  The screened electron-electron interaction in a multi-band electron system is
calculated within the random phase approximation and in the tight-binding
representation. The obtained dielectric matrix contains, beside the usual
site-site correlations, also the site-bond and bond-bond correlations, and thus
includes all physically relevant polarization processes. The arguments are
given that the bond contributions are negligible in the long wavelength limit.
We analyse the system with two non-overlapping bands in this limit, and show
that the corresponding dielectric matrix reduces to a $2\times2$ form. The
intra-band and inter-band contributions are represented by diagonal matrix
elements, while the off-diagonal elements contain the mixing between them. The
latter is absent in insulators but may be finite in conductors. Performing the
multipole expansion of the bare long-range interaction, we show that this
mixing is directly related to the symmetry of the atomic orbitals participating
in the tight-binding electronic states. In systems with forbidden atomic
dipolar transitions, the intra-band and inter-band polarizations are separated.
However, when the dipolar transitions are allowed, the off-diagonal elements of
the dielectric matrix are of the same order as diagonal ones, due to a finite
monopole-dipole interaction between the intra-band and inter-band charge
fluctuations.
