We formulate and implement Helical Density Functional Theory (Helical DFT) — a self-consistent first principles simulation method for nanostructures with helical symmetries. Such materials are well represented in all of nanotechnology, chemistry and biology, and prominent examples include nanotubes, nanosprings, nanowires, miscellaneous chiral structures and important proteins. The overwhelming preponderance of such helical structures in all of science and engineering and the likelihood of these systems being associated with exotic materials properties, provides the motivation to develop systematic and predictive tools for their study.

Following this line of thought, we develop a mathematical and computational framework in this contribution, that allows helical structures to be studied ab initio, using Kohn–Sham theory. We first show that the electronic states in helical structures can be characterized by means of special solutions to the single electron problem called helical Bloch waves. We rigorously demonstrate the existence and completeness of such solutions, and then describe how they can be used to reduce the Kohn–Sham Density Functional Theory (KS-DFT) equations for helical structures to a suitable fundamental domain. Next, we develop a symmetry-adapted finite-difference strategy in helical coordinates to discretize the governing equations, and obtain a working realization of our proposed approach. We verify the accuracy and convergence properties of our numerical implementation through examples. Finally, we employ Helical DFT to study the properties of zigzag and chiral single wall black phosphorus (i.e., phosphorene) nanotubes. Specifically, we use our simulations to evaluate the torsional stiffness of a zigzag nanotube ab initio. Additionally, we observe an insulator-to-metal-like transition in the electronic properties of this nanotube as it is subjected to twisting. We also find that a similar transition can be effected in chiral phosphorene nanotubes by means of axial strains. The strong dependence of the band gap of these materials on various modes of strain suggests their possible use as nanomaterials with tunable electronic and transport properties. Notably, self-consistent ab initio simulations of this nature are unprecedented and well outside the scope of any other systematic first principles method in existence. We end with a discussion on various future avenues and applications.

我们制定并实施了螺旋密度泛函理论（Helical DFT）——一种具有螺旋对称性的纳米结构的自洽第一原理模拟方法。这些材料在所有纳米技术、化学和生物学中都有很好的代表，突出的例子包括纳米管、纳米弹簧、纳米线、各种手性结构和重要的蛋白质。在所有科学和工程中这种螺旋结构的压倒性优势以及这些系统与奇异材料特性相关的可能性，为开发系统和预测工具提供了动力。

遵循这一思路，我们在此贡献中开发了一个数学和计算框架，允许使用 Kohn-Sham 理论从头研究螺旋结构。我们首先表明螺旋结构中的电子态可以通过称为螺旋布洛赫波的单电子问题的特殊解来表征. 我们严格证明了此类解的存在性和完整性，然后描述了如何将它们用于将螺旋结构的 Kohn-Sham 密度泛函理论 (KS-DFT) 方程简化为合适的基本域。接下来，我们在螺旋坐标中开发了一种对称适应有限差分策略来离散化控制方程，并获得我们提出的方法的工作实现。我们通过示例验证了我们的数值实现的准确性和收敛性。最后，我们采用螺旋 DFT 来研究锯齿形和手性单壁黑磷（即磷烯）纳米管的性质。具体来说，我们使用我们的模拟来评估锯齿形纳米管ab initio的扭转刚度. 此外，我们观察到这种纳米管在受到扭曲时的电子特性发生了类似绝缘体到金属的转变。我们还发现通过轴向应变可以在手性磷烯纳米管中实现类似的转变。这些材料的带隙对各种应变模式的强烈依赖性表明它们可能用作具有可调电子和传输特性的纳米材料。值得注意的是，这种性质的自洽从头模拟是前所未有的，并且远远超出了现有的任何其他系统的第一原理方法的范围。我们以对各种未来途径和应用的讨论结束。