  In a molecule subjected to no external fields, motion of nuclei is governed
by a function V of nuclear coordinates. This function (potential energy) is a
sum of two terms: Coulomb repulsion between nuclei and the electronic effective
potential E which results from the Born-Oppenheimer approximation. In this
paper we have presented a group of coordinate transformations which are the
symmetries of functions E and V. We showed that the formula for dynamical
representation, which has fundamental importance in the symmetry analysis of
normal modes of a molecule, follows from these symmetries. In addition, we
proved that every molecule has at least one normal mode belonging to the
totally symmetric (and therefore Raman-active) irreducible representation of
the point group of that molecule. Next, we used symmetries of V and E to
analyze possible shapes of some molecule types. We applied both
Abud-Sartori-Michel theorem and symmetry adapted expansion of the electronic
effective potential around united atom. Finally, we derived an approximate
relation which predicts order of magnitude of the vibration frequency from the
bond length in a diatomic molecule.
