  Motivated by a recent paper of Pennisi (2021) in the relativistic framework,
we revisit the previous approach of two hierarchies of moments critically and
propose a new natural physical hierarchy of moments to describe classical
rarefied non-polytropic polyatomic gas in the framework of Molecular Rational
Extended Thermodynamics. The differential system of the previous approach is
proved to be a \emph{principal subsystem} of the present one. The main idea is
that at the molecular level, the total energy is the kinetic energy plus the
energy of internal mode due to the rotation and vibration, and the increasing
moments contain this total energy as power in which the power index increases
with the number of tensorial indexes. In particular, we consider the case of
$15$ moments, and we close the system using the variational method of the
Maximum Entropy Principle. We prove the convexity of entropy and the
possibility to put the system in symmetric form. This more rich kinetic
framework may be interesting also as possible applications to biomathematics or
other fields in which kinetic models were applied recently with success.
