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arXiv:physics/0510232 · Physical Review A

Long-range interactions between a He(2S3) atom and a He(2P3) atom for like isotopes

Junlang Zhang, Z.-C. Yan, D. Vrinceanu, J. F. Babb, H. R. Sadeghpour

Published Feb 9, 2006• 23 citations
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Abstract

For the interactions between a $\mathrm{He}(2\phantom{\rule{0.2em}{0ex}}^{3}S)$ atom and a $\mathrm{He}(2\phantom{\rule{0.2em}{0ex}}^{3}P)$ atom for like isotopes, we report perturbation theoretic calculations using accurate variational wave functions in Hylleraas coordinates of the coefficients determining the potential energies at large internuclear separations. We evaluate the coefficient ${C}_{3}$ of the first order resonant dipole-dipole energy and the van der Waals coefficients ${C}_{6}$, ${C}_{8}$, and ${C}_{10}$ for the second order energies arising from the mutual perturbations of instantaneous electric dipole, quadrupole, and octupole interactions. We also evaluate the leading contribution to the third-order energy. We establish definitive values including treatment of the finite nuclear mass for the $^{3}\mathrm{He}(2\phantom{\rule{0.2em}{0ex}}^{3}S)\text{\ensuremath{-}}^{3}\mathrm{He}(2\phantom{\rule{0.2em}{0ex}}^{3}P)$ and $^{4}\mathrm{He}(2\phantom{\rule{0.2em}{0ex}}^{3}S)\text{\ensuremath{-}}^{4}\mathrm{He}(2\phantom{\rule{0.2em}{0ex}}^{3}P)$ interactions.

