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...
We consider semi-classical macroscopic electrodynamics that is translationally invariant (independent of the choice of an arbitrary, implicit set of coordinate origins for molecule-fixed axes) for linear, homogeneous, anisotropic media interacting with harmonic, plane electromagnetic waves. We extend a previous formulation at electric octopole-magnetic quadrupole order to include media comprising magnetic molecules (those possessing both time-even and time-odd properties). This requires two additional invariant, time-odd molecular polarizabilities. Overall, the electrodynamics depends on 10 invariant polarizabilities—5 time even (one each of electric dipole and electric quadrupole–magnetic dipole order, and three of electric octopole-magnetic quadrupole order) and 5 time odd (one, two, and two, respectively)—that are required for the description of linear transmission and reflection phenomena, and material constants. The two additional time-odd polarizabilities account for certain predicted effects, and one of them contributes to the inverse ac permeability of magnetic media. The results are presented in a form that is suitable for numerical computation.
Articles you may be interested inTopology of surfaces for molecular Stark energy, alignment, and orientation generated by combined permanent and induced electric dipole interactionsWe consider semi-classical multipole theory for non-magnetic molecules interacting with harmonic plane electromagnetic waves, to electric octopole-magnetic quadrupole order and relative to an arbitrary set of molecular coordinate origins {O n }. Spatial averaging of expectation values of induced molecular multipole moments produces a macroscopic theory for linear, homogeneous, anisotropic media that has three shortcomings: it is only partially invariant with respect to {O n }, it is ambivalent on the Post constraint (equality of the traces of the magnetoelectric tensors), and it yields non-unique dynamic response fields D and H. To remedy these, we present a fully invariant theory that is consistent (affirmative) on the Post constraint, and is based on five time-even, invariant molecular polarizability tensors (one each of electric dipole and electric quadrupole-magnetic dipole order, and three of electric octopolemagnetic quadrupole order). As in previous work on linear phenomena, translational invariance is achieved through the Van Vleck-Buckingham condition. Uniqueness of the invariant response fields is demonstrated, based on linear independence of molecular polarizability tensors at each multipole order above electric dipole. Our results are compared with previously published expressions for two invariant polarizabilities. C 2012 American Institute of Physics. [http://dx.This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 132.174.255.116 On: Tue, 11 Aug 2015 00:03:48 073518-2 de Lange, Raab, and Welter J. Math. Phys. 53, 073518 (2012)polarizabilities it provides -we consider the theory up to this order. We restrict ourselves to effects induced by harmonic, plane electromagnetic waves in linear, homogeneous, anisotropic media. At electric dipole order the situation is straightforward. If the molecules are non-magnetic (that is, they possess only time-even properties 12 ), then there is just one molecular polarizability tensor, the dynamic electric dipole-electric dipole polarizability α ij . Because the quantum-mechanical expression for α ij is translationally invariant (independent of the choice of molecular coordinate origin relative to which it is evaluated), 2 so are all molecular and electromagnetic properties expressed in terms of it. There is no contribution to the magnetoelectric tensors (see Sec. VII), and the response fields given by (39) and (40) cannot be transformed within this order. 13 For magnetic molecules there are also contributions from the time-odd counterpart α ij of α ij , and the same conclusions hold. 13 At electric quadrupole-magnetic dipole order the semi-classical theory is only partially invariant, and a fully invariant theory has been presented in a previous paper. 4 For non-magnetic mole...
It is shown that the Coulomb problems associated with the Schrödinger, Klein–Gordon, and Dirac equations are shape invariant. This property is used to obtain the energy eigenvalues and the normalized coordinate-space eigenfunctions for bound states of these problems.
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