  A large number of candidate binary stars with apparent acceleration on the
sky has emerged from analysis of astrometric data collected by the Hipparcos,
Tycho-2, and Gaia space missions. Although the apparent acceleration can serve
as a relatively reliable indicator of binarity, it provides scarce information
about the orbital and physical parameters of the components. With an emphasis
on the search for stellar-mass black holes and neutron stars hidden in binary
systems, we start a broader effort to characterize the most promising
candidates using follow-up ground-based observations. Accurate quantification
of orbital and physical parameters of systems with dim or invisible companions
requires combination of Hipparcos, Gaia, and precision spectroscopic
measurements. In this paper, we review the necessary steps in this
implementation and describe the improved Hipparcos-Gaia sample of long-term
astrometric accelerations which includes correction of sky-correlated
systematic errors using the vector spherical decomposition method. As an
example, we study one Hipparcos star with a large acceleration, HIP 44842,
where the companion is revealed to be a normal main sequence star.
