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 [95], 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

(7.11) |

where is the kinetic energy operator of the electrons, and are the electron-electron and electron-ion interactions respectively. Following the notation of Horsfield

(7.12) |

where is a single particle (doubly occupied) eigenfunction, and is the corresponding eigenvalue. It has to be mentioned that the

(7.13) |

where

Taking into account the orthonormality of the eigenstates, the
eigenvalues and eigenstates of the Hamiltonian are therefore found by
solving the matrix equation

(7.14) |

where

(7.15) |

are the matrix elements and

(7.16) |

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 [90]. This is the *minimal basis set* approximation. Thus, for instance, only the 2*s*(one orbital) and 2*p* (three orbitals: *p*_{x},*p*_{y}, and *p*_{z}) orbitals
are considered in the case of diamond and 3*s* and 3*p* orbitals for
silicon, to describe the occupied (valance) bands. For these two
materials there are 16 possible hopping integrals. However, it can be
shown
[97] 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 [98]. 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.*
[101] shown that the cohesive energy of the crystal can then
be written in the form

(7.17) |

where is the band energy, is the repulsive potential, given as a sum of pair potentials, and is the total energy of the free atoms. are the eigenvalues obtained from the diagonalization of the Hamiltonian matrix. Within the adiabatic approximation, the electrons are assumed to be in their ground state, so that all the states below the Fermi level are occupied, and the summation that appears in the band energy is made over these occupied

The form of the repulsive energy
proposed by Xu *et. al.* [102] and used in the present research is

(7.18) |

where