The D₂ point group of the Two-Rotors Model and scissors modes of negative parity

The intrinsic Hamiltonian of the Two-Rotors Model as a quantized classical system in which the rotors are abstract rigid bodies has a $D_2$ point symmetry and then its eigenstates come in quadruplets. Only one member of each of the two lowest quadruplets was investigated so far, the ground state and the scissors mode. I determine the whole quadruplets which turn out to contain states of negative parity. When the rotors are made of particles, the $D_2$ symmetry is broken. The actual existence of the new states of the multiplets in atomic nuclei depends on the existence of excited states of the neutron and proton fluids separately odd under inversion of the intrinsic coordinates. The energies of the new states are $\omega +e,\omega +2e$ when one or both rotors have negative parity respectively, where ω is the scissors excitation energy and e is the excitation energy of a single rotor with negative parity. Nonvanishing transitions occur only between states whose energies differ by ω.