Electric dipole gauge

This approach clearly distinguishes two ranges of interaction At the scale r < A, where the electrostatic interaction dominates, and at the scale r > /, where the retardation effects dominate. This scale property justifies the separation, implicit in the Coulomb gauge, between instantaneous terms and retarded terms. However, the electric-dipole gauge shows that these two distinct aspects of the electromagnetic interaction are physically undissoci-able, even though it is possible in many problems to omit retardation effects. 

Using the electric-dipole gauge (Section I.A.l.c), the intermolecular interactions have the additive form on the molecular charges given by... 

Then we have the same electric impulse in both cases. This gives the same electric gauge vector potential, 4,2. However, the Lorenz gauge potentials are quite different. For the electric dipole in Section VI, both 42 and 2 are zero. For the toroidal antenna equivalent electric dipole in Section VII, while 2 is zero, 42 is non zero. How then are these two cases different Within the gauge condition... 

So now we have the question poased in an interesting form. There are two quite different kinds of antennas, both of which produce electric dipole fields, but different Lorenz potentials, one emphasizing the vector potential and the other, the scalar potential. In a classical electromagnetic sense, one cannot distinguish these two cases by measurements of the fields (the measurable quantities) at distances away from the source region. The gauge invariance of QED implies the same in quantum sense. 

So our choices of the two antennas is not unique for separately emphasizing the Lorenz vector and scalar potentials. All that is required is for the two to have the same exterior fields (say, electric dipole fields, or more general multipole fields) with different potentials (related by the gauge condition). In a classical electromagnetic sense, these antennas cannot be distinguished by exterior measurements. This is a classical nonuniqueness of sources. In a QED sense, the same is the case due to gauge invariance in its currently accepted form. 

Ferraro and coll, used canonical transformation of the Hamiltonian to resolve the average optical rotatory power of a molecule into atomic contributions, based on the acceleration gauge for the electric dipole, and/or the torque formalism [151], This method has been applied to the study of the conformational profile of the optical rotatory poser of hydrogen peroxide and hydrazine [152]. 

In several cases, the polarizability distribution can be found by chemical intuition. For instance, in the case of naphthalene, which is made up of two identical fragments, the polarizability can be decomposed into two equivalent parts. Also, group or atom contributions can be deduced from a variety of schemes such as Stone s approach [74], the theory of atoms in molecules [75], the localization of molecular orbitals into chemical functions [76], atom/ bond additivity [77], the use of the acceleration gauge for the electric dipole operator [78], quantum mechanically determined induction energies [79], or calculated molecular quadrupole polarizabilities and their derivatives with respect to molecular deformations [80]. Several of these models consider charge... 

D.H. Kobe, Gauge-invariant resolution of the controversy over length versus velocity forms of the interaction with electric dipole radiation, Phys. Rev. A 19 (1979) 205. 

D. H. Kobe, Gauge transformations and the electric dipole approximation. Am. J. Phys. 50 (1982) 128. 

We shall here choose the electric-dipole approximation, but comment on the use of the Coulomb gauge below. 

In the length gauge, Eq. (2.122), the operator could be the electric dipole or quadrupole operator, defined in Appendix A. It depends on coordinates and momenta of the electrons but it is independent of time, whereas we assume that the time-dependent field. F. (t) does not depend on any electronic variables. The subscript p - again denotes components of a tensor of appropriate rank. On the other hand, in the velocity gauge, Eq. (2.125), the operator is equal to the total canonical... 

In order to derive a quantum mechanical expression for the frequency-dependent polarizability we can make use of time-dependent response theory as described in Section 3.11. We need therefore to evaluate the time-dependent expectation value of the electric dipole operator (4 o(i (f)) Pa o( (t))) in the presence of a time-dependent electric field, Eq. (7.11). Employing the length gauge, Eqs. (2.122) - (2.124), which implies that the time-dependent electric field enters the Hamiltonian via the scalar potential in Eq. (2.105), the perturbation Hamilton operator for the periodic and spatially uniform electric field of the electromagnetic wave is given as... 

As an example, consider again the interaction between electrons and a uniform electric field. In O section The Linear Response Function, the electric dipole polarizibality was identified using the length gauge perturbation operator... 

While O Eq. 5.11 is called the length gauge or dipole-length gauge expression, O Eq. 5.45 is often called the mixed gauge form since it involves both the electric dipole operator and the momentum operator. The length and mixed gauge polarizabilities are equivalent due to the equation of motion... 

It can be shown that the trace of the tensor G, and hence the computed optical rotation of a sample of randomly oriented chiral molecules, is independent of the origin provided that the linear response function satisfies O Eq. 5.46 and that the commutator of O Eq. 5.47 is fulfilled. Consequently, approximate linear response calculations of the length gauge optical rotation depend on the chosen coordinate origin. On the other hand, the trace of the velocity gauge formulation of the electric dipole - magnetic dipole polarizability... 

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