大学化学 >> 2022, Vol. 37 >> Issue (7): 2110068.doi: 10.3866/PKU.DXHX202110068

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直链和单环共轭多烯本征行列式递推公式及本征能级的推导

张冬菊(), 蔡欣懿, 宋其圣()   

  • 收稿日期:2021-10-22 录用日期:2022-12-07 发布日期:2022-02-22
  • 通讯作者: 张冬菊,宋其圣 E-mail:zhangdj@sdu.edu.cn;sqs@sdu.edu.cn
  • 作者简介:宋其圣, Email: sqs@sdu.edu.cn
    张冬菊, Email: zhangdj@sdu.edu.cn
  • 基金资助:
    国家自然科学基金(21773139)

Derivations of the Recurrence Formula for Eigendeterminants and Eigenenergy Levels of Linear and Monocyclic Conjugated Polyenes

Dongju Zhang(), Xinyi Cai, Qisheng Song()   

  • Received:2021-10-22 Accepted:2022-12-07 Published:2022-02-22
  • Contact: Dongju Zhang,Qisheng Song E-mail:zhangdj@sdu.edu.cn;sqs@sdu.edu.cn

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摘要：

休克尔(Hückel)分子轨道法是描述共轭体系π电子运动的近似模型，《结构化学》教科书中通常直接给出其本征行列式和其本征能级的递推公式，缺乏推导过程。本文用数学归纳法推导了直链和单环共轭多烯本征多项的递推公式，给出了对应本征能级的通式、特点和规律，适于未掌握群论和图论等数学工具的学生学习使用。

关键词: 共轭多烯, 本征行列式, 递推公式, 本征能级

Abstract:

The Hückel molecular orbital method is an approximate model to describe the interactions of π-electrons in a conjugate system. In structural chemistry textbooks, the recursive formulas for the eigendeterminants and eigenenergy levels of conjugated π-electrons systems have been directly provided, without derivation. Based on mathematical induction, this work shows how these recursive formulas are derived for linear and monocyclic conjugated polyenes, and clarifies the characteristics and regularity of the eigenenergy levels. This information is suitable for students who lack mathematical knowledge with regard to topics such as the group theory and graph theory.

Key words: Conjugated polyenes, Eigendeterminants, Recurrence formula, Eigenenergy levels