A Density-Based Continuous Local Symmetry Measure

Abstract

Although continuous symmetry theory has attracted increasing attention in modern chemistry, local symmetry remains under-investigated. As a consequence, the relationship between symmetry and chemical behavior is often obscured, limiting the practical use of fuzzy symmetry measures. In this study, we introduce a novel framework for evaluating local symmetry based on electron density localization, and present continuous symmetry representations for several representative molecules. Our approach not only quantitatively captures global symmetry, but also reveals distinctive features of symmetry in a local chemical environment. The related concept, local chirality or chirotopicity, is also discussed. Overall, the proposed local symmetry and chirality measures provide valuable insights into molecular structure and structure-property relationships.

Figures (10)

• Figure 1: Contour plot of local plane symmetry at the $\alpha$ carbon of 1-pentanol [$S_{\ce{C_{\alpha}}}(\sigma)$] as a function of polar and azimuthal angles of the $\sigma$ plane.
• Figure 2: Local reflection symmetry at carbon centers, $S_{\text{C}n}(\sigma)$, computed with varying radial extent $R$. Vertical dashed lines indicate selected internuclear distances.
• Figure 3: Local reflection symmetry at carbon centers, $S_{\text{C}n}(\sigma)$, with $R$ fixed at 2 a.u.
• Figure 4: Diprotonated octamethyltetraphenylporphyrin (H2[OMTPP]) framework and dimethoxyphenyl-/substituted analogues.
• Figure 5: Local $S_4$ symmetry at the molecular center, $S_O(S_4)$, computed with varying radial extent $R$.