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Molecular Physics

Critical binding of diatomic molecules

John Lekner

Published Mar 1, 1972• 34 citations
Publisher

Abstract

The behaviour of two bodies that are just bound or nearly bound is discussed. A class of potentials is given for which Schrödinger's equation has exact solutions at critical binding (zero binding energy). This class includes the known solution for the 6–10 potential. For a general potential characterized by a coupling parameter α, it is shown that the bound state energy tends to zero as -(α - α0)2, where α0 is the critical value of the coupling parameter. Small energy scattering of atoms which are near critical binding (e.g. helium atoms) is examined. It is shown that determination of the total cross-section up to terms of order k 2 is in principle sufficient to distinguish between bound and virtual states of the diatomic molecule.

DOI
10.1080/00268977200100601

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Physical Review Letters•Jan 10, 1972•14 citations
The behaviour of two bodies that are just bound or nearly bound is discussed. A class of potentials is given for which Schrödinger's equation has exact solutions at critical binding (zero binding energy). This class includes the known solution for the 6–10 potential. For a general potential characterized by a coupling parameter α, it is shown that the bound state energy tends to zero as -(α - α0)2, where α0 is the critical value of the coupling parameter. Small energy scattering of atoms which are near critical binding (e.g. helium atoms) is examined. It is shown that determination of the total cross-section up to terms of order k 2 is in principle sufficient to distinguish between bound and virtual states of the diatomic molecule.
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 RRR. For example, the helium dimer 4He2{{}^{4}\mathrm{He}}_{2}4He2​ 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 sss-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 4He2{{}^{4}\mathrm{He}}_{2}4He2​, we present results for 9Be2,20Ne2{{}^{9}\mathrm{Be}}_{2},\phantom{\rule{0.28em}{0ex}}{{}^{20}\mathrm{Ne}}_{2}9Be2​,20Ne2​, and KRb.
Close-coupling calculations have been performed to investigate the formation of three-particle bound states in a model collinear system. A generalization of Levinson's theorem (1966) is employed to determine the critical potential strengths necessary for the existence of the triatomic states. The system consists of an atom interacting through a Morse potential with a diatomic molecule bound by a harmonic potential. Comparisons are made with classical phase-space and S-matrix theory and with quantum-mechanical variational and perturbation calculations. When only a few bound states are involved it is shown that the quantal approximations are more accurate than the classical results.
We examine the existence of stable bound states of some diatomic molecular ions where the neutral molecule itself is either unbound or barely bound. Two bound states have been found for the ${\mathrm{HeH}}^{\ensuremath{-}}$ ion even though HeH is known to be unbound. We have found no bound states for ${\mathrm{He}}_{2}^{\ensuremath{-}}$ even though 4He2{}^{4}{\mathrm{He}}_{2}4He2​ is known to have one bound state. The predicted binding energies for ${\mathrm{HeH}}^{\ensuremath{-}}$ are of the order of 1 K and depend sensitively on the isotope combinations. Other weakly bound negative ions such as ${\mathrm{NeH}}^{\ensuremath{-}}$ and ${\mathrm{ArH}}^{\ensuremath{-}}$ have also been investigated. These weakly bound molecular ions are good candidates for cold collision experiments.
The best available intermolecular potential for helium by Aziz, Nain, Carley, Taylor, and McConville is shown here for the first time to have a 4He2 bound state. Two numerical analyses, namely, eigenvalue solution and effective range theory, are used to support this conclusion. Unlike usual chemically bound species, the binding energy of this very weakly bound level is found to be only 8.3×10−4 K, which is four orders of magnitude smaller than the potential well depth ε = 10.8 K. The scattering length for He+He collisions, determined from effective range theory, is used to calculate the elastic cross section in the very low energy limit. The results (1.878×105 Å2 for 4He+4He and 6.035×102 Å for 3He+3He) are consistent with measurements at the lowest velocities yet attained. In terms of the estimated uncertainties of the parameters of the potential of Aziz and co-workers, it is shown that it is very likely that a bound state of the 4He2 molecule does in fact exist.
Within the framework of Faddeev theory we calculate the binding energy of the lowest bound state of three identical particles. Excluding three-body potentials, the particles are assumed to interact mutually via a two-body Morse potential. Finally, our model calculations are applied to a system formed of three helium atoms. Using a Morse potential which is fitted to the He-He interaction potential calculated by Bertoncini and Wahl, our calculation predicts a stable triatomic helium molecule with a binding energy of 0.4\ifmmode^\circ\else\textdegree\fi{}K, even though a diatomic helium molecule does not exist for the same well depth.