Gauge invariance

Gauge Invariance. An additional complication arises in the case of magnetic properties not present in the electrical case. Because V-B=0, B can be written B= V X /4. However, there are an infinite number of ways of defining A since 

In practice, calculations of z are based on the uncoupled Hartree-Fock, the finite field, and the self-consistent perturbation methods. Some workers use gauge-invariant atomic orbitals (GIAOs). A full review of the gauge invariance of SCF wavefunctions has been given by Epstein.  

Stevens and Lipscomb s method, for diatomics, avoids the self-consistency error by ivriting the first-order perturbed wavefunction in terms of the n-orbitals, since the ground-state wavefunction involves only a-orbitals. Thus for linear molecules the 

Moccia et have calculated magnetic susceptibilities for some small poly- 

Both of the previous methods calculate and separately. These are of opposite sign and numerically of the same order of magnitude, and so errors in both could make the total error in the sum large. It is therefore desirable to calculate the 

Ditchfield, in Molecular Structure and Properties , ed. G. Allen, MTP International Review of Science, Physical Chemistry Series 1, Volume 2, Butterworths, London, and University Park Press, Baltimore, 1972. 

4 Gauge Invariance. - The electric and magnetic fields are measinable physical quantities and can be regarded as directly responsible for effects on the molecxfles. Given well defined fields there will be well defined molecular responses. The hamiltonians, on the other hand, are written in terms of the potentials, V and A and these are not uniquely defined. The fields are given by. 

I mentioned above the magnetic vector potential A. This is given in the static case by 

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). 

London (1937) first made use of these functions in connection with ring currents in aromatic hydrocarbons. A key paper for the use of GIAO is that of Ditch-field. 

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

In the case of a scalar field, the irreducible matrix D is a unit matrix, and drops out of. I1. For rotation through an angle S9t about the Cartesian axis ek, the rotational submatrix of the Lorentz matrix is given by Xkx = ()Hkekl]x], where el]k is the totally antisymmetric Levi-Civita tensor. For the one-electron Schrodinger field f, Noether s theorem defines three conserved components of a spatial axial vector, 

For the Dirac bispinor, the irreducible representation matrix Dab for each helicity component is a Pauli spin matrix a multiplied by ti/2. Then 

In classical electrodynamics, the field equations for the Maxwell field A/( depend only on the antisymmetric tensor which is invariant under a gauge transformation A/l A/l + ticduxix), where x is an arbitrary scalar field in space-time. Thus the vector field A/( is not completely determined by the theory. It is customary to impose an auxