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+ 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 in the potential energy for the three well-separated constituents, where
n refers to the corresponding order in inverse powers of distance, obtaining the additive interaction coefficients
C4,C6,C7,C8,C9 and the nonadditive interaction coefficients
C7,C9. The obtained coefficients
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 and
7Li.
We theoretically investigate long-range interactions between an excited
L-state He atom and two identical
S-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 and ${n}_{0}^{\ensuremath{'}}$ principal quantum numbers, $\ensuremath{\lambda}=1$ or 3 the spin multiplicity, and
L the orbital angular momentum of a He atom. Using degenerate perturbation theory for the energies up to second-order, we evaluate the coefficients
C3 of the first-order dipolar interactions and the coefficients
C6 and
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 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 or
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
P 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.
We report a theoretical study on the long-range additive and nonadditive potentials for a three-body hybrid atom-atom-ion system composed of one ground-
S-state Li atom, one excited-
P-state Li atom, and one ground-
S-state
Li+ ion, $\mathrm{Li}(2{\phantom{\rule{0.16em}{0ex}}}^{2}S)\text{\ensuremath{-}}\mathrm{Li}(2{\phantom{\rule{0.16em}{0ex}}}^{2}P)\text{\ensuremath{-}}{\mathrm{Li}}^{+}(1{\phantom{\rule{0.16em}{0ex}}}^{1}S$). The interaction coefficients are evaluated with highly accurate wave functions calculated variationally in Hylleraas coordinates. For this hybrid system the three-body nonadditive interactions (appearing in second order) induced by the energy degeneracy and enhanced by the induction effect of the
Li+ ion through the internal electric field can be strong and even stronger than the two-body additive interactions at the same order. We find that for particular geometries the two-body additive interactions of the system sum to zero, leaving only three-body nonadditive interactions, thus making the present system potentially a platform to explore quantum three-body effects. We also extract by first principles the leading coefficients of the long-range electrostatic, induction, and dispersion energies of
Li2+ electronic states correlating to ${\text{Li}}^{+}(1{\phantom{\rule{0.16em}{0ex}}}^{1}S)\ensuremath{-}\text{Li}(2{\phantom{\rule{0.16em}{0ex}}}^{2}P)$. The results should be especially valuable for the exploration of schemes to create trimers with ultracold atoms and ions in optical lattices.
For the interactions between a
He(23S) atom and a
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 of the first order resonant dipole-dipole energy and the van der Waals coefficients
C6,
C8, and
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 $^{3}\mathrm{He}(2\phantom{\rule{0.2em}{0ex}}^{3}S)\text{\ensuremath{-}}^{3}\mathrm{He}(2\phantom{\rule{0.2em}{0ex}}^{3}P)$ 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.
We study the nonadditive part of the long-range interaction among three alkali-metal atoms in their ground states. Using nondegenerate perturbation theory, up to the third order, we have computed the dispersion coefficients C for three alkali-metal atoms interacting via their electric dipole moments. Both heteronuclear and homonuclear cases are considered. The numerical values for the C coefficients suggest that such three-body dipole interaction effects may not be neglected in the description of the long-range surface potential interaction for alkali-metal atoms. Furthermore, we show that approximate formulas for C of Midzuno and Kihara [J. Phys. Soc. Jpn. 11, 1045 (1956)] give results in excellent agreement with results of our calculations. Comparisons are given of our results with results of others and with results of other approximate formulas. The effect on our results for the C coefficient of uncertainties in the experimental values of the static dipole polarizabilities, on which our results depend, is also analyzed.
In this work, the ground-state energy of the lithium atom is systematically investigated using both time-independent perturbation theory and the variational method to provide a comprehensive pedagogical analysis of many-body atomic systems. The unperturbed Hamiltonian is initially constructed by neglecting electron-electron interactions, treating the system as three independent hydrogen-like electrons to yield a zeroth-order energy baseline of -275.51 eV. The antisymmetric fermionic nature of the exact wave function is rigorously enforced through the Slater determinant formalism. First-order perturbation theory is applied to evaluate static inter-electronic repulsion using exact Coulomb and exchange integrals, refining the energy state to -192.01 eV. To account for dynamical electronic correlation, second-order perturbation theory is computed numerically for virtual single-electron s-orbital transitions, leading to a total perturbative energy of -196.36 eV. A brief discussion of two-electron excitations is also included to encapsulate further physical realism within the framework. Furthermore, a non-orthogonal two-parameter variational approach is employed to model the shell-specific shielding effect. By optimizing the effective nuclear charges, the variational method establishes a superior upper bound energy of -201.187 eV. The results of both methods are comprehensively contrasted against each other and the reference baseline to provide critical insights into the nature of electron correlation and screening in multi-electron atoms.