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Physical Review A

Cold magnetically trapped 2Dg scandium atoms. I. Interaction potential

Tijs Karman, Xi Chu, Gerrit C. Groenenboom

Published Nov 5, 2014• 10 citations
Publisher

Abstract

We present a first principles description of the interaction of two ground-state scandium atoms. Scandium has a ${{}^{2}D}_{g}$ ground state. Thirty molecular states correlate to the lowest dissociation limit of the dimer. In the short range, potential energy curves are calculated using second-order $n$-electron valence state perturbation theory. The first-order long-range interaction is calculated at the complete active space self-consistent field level. We determine the second-order long-range dispersion interaction from atomic dynamic polarizabilities at imaginary frequencies. These polarizabilities are calculated using time-dependent density functional theory. We merge the short-range approach with the long-range model to obtain a physical description of the 30 potential energy curves correlating to the ${{}^{2}D}_{g}+{{}^{2}D}_{g}$ limit. Diabatic potentials are presented that can be used in quantum scattering calculations, in order to study Zeeman relaxation of ultracold scandium atoms.

DOI
10.1103/physreva.90.052701

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Cold magnetically trapped 2Dg scandium atoms. II. Scattering dynamics

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Cold collisions of ground-state calcium atoms in a laser field: A theoretical study

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We present a first principles description of the interaction of two ground-state scandium atoms. Scandium has a 2Dg{{}^{2}D}_{g}2Dg​ ground state. Thirty molecular states correlate to the lowest dissociation limit of the dimer. In the short range, potential energy curves are calculated using second-order nnn-electron valence state perturbation theory. The first-order long-range interaction is calculated at the complete active space self-consistent field level. We determine the second-order long-range dispersion interaction from atomic dynamic polarizabilities at imaginary frequencies. These polarizabilities are calculated using time-dependent density functional theory. We merge the short-range approach with the long-range model to obtain a physical description of the 30 potential energy curves correlating to the 2Dg+2Dg{{}^{2}D}_{g}+{{}^{2}D}_{g}2Dg​+2Dg​ limit. Diabatic potentials are presented that can be used in quantum scattering calculations, in order to study Zeeman relaxation of ultracold scandium atoms.
The binary collision dynamics of 2Dg,3/2{}^{2}{D}_{g,3/2}2Dg,3/2​ ground state scandium atoms is studied from first principles. We employ 30 coupled diabatic ab initio potentials in a coupled-channels study of the scattering dynamics of cold and ultracold scandium atoms in external magnetic fields. Due to the long-ranged magnetic dipolar interaction, the field dependence of the cross section does not follow the threshold laws derived by Volpi and Bohn [Phys. Rev. A 65, 052712 (2002)]. In the field-free case, the near-threshold cross section is independent of the collision energy, and hence the cross section does not follow the well-established Wigner threshold laws. The observed threshold behavior is explained in the Born approximation. For energies above $1\phantom{\rule{4pt}{0ex}}\ensuremath{\mu}\mathrm{K}$, inelastic collisions are driven by the anisotropic nonrelativistic electronic interaction. For energies below $100\phantom{\rule{4pt}{0ex}}\ensuremath{\mu}\mathrm{K}$, the ratio of elastic-to-inelastic collisions is likely to be favorable for evaporative cooling. Both anisotropy in the long-range interaction and in the short-range potential contribute to large cross sections for inelastic collisions at higher energies and lead to a small ratio of elastic-to-inelastic collisions. This is in agreement with the large rates for Zeeman relaxation of submerged-shell atoms observed experimentally. The effect of the uncertainty in the ab initio potential is sampled by scaling the reduced mass and is found to have little influence on the conclusions drawn from this work.
There is at present great interest in the properties of ultracold molecules. Molecules are created in traps in excited rovibrational states and any vibrational relaxation results in the trap loss. This thesis provides a theoretical study of interactions and collisions in the spin-polarized lithium -b lithium dimer system at ultralow energies. Potential energy surface of the electronic quartet ground state of lithium trimer is generated ab initio using the CCSD(T) method and represented by an IMLS/Shepard fit. Long-range nonadditive interactions are modelled using a symmetric global form with coefficients taken from a fit to the atom-molecule dispersion coefficients. The surface allows barrierless atom-exchange reactions. It has a global minimum of ≈ 4000 cm(^-1) at equilateral geometries with r(_e) = 3.1 Å. The nonadditive interactions are very strong near equilibrium. They increase the well depth by a factor of 4 and reduce the interatomic distance by ≈ 1 Å. Another surface of À symmetry in C(_s) meets the ground state surface at linear geometries at short range. Part of the seam, near D(_ooh) geometries, is in an energetically accessible region for cold collisions. Inside the seam, the lowest À surface correlates with (^4)II rather than (^4)Σ state. Inelastic and reactive collisions are investigated using a quantum mechanical coupled channel method in hyperspherical coordinates. Bosonic and fermionic systems in the spin-stretched states are considered. The inelastic rate coefficients from the rovibrationally excited states of dimer at ultralow collision energies are large, often above 10-(^-10) cm(^3)s(^-1) The elastic cross sections are ≈ 3 orders of magnitude lower at 1 nK. Atom-molecule mixtures, at the densities found in Bose-Einstein condensates of alkali metal atoms that were recently produced, would last only a fraction of a second. Classical Langevin model describes semi-quantitatively the energy dependence of inelastic cross sections above ≈ 50 mK. No systematic differences between the bosonic and fermionic systems were found. Sensitivity of the results on potential was investigated. Reactions in isotopic mixtures of lithium may be exothermic even from the molecular ground state. The reactive rate coefficients are 1 - 2 orders of magnitude smaller than those in systems involving an initially vibrationally excited dimer.
