We present theoretical models to study the localized electronic surface states in metallic structures. The materials under study have been chosen with different types of cubic meshes, fcc, sc and bcc. The calculation method used is closely related to the Linear Combination of Atomic Orbitals (LCAO) in the tight-binding method. We consider three cases: each of the atoms is described by a single atomic orbital of [Formula: see text]-, [Formula: see text]- and [Formula: see text]-type orbitals. In order to solve the rectangular secular equations of the systems under study, the phase field matching method is involved. In particular, we apply our approach to calculate the localized electronic surface states of some metals: (i) Chromium and Silver having, respectively, bcc and fcc structure and described as [Formula: see text]-type orbital. (ii) Nickel with sc crystallization and described by [Formula: see text]-type orbital. (iii) Palladium (Pd) given in fcc crystallization and described by [Formula: see text]-type orbital. The obtained results illustrate spatial edge effects between the bulk modes and the localized electronic states of the metallic surfaces over the three orientations of high symmetry path. We observe many localized states above and below the bulk band range. In addition, the relaxation effect on the surface layer has been investigated to compute the localized electronic surface state in this case and illustrate the lift of the degeneracy compared to the first calculations based on an ordered surface. The spacing geometry caused by the relaxation on the surface has been determined by using the Molecular dynamic algorithm and Morse interatomic potential.
