  We present a symmetry-adapted real-space formulation of Kohn-Sham density
functional theory for cylindrical geometries and apply it to the study of large
X (X=C, Si, Ge, Sn) nanotubes. Specifically, starting from the Kohn-Sham
equations posed on all of space, we reduce the problem to the fundamental
domain by incorporating cyclic and periodic symmetries present in the angular
and axial directions of the cylinder, respectively. We develop a high-order
finite-difference parallel implementation of this formulation, and verify its
accuracy against established planewave and real-space codes. Using this
implementation, we study the band structure and bending properties of X
nanotubes and Xene sheets, respectively. Specifically, we first show that
zigzag and armchair X nanotubes with radii in the range 1 to 5 nm are
semiconducting. In particular, we find an inverse linear dependence of the
bandgap with respect to the radius for all nanotubes, other than the armchair
and zigzag type III carbon variants, for which we find an inverse quadratic
dependence. Next, we exploit the connection between cyclic symmetry and uniform
bending deformations to calculate the bending moduli of Xene sheets in both
zigzag and armchair directions. We find Kirchhoff-Love type bending behavior
for all sheets, with graphene and stanene possessing the largest and smallest
moduli, respectively. In addition, other than graphene, the sheets demonstrate
significant anisotropy, with larger bending moduli along the armchair
direction. Finally, we demonstrate that the proposed approach has very good
parallel scaling and is highly efficient, enabling ab initio simulations of
unprecedented size for systems with a high degree of cyclic symmetry. In
particular, we show that even micron-sized nanotubes can be simulated with
modest computational effort.
