Out-of-plane symmetry constraints for in-plane bending vibrations

Whenever the dipole moment derivatives of out-of-plane bending vibrations in planar molecules are described in terms of atomic charges and their changes (either called charge transfer , C T , or charge fluxes , C F ), a constraint arises from the symmetry of the normal coordinate, requesting the charge transfer term to vanish, i.e., C T = 0 . This was first reported in 1989 by Dinur & Hagler and no exception has been found to this date. In this work, we show that the aforementioned symmetry constraints in planar molecules are, in fact, a special case within a more general one in which a given vibration is symmetric with respect to a given molecular plane of symmetry. This general case does not depend on the vibration being out-of-plane or in-plane bendings, nor on the molecule being planar or not. Nonetheless, whereas in the general case only the atoms lying on a symmetry plane will show C T = 0 , for out-of-plane vibrations of planar molecules all atoms will show C T = 0 simultaneously because all atoms belong to that plane, making the overall C T = 0 too. The argument is grounded on symmetry elements and their mathematical properties, while numerical examples are also presented and discussed.