  Size extensivity, defined as the correct scaling of energy with system size,
is a desirable property for any many-body method. Traditional CI methods are
not size extensive hence the error increases as the system gets larger. Coupled
electron pair approximation (CEPA) methods can be constructed as simple
extensions of truncated configuration interaction (CI) that ensures size
extensivity. One of the major issues with the CEPA and its variants is that
singularities arise in the amplitude equations when the system starts to be
strongly correlated. In this work, we extend the traditional Slater
determinant-based coupled electron pair approaches like CEPA-0, averaged
coupled-pair functional (ACPF) and average quadratic coupled-cluster (AQCC) to
a new formulation based on tensor product states (TPS). We show that a TPS
basis can often be chosen such that it removes the singularities that commonly
destroy the accuracy of CEPA-based methods. A suitable TPS representation can
be formed by partitioning the system into separate disjoint clusters and
forming the final wavefunction as the tensor product of the many body states of
these clusters. We demonstrate the application of these methods on simple bond
breaking systems such as CH$_4$ and F$_2$ where determinant based CEPA methods
fail. We further apply the TPS-CEPA approach to stillbene isomerization and few
planar $\pi-$conjugated systems. Overall the results show that the TPS-CEPA
method can remove the singularities and provide improved numerical results
compared to common electronic structure methods.
