Three-Body Dispersion Potentials Involving Electric Octupole Coupling

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.

Effective Hamiltonians in Nonrelativistic Quantum Electrodynamics

“…From Equation (12), it is easy to see that, in order that the quantity above be nonvanishing, it is necessary that the total number of photons in {m} and {n} (that is, the photon number summed over all field modes) must be equal or differ by two. There are thus three possibilities: (i) {n} = {m}, that is elements diagonal in the photon space state; (ii) the difference in {n} and {m} is all in a single mode ( k λ), and thus m k λ = n k λ ± 2, with all other modes containing the same number of photons; (iii) the difference is by one photon in each of the two modes ( k λ) and ( k λ), with ( k λ) = n k λ ± 1 and ( k λ) = n k λ ± 1 (upper or lower sign for both modes), while all other modes in {m} and {n} have the same photon number.…”

“…We now consider the off-diagonal elements of the effective Hamiltonian between atom-field states, which we can obtain from (11) and (12); as mentioned at the end of the previous section, in this case the total number of photons in the two states must differ by two. We find…”

“…In such cases, the number of relevant Feynman diagrams rapidly grows with the perturbative order, with consequent increasing complexity of the calculations. For this reason, the possibility of finding approximated effective Hamiltonians allowing for the simplification of calculations is very important, and, hopefully, this will also yield a transparent interpretation and physical insights into the relevant physical processes involved [6,[10][11][12][13][14]. Effective Hamiltonians can be also used in dynamical (time-dependent) nonequilibrium situations [15].…”

Effective Hamiltonians in Nonrelativistic Quantum Electrodynamics

“…From Equation (12), it is easy to see that, in order that the quantity above be nonvanishing, it is necessary that the total number of photons in {m} and {n} (that is, the photon number summed over all field modes) must be equal or differ by two. There are thus three possibilities: (i) {n} = {m}, that is elements diagonal in the photon space state; (ii) the difference in {n} and {m} is all in a single mode ( kλ ), and thus mkλ = nkλ ± 2, with all other modes containing the same number of photons; (iii) the difference is by one photon in each of the two modes ( kλ ) and ( kλ ), with ( kλ ) = nkλ ± 1 and ( kλ ) = n kλ ± 1 (upper or lower sign for both modes), while all other modes in {m} and {n} have the same photon number.…”

Dispersion Interactions between Neutral Atoms and the Quantum Electrodynamical Vacuum

“…While the two-body interaction is always attractive, the threebody component can be attractive or repulsive according to the geometrical configuration of the three atoms[67]. Three-body interactions between molecules involving higher-order multipoles have been also considered in the literature[68][69][70].VII. TWO-AND THREE-BODY DISPERSION INTERACTIONS AS A CONSE-QUENCE OF VACUUM FIELD FLUCTUATIONSAn important point when dealing with van der Waals and Casimir-Polder dispersion interactions is the formulation of physical models aiming to explaining their origin, stressing quantum aspects, and giving physical insights of the processes involved.…”