The model shown in the preceding section enables the calculation of the energy bands of a system of well defined configuration. Thus, the parameters suitable for the diamond lattice, for instance, cannot be used for calculations in graphite. To describe systems with a wide variety of coordinations with the same tight binding parameters, a total energy scheme has to be employed, which accounts for interactions other than the single-electron one's, with an explicit dependency on the interatomic distances.
The tight binding model has been developed on the basis of two major approximations. The first to be considered is the adiabatic approximation , which is based on the fact that electrons move typically faster than the ions. The latter can thus be considered in their ground state at any moment for a particular instantaneous ionic configuration, and the electronic and ionic degrees of freedom can therefore be separated. The second approximation consists in reducing the N-body problem to a one-electron scheme, where each electron moves independently of the others, and experiences an effective interaction due to the other electrons and to the ions. Within these approximations, the one-particle electronic part of the total Hamiltonian can be written in the form
Taking into account the orthonormality of the eigenstates, the
eigenvalues and eigenstates of the Hamiltonian are therefore found by
solving the matrix equation
The off-diagonal matrix elements , for , are called hopping integrals, and the on-site elements are the atomic orbital energies. In the tight binding approach, these hopping integrals and the on-site matrix elements are constants to be fitted on the basis of the following approximations (besides the two-center approximation introduced above):
(i) Only atomic orbitals whose energy is close to that of the energy bands we are interested in, are used . This is the minimal basis set approximation. Thus, for instance, only the 2s(one orbital) and 2p (three orbitals: px,py, and pz) orbitals are considered in the case of diamond and 3s and 3p orbitals for silicon, to describe the occupied (valance) bands. For these two materials there are 16 possible hopping integrals. However, it can be shown  that only hopping integrals between orbitals with the same angular momentum about the bond axis, are non-vanishing. There remain therefore just four nonzero hopping integrals, labeled (), (), (), and (). stands for orbitals with 0 angular momentum about the bond axis and for orbitals with angular momentum 1. The dependence of these hopping integral in the distance between the atoms will be considered further.
(ii) We consider only hopping integrals between two atoms separated by a distance shorter than a suitable cutoff. Obviously, to reduce the number of parameters to be fitted, a cutoff which includes the nearest neighbors is appropriate. However, the orthogonalized functions (Löwdin) extend further than those (non-orthogonal) from which they are derived, because the orthogonalization procedure involves orbitals from nearby atoms. Thus, interactions extending beyond first nearest neighbors have to be taken into account when an orthogonal basis is used.
Considering the approximations above, the off-diagonal elements of the Hamiltonian matrix (for ) are fitted to electronic band structure of the equilibrium crystal phase, as calculated by more accurate first-principle models . Sets of hopping integrals can thus be obtained for each crystalline structure considered.
On the basis of the Harris-Foulkes formulation of the density functional theory [99,100], Sutton et. al.  shown that the cohesive energy of the crystal can then be written in the form
The form of the repulsive energy
proposed by Xu et. al.  and used in the present research is