In this paper, we develop a general but very simple mathematical foundation for the predefined coefficient graphical method of Hückel molecular orbital theory (HMO). We first present the general solution for the recurrence relation of the coefficients of Hückel molecular orbitals (MOs). Subsequently, for all the three unbranched hydrocarbons, i.e., open-chain, cyclic Hückel and Möbius 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 Hückel MOs in an elegant fashion without even knowing the actual expressions. In fact, we can use the symmetric properties to derive the Hückel 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.

中文翻译：

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赫克尔分子轨道的图形表示

在本文中，我们为Hückel分子轨道理论（HMO）的预定义系数图形方法开发了通用但非常简单的数学基础。我们首先介绍Hückel分子轨道（MOs）系数的递归关系的一般解。随后，对于所有三种非支链烃，即开链，环状赫克尔和莫比乌斯多烯，探索了不同的边界条件以获得MOs及其能级。递归关系的解析延续（其中一个域从积分扩展到实数）使我们能够以优雅的方式分析HückelMO的对称性质，甚至不知道实际的表达式。事实上，我们可以利用对称性来推导非支链烃和某些支链烃（如萘）的HückelMO。因此，这项工作还为在高级物理化学课程中向学生展示HMO模型提供了一种教学替代方法。最后，图形化方法可能是学生理解HMO理论的良好记忆工具。