First-Order Quantum Correction in Coherent State Expectation Value of
  Loop-Quantum-Gravity Hamiltonian
  Given the non-graph-changing Hamiltonian $\widehat{H[N]}$ in Loop Quantum
Gravity (LQG), $\langle\widehat{H[N]}\rangle$, the coherent state expectation
value of $\widehat{H[N]}$, admits an semiclassical expansion in $\ell^2_{\rm
p}$. In this paper, as presenting the detailed derivations of our previous work
arXiv:2012.14242, we explicitly compute the expansion of
$\langle\widehat{H[N]}\rangle$ to the linear order in $\ell^2_{\rm p}$ on the
cubic graph with respect to the coherent state peaked at the homogeneous and
isotropic data of cosmology. In our computation, a powerful algorithm is
developed, supported by rigorous proofs and several theorems, to overcome the
complexity in the computation of $\langle \widehat{H[N]} \rangle$.
Particularly, some key innovations in our algorithm substantially reduce the
complexity in computing the Lorentzian part of $\langle\widehat{H[N]}\rangle$.
Additionally, some quantum correction effects resulting from
$\langle\widehat{H[N]}\rangle$ in cosmology are discussed at the end of this
paper.
