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arXiv:2101.08701 · Physical Review Research

Quantum-mechanical force balance between multipolar dispersion and Pauli repulsion in atomic van der Waals dimers

Ornella Vaccarelli, Dmitry V. Fedorov, Martin Stöhr, Alexandre Tkatchenko

Published Aug 20, 2021• 18 citations
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Abstract

The structure and stability of atomic and molecular systems with van der Waals (vdW) bonding are often determined by the interplay between attractive dispersion interactions and repulsive interactions caused by electron confinement. Arising due to different mechanisms---electron correlation for dispersion and the Pauli exclusion principle for exchange repulsion---these interactions do not appear to have a straightforward connection. In this paper, we use a coarse-grained approach for evaluating the exchange energy for two coupled quantum Drude oscillators and investigate the mutual compensation of the attractive and repulsive forces at the equilibrium distance within the multipole expansion of the Coulomb potential. This compensation yields a compact formula relating the vdW radius of an atom to its multipole polarizabilities, ${R}_{\mathrm{vdW}}={A}_{l}^{\phantom{\rule{0.16em}{0ex}}}\phantom{\rule{0.16em}{0ex}}{\ensuremath{\alpha}}_{l}^{2/7(l+1)}$, where $l$ is the multipole rank and ${A}_{l}$ is a conversion factor. Such a relation is compelling because it connects an electronic property of an isolated atom (atomic polarizability) with an equilibrium distance in a dimer composed of two closed-shell atoms. We assess the accuracy of the revealed formula for noble-gas, alkaline-earth, and alkali atoms and show that ${A}_{l}$ can be assumed to be universal constants. Besides a seamless definition of vdW radii, the proposed relation can also be used for the efficient determination of atomic multipole polarizabilities solely based on the corresponding dipole polarizability and the vdW radius. Finally, our work provides a basis for the construction of efficient and minimally empirical interatomic potentials by combining multipolar interatomic exchange and dispersion forces on an equal footing.

