  This work establishes the algebraic structure of the Kohn-Sham equations to
be solved in a density formulation of electron and phonon dynamics, including
the superconducting order parameter. A Bogoliubov transform is required to
diagonalize both the fermionic and bosonic Kohn-Sham Hamiltonians since they
both represent a non-interacting quantum field theory. The Bogoliubov transform
for phonons is non-Hermitian in the general case, and the corresponding
time-evolution is non-unitary. Several sufficient conditions for ensuring that
the bosonic eigenvalues are real are provided and a practical method for
solving the system is described. Finally, we produce a set of approximate
mean-field potentials which are functionals of the electronic and phononic
density matrices and depend on the electron-phonon vertex.
