  For an oriented diagram of a link $L$ in the 3-sphere, Cho and Murakami
defined the potential function whose critical point, slightly different from
the usual sense, corresponds to a boundary parabolic
$\mathrm{PSL}(2,\mathbb{C})$-representation of $\pi_1(S^3 \setminus L)$. They
also showed that the volume and Chern-Simons invariant of such a representation
can be computed from the potential function with its partial derivatives. In
this paper, we extend the potential function to a
$\mathrm{PSL}(2,\mathbb{C})$-representation that is not necessarily boundary
parabolic. Under a mild assumption, it leads us to a combinatorial formula for
computing the volume and Chern-Simons invariant of a
$\mathrm{PSL}(2,\mathbb{C})$-representation of a closed 3-manifold.
