  $^{12}$C$(\alpha, \gamma)^{16}$O radiative-capture process is a key reaction
to produce the element of oxygen in stars. Measuring the cross section near the
Gamow window is extremely hard because it is too small. To make a theoretical
contribution towards resolving the long-standing problem, I present a
microscopic formulation that aims at providing all materials needed to
calculate the cross section. The states of $^{12}$C and $^{16}$O relevant to
the reaction are respectively described with fully microscopic 3
$\alpha$-particle and 4 $\alpha$-particle configurations, in which the relative
motion among the $\alpha$ particles is expanded in terms of correlated Gaussian
basis functions. The configuration space has the advantage of being able to
well describe the reduced $\alpha$-width amplitudes of the states of $^{16}$O.
Both electric dipole and electric quadrupole transitions are responsible for
the radiative-capture process. The $\alpha$ particle is described with a
$(0s)^4$ configuration admixed with a small amount of an isospin $T=1$ impurity
component, which is crucially important to account for the isovector electric
dipole transition. The isoscalar electric dipole operators are also taken into
account up to the first order beyond the long-wavelength approximation. All the
necessary ingredients are provided to make the paper self-contained and ready
for numerical computations.
