  The exchange interaction between electrons located at different randomly
distributed impurities is studied for small density of impurities. The
singlet-triplet splitting 2J(R) is calculated for two Coulomb centers at a
distance R. Interpolated formulas are found which work for all distances R from
zero to infinity. The data from atomic physics are used for the interpolation
in three-dimensional case. For two-dimensional case the original calculations
are performed to find asymptotic behavior of the splitting at large R, the
splitting for the ``two-dimensional helium atom'' (R=0) and the splitting at
$R=a_B$, where $a_B$ is the effective Bohr radius. The spin structure of
impurity band is described by the Heisenberg Hamiltonian. The ground state of a
system consists of localized singlets. The new results are obtained for the
distribution of the singlet pairs in the ground state. These results are exact
at low density. The problem is reduced to a non-trivial geometric problem which
is solved in the mean field approximation and by computer modeling. The density
of free electrons is found as a function of temperature and the distribution
function of the singlet-triplet transitions energies is calculated. Both
functions are given in an analytical form.
