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Physical Review A

Relation between properties of long-range diatomic bound states

V. Špirko, Stephan P. A. Sauer, Krzysztof Szalewicz

Published Jan 23, 2013• 11 citations
Publisher

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.

DOI
10.1103/physreva.87.012510

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The Morse/Long Range (MLR) potential has become one of the most reliable and highly \nused potential energy functions for diatomic molecules. It includes the theoretical long range \nbehaviour that diatomic molecules are known to exhibit as they approach the dissociation \nlimit. Heavy alkali metals with adjacent electronic states often exhibit strong coupling \nbetween the spin and orbital angular momentum. The ground state X¹Σg⁺ and the lowest \nlying triplet state aᶟΣᵤ⁺ of Cs₂ exhibit such coupling effects and as a result, modeling the \nhighest vibrational states of these states is a non-trivial problem. Utilizing scattering length \nvalues obtained from published analysis of 60 Feshbach resonances, the correct form of the \npotential energy function was determined. Moreover, the scattering length values were used \nto determine the correct leading dispersion coefficient that describes the true form of the \nlong-range potential energy functions. All previous attempts to determine global potential \nenergy functions for these states have considered only the optical spectroscopic data. This is \nthe first ever effort attempting to use scattering lengths determined from cold atom collision \nexperiments in a combined analysis with conventional spectroscopic data.

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Long-range states of diatomic molecules have average values of internuclear separations \ensuremath⟨R\ensuremath⟩\ensuremath{\langle}R\ensuremath{\rangle}\ensuremath⟨R\ensuremath⟩ 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⟨R\ensuremath⟩\ensuremath{\langle}R\ensuremath{\rangle}\ensuremath⟨R\ensuremath⟩ of about 50 \AA{}. We show that the properties of these states, such as \ensuremath⟨R\ensuremath⟩\ensuremath{\langle}R\ensuremath{\rangle}\ensuremath⟨R\ensuremath⟩, 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.
We investigate the long-range, two-body interactions between rubidium and potassium atoms in highly excited (n=70)(n=70)(n=70) Rydberg states. After establishing properly symmetrized asymptotic basis states, we diagonalize an interaction Hamiltonian consisting of the standard Coulombic potential expansion and atomic fine structure to calculate electronic potential energy curves. We find that when both atoms are excited to either the 70s70s70s state or the 70p70p70p state, both the $\mathrm{\ensuremath{\Omega}}={0}^{+}$ symmetry interactions and the $\mathrm{\ensuremath{\Omega}}={0}^{\ensuremath{-}}$ symmetry interactions demonstrate a deep potential well capable of supporting many bound levels; the sizes of the corresponding dimer states are of the order of 2.25 $\ensuremath{\mu}\mathrm{m}$. We establish nnn-scaling relations for the equilibrium separation Re{R}_{e}Re​ and the dissociation energy De{D}_{e}De​ and find these relations to be consistent with similar calculations involving the homonuclear interactions between rubidium and cesium. We discuss the specific effects of $\ensuremath{\ell}$ mixing and the exact composition of the calculated potential well via the expansion coefficients of the asymptotic basis states. Finally, we apply a Landau-Zener treatment to show that the dimer states are stable with respect to predissociation.
Simple analytical expressions are derived for the bound-state energy as well as for the resonance energy and width in the case of the Gaussian potential. These accurate formulas may find applications in nuclear physics, in studies of low-energy electron–atom collisions, and in simulations of few-body properties in Bose–Einstein condensates and Fermi gases. An important feature of the present approach is that the formulas rely solely on the knowledge of a set of real bound-state energies. In addition, an analogous expression is obtained for the s -wave scattering length. This formula is constructed so as to yield highly accurate values of the scattering length in the ground-state region, where the scattering length becomes large.
We introduce a computational method developed for study of long-range molecular Rydberg states of such systems that can be approximated by two electrons in a model potential of the atomic cores. Only diatomic molecules are considered. The method is based on a two-electron RRR-matrix approach inside a sphere centered on one of the atoms. The wave function is then connected to a Coulomb region outside the sphere via a multichannel version of the Coulomb Green's function. This approach is put into a test by its application to a study of Rydberg states of the hydrogen molecule for internuclear distances RRR from 20 to 400 bohrs and energies corresponding to nnn from 3 to 22. The results are compared with previous quantum chemical calculations (lower quantum numbers nnn) and computations based on contact-potential models (higher quantum numbers nnn).
The Morse/Long Range (MLR) potential has become one of the most reliable and highly \nused potential energy functions for diatomic molecules. It includes the theoretical long range \nbehaviour that diatomic molecules are known to exhibit as they approach the dissociation \nlimit. Heavy alkali metals with adjacent electronic states often exhibit strong coupling \nbetween the spin and orbital angular momentum. The ground state X¹Σg⁺ and the lowest \nlying triplet state aᶟΣᵤ⁺ of Cs₂ exhibit such coupling effects and as a result, modeling the \nhighest vibrational states of these states is a non-trivial problem. Utilizing scattering length \nvalues obtained from published analysis of 60 Feshbach resonances, the correct form of the \npotential energy function was determined. Moreover, the scattering length values were used \nto determine the correct leading dispersion coefficient that describes the true form of the \nlong-range potential energy functions. All previous attempts to determine global potential \nenergy functions for these states have considered only the optical spectroscopic data. This is \nthe first ever effort attempting to use scattering lengths determined from cold atom collision \nexperiments in a combined analysis with conventional spectroscopic data.