  A van der Waals (vdW) density functional was implemented in the mixed basis
approach previously developed for studying two dimensional systems, in which
the vdW interaction plays an important role. The basis functions here are taken
to be the localized B-splines for the finite non-periodic dimension and plane
waves for the two periodic directions. This approach will significantly reduce
the size of the basis set, especially for large systems, and therefore is
computationally efficient for the diagonalization of the Kohn-Sham Hamiltonian.
We applied the present algorithm to calculate the binding energy for the
two-layer graphene case and the results are consistent with data reported
earlier. We also found that, due to the relatively weak vdW interaction, the
charge density obtained self-consistently for the whole bi-layer graphene
system is not significantly different from the simple addition of those for the
two individual one-layer system, except when the interlayer separation is close
enough that the strong electron-repulsion dominates. This finding suggests an
efficient way to calculate the vdW interaction for large complex systems
involving the Moire pattern configurations.
