Exploring the Electronic Potential of Effective Tight$-$Binding Hamiltonians

The linear combination of atomic orbitals (LCAO) is a standard method for studying solids and molecules, it is also known as the tight$-$binding (TB) method. In most of the implementations only the basis set and the coupling constants are provided, without the explicit definition of kinetic and potential energy operators. The tight$-$binding scheme is, nonetheless, capable of providing an accurate description of properties such as the electronic bands and elastic constants for many materials. However, for some applications, the knowledge of the underlying electronic potential associated with the tight$-$binding hamiltonian might be important to guarantee that the actual physics is preserved by the semiempirical scheme. In this work the electronic potentials that arise from the use of tight$-$binding effective hamiltonians it is explored. The formalism is applied to the extended H\"{u}ckel tight$-$binding (EHTB) hamiltonian, which is a two$-$center Slater$-$Koster approach that makes explicit use of the overlap matrix.