## Invariant multipole theory of induced macroscopic fields in homogenous dielectrics.

##### Abstract

A harmonic plane electromagnetic wave incident on a molecule distorts its charge distribution, thereby
producing an infinite series of induced multipole moments expressed in terms of contributions that
are due to the electric and magnetic fields E and B, and their space and time derivatives. For a linear
dependence of an induced moment on a particular field property, as treated in this thesis, the constant
of proportionality is essentially the corresponding molecular polarizability. Each polarizability is of
a definite multipole order (electric dipole, electric quadrupole–magnetic dipole, electric octopole–
magnetic quadrupole, etc.). The contribution of each multipole term to a physical property diminishes
rapidly with increasing multipole order. In general, the moments and polarizabilities are dependent
on an arbitrary choice of molecular coordinate origin, relative to which the positions of molecular
constituents are referred.
Electromagnetic observables are expressible, in part, in terms of contributions of the polarizabilities
of the same multipole order. The aim of multipole theory is to explain effects to the lowest
relevant multipole order, since higher-order contributions are negligible. A necessary criterion for
such a theory is that it be independent of the choice of molecular coordinate origin. Van Vleck [1]
introduced this condition, and Buckingham [2] and others [3, 4] have used it as a standard test of the
theory.
The macroscopic continuum theory of electromagnetics, as embodied in Maxwell’s macroscopic
equations, involves molecular properties and electromagnetic fields averaged over a sampling volume
of dimensions much smaller than the wavelength of the fields and much larger than molecular
dimensions [5]. This averaging entails specifying a set of molecular coordinate origins.
The multipole expressions for the macroscopic induced bound charge and current densities and the
propagation equation are origin independent in part due to cancellation of their origin dependences
among terms of the same multipole order — the so-called Van Vleck–Buckingham condition [6].
The multipole expressions for the dynamic response fields, D(E,B) and H(E,B), above electric
dipole order depend on origin, and thus the theory is only partially invariant. To obtain a consistent
invariant multipole theory of induced macroscopic fields up to electric octopole–magnetic quadrupole
order, origin-independent expressions corresponding to the molecular polarizabilities are determined.
When used in place of the molecular polarizabilities, these invariant expressions leave the originindependent
aspects of the theory unchanged, and yield physically acceptable expressions for the
macroscopic fields. The resulting theory is fully invariant for both transmission and reflection.
The procedure to determine invariant polarizabilities requires manipulations of expressions involving
Cartesian tensors up to rank four, contracted with isotropic tensors up to rank eight, at electric
octopole–magnetic quadrupole order. The algebraic software package mathematica was used to
facilitate the evaluation of these expressions.