The Full Configuration Interaction (FCI) method in electronic structure theory has factorial scaling with the number of orbitals, and its use on chemically interesting systems for describing electron correlation is therefore currently restricted.

In recent years, various non-linear parameterisations have been proposed to truncate the FCI wave function and overcome this scaling restriction. Such methods build on so-called contracted CI using segmented many-electron basis functions, and include Graphically Contracted Functions (GCF) and Matrix Product States (MPS).

The purpose of this thesis is to investigate the theoretical foundations of these non-linear methods. Their behaviour in the FCI limit is investigated under global symmetries $U(u) \otimes SU(2)$ and $U(n)$, where it is shown that a rigorous analytical solution for the non-linear parameters exists in both cases. Such parameters that might be optimised can therefore be determined analytically.

The behaviour of segmented basis functions is also investigated following truncation of the wave function away from the FCI limit. Analytical and numerical results are presented, showing that truncated non-linear parameterisation leads to an upper bound for the FCI ground state energy.

Finally using the non linear parameterisations a Unitary Group Approach $U(n)$ Tensor Network is proposed. Where both the tensor network state and operator are constructed using $SU(2)$ angular momentum coupling in the form of Jucy's diagrams.