  We present a detailed analysis of various tensor network parameterizations
within the Complete Graph Tensor Network States (CGTNS) approach. We extend our
2-site CGTNS scheme by introducing 3-site correlators. For this we devise three
different strategies. The first relies solely on 3-site correlators and the
second on 3-site correlators added on top of the 2-site correlator ansatz. To
avoid an inflation of the variational space introduced by higher-order
correlators, we limit the number of higher-order correlators to the most
significant ones in the third strategy. Approaches for the selection of these
most significant correlators are discussed. The sextet and doublet spin states
of the spin-crossover complex manganocene serve as a numerical test case. In
general, the CGTNS scheme achieves a remarkable accuracy for a significantly
reduced size of the variational space. The advantages, drawbacks, and
limitations of all CGTNS parameterizations investigated are rigorously
discussed.
