\begin{equation}
A_{ai;bj} = \langle 0 \vert [\hat{q}^{\dagger}_{ai},[\hat{q}_{bj},\hat{f}_0 + \hat{v}^{\mathrm{PE}}]] \vert 0 \rangle +
  \langle 0 \vert [\hat{q}^{\dagger}_{ai},\hat{v}^{\mathrm{ind}}]] \vert 0 \rangle \label{Hessian_2a} \\
B_{ai;bj} = \langle 0 \vert [\hat{q}_{ai},[\hat{q}_{bj},\hat{f}_0 + \hat{v}^{\mathrm{PE}}]] \vert 0 \rangle +
  \langle 0 \vert [\hat{q}_{ai},\hat{v}^{\mathrm{ind}}] \vert 0 \rangle .
\end{equation}