  The Gaussian expansion method (GEM) is extensively applied to the
calculations in the random-phase approximation (RPA). We adopt the
mass-independent basis-set that has been tested in the mean-field calculations.
By comparing the RPA results with those obtained by several other available
methods for Ca isotopes, using a density-dependent contact interaction and the
Woods-Saxon single-particle states, we confirm that energies, transition
strengths and widths of their distribution are described by the GEM bases to
good precision, for the $1^-$, $2^+$ and $3^-$ collective states. The GEM is
then applied to the self-consistent RPA calculations with the finite-range
Gogny D1S interaction. The spurious center-of-mass motion is well separated
from the physical states in the $E1$ response, and the energy-weighted sum
rules for the isoscalar transitions are fulfilled reasonably well. Properties
of low-energy transitions in $^{60}$Ca are argued in some detail.
