  Making and using polaritonic states (i.e., hybrid electron-photon states) for
chemical applications have recently become one of the most prominent and active
fields that connects the communities of chemistry and quantum optics. Modeling
of such polaritonic phenomena using ab initio approaches calls for new
methodologies, leading to the reinvention of many commonly used electronic
structure methods, such as Hartree-Fock, density functional, and coupled
cluster theories. In this work, we explore the formally exact diffusion quantum
Monte Carlo approach (DQMC) to obtain numerical solutions to the polaritonic
ground state during the dissociation of the H$_2$ molecular system. We examine
various electron-nuclear-photon properties throughout the dissociation, such as
changes to the minimum of the cavity Born-Oppenheimer surface, the localization
of the electronic wavefunction, and the average mode occupation. Finally, we
directly compare our results to that obtained with state-of-the-art, yet
approximate, polaritonic coupled cluster approaches.