State-of-the-art ab initio techniques have been applied to compute the potential-energy curves for the ground $X{}^{1}{\ensuremath{\Sigma}}_{g}^{+}$ and excited ${}^{1}{\ensuremath{\Pi}}_{g}(4s3d)$ states of the calcium dimer in the Born-Oppenheimer approximation. The weakly bound ground state was calculated by symmetry-adapted perturbation theory, while the strongly bound excited state was computed using a combination of the linear-response theory within the coupled-cluster singles and doubles framework for the core-valence electronic correlation and of the full configuration interaction for the valence-valence correlation. The ground-state potential has been corrected by considering the relativistic terms resulting from the first-order many-electron Breit theory, and the retardation corrections. The magnetic electronic transition dipole moment governing the ${}^{1}{\ensuremath{\Pi}}_{g}{\ensuremath{\leftarrow}}^{1}{\ensuremath{\Sigma}}_{g}^{+}$ transitions has been obtained as the first residue of the polarization propagator computed with the coupled-cluster method restricted to single and double excitations. The computed energies and transition moments have been analytically fitted and used in the dynamical calculations of the rovibrational energy levels, ground-state scattering length, photoassociation intensities at ultralow temperatures, and spontaneous emission coefficients from the ${}^{1}{\ensuremath{\Pi}}_{g}(4s3d)$ to the $X{}^{1}{\ensuremath{\Sigma}}_{g}^{+}$ state. The spectroscopic constants of the theoretical ground-state potential are in a good agreement with the experimental values derived from the Fourier-transform spectra [O. Allard et al., Eur. Phys. J. D (to be published)]. The theoretical s-wave scattering length for the ground state is a=44bohrs,a=44\mathrm{bohrs},a=44bohrs, suggesting that it should be possible to obtain a stable Bose-Einstein condensate of calcium atoms. Finally, the computed photoassociation intensities and spontaneous emission coefficients suggest that it should be possible to obtain cold calcium molecules by photoassociation of cold atoms to the first ${}^{1}{\ensuremath{\Pi}}_{g}$ state followed by a spontaneous emission to the ground state.
In order to study ultracold charge-transfer processes in hybrid atom-ion traps, we have mapped out the potential-energy curves and molecular parameters for several low-lying states of the Rb, Yb+{}^{+}+ system. We employ both a multireference configuration interaction and a full configuration interaction (FCI) approach. Turning points, crossing points, potential minima, and spectroscopic molecular constants are obtained for the lowest five molecular states. Long-range parameters, including the dispersion coefficients, are estimated from our ab initio data. The separated-atom ionization potentials and atomic polarizability of the ytterbium atom (${\ensuremath{\alpha}}_{d}=128.4$ atomic units) are in good agreement with experiment and previous calculations. We present some dynamical calculations for (adiabatic) scattering lengths for the two lowest (Yb, Rb+{}^{+}+) channels that were carried out in our work. However, we find that the pseudopotential approximation is rather limited in validity and only applies to nK temperatures. The adiabatic scattering lengths for both the triplet and singlet channels indicate that both are large and negative in the FCI approximation.
We have performed three theoretical simulations relevant for describing collisions among laser-cooled silver atoms and for the formation of Ag 2 molecules from these colliding atoms. Firstly, we determined the relativistic electronic structure of Ag 2 molecules in ground and low-lying excited states. Secondly, we computed rotational and vibrational levels of the ground and excited electronic states as well as rovibrationally averaged electric transition dipole moments. Using this knowledge, we analyzed a simplified quantum-mechanical model of the one-photon photoassociation process to form electronically excited Ag 2 from microkelvin Ag atoms and make predictions for lineshapes and saturation effects as functions of laser frequency and intensity. Finally and thirdly, we performed coupled-channels calculations, numerical solutions of sets of coupled radial Schrödinger equations of ultracold ground-state Ag collisions in an external magnetic field. These calculations include the effects of two Born-Oppenheimer potentials as well as hyperfine Fermi-contact and Zeeman interactions. We discuss the expected range of s -wave scattering lengths as well as strengths and distribution of Fano-Feshbach resonances as a function of the magnetic-field strength for the Ag 107 and Ag 109 isotopes. We highlight the periodicity of the scattering length with small changes in the depths of the Born-Oppenheimer potentials. The Fano-Feshbach resonances can be used to magneto-associate ultracold Ag atoms into weakly bound ground-state Ag 2 dimers.
The potential energy curve for the ground-state of radium dimer (Ra2) is provided by means of atomic and molecular relativistic coupled cluster calculations. The short-range part of this curve is defined by an equilibrium bond length of 5.324 Å, a dissociation energy of 897 cm(-1), and a harmonic vibrational frequency of 20.5 cm(-1). The asymptotic behavior at large interatomic distances is characterized by the van der Waals coefficients C6 = 5.090 × 10(3), C8 = 6.978 × 10(5), and C10 = 8.786 × 10(7) atomic units. The two regions are matched in an analytical potential to provide a convenient representation for use in further calculations, for instance, to model cold collisions between radium atoms. This might become relevant in future experiments on ultracold, optically trapped, radioactive radium atoms that are used to search for a permanent electric dipole moment.