We discuss an algebraic treatment of four-body clusters which includes both continuous and discrete symmetries. In particular, tetrahedral configurations with Td symmetry are analyzed with respect to the energy spectrum, transition form factors and B(EL) values. It is concluded that the low-lying spectrum of 16O can be described by four α particles at the vertices of a regular tetrahedron, not as a rigid structure but rather a more floppy structure with relatively large rotation–vibration interactions and Coriolis forces.

We calculate single-particle levels in potentials with Z2 (dumbbell), D3h (triangle) and Td (tetrahedral) symmetry, appropriate to the α-cluster structure of 8Be, 12C and 16O respectively. We suggest that these can be used to study, within the framework of a cluster shell model (CSM), kα+x nucleon structures, with k=2,3,4 and x=1,2,…, in particular the single particle (x=1) structures 9Be, 9B; 13C, 13N; 17O, 17F.

The concepts of quantum phase transition (QPT) and excited state quantum phase transition (ESQPT) are applied to the study of bending motion in non-rigid molecules. Two key signatures of QPT and ESQPT, the Birge-Sponer (or anharmonicity) plot and the monodromy plot are discussed. A detailed analysis of bending motion in a series of molecules (HCNO, DCNO, BrCNO, ClCNO, CH3NCS, GeH3NCO, CH3NCO, OCCCS, NCCNO, NCNCS) spanning the rigidly-linear to rigidly-bent transition is presented, and potential functions in terms of angle variables are constructed. The effect of QPT and ESQPT on the rotational constant B and on rotation–vibration interactions through the splitting of the J = 1, Ka = ±1 components is evaluated and, finally, some anomalous features of mass scaling in non-rigid molecules are identified.Highlights► Simple algebraic Hamiltonian models bent-linear geometric transition. ► 2-dim vibron model can describe bending in small, non-rigid molecules. ► Quantum monodromy is an excited state quantum phase transition.