immediately faced with a twofold basic problem: How can we describe the embedding of the system block within the superblock and how can one include the boundary conditions during a RG step. In the framework of DMRG these problems are overcome by focusing on one particular state, the [*target state*]{} $\Ket{\psi}$, which is the ground state of the superblock Hamiltonian obtained by diagonalization. By using a complete set of eigenstates of the system block $\left\{\Ket{\psi^{\text{system}}_m},\; m=1,\dots ,{l_{\text{system}}}\right\}$ and a complete set of eigenstates for the environment represented by the superblock $\left\{\Ket{\psi^{\text{environment}}_n},\; n=1,\dots ,{l_{\text{environment}}}\right\}$ we decompose the target state $\Ket{\psi}$ according to $$\label{dmrg3}
 \Ket{\psi}\,=\,\sum_{m}^{l_{\text{system}}}\,\sum_{n}^{l_{\text{environment}}}
 \; c_{m,n}\,\Ket{\psi^{\text{system}}_m}\,
  \otimes\,\Ket{\psi^{\text{environment}}_n}\,.$$ We are interested in those states, which lead to an optimal representation of the target state $|\psi >$ in a “truncated” basis. Of course in this way we loose the exactness of relation (\[dmrg3\]) and we therefore denote the new result as an [*optimal approximation*]{} expressed as $$\label{dmrg4}
 \Ket{\psi}\,\approx\,\Ket{\psi^{\text{opt}}}\,=\,\sum_{p}^{l_{\text{opt}}}\,
    \sum_n^{l_{\text{environment}}}\; \gamma_{p,n}\,
     \Ket{\psi^{\text{opt}}_p}\,\otimes\,\Ket{\psi^{\text{environment}}_n}\,,$$ where the [*optimal states*]{} $\left\{\Ket{\psi^{\text{opt}}_p},\; p=1,\dots ,l_{\text{opt}}
  <{l_{\text{system}}}\right\}$ are defined in terms of the original system block states by $$\label{dmrg5}
 \Ket{\psi^{\text{opt}}_p}\,=\,\sum_{m}^{l_{\text{system}}}\;
  {\alpha}_{m,p}^{\text{opt}}\,\Ket{\psi^{\text{system}}_m}$$ with some coefficients ${\alpha}_{m,p}^{\text{opt}}$. The coefficients $\gamma_{p,n}$ in (\[dmrg4\]) can be determined by examining the [*reduced density matrix*]{} of the 