DOI
10.1103/physreva.73.022710

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For the interactions between a He(23S)\mathrm{He}(2\phantom{\rule{0.2em}{0ex}}^{3}S)He(23S) atom and a He(23P)\mathrm{He}(2\phantom{\rule{0.2em}{0ex}}^{3}P)He(23P) atom for like isotopes, we report perturbation theoretic calculations using accurate variational wave functions in Hylleraas coordinates of the coefficients determining the potential energies at large internuclear separations. We evaluate the coefficient C3{C}_{3}C3​ of the first order resonant dipole-dipole energy and the van der Waals coefficients C6{C}_{6}C6​, C8{C}_{8}C8​, and C10{C}_{10}C10​ for the second order energies arising from the mutual perturbations of instantaneous electric dipole, quadrupole, and octupole interactions. We also evaluate the leading contribution to the third-order energy. We establish definitive values including treatment of the finite nuclear mass for the 3He(23S)\ensuremath-3He(23P)^{3}\mathrm{He}(2\phantom{\rule{0.2em}{0ex}}^{3}S)\text{\ensuremath{-}}^{3}\mathrm{He}(2\phantom{\rule{0.2em}{0ex}}^{3}P)3He(23S)\ensuremath-3He(23P) and 4He(23S)\ensuremath-4He(23P)^{4}\mathrm{He}(2\phantom{\rule{0.2em}{0ex}}^{3}S)\text{\ensuremath{-}}^{4}\mathrm{He}(2\phantom{\rule{0.2em}{0ex}}^{3}P)4He(23S)\ensuremath-4He(23P) interactions.
For helium, long-range interaction coefficients C3{C}_{3}C3​, C6{C}_{6}C6​, C8{C}_{8}C8​, C9{C}_{9}C9​, and C10{C}_{10}C10​ for all $S\text{\ensuremath{-}}S$ and $S\text{\ensuremath{-}}P$ pairs of the energetically lowest five states (11S,23S,21S,23P,(1\phantom{\rule{0.2em}{0ex}}^{1}S,\phantom{\rule{0.2em}{0ex}}2\phantom{\rule{0.2em}{0ex}}^{3}S,\phantom{\rule{0.2em}{0ex}}2\phantom{\rule{0.2em}{0ex}}^{1}S,\phantom{\rule{0.2em}{0ex}}2\phantom{\rule{0.2em}{0ex}}^{3}P,(11S,23S,21S,23P, and 21P)2\phantom{\rule{0.2em}{0ex}}^{1}P)21P) are calculated precisely using correlated wave functions in Hylleraas coordinates. Finite nuclear isotope mass effects are included.
We theoretically investigate long-range interactions between an excited LLL-state He atom and two identical SSS-state He atoms for the cases of the three atoms all in spin-singlet states or all in spin-triplet states, denoted by He(${n}_{0}^{\ensuremath{\lambda}}S$)-He(${n}_{0}^{\ensuremath{\lambda}}S$)-He(${n}_{0}^{\ensuremath{'}\ensuremath{\lambda}}L$), with n0{n}_{0}n0​ and ${n}_{0}^{\ensuremath{'}}$ principal quantum numbers, $\ensuremath{\lambda}=1$ or 3 the spin multiplicity, and LLL the orbital angular momentum of a He atom. Using degenerate perturbation theory for the energies up to second-order, we evaluate the coefficients C3{C}_{3}C3​ of the first-order dipolar interactions and the coefficients C6{C}_{6}C6​ and C8{C}_{8}C8​ of the second-order additive and nonadditive interactions. Both the dipolar and dispersion interaction coefficients, for these three-body degenerate systems, show dependences on the geometrical configurations of the three atoms. The nonadditive interactions start to appear in second-order. To demonstrate the results and for applications, the obtained coefficients Cn{C}_{n}Cn​ are evaluated with highly accurate variationally generated nonrelativistic wave functions in Hylleraas coordinates for $\text{He}(1{\phantom{\rule{0.16em}{0ex}}}^{1}S)\text{\ensuremath{-}}\text{He}(1{\phantom{\rule{0.16em}{0ex}}}^{1}S)\text{\ensuremath{-}}\text{He}(2{\phantom{\rule{0.16em}{0ex}}}^{1}S)$, $\text{He}(1{\phantom{\rule{0.16em}{0ex}}}^{1}S)\text{\ensuremath{-}}\text{He}(1{\phantom{\rule{0.16em}{0ex}}}^{1}S)\text{\ensuremath{-}}\text{He}(2{\phantom{\rule{0.16em}{0ex}}}^{1}P)$, $\text{He}(2{\phantom{\rule{0.16em}{0ex}}}^{1}S)\text{\ensuremath{-}}\text{He}(2{\phantom{\rule{0.16em}{0ex}}}^{1}S)\text{\ensuremath{-}}\text{He}(2{\phantom{\rule{0.16em}{0ex}}}^{1}P)$, and $\text{He}(2{\phantom{\rule{0.16em}{0ex}}}^{3}S)\text{\ensuremath{-}}\text{He}(2{\phantom{\rule{0.16em}{0ex}}}^{3}S)\text{\ensuremath{-}}\text{He}(2{\phantom{\rule{0.16em}{0ex}}}^{3}P)$. The calculations are given for three like nuclei for the cases of hypothetical infinite mass He nuclei, and of real finite mass 4He^{4}\mathrm{He}4He or 3He^{3}\mathrm{He}3He nuclei. The special cases of the three atoms in equilateral triangle configurations are explored in detail, and for the cases in which one of the atoms is in a PPP state, we also present results for the atoms in an isosceles right triangle configuration or in an equally spaced collinear configuration. The results can be applied to construct potential energy surfaces for three helium atom systems.
