With the aim of including small amplitude quantum nuclear dynamics in solid-state calculations, we derive a set of equations by applying Wick's theorem to the square of the Fröhlich Hamiltonian. These are noninteracting fermionic and bosonic Hamiltonians with terms up to quadratic order in the field operators. They depend on one another's density matrices and are therefore to be solved self-consistently. A Bogoliubov transformation is required to diagonalize both the fermionic and bosonic Hamiltonians since they represent noninteracting quantum field theories with an indefinite number of particles. The Bogoliubov transform for phonons is non-Hermitian in the general case, and the corresponding time evolution is nonunitary. Several sufficient conditions for ensuring that the bosonic eigenvalues are real are provided. The method was implemented in an all-electron code and shown to correctly predict the renormalization of the Kohn-Sham band gap of diamond and silicon due to the electron-phonon interaction. The theory also verifies that niobium and MgB2 are phonon-mediated superconductors and predicts the existence and magnitude of their superconducting gaps. Lastly, we confirm that copper is not a superconductor even at zero temperature.
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