  A mixed basis approach based on density functional theory is extended to
one-dimensional(1D) systems. The basis functions here are taken to be the
localized B-splines for the two finite non-periodic dimensions and the plane
waves for the third periodic direction. This approach will significantly reduce
the number of the basis and therefore is computationally efficient for the
diagonalization of the Kohn-Sham Hamiltonian. For 1D systems, B-spline
polynomials are particularly useful and efficient in two-dimensional spatial
integrations involved in the calculations because of their absolute
localization. Moreover, B-splines are not associated with atomic positions when
the geometry structure is optimized, making the geometry optimization easy to
implement. With such a basis set we can directly calculate the total energy of
the isolated system instead of using the conventional supercell model with
artificial vacuum regions among the replicas along the two non-periodic
directions. The spurious Coulomb interaction between the charged defect and its
repeated images by the supercell approach for charged systems can also be
avoided. A rigorous formalism for the long-range Coulomb potential of both
neutral and charged 1D systems under the mixed basis scheme will be derived. To
test the present method, we apply it to study the infinite carbon-dimmer chain,
graphene nanoribbon, carbon nanotube and positively-charged carbon-dimmer
chain. The resulting electronic structures are presented and discussed in
details.