The long-range interactions among two- or three-atom systems are of considerable importance in the cold and ultracold research areas for many-body systems. For an ion and an atom, the long-range interaction potential is dominated by the induction (or polarization) potential resulting from the (classical) effect of the ion's electric field on the atom and the leading term of the induction potential is much stronger than the (quantum mechanical) dispersion (or van der Waals) interaction. The present paper focuses on the long-range interaction of the $\mathrm{Li}(2\phantom{\rule{0.16em}{0ex}}^{2}S)\ensuremath{-}\mathrm{Li}(2\phantom{\rule{0.16em}{0ex}}^{2}S)\ensuremath{-}{\mathrm{Li}}^{+}(1\phantom{\rule{0.16em}{0ex}}^{1}S)$ system, to see what changes this induction effect (originating in the electric field of the Li+{\mathrm{Li}}^{+}Li+ ion) yields in the long-range additive and nonadditive interactions of this three-body system. Using perturbation theory for energies, we evaluate the coefficients Cn{C}_{n}Cn​ in the potential energy for the three well-separated constituents, where nnn refers to the corresponding order in inverse powers of distance, obtaining the additive interaction coefficients C4,C6,C7,C8,C9{C}_{4}, {C}_{6}, {C}_{7}, {C}_{8}, {C}_{9}C4​,C6​,C7​,C8​,C9​ and the nonadditive interaction coefficients C7,C9{C}_{7}, {C}_{9}C7​,C9​. The obtained coefficients Cn{C}_{n}Cn​ are calculated with highly accurate variationally generated nonrelativistic wave functions in Hylleraas coordinates. Our calculations may be of interest for the study of three-body recombination and for constructing precise potential energy surfaces. We also provide precise evaluations of the long-range potentials for the two-body $\mathrm{Li}(2\phantom{\rule{0.16em}{0ex}}^{2}S)\ensuremath{-}{\mathrm{Li}}^{+}(1\phantom{\rule{0.16em}{0ex}}^{1}S)$ system. For both the two-body and three-body cases, we provide results for the like-nuclei cases of 6Li^{6}\mathrm{Li}6Li and 7Li^{7}\mathrm{Li}7Li.
General formulas for calculating the several leading long-range interactions among three identical atoms where two atoms are in identical SSS states and the other atom is in a PPP state are obtained using perturbation theory for the energies up to second order. The first-order (dipolar) interactions depend on the geometrical configurations of the three atoms. In second order, additive and nonadditive dispersion interactions are obtained. The nonadditive interactions depend on the geometrical configurations in marked contrast to the case where all three atoms are in identical SSS states, for which the nonadditive (also known as triple-dipole or as Axilrod-Muto-Teller) dispersion interactions appear at the third order. The formalism is demonstrated by the calculation of the coefficients for the $\mathrm{Li}(2\phantom{\rule{0.16em}{0ex}}^{2}S)\ensuremath{-}\mathrm{Li}(2\phantom{\rule{0.16em}{0ex}}^{2}S)\ensuremath{-}\mathrm{Li}(2\phantom{\rule{0.16em}{0ex}}^{2}P)$ system using variationally generated atomic lithium wave functions in Hylleraas coordinates. The present dipolar coefficients and additive and nonadditive dispersion coefficients may be useful in constructing precise potential energy surfaces for this three lithium atom system.
We evaluate the electric-dipole and electric-quadrupole static and dynamic polarizabilities for the 6s21S06{s}^{2\phantom{\rule{0.16em}{0ex}}1}{S}_{0}6s21S0​, 6s6p3P0o6s6p{\phantom{\rule{0.16em}{0ex}}}^{3}{P}_{0}^{o}6s6p3P0o​, and 6s6p3P1o6s6p{\phantom{\rule{0.16em}{0ex}}}^{3}{P}_{1}^{o}6s6p3P1o​ states and estimate their uncertainties. A first-principles relativistic method is developed for an accurate calculation of the van der Waals coefficients of dimers involving excited-state atoms with strong decay channel to the ground state. This method is used for evaluation of the long-range interaction coefficients of particular experimental interest, including the C6{C}_{6}C6​ coefficients for the Yb-Yb 1S0+3P0,1o{}^{1}{S}_{0}+{\phantom{\rule{0.16em}{0ex}}}^{3}{P}_{0,1}^{o}1S0​+3P0,1o​ and 3P0o+3P0o{}^{3}{P}_{0}^{o}+{\phantom{\rule{0.16em}{0ex}}}^{3}{P}_{0}^{o}3P0o​+3P0o​ dimers and C8{C}_{8}C8​ coefficients for the 1S0+1S0{}^{1}{S}_{0}+{\phantom{\rule{0.16em}{0ex}}}^{1}{S}_{0}1S0​+1S0​ and 1S0+3P1o{}^{1}{S}_{0}+{\phantom{\rule{0.16em}{0ex}}}^{3}{P}_{1}^{o}1S0​+3P1o​ dimers.
The electric dipole, quadrupole, and octupole polarizabilities of a helium atom and the associated dispersion coefficients for the interaction of two helium atoms are calculated for the atomic n=2n=2n=2 singlet and triplet states, respectively, He(21S)\mathrm{He}(2{}^{1}S)He(21S) and He(23S)\mathrm{He}(2{}^{3}S)He(23S). The calculations utilize highly accurate, variationally determined, wave functions composed of doubled basis sets. The effects of retardation due to the finite speed of light are calculated for the dispersion potentials of two He(21S)\mathrm{He}(2{}^{1}S)He(21S) or two He(23S)\mathrm{He}(2{}^{3}S)He(23S) atoms and for the atom-surface interaction potentials of a He(21S)\mathrm{He}(2{}^{1}S)He(21S) or a He(23S)\mathrm{He}(2{}^{3}S)He(23S) atom with a perfectly conducting or a dielectric wall.