The *band* model is applied to perfect crystals with translational
symmetry, to find the electronic structure of the system. Following
the work of Slater and Koster [90], the electronic wave
functions are approximated by linear combinations of atomic orbitals,
in term of Bloch sums, used as basis functions. Starting with an
atomic orbital
,
located on an atom
at ,
the Bloch sum is written as

Here,

= | (7.8) | ||

= |

This can be further simplified since one of the two sums gives exactly

(7.9) |

Further simplifications are needed to calculate the integrals above.
Slater and Koster [90] proposed to neglect the three-center
integrals with respect to the two-center integrals. The three-center
integrals involve two orbitals located at different atoms, and a
potential part of the Hamiltonian due to a third atom. Thus, in the
Slater and Koster's approach, only the potential (considered
spherical) due to the two atoms upon which the orbitals are located, is
taken into account. The integrals can therefore be replaced by
parameters which depend on the internuclear distance
,
and on the symmetry of the orbitals involved. For
example, if
had the symmetry of a *p* orbital, it
could be expressed as a linear combination of
and
,
where
and
refer to the component of the
angular momentum around the axis between the two atoms.

The Hamiltonian matrix elements can therefore be written

(7.10) |

where