Image potential states from the van der Waals density functional

The image potential state is one of the fundamental surface electronic states and has a great relevance to many surface phenomena, but its accurate description is a great challenge for the semilocal density functional. Here, we use the nonlocal van der Waals density functional to describe the image potential states of graphene, graphite, and carbon nanotubes. We found that although it does not yield the correct image potential outside the surface, the van der Waals density functional improves the description of image potential states because of the nonlocal correlation potential. Our study demonstrates the usefulness of the van der Waals density functional to study the surface electronic properties.