  A recently proposed theoretical framework for the description of electronic
quantum friction for constant-velocity nuclear projectiles traversing periodic
crystals is here implemented using a local basis representation. The theory
requires a change of reference frame to that of the projectile, and a basis set
transformation for the target basis functions to a gliding basis is presented,
which is time-periodic but does not displace in space with respect to the
projectile, allowing a local-basis Floquet impurity-scattering formalism to be
used. It is illustrated for a one-dimensional single-band tight-binding model,
as the simplest paradigmatic example, displaying the qualitative behaviour of
the formalism. The time-dependent non-orthogonality of the gliding basis
requires care in the proper (simplest) definition of a local projectile
perturbation. The Fermi level is tilted with a slope given by the projectile
velocity, which complicates integration over occupied states. It is solved by a
recurrent application of the Lippmann-Schwinger equation, in analogy with
previous non-equilibrium treatment of electron ballistic transport. Aiming
towards a first-principles mean-field-like implementation, the final result is
the time-periodic particle density in the region around the projectile,
describing the stroboscopically stationary perturbation cloud around the
projectile, out of which other quantities can be obtained, such as the
electronic stopping power.
