in this Hamiltonian is the interaction of each monomer with the complex chemical potential field $ i w \left({\bf r},{\bf u} \right)$. We can simply interpret $w \left({\bf r},{\bf u} \right)$ as the response of test polymer chain to the potential of all the interactions involved in the system. The second term, $Q\left[ i w\right]$ is the normalized partition function of a wormlike chain which is the entropic term [@Glenn] $$Q\left[ i w\right]=\frac{\int{\mathcal{D}{\bf r}}\;\exp \left(-\beta U_{0}\left[{\bf u}  \right] -\beta U_{1}\left[{\bf r},i w  \right] \right)  }{ \int{\mathcal{D}{\bf r}}\;\exp \left(-\beta U_{0}\left[{\bf u}  \right]  \right) }$$

where $U_{0}$ is the bonded potential between each monomer and $U_{1}$ is the interaction of each monomer with the complex chemical potential field $i w \left({\bf r},{\bf u} \right)$.

Due to the nature of the liquid crystals, the non-bonded interaction between the segments, the third term in Eq.(\[eq:Hamiltonian\]), should depend on the orientation of the segments, $v \left({\bf u},{\bf u^{\prime}} \right) \propto \lvert {\bf u}\times{\bf u}^{\prime} \rvert$, which is the Onsager model for LCPs [@Onsager]. The entropic term depends on the bonded potential for each chain, by increasing the stiffness, the translational entropy loss increases which implies that in the limiting cases of flexible and rod polymers, 