Graphical Representation of Huckel Molecular Orbitals

In this paper, we develop a general but very simple mathematical foundation for the predefined coefficient graphical method of Huckel molecular orbital theory (HMO). We first present the general solution for the recurrence relation of the coefficients of Mickel molecular orbitals (MOs). Subsequently, for all the three unbranched hydrocarbons, i.e., open-chain, cyclic Huckel and Mobius polyenes, different boundary conditions are explored for obtaining the MOs and their energy levels. The analytic continuation of the recurrence relation, in which one extends the domain from integral to real, allows us to analyze the symmetric properties of Huckel MOs in an elegant fashion without even knowing the actual expressions. In fact, we can use the symmetric properties to derive the Huckel MOs of the unbranched hydrocarbons and some branched hydrocarbons such as naphthalene. Consequently, this work also provides a pedagogical alternative to present the HMO model for students in an advanced physical chemistry course. Finally, the graphical approach could be a good mnemonic device for students' comprehension of the HMO theory.