Stimulated by recent progress in atom manipulation technology, the electronic properties of periodic structures artificially created with atoms on a substrate surface are studied, where constituent atoms are isolated from substrate atoms and interact with one another through neighboring-atom interactions. By reducing the lattice constant from infinity, the neighboring-atom interaction is gradually turned on, and discrete atomic states broaden to form energy bands. Band structures of a simple one-dimensional atomic chain, a two-dimensional square array, and two parallel atomic chains formed by Si are calculated as a function of Lattice constant using the tight-binding theory with universal parameters. For practical lattice constants, these Si structures are all metallic due to the existence of the pi band, within which the Fermi energy lies; however, at very low spacings, possible for carbon, the double chain can become insulating. For group-Il elements such as Mg, the rr band and the conduction band are empty while the valence band is fully occupied. The band-gap variation with lattice constant is reflected in the electronic properties directly : e.g., a Mg atomic chain is insulating for lattice constants greater than or less than 4.2 Angstrom, at which the band gap disappears and the chain becomes metallic.