A Geometric Method for Determining the Huckel Molecular Orbital Energy Levels of Open-Chain, Fully Conjugated Molecules A. D. Baker The City University of New York, Queens College, Flushing, NY 11367 M. D. Baker University of Toronto. Toronto. Ontario. Canada M5S 1Al Texts dealine with the aoolication of elementarv. auantum .. . mechanics to organic systems generally include a description of the Frost circle mnemonic for rnlculatine the H u c k ~ mol lecular orbital energy levels of cyclic conjugked polyenes.1-5 The use of this mnemonic simplifies the time-consuming task of creating and solving the appropriate secular equations. Curiously, there is lacking in these texts any description of nmemonic devices for similar calculations on open-chain ~ v s t e r n s We . ~ describe here such a device which is straiehtforward and which uses aline drawing of the molecule in settine UD the eeometric mnemonic. 'i'he procehure is to draw a circle of diameter 4P and center a.exactlv as would he done when usinc the Frost circle. Inside the circlk, a line drawing of the openychain molekule is constructed following the rules: (1) The line drawing should reflect the number of C-C bonds in for the ally1 the open-chain molecule, e.g. I for ethylene system, and for butadiene. (See figure and steps below.)
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' Carey, F. A,. and Sundberg. R. J., "Advanced Organic Chemistry."
(Part A: Structure and Mechanism), Plenum. New York. 1977. Lowry. T. H., and Richardson. K. S., "Mechanism and Theory in Organic Chemistry." 2nd ed.. Harper and Row. New York. 1981. Roberts, J. D.. and Caserio. M. C., "Basic Principles of Organic Chemistry," 2nd ed., Benjamin. 1977. Hirsch, J. A,, "Concepts in Theoretical Organic Chemistry." Allyn and Bacon. Newton, MA, 1974. Coulson. C. A.. O'Leary. B., and Mallion. R. El.. "Huckel Theory for hganic Chemists," Academic Press, New Ywk, 1978, (a rewnsbuction by O'Leary and Mallion of Coulson's lectures to Oxford undergraduates). Nevertheless. Frost and Muslin 1J. Chern. Phvs.:21. 572.11953)l have pointed out that the Huckel molkcular oroitaienkr& level; for 4 +atom. open-chainconjugated system are identical to some of the levels of an (n 4btom. cyclic-conjugated molecule. On this basis a modification of the Frost circle approach to open-chain systems is possible. For teaching purposes, however, this is not particularly useful since it requires a knowledge of the Huckel results for cyclic systems to solve problems involving open-chain systems, while it is natural and the usual oractice to consider the latter Woe ., of svstem first. The mne monic presented in thts present paper avoids h s difficulty, requwng no prior knowledge of resulrs for cyclic molecules.
12) . . The line drawine should he svmmetrie with remeet to the horiruntal axis of the circle. (3) The term in^ ofthe linedrawing should be equidistant from the top and bottom of the circle. (4) Each side ot the line draw in^ should he of equal len~qh, and this lrnrth should alw lw thp same nu thedistance from either the topor bottom of the circle to the termini of the line drawing (i.e., the dashed lines and the heavy lines in the figure are ofthe same length). The energy levels of the system are now given by the points at which the line drawing touches the circle. T h e success of the method is easily established by simple trigonometry. The orhital energy levels are given by E=a+m,p
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where mj = (2 c o s j ~ ) l ( n 1) for j = 1,2,. . . n and a and fl have their usual designations as the Coulomb and resonance integrals, in agreement with values found by solution of the appropriate secular equations. Acknowledgment . We are grateful to W. F. Berkowitz of Queens College for helpful discussion and suggestions.
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Journal of Chemical Education
The W e t molecular wbnal energy levels tor &ylene. me alty lc sywm, and fabmdnene. follovmg the geamen c m e w descr beo n the ten
