Embedding on to a one-dimensional crystal

A simple expression is derived for the band structure of a one-dimensional periodic potential in terms of two solutions of the Schroedinger equation within the unit cell, one with a zero-derivative boundary condition on the left-hand end of the cell and the other with zero derivative on the right-hand end. From this starting point, a new expression is derived for the embedding potential - this can be added to the Hamiltonian for the surface region of a crystal to replace the semi-infinite substrate, in a one-dimensional approximation. The results are demonstrated in calculations of the band structure and embedding potential for Al in the [001] direction, and the surface electronic structure of the Al(001) surface.