Here P y includes the derivatives of orbital rotations. The first term represents the MM charges and dipoles induced by the seond-order QM potentials and fields with respect to QM nuclear coordinate changes, while the second term is the interaction between first-order QM potentials and fields and MM induced charges and dipoles from another first-order QM potentials and fields perturbed by QM atom displacements. After the construction of the Hessian matrix in the mass-weighted coordinates, and projecting the rotational and translational degrees of freedom out of this harmonic force constant matrix, one could obtain the normal modes and vibrational frequencies by diagonalizing the projected matrix. Note that here the Hessian matrix only includes the QM-QM block, and hence, it is actually partial Hessian within the QM region.