  Chirality-induced spin selectivity (CISS) refers to an ability to induce a
spin polarization of an electron transmitted through chiral materials. An
important experimental observation is that incredibly large spin polarization
is realized at room temperature even for organic molecules that have weak
spin-orbit coupling (SOC), although SOC is the only interaction that can
manipulate the electrons' spins in the setups. Therefore, the mechanism of the
CISS needs to be constructed in a way insensitive to or enhancing the magnitude
of the SOC strength. In this paper, we describe a theoretical study of CISS
with a model chiral molecule that belongs to the point group $\mathrm{C}_3$. In
this molecule, electronic translational and rotational degrees of freedom for
an injected electron are coupled to one another via the nuclear vibrational
mode with a pseudo Jahn-Teller effect. By properly taking the molecular
symmetry as well as the time-reversal symmetry into account and classifying the
molecular ground states by their angular- and spin-momentum quantum numbers, we
show that the chiral molecule can act as an efficient spin filter. The
efficiency of this spin filtering can be nearly independent of the SOC strength
in this model, while it well exceeds the spin polarization relying solely on
the SOC. The nuclear vibrations turned out to have the role of not only
mediating the translation-rotation coupling, but also enhancing the
spin-filtering efficiency.
