  In light of the recently published complete set of statistically correct
Gronbech-Jensen (GJ) methods for discrete-time thermodynamics, we revise a
differential operator splitting method for the Langevin equation in order to
comply with the basic GJ thermodynamic sampling features, namely the Boltzmann
distribution and Einstein diffusion, in linear systems. This revision, which is
based on the introduction of time scaling along with flexibility of a
discrete-time velocity attenuation parameter, provides a direct link between
the ABO splitting formalism and the GJ methods. This link brings about the
conclusion that any GJ method has at least weak second order accuracy in the
applied time step. It further helps identify a novel half-step velocity, which
simultaneously produces both correct kinetic statistics and correct transport
measures for any of the statistically sound GJ methods. Explicit algorithmic
expressions are given for the integration of the new half-step velocity into
the GJ set of methods. Numerical simulations, including quantum-based molecular
dynamics (QMD) using the QMD suite LATTE, highlight the discussed properties of
the algorithms as well as exhibit the direct application of robust, time step
independent stochastic integrators to quantum-based molecular dynamics.
