Abstract
Long-range states of diatomic molecules have average values of internuclear separations $\ensuremath{\langle}R\ensuremath{\rangle}$ at least one order of magnitude larger than the equilibrium value of $R$. For example, the helium dimer ${{}^{4}\mathrm{He}}_{2}$ has a single bound state with $\ensuremath{\langle}R\ensuremath{\rangle}$ of about 50 \AA{}. We show that the properties of these states, such as $\ensuremath{\langle}R\ensuremath{\rangle}$, the dissociation energy, or the $s$-wave scattering length, can be related by simple, yet very accurate, formulas if a potential-energy curve is known. By examining a range of ab initio and empirical helium dimer potentials as well as scaling these potentials, we found that the formulas remain accurate, even if very approximate potentials were used. In addition to ${{}^{4}\mathrm{He}}_{2}$, we present results for ${{}^{9}\mathrm{Be}}_{2},\phantom{\rule{0.28em}{0ex}}{{}^{20}\mathrm{Ne}}_{2}$, and KRb.
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