  We formulate and implement 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 are
expected to be associated with unprecedented material properties. We rigorously
demonstrate the existence and completeness of special solutions to the single
electron problem for helical nanostructures, called helical Bloch waves. We
describe how the Kohn-Sham Density Functional Theory equations for a helical
nanostructure can be reduced to a fundamental domain with the aid of these
solutions. A key component in our mathematical treatment is the definition and
use of a helical Bloch-Floquet transform to perform a block-diagonalization of
the Hamiltonian in the sense of direct integrals. We develop a symmetry-adapted
finite-difference strategy in helical coordinates to discretize the governing
equations, and obtain a working realization of the 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. 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. 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.
