# Three-Body Dispersion Potentials Involving Electric Octupole Coupling

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Molecular QED Calculation of the 3-Body Dispersion Potential

**171**, 1 (1993) and in Ref [22]. Higher-order multipole terms may be derived in a similar manner. Because Equation (11) represents an effective two-photon coupling vertex, second rather than fourth-order perturbation theory could be employed together with two instead of twelve time-ordered diagrams to yield the pair dispersion energy shift. Even greater advantages accrue on using the interaction Hamiltonian (11) to compute the potential between three atoms or molecules [22]. Only six topologically distinct diagrams are required to be summed over at third-order of perturbation theory using the formula

## 3. DD-DD-DO Energy Shift

^{3}term, with ${\mathrm{e}}^{-3uR}~1$ for uR << 1, and using result (A15) from Appendix C, we see that Equation (30) results in an R

^{−8}near-zone limiting dependence on separation distance for an equilateral triangle arrangement:

^{−12}. The potentials for the collinear arrangement are positive in sign.

## 4. DD-DO-DO Dispersion Potential

^{4}term and using the integral result (A16). This gives a potential with an R

^{−9}short-range dependence,

^{−14}behaviour, and for which the polarisabilities are static.

## 5. DO-DO-DO Interaction Energy

^{5}/243, yielding a near-zone asymptote

^{−10}behaviour.

^{6}, the u-integral in Equation (55) is evaluated using Equation (32) to give

^{−16}dependence.

## 6. DD-DO-OO Dispersion Potential

^{2}/R

^{2}in each integral term of Equation (63) ensures the potential is entirely retarded, containing no uR-independent terms, as expected since the mixed dipole-octupole polarisability of B is independent of the octupole weight-3 term. A form applicable at very short range may be obtained on retaining the u-independent term in the second integral of Equation (63) and using the integral result (A14). This is found to be

## 7. Summary

^{−9}, where each replacement of a dipole with an octupole leads to a factor of the order (a/R)

^{2}<< 1, where a represents the extent of the electronic wave function. On top of this, the absence of true static terms leads to factors (kR)

^{m}<< 1, where m is zero or a positive integer and k is the wave number of the radiation exchanged between the molecules. Note that DO-DO and DO-DO-DO interactions are special cases where the exact balance between (a/R)

^{4}and (a/R)

^{6}, respectively with (kR)

^{4}and (kR)

^{6}leads to an additional factor R

^{−1}arising from a Casimir–Polder type integral. The emerging power laws for pair and three-body interactions are shown in Table 1.

^{−10}behaviour. Again, the unexpected behaviour stems from an additional small factor (kR)

^{3}, as shown in Table 1. In this context it is useful to remark that short- and long-range expansions of the Casimir–Polder dispersion potential, as well as all correction terms up to second order in the fine structure constant have been performed from consideration of the orbit-orbit contribution due to the Breit-Pauli Hamiltonian, including relativistic effects [38,39], and compared with recent molecular QED calculations [30].

_{0}is the Bohr radius, and the transition energy to be of the order of one Rydberg, the ratio $\Delta {E}_{3}/\Delta {E}_{2}$ is unity at distances of around 3a

_{0}, with $\Delta {E}_{3}$ increasing in importance at larger distances.

## Funding

## Conflicts of Interest

## Appendix A. Wave Vector Integrals Involving Single Polarisability

^{2}>> a

^{2}, and which may be evaluated using Equation (32). Hence on collecting results,

## Appendix B. Two Polarisabilities

^{2}>> a

^{2}, b

^{2}has been made.

## Appendix C. Three Polarisabilities

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**Figure 1.**One of the six possible time-ordered sequences containing effective two-photon interaction vertices.

**Table 1.**Short-range dependences of dispersion potentials: Near-zone limiting behaviour of various equally displaced two- and three-body dispersion energy shifts involving electric dipole (D), quadrupole (Q) and octupole (O) couplings.

Multipole Coupling | Near-Zone Power Law | Ref. |
---|---|---|

DD-DD | R^{−6} | [9,10,11] |

DD-QQ | R^{−6} × (a/R)^{2} × (kR)^{0}~R^{−8} | [29] |

DD-DO | R^{−6} × (a/R)^{2} × (kR)^{2}~ R^{−6} | [30] |

DO-DO | R^{−6} × (a/R)^{4} × (kR)^{4} × R^{−1}~R^{−7} | [30] |

DD-OO | R^{−6} × (a/R)^{4} × (kR)^{3}~R^{−7} | [29] |

DD-DD-DD | R^{−9} | [26] |

DD-DD-QQ | R^{−9} × (a/R)^{2} × (kR)^{0}~R^{−11} | [14] |

DD-DD-DO | R^{−9} × (a/R)^{2} × (kR)^{3}~R^{−8} | |

DD-QQ-QQ | R^{−9} × (a/R)^{4} × (kR)^{0}~R^{−13} | [14] |

DD-DO-DO | R^{−9} × (a/R)^{4} × (kR)^{4}~R^{−9} | |

DO-DO-DO | R^{−9} × (a/R)^{6} × (kR)^{6} × R^{−1}~R^{−10} | |

DD-DO-OO | R^{−9} × (a/R)^{6} × (kR)^{0}~R^{−15} |

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Buhmann, S.Y.; Salam, A.
Three-Body Dispersion Potentials Involving Electric Octupole Coupling. *Symmetry* **2018**, *10*, 343.
https://doi.org/10.3390/sym10080343

**AMA Style**

Buhmann SY, Salam A.
Three-Body Dispersion Potentials Involving Electric Octupole Coupling. *Symmetry*. 2018; 10(8):343.
https://doi.org/10.3390/sym10080343

**Chicago/Turabian Style**

Buhmann, Stefan Yoshi, and A. Salam.
2018. "Three-Body Dispersion Potentials Involving Electric Octupole Coupling" *Symmetry* 10, no. 8: 343.
https://doi.org/10.3390/sym10080343