Arrow diagram approach to nonorthogonal electron group functions in extended systems

For nonorthogonal electron group functions, the arrow diagram (AD) method (Kantorovich and Zapol 1992 J. Chem. Phys. 96 8420; 1992 J. Chem. Phys. 96 8427) provides a convenient procedure for calculating matrix elements <Ψ-vertical bar O-circonflex-vertical barΨ> of arbitrary symmetrical operators O-circonflex. The total wavefunction of the system Ψ A-circonflexΠlΦl is represented as an antisymmetrized product of nonorthogonal many-electron group functions ΦI of each group I in the system. For extended (e.g.?infinite) systems the calculation of the mean value of an operator is ill defined, however, as it requires that each term of the diagram expansion be divided by the normalization integral S=<Ψ-vertical barΨ> which is given by an AD expansion as well. In this work, we cast the mean value of a symmetrical operator in a form of an AD expansion which is a linear combination of linked ADs. By analysing an exactly solvable one-dimensional Hartree-Fock problem, we find that pre-factors, attached to every linked AD in the linear combination, can be expanded in a power series with respect to overlap. A general method of calculating these pre-factors in a form of a power series expansion with respect to overlap is suggested. This advance makes the AD theory applicable to extended systems, and allows one to calculate the mean value of an arbitrary symmetrical operator correct up to the desired order of overlap within the group function theory. In particular, we derive the effective Hamiltonian of a quantum cluster surrounded by overlapping group functions (e.g. bonds) in the environment region which is correct up to the second order with respect to overlap (an embedding problem)