Rotational-vibrational relative equilibria and the structure of quantum energy spectrum of the tetrahedral molecule P ₄

Abstract:

We find relative equilibria (RE) of the rotating and vibrating tetrahedral molecule P₄ and study the correspondence of these RE's to the extremal quantum states in the vibration-rotation multiplet and to the extrema of the semi-quantum rotational energy surfaces obtained for a number of excited vibrational states. To compute the energy of RE's we normalize the full rotation-vibration Hamiltonian H of P₄ in the approximation of nonresonant modes ν ^E ₂ and ν ^F_2 ₃ and find the stationary points of the resulting normal form (known as reduced effective Hamiltonian H _eff ) which is defined on the reduced phase space CP ₂ × CP ₁ × S ₂ . Most of these points are fixed points of the symmetry group action on CP ₂ × CP ₁ × S ₂ . To explain our results in more detail we introduce numerical values of the parameters of H, such as the cubic force constants, using an atom-atom harmonic potential with one adjustable parameter. This simple model gives correct qualitative description of the rotational structure of the lowest excited vibrational states ν ₂ , ν ₃ and ν ₂ + ν ₃ of P₄.