  The coupling of electrons and phonons is governed wisely by the symmetry
properties of the crystal structures. In particular, for two-dimensional (2D)
systems, it has been suggested that the electrons do not couple to phonons with
pure out-of-plane distortion, as long as there is a $\sigma_h$ symmetry. We
show that such a statement is correct when constituents of the unit-cell layer
are only located in the $\sigma_h$ symmetric plane; a prominent example of such
a system is graphene. For those 2D crystals in which atoms are vertically
located away from the horizontal symmetric plane (e.g., 1H transition metal
dichalcogenides), acoustic flexural modes do not couple to the electrons up to
linear order, while optical flexural phonons, which preserve $\sigma_h$
symmetry, do couple with the electrons. Our conclusions are supported by an
analytic argument together with numerical calculations using density functional
perturbation theory.
