  This article describes a method for calculating higher order or nonadiabatic
corrections in Born-Oppenheimer theory and its interaction with the
translational degrees of freedom. The method uses the Wigner-Weyl
correspondence to map nuclear operators into functions on the classical phase
space and the Moyal star product to represent operator multiplication on those
functions. The result is a power series in $\kappa^2$, where $\kappa
=(m/M)^{1/4}$ is the usual Born-Oppenheimer parameter. The lowest order term is
the usual Born-Oppenheimer approximation while higher order terms are
nonadiabatic corrections. These are needed in calculations of electronic
currents, momenta and densities. The method was applied to Born-Oppenheimer
theory by Littlejohn and Weigert (1993), in a treatment that notably produced
the correction $K_{22}$ to the Born-Oppenheimer Hamiltonian (see {\em infra}).
Recently Matyus and Teufel (2019) have applied an improved and more elegant
version of the method to Born-Oppenheimer theory, and have calculated the
Born-Oppenheimer Hamiltonian for multiple potential energy surfaces to order
$\kappa^6$. One of the shortcomings of earlier methods is that the separation
of nuclear and electronic degrees of freedom takes place in the context of the
exact symmetries (for an isolated molecule) of translations and rotations, and
these need to be a part of the discussion. This article presents an independent
derivation of the Moyal expansion in molecular Born-Oppenheimer theory, with
special attention to the translational degrees of freedom. We show how
electronic currents and momenta can be calculated within the framework of Moyal
perturbation theory; we derive the transformation laws of the electronic
Hamiltonian, the electronic eigenstates, and the derivative couplings under
translations.