DOI
10.1103/physrevresearch.3.033181

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The structure and stability of atomic and molecular systems with van der Waals (vdW) bonding are often determined by the interplay between attractive dispersion interactions and repulsive interactions caused by electron confinement. Arising due to different mechanisms---electron correlation for dispersion and the Pauli exclusion principle for exchange repulsion---these interactions do not appear to have a straightforward connection. In this paper, we use a coarse-grained approach for evaluating the exchange energy for two coupled quantum Drude oscillators and investigate the mutual compensation of the attractive and repulsive forces at the equilibrium distance within the multipole expansion of the Coulomb potential. This compensation yields a compact formula relating the vdW radius of an atom to its multipole polarizabilities, RvdW=Al\ensuremathαl2/7(l+1){R}_{\mathrm{vdW}}={A}_{l}^{\phantom{\rule{0.16em}{0ex}}}\phantom{\rule{0.16em}{0ex}}{\ensuremath{\alpha}}_{l}^{2/7(l+1)}RvdW​=Al​\ensuremathαl2/7(l+1)​, where lll is the multipole rank and Al{A}_{l}Al​ is a conversion factor. Such a relation is compelling because it connects an electronic property of an isolated atom (atomic polarizability) with an equilibrium distance in a dimer composed of two closed-shell atoms. We assess the accuracy of the revealed formula for noble-gas, alkaline-earth, and alkali atoms and show that Al{A}_{l}Al​ can be assumed to be universal constants. Besides a seamless definition of vdW radii, the proposed relation can also be used for the efficient determination of atomic multipole polarizabilities solely based on the corresponding dipole polarizability and the vdW radius. Finally, our work provides a basis for the construction of efficient and minimally empirical interatomic potentials by combining multipolar interatomic exchange and dispersion forces on an equal footing.
We study the van der Waals interaction between Rydberg alkali-metal atoms with fine structure (n2Lj{n}^{2}{L}_{j}n2Lj​; $L\ensuremath{\le}2$) and heteronuclear alkali-metal dimers in the ground rovibrational state ($X^{1}\mathrm{\ensuremath{\Sigma}}^{+}$; v=0,J=0v=0, J=0v=0,J=0). We compute the associated C6{C}_{6}C6​ dispersion coefficients of atom-molecule pairs involving 133Cs^{133}\mathrm{Cs}133Cs and 85Rb^{85}\mathrm{Rb}85Rb atoms interacting with KRb, LiCs, LiRb, and RbCs molecules. The obtained dispersion coefficients can be accurately fitted to a state-dependent polynomial O(n7)O({n}^{7})O(n7) over the range of principal quantum numbers $40\ensuremath{\le}n\ensuremath{\le}150$. For all atom-molecule pairs considered, Rydberg states n2Sjn^{2}S_{j}n2Sj​ and n2Pjn^{2}P_{j}n2Pj​ result in attractive 1/R61/{R}^{6}1/R6 potentials. In contrast, n2Djn^{2}D_{j}n2Dj​ states can give rise to repulsive potentials for specific atom-molecule pairs. The interaction energy at the LeRoy distance approximately scales as ${n}^{\ensuremath{-}5}$ for n>40n>40n>40. For intermediate values of $n\ensuremath{\lesssim}40$, both repulsive and attractive interaction energies of the order of 10--$1000\phantom{\rule{0.28em}{0ex}}\ensuremath{\mu}\mathrm{K}$ can be achieved with specific atomic and molecular species. The accuracy of the reported C6{C}_{6}C6​ coefficients is limited by the quality of the atomic quantum defects, with relative errors $\mathrm{\ensuremath{\Delta}}{C}_{6}/{C}_{6}$ estimated to be no greater than 1% on average.
The random-phase approximation with exchange (RPAE) is used with a BBB-spline basis to compute dynamic dipole polarizabilities of noble-gas atoms and several other closed-shell atoms (Be, Mg, Ca, Zn, Sr, Cd, and Ba). From these, values of the van der Waals C6{C}_{6}C6​ constants for positronium interactions with these atoms are determined and compared with existing data. After correcting the RPAE polarizabilities to fit the most accurate static polarizability data, our best predictions of C6{C}_{6}C6​ for Ps--noble-gas pairs are expected to be accurate to within 1%, and to within a few percent for the alkaline-earth metals. We also used accurate dynamic dipole polarizabilities from the literature to compute the C6{C}_{6}C6​ coefficients for the alkali-metal atoms. Implications of increased C6{C}_{6}C6​ values for Ps scattering from more polarizable atoms are discussed.
With the use of second-order perturbation theory in the long-range interatomic interaction for the degenerate states of two Rydberg atoms we have obtained a general formula for the dependence of atomic interaction energy on the interatomic distance R in the presence of the Förster resonance. Inside of the ‘Förster sphere’ ( R < R F ) this dependence transforms to the formula for electric dipole interaction energy Δ E d − d = C 3 / R 3 and for R > R F it transforms to the formula for the van der Waals interaction energy Δ E VdW = − C 6 / R 6 . The van der Waals constant C 6 is represented as an expansion in terms of irreducible components which define the dependence on the interatomic axis orientation relative to the quantisation axis of projections M of the total angular momentum J . The numerical values of the irreducible components of tensor C 6 were calculated for rubidium atoms in the same Rydberg states | nlJM 〉 with large quantum numbers n . We present the calculated resonance interaction energy of two rubidium atoms in the states |43D 5/2 M 〉, whose total energy exceeds by only 8 MHz the total energy of one of the atoms in the state |45P 3/2 M 〉 and of the other in the state |41F 7/2 M 〉.
The van der Waals force between a single atom and a rod-like molecule is investigated by introducing a one-dimensional dipole–dipole interaction atomic model. The force between a single atom and its nearest neighbor atom in a rod-like molecule is related to the correlation between them, which is obtained by solving the problem of coupled quantum oscillators. The dependence of the correlation on the coupling constant between the single atom and atoms in the molecule are presented for different molecular weight. Furthermore, the transition of the force from attractive to repulsive in the excited state is considered with regard to the correlation.
van der Waals interactions between atoms and dielectric surfaces are reinvestigated. To describe the nonretarded interaction potential between a dispersive dielectric surface and an atom in an arbitrary internal energy state, we derive a general expression in terms of an integral, over real frequency, of the combined atom and surface polarizabilities. It is shown that, for excited atoms, the expression is equivalent to the one obtained by Wylie and Sipe [Phys. Rev. A 32, 2030 (1985)]. We thus demonstrate how to extend this approach to excited atoms interacting with birefringent dielectrics. For isotropic dielectrics, a method of integration in closed form allows us to derive an approximate formula for the van der Waals interaction constant in terms of resonance frequencies and oscillator strengths of both the atom and the dielectric. Frequency-dependent ``dielectric reflection'' coefficients are introduced for virtual atomic dipole couplings either in absorption or in emission. In absorption, the reflection coefficient is always positive and smaller than unity. In emission, it may take arbitrary values, positive or negative (corresponding to van der Waals repulsion). Such a behavior is shown to be related to resonant excitation exchange between the atomic system and the dielectric medium, when an atomic transition frequency gets in resonance with a dielectric absorption band. Numerical calculations performed for the cesium-sapphire system are shown to be in good agreement with data obtained by selective-reflection spectroscopy. Finally, experimental tests of the birefringent character of the sapphire response are discussed.