  Emission line profiles of tracer molecule H$_2$CO 140 GHz transition from
gravitational core collapsing clouds in the dynamic process of forming
protostars are calculated, using a simple ray-tracing radiative transfer model.
Three self-similar dynamic inside-out core collapse models -- the conventional
polytropic model, the empirical hybrid model and the isothermal model -- for
star-forming molecular clouds are examined and compared. The isothermal model
cannot produce observed asymmetric double-peak molecular line profiles. The
conventional polytropic model, which gives flow velocity, mass density and
temperature profiles self-consistently, can produce asymmetric double-peak line
profiles for a core collapsing cloud. In particular, the blue peak is stronger
than the red peak, consistent with a broad class of molecular line profile
observations. The relative strengths of the blue and red peaks within a
molecular line profile are determined by the cloud temperature gradient. The
conventional polytropic model can be utilized to produce molecular line-profile
templates, for extracting dynamical information from line spectra of molecular
globules undergoing a gravitational core collapse. We show a sample fit using
the 140 GHz H$_2$CO emission line from the central region of the molecular
globule B335 by our model with $\gamma=1.2$. The calculation of line profiles
and fitting processes also offer a scenario to estimate the protostellar mass,
the kernel mass accretion rate, and the evolution time scale of a core
collapsing cloud. Our model can be readily adapted to other tracer molecules
with more or less constant abundances in star-forming clouds.
