A potential function is computed for the lithium ion core, and is used to calculate the cohesive energy of metallic lithium, which is found to be in good agreement with experiment. Eigenvalues for conduction electrons in metallic lithium are calculated for points of high symmetry in the Brillouin zone, using the improved cellular method of Howarth and Jones, in which boundary conditions are applied at a large number of points on the surface of the atomic polyhedron; the merits of this type of boundary condition are discussed. The energy gap at the centre of a face of the zone is found to be 2.57 eV, the lowest state at that point having a p-like symmetry, and the interpretation of the soft x-ray emission spectrum of lithium is discussed in the light of this result.