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We formulate and implement Cyclic Density Functional Theory (Cyclic DFT) -- aself-consistent first principles simulation method for nanostructures withcyclic symmetries. Using arguments based on Group Representation Theory, werigorously demonstrate that the Kohn-Sham eigenvalue problem for such systemscan be reduced to a fundamental domain (or cyclic unit cell) augmented withcyclic-Bloch boundary conditions. Analogously, the equations of electrostaticsappearing in Kohn-Sham theory can be reduced to the fundamental domainaugmented with cyclic boundary conditions. By making use of this symmetry cellreduction, we show that the electronic ground-state energy and theHellmann-Feynman forces on the atoms can be calculated using quantities definedover the fundamental domain. We develop a symmetry-adapted finite-differencediscretization scheme to obtain a fully functional numerical realization of theproposed approach. We verify that our formulation and implementation of CyclicDFT is both accurate and efficient through selected examples.  The connection of cyclic symmetries with uniform bending deformationsprovides an elegant route to the ab-initio study of bending in nanostructuresusing Cyclic DFT. As a demonstration of this capability, we simulate theuniform bending of a silicene nanoribbon and obtain its energy-curvaturerelationship from first principles. A self-consistent ab-initio simulation ofthis nature is unprecedented and well outside the scope of any other systematicfirst principles method in existence. Our simulations reveal that the bendingstiffness of the silicene nanoribbon is intermediate between that of grapheneand molybdenum disulphide. We describe several future avenues and applicationsof Cyclic DFT, including its extension to the study of non-uniform bendingdeformations and its possible use in the study of the nanoscale flexoelectriceffect.

Cyclic Density Functional Theory : A route to the first principles simulation of bending in nanostructures