On the transition from microscopic to macroscopic electrodynamics

Abstract

Microscopic and macroscopic polarization within a combined quantum mechanics and molecular mechanics model J. Chem. Phys. 122, 034103 (2005); 10.1063/1.1831271Density-functional theory calculations of optical rotatory dispersion in the nonresonant and resonant frequency regions Implicit in the change from microscopic electrodynamics to a macroscopic, multipole theory is a set of molecule-fixed coordinate systems -and hence an arbitrary set of molecular origins {O n } -relative to which the positions of molecular constituents are specified. We examine the extent to which this theory satisfies a Van Vleck-Buckingham-type translational invariance with respect to the choice of {O n } in a linear, homogeneous, anisotropic medium. For contributions above electric dipole order, the theory is only partially invariant, and therefore incomplete: the corresponding macroscopic Maxwell equations yield unphysical results for certain phenomena. We propose a fully invariant formulation, based on the use of invariant molecular polarizability tensors in the quantum-mechanical expressions for expectation values of molecular multipole moments induced by harmonic, plane electromagnetic waves. We show that expressions for the invariant polarizabilities can be discerned from the partially invariant theory, and we discuss the uniqueness of our procedure. C 2012 American Institute of Physics. [doi:10.1063/1.3677767] the theory describes origin-independent observables is encapsulated in the following statement:To each multipole order (that is, electric dipole; electric quadrupole-magnetic dipole; electric octopole-magnetic quadrupole; etc), the theory forms a linear combination of various origin-dependent polarizabilities 26 in such a way that the overall expression for an observable is invariant due to cancellation − among the terms of that order − of changes produced by a shift in the molecular coordinate origin.(In view of its antecedence, it seems appropriate to refer to (II) as the Van Vleck-Buckingham condition.) These studies present a satisfying picture: theoretical expressions for observables that are expected to be origin independent are found to be so, and consequently their experimental values can be quoted without reference to a molecular coordinate origin. 27 However, we have shown that a notable exception occurs in the case of the macroscopic electromagnetic response fields D and H for a linear, homogeneous, anisotropic medium interacting with harmonic, plane electromagnetic waves. Here, the dynamic material constants (the permittivity, inverse permeability, and magnetoelectric coefficients) in the constitutive relations D(E,B) and H(E,B) are macroscopic observables whose experimental values are not linked to any coordinate origin. On the other hand, a direct calculation yields multipole expressions for these observables that are origin dependent 28, 29 (see also Secs. V and VII). This unacceptable result produces other unphysical features such as origin dependence of the energy flow 30 (the time average of the instantaneou...

Multipole theory and the Hehl–Obukhov decomposition of the electromagnetic constitutive tensor

“…11,12,[15][16][17] That is, α i j and α ′ i j are origin independent, but G i j , G ′ i j , a i jk , . .…”

“…15,16 Therefore, (I) and (II) apply also to the evaluation of V (2) i j and S i jk . However, the molecular tensorsw…”

“…(Quantummechanical formulae for these and higher-order polarizabilities, together with their intrinsic symmetries and other properties, are available in the literature. 11,12,[15][16][17] ) Then, in (11) and (12), 16,18 …”

“…15,16 Five are time even: the electric dipole-electric octopole polarizability (b i jkℓ ), electric quadrupole-electric quadrupole polarizability (d i jkℓ ), electric dipole-magnetic quadrupole polarizability (H ′ i jk ), electric quadrupole-magnetic dipole polarizability (L ′ i jk ), and the magnetizability ( χ i j ), which consist of para-and dia-magnetic parts. The other five are their time-odd counterparts, denoted b ′ i jkℓ , d ′ i jkℓ , H i jk , L i jk , and χ ′ i j .…”

“…The response fields can also be determined from their expansions in terms of macroscopic multipole moment densities 11,12,16 -see (68) and (69). This does not necessarily yield the same solutions as the preceding method, in which case it is required to distinguish between them on physical grounds (see Sec.…”