Gauge invariant

In the same section, we also see that the source of the appropriate analytic behavior of the wave function is outside its defining equation (the Schibdinger equation), and is in general the consequence of either some very basic consideration or of the way that experiments are conducted. The analytic behavior in question can be in the frequency or in the time domain and leads in either case to a Kramers-Kronig type of reciprocal relations. We propose that behind these relations there may be an equation of restriction, but while in the former case (where the variable is the frequency) the equation of resh iction expresses causality (no effect before cause), for the latter case (when the variable is the time), the restriction is in several instances the basic requirement of lower boundedness of energies in (no-relativistic) spectra [39,40]. In a previous work, it has been shown that analyticity plays further roles in these reciprocal relations, in that it ensures that time causality is not violated in the conjugate relations and that (ordinary) gauge invariance is observed [40]. 

One can define a phase that is given as an integral over the log of the amplitude modulus and is therefore an observable and is gauge invariant. This phase [which is unique, at least in the cases for which Eq. (9) holds] differs from other phases, those that are, for example, a constant, the dynamic phase or a gauge-transformation induced phase, by its satisfying the analyticity requirements laid out in Section I.C.3. 

Starting from a completely different angle, namely, the nuclear Lagrangean and the requirement of local gauge invariance, we have shown in Section IV.B... 

RPA, and CPHF. Time-dependent Hartree-Fock (TDFIF) is the Flartree-Fock approximation for the time-dependent Schrodinger equation. CPFIF stands for coupled perturbed Flartree-Fock. The random-phase approximation (RPA) is also an equivalent formulation. There have also been time-dependent MCSCF formulations using the time-dependent gauge invariant approach (TDGI) that is equivalent to multiconfiguration RPA. All of the time-dependent methods go to the static calculation results in the v = 0 limit. 

TDGI (time-dependent gauge-invariant) ah initio method used for computing nonlinear optical properties... 

Magnetic properties should be independent of the choice of coordinate origin. The term choice of origin is often translated into choice of gauge, and so we say that physical properties should be gauge-invariant (for a discussion, see Hameka, 1965). 

Self-Consistent Perturbation Theory of Diamagnetism 1. A Gauge-Invariant... 

Gauge invariance requires that if we replace a°(k) by a ( ) + kuA(k), the matrix element remain unchanged. Stated differently, if al(k) is of the form k times a function of k, then must vanish hence... 

Quantization of radiation field in terms of field intensity operators, 562 Quantum electrodynamics, 642 asymptotic condition, 698 gauge invariance in relation to operators inducing inhomogeneous Lorentz transformations, 678 invariance properties, 664 invariance under discrete transformations, 679... 

The variation condition 6At = 0 can be independently imposed for variations of and its adjoint. The condition of gauge invariance requires that trial functions have the form... 

The frequency dependence is taken into accoimt through a mixed time-dependent method which introduces a dipole-moment factor (i.e. a polynomial of first degree in the electronic coordinates ) in a SCF-CI (Self Consistent Field with Configuration Interaction) method (3). The dipolar factor, ensuring the gauge invariance, partly simulates the molecular basis set effects and the influence of the continuum states. A part of these effects is explicitly taken into account in an extrapolation procedure which permits to circumvent the sequels of the truncation of the infinite sum-over- states. 

By omitting time-dependent terms, as in the preceding paragraph, the liP ) function may be read as the sum of the unperturbed wavefunction ) and a term which is the product of this function by a linear combination of the electronic coordinates, i.e. the Kirkwood s j) function. Thus, the (r) dipolar factor ensures gauge-invariance. But the role of the dipolar factor g f) in this mixed method is essential on the following point its contribution in the a computation occurs in a complementary (and sometimes preponderant) way to that calculated only from the n) excited states, the number of which is unavoidably limited by the computation limits. But before discussing their number, we have to comment the description of these states. 

The s-wave contribution to the photo ionization from the 3a3p level is plotted in figure 3 and shows a quite satisfactory gauge invariance. Its peak value is in excellent agreement with that yielded by our previous STOCOS ealeulations, 346 Mb (3). 

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