A new generalization of the Casimir–Polder potential to higher electric multipole polarizabilities

Abstract

An elementary method for calculating retarded interaction energies for a pair of molecules with electric multipole polarizabilities of arbitrary order is presented. In the multipolar framework of quantum electrodynamics, the interaction energy is viewed as arising from two-photon exchange and calculated using fourth order perturbation theory. It is shown how the energy may be expressed in terms of derivatives of the Casimir–Polder formula written in a special form as an integral over imaginary frequency. Explicit formulas are presented for (a) an electric dipole polarizable molecule interacting with an electric quadrupole polarizable molecule, (b) an electric dipole polarizable molecule interacting with an electric octupole polarizable molecule, and (c) an electric quadrupole polarizable molecule interacting with another electric quadrupole polarizable molecule. The results are expressed in terms of reducible and irreducible components of multipole moments. For case (b) it is shown that in addition to the weight-3 components of the octupole moments, weight-1 components contribute to the interaction energy. For cases (a) and (c) traceless components (weight-2) of the quadrupole moments contribute. The results are compared with other calculations and discrepancies pointed out.