  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.
