  We follow the evolution of the Ionization Potential (IP) for the paradigmatic
quasi-one-dimensional trans-acetylene family of conjugated molecules, from
short to long oligomers and to the infinite polymer trans-poly-acetylene (TPA).
Our results for short oligomers are very close to experimental available data.
We find that the IP varies with oligomer length and converges to the given
value for TPA with a smooth, coupled inverse-length-exponential behavior. Our
prediction is based on an "internally-consistent" scheme to adjust the exchange
mixing parameter $\alpha$ of the PBEh hybrid density functional, so as to
obtain a description of the electronic structure consistent with the
quasiparticle approximation for the IP. This is achieved by demanding that the
corresponding quasiparticle correction, in the GW@PBEh approximation, vanishes
for the IP when evaluated at PBEh($\alpha^{ic}$). We find that $\alpha^{ic}$ is
also system-dependent and converges with increasing oligomer length, allowing
to capture the dependence of IP and other electronic properties.
