In the tight-binding method the basis functions are atomic like orbitals. The absolute minimum basis set is usually composed of the states of the outer shell. For instance for Si one would take four orbitals (adopting the bra and ket notation) :
and for transition metals nine :
The number of basis functions per atoms will be noted 
.
If the number of atoms in the system is 
then the dimension
of the space spanned by the basis set is 
.
Any state is expressed as a linear combination of these atomic-like
orbitals 
:
where the index 
runs over atoms and orbitals.
In all generality we will assume that the basis set is not orthonormal :
In the empirical tight-binding method (ETBM) the basis functions are not know explicitly. Instead the matrix elements of the Hamiltonian and the overlap are given :
where the label 
refers to the type
of the ion associated with which orbital 
.
The aim of any of the methods presented here is to calculate the electronic structure of the system or in other words to find the electronic ground state.
The most straightforward approach to this problem
is to calculate the eigenvalues 
and eigenstates
of the Hamiltonian operator :
The eigenvalues and the eigenvectors of the Hamiltonian contain all the information we need to calculate the total-energy and the forces on the atoms.
The number of electrons is given by
where 
is some broadening function, in our case we
will use the Fermi function 
, and 
is
the width of the broadening. The quantity 
is called the
occupation of state i. For 
0 the Fermi
function tends to the step function 
and =0 if
> 
and =1 if
< . The factor two accounts for the
spin degeneracy (each state is occupied by two electrons). Equation
(7) should be regarded as the definition of the Fermi
energy which solves the equation
.
The electronic energy is expressed as the sum of the eigenenergies of the Hamiltonian :
where
Deriving the forces from the eigenstates is done via the Hellman-Feynman theorem. For simplicity we will consider an insulator for which the force acting on atom I is simply :
The molecular dynamics-method (MD) is simply a method for integrating the classical equations of motion of the atoms, which given by :
The simulations proceeds in four steps :
