Binding energy and scattering length for diatomic systems

doi:10.1088/0953-4075/37/21/004 We discuss the relationship between scattering length and binding energy for a diatomic system with long-range van der Waals interaction. The well-known relation, 0 = −(h̄2/2µ)(1/a0)2, is generalized to much higher orders for the s wave. Analytic results for the binding energies of p and higher angular momentum states are also presented. A least-bound state of a diatomic molecule, such as the least-bound s state or the least-bound p state (for spin-aligned fermions), plays a fundamental role in cold-atom physics. It is the dominant final state for the three-body recombination process (see, e.g. [1, 2]) that determines the lifetime of a condensate. It is also the state in which large samples of long-range molecules [3, 4] may be formed by taking advantage of a Feshbach resonance [5–16]. The understanding of the properties of this state, such as its energy, and how they relate to the scattering length, is both useful experimentally and a necessary ingredient of any theoretical formulation. While the binding energy may be easily computed numerically, such results are difficult to