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A detailed understanding of strong matter-photon interactions requiresfirst-principle methods that can solve the fundamental Pauli-Fierz Hamiltonianof non-relativistic quantum electrodynamics efficiently. A possible way toextend well-established electronic-structure methods to this situation is toembed the Pauli-Fierz Hamiltonian in a higher-dimensional light-matter hybridauxiliary configuration space. In this work we show the importance of theresulting hybrid Fermi-Bose statistics of the polaritons, which are the newfundamental particles of the ``photon-dressed'' Pauli-Fierz Hamiltonian forsystems in cavities. We show that violations of these statistics can lead tounphysical results. We present an efficient way to ensure the proper symmetryof the underlying wave functions by enforcing representability conditions onthe dressed one-body reduced density matrix. We further present a generalprescription how to extend a given first-principles approach to polaritons andas an example introduce polaritonic Hartree-Fock theory. While being asingle-reference method in polariton space, polaritonic Hartree-Fock is amulti-reference method in the electronic space, i.e. it describes electroniccorrelations. We also discuss possible applications to polaritonic QEDFT. Weapply this theory to a lattice model and find that the more delocalized thebound-state wave function of the particles is, the stronger it reacts tophotons. The main reason is that within a small energy range many states withdifferent electronic configurations are available as opposed to a stronglybound (and hence energetically separated) ground-state wave function. Thisindicates that under certain conditions coupling to the quantum vacuum of acavity can indeed modify ground state properties.

Light-matter hybrid-orbital-based first-principles methods: the influence of the polariton statistics