Abstract
Non-pairwise additive three-body dispersion potentials dependent upon one or more electric octupole moments are evaluated using the theory of molecular quantum electrodynamics. To simplify the perturbation theory calculations, an effective two-photon interaction Hamiltonian operator is employed. This leads to only third-order theory being required to evaluate energy shifts instead of the usual sixth-order formula, and the summation over six time-ordered sequences of virtual photon creation and annihilation events. Specific energy shifts computed include DD-DD-DO, DD-DO-DO, DO-DO-DO, and DD-DO-OO terms, where D and O are electric dipole and octupole moments, respectively. The formulae obtained are applicable to an arbitrary arrangement of the three particles, and we present explicit results for the equilateral triangle and collinear configurations, which complements the recently published DD-DD-OO potential. In this last case it was found that the contribution from the octupole weight-1 term could be viewed as a higher-order correction to the triple-dipole dispersion potential DD-DD-DD. In a similar fashion the octupole moment is decomposed into its irreducible components of weights-1 and -3, enabling insight to be gained into the potentials obtained in this study. Dispersion interaction energies proportional to mixed dipole-octupole polarisabilities, for example, are found to depend only on the weight-1 octupole moment for isotropic species and are retarded. Additional approximations are necessary in the evaluation of wave vector integrals for these cases in order to yield energy shifts that are valid in the near-zone.
Highlights
It is well-known that the contribution to the total interaction energy arising from the non-pairwise additive three-body van der Waals dispersion potential is very small [1]
A systematic study has been peformed of dispersion interactions between three molecules when the effects of electric octupole coupling have been accounted for, supplementing a previously published result involving two electric dipole polarisable species, and a third that is pure electric octupole polarisable
This has been carried out using the theory of molecular quantum electrodynamics (QED), in which the electromagnetic field is quantised and interactions between non-relativistic microscopic particles take place via the exchange of one or more virtual photons
Summary
It is well-known that the contribution to the total interaction energy arising from the non-pairwise additive three-body van der Waals dispersion potential is very small [1]. Of historical interest is that the non-retarded result for atoms in the ground state, obtained via third-order perturbation theory and static dipolar coupling operators, was first computed by Axilrod and Teller [26], and by Muto [27] Their energy shift exhibited inverse cubic separation distance dependence on each inter-particle displacement, and inverse ninth power law behaviour in the case of an equilateral triangle set up. In this paper we aim to study dispersion forces among three particles in which one or more entities is described by mixed electric dipole-octupole polarisability This quantity is non-vanishing for all molecules but is zero for atoms that undergo transitions via these two multipole moments from the ground state to the same virtual electronic level. Useful integrals required to obtain asymptotically limiting forms of energy shifts dependent upon one, two or three atomic polarisabilities applicable at short-range are given in the Appendices
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