Arrow diagram method based on overlapping electronic groups: corrections to the linked AD theorem

Arrow diagram (AD) method (L. Kantorovich and B. Zapol, J. Chem. Phys. \textbf{96}, 8420 (1992); \emph{ibid}, 8427) provides a convenient means of systematic calculation of arbitrary matrix elements, $<\Psi|\hat{O}|\Psi>$, of symmetrical operators, $\hat{O}$, in quantum chemistry when the total system wavefunction $\Psi$ is represented as an antisymmetrised product of overlapping many-electron group functions, $\Phi_{A}$, corresponding to each part (group) $A$ of the system: $\Psi=\hat{A}\prod_{A}\Phi_{A}$. For extended (e.g. infinite) systems the calculation is somewhat difficult, however, as mean values of the operators require that each term of the diagram expansion is to be divided by the normalisation integral $S=<\Psi|\Psi>$, which is given by an AD expansion as well. A linked AD theorem suggested previously (L. Kantorovich, Int. J. Quant. Chem. \textbf{76}, 511 (2000)) to deal with this problem is reexamined in this paper using a simple Hartree-Fock problem of a one-dimensional ring of infinite size which is found to be analytically solvable. We find that corrections to the linked AD theorem are necessary in a general case of a finite overlap between different electronic groups. A general method of constructing these corrections in a form of a power series expansion with respect to overlap is suggested. It is illustrated on the ring model system.