecticut 06520 ~~
-,.~, ~
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Volume 51. Number 7. July 1974
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471
