  A mixed basis approach based on density functional theory is employed for low
dimensional systems. The basis functions are taken to be plane waves for the
periodic direction multiplied by B-spline polynomials in the non-periodic
direction. B-splines have the following advantages:(1) the associated matrix
elements are sparse, (2) B-splines possess a superior treatment of derivatives,
(3) B-splines are not associated with atomic positions when the geometry
structure is optimized, making the geometry optimization easy to implement.
With this mixed basis set we can directly calculate the total energy of the
system instead of using the conventional supercell model with a slab sandwiched
between vacuum regions. A generalized Lanczos-Krylov iterative method is
implemented for the diagonalization of the Hamiltonian matrix. To demonstrate
the present approach, we apply it to study the C(001)-(2x1) surface with the
norm-conserving pseudopotential, the n-type delta-doped graphene, and graphene
nanoribbon with Vanderbilts ultra-soft pseudopotentials. All the resulting
electronic structures were found to be in good agreement with those obtained by
the VSAP code, but with a reduced number of basis.
