Rotational and rotationless states of weakly bound molecules

By making use of the quantization rule of Raab and Friedrich [Phys. Rev. A 78, 022707 (2008)], we derive simple and accurate formulae for the number of rotational states supported by a weakly bound vibrational level of a diatomic molecule and the rotational constants of any such levels up to the threshold, and provide a criterion for determining whether a given weakly bound vibrational level is rotationless. The results depend solely on the long-range part of the molecular potential and are applicable to halo molecules.

Figure 1

(Color online) A schematic illustrating the role of the centrifugal term in the effective potential (12); the energy splittings have been exaggerated. Shown is the position of a rotationless vibrational level $v\left(J=0\right)$ (dashed line), as well as the position of the same level when pushed up by the centrifugal term to threshold $v\left(J=J^{\ast }\right)=v_{\text{th}}\left(J=0\right)$ (full line at threshold). When the rotational angular momentum $J$ exceeds the critical value $J^{\ast }$, the centrifugal term pushes the vibrational level above threshold $v\left(J>J^{\ast }\right)$ (full line above threshold), thus leading to dissociation.