  We present the Copupled Cluster (CC) method and the Density matrix
Renormalization Grooup (DMRG) method in a unified way, from the perspective of
recent developments in tensor product approximation. We present an introduction
into recently developed hierarchical tensor representations, in particular
tensor trains which are matrix product states in physics language. The discrete
equations of full CI approximation applied to the electronic Schr\"odinger
equation is casted into a tensorial framework in form of the second
quantization. A further approximation is performed afterwards by tensor
approximation within a hierarchical format or equivalently a tree tensor
network. We establish the (differential) geometry of low rank hierarchical
tensors and apply the Driac Frenkel principle to reduce the original
high-dimensional problem to low dimensions. The DMRG algorithm is established
as an optimization method in this format with alternating directional search.
We briefly introduce the CC method and refer to our theoretical results. We
compare this approach in the present discrete formulation with the CC method
and its underlying exponential parametrization.
