  We introduce a diagrammatic quantum field formalism for the evaluation of
normalized expectation values of operators, and suitable for systems with
localized electrons. It is used to develop a convergent series expansion for
the energy in powers of overlap integrals of single-particle orbitals. This
method gives intuitive and practical rules for writing down the expansion to
arbitrary order of overlap, and can be applied to any spin configuration and to
any dimension. Its applicability for systems with well localized electrons has
been illustrated with examples, including the two-dimensional Wigner crystal
and spin-singlets in the low-density electron gas.
