We present a deep-atmosphere formulation for HOMME, the atmospheric dynamical core of E3SM. HOMME presently uses traditional and shallow-fluid approximations. The approximations omit the cosine-Coriolis terms and a few small metric terms, and the radius of the domain is considered constant independently of height. The deep-atmosphere  formulation relaxes these assumptions; it is a natural extension of the current formulation of HOMME, which is based on a Hamiltonian approach for a pressure-based vertical coordinate. We discuss analytical and numerical aspects of energy conservation, choices for the top boundary condition, physics-dynamics coupling, and implicit-explicit Runge-Kutta time-stepping for the new formulation.