  We employ tensor network methods for the study of the seniority quantum
number - defined as the number of unpaired electrons in a many-body wave
function - in molecular systems. Seniority-zero methods recently emerged as
promising candidates to treat strong static correlations in molecular systems,
but are prone to deficiencies related to dynamical correlation and dispersion.
We systematically resolve these deficiencies by increasing the allowed
seniority number using tensor network methods. In particular, we investigate
the number of unpaired electrons needed to correctly describe the binding of
the neon and nitrogen dimer and the $D_{6h}$ symmetry of benzene.
