  The thermodynamic limit and boundary energy of the isotropic spin-1
Heisenberg chain with non-diagonal boundary fields are studied. The finite size
scaling properties of the inhomogeneous term in the $ T-Q $ relation at the
ground state are calculated by the density matrix renormalization group. Based
on our findings, the boundary energy of the system in the thermodynamic limit
can be obtained from Bethe ansatz equations of a related model with parallel
boundary fields. These results can be generalized to the $SU(2)$ symmetric high
spin Heisenberg model directly.
