  In this paper we analyze new approximations of the Green's function coupled
cluster (GFCC) method where locations of poles are improved by extending the
excitation level of inner auxiliary operators. These new GFCC approximations
can be categorized as GFCC-i($n,m$) method, where the excitation level of the
inner auxiliary operators ($m$) used to describe the ionization potentials and
electron affinities effects in the $N$$-$1 and $N$+1 particle spaces is higher
than the excitation level ($n$) used to correlate the ground-state coupled
cluster wave function for the $N$-electron system. Furthermore, we reveal the
so-called "$n$+1" rule in this category (or the GFCC-i($n$,$n$+1) method),
which states that in order to maintain size-extensivity of the Green's function
matrix elements, the excitation level of inner auxiliary operators
$X_p(\omega)$ and $Y_q(\omega)$ cannot exceed $n$+1. We also discuss the role
of the moments of coupled cluster equations that in a natural way assures these
properties. Our implementation in the present study is focused on the first
approximation in this GFCC category, i.e. the GFCC-i(2,3) method. As our first
practice, we use the GFCC-i(2,3) method to compute the spectral functions for
the N$_2$ and CO molecules in the inner and outer valence regimes. In
comparison with the GFCCSD results, the computed spectral functions from the
GFCC-i(2,3) method exhibit better agreement with the experimental results and
other theoretical results, particularly in terms of providing higher resolution
of satellite peaks and more accurate relative positions of these satellite
peaks with respect to the main peak positions.
