Abstract：
Chirality is an essential geometric property unifying small molecules, biological macromolecules, inorganic nanomaterials, biological microparticles, and many other chemical structures. Numerous chirality measures have attempted to quantify this geometric property of mirror asymmetry and to correlate these measures with physical and chemical properties. However, their utility has been widely limited because these correlations have been largely notional. Furthermore, chirality measures also require prohibitively demanding computations, especially for chiral structures comprised of thousands of atoms. Acknowledging the fundamental problems with quantification of mirror asymmetry, including the ambiguity of sign-variable pseudoscalar chirality measures, we revisit this subject because of the significance of quantifying chirality for quantitative biomimetics and describing the chirality of nanoscale materials that display chirality continuum and scale-dependent mirror asymmetry. We apply the concept of torsion within the framework of differential geometry to the graph theoretical representation of chiral molecules and nanostructures to address some of the fundamental problems and practical limitations of other chirality measures. Chiral gold clusters and other chiral structures are used as models to elaborate a graph-theoretical chirality (GTC) measure, demonstrating its applicability to chiral materials with different degrees of chirality at different scales. For specific cases, we show that GTC provides an adequate description of both the sign and magnitude of mirror asymmetry. The direct correlations with macroscopic properties, such as chiroptical spectra, are enhanced by using the hybrid chirality measures combining parameters from discrete mathematics and physics. Taking molecular helices as an example, we established a direct relation between GTC and optical activity, indicating that this chirality measure can be applied to chiral metamaterials and complex chiral constructs.
摘要：
手性是统一小分子的基本几何性质，生物大分子，无机纳米材料,生物微粒，和许多其他化学结构。许多手性措施已经尝试量化镜像不对称性的这种几何特性，并将这些措施与物理和化学性质相关联。然而,它们的效用受到了广泛的限制，因为这些相关性在很大程度上是名义上的。此外，手性措施还需要非常苛刻的计算，尤其是由数千个原子组成的手性结构。认识到镜像不对称量化的基本问题，包括符号变量伪标量手性度量的模糊性，我们重新审视这个主题，因为量化手性对定量仿生和描述显示手性连续和尺度依赖性镜像不对称性的纳米级材料的手性的重要性。我们将微分几何框架内的扭转概念应用于手性分子和纳米结构的图论表示，以解决其他手性措施的一些基本问题和实际局限性。手性金簇和其他手性结构被用作模型来阐述图论手性（GTC）测量，证明了其在不同尺度下对具有不同手性程度的手性材料的适用性。具体案例,我们证明了GTC对镜像不对称性的符号和大小都提供了足够的描述。与宏观性质的直接相关，比如手性光谱，通过使用结合离散数学和物理学参数的混合手性度量来增强。以分子螺旋为例，我们建立了GTC和光学活性之间的直接关系，这表明这种手性测量可以应用于手性超材料和复杂的手性结构。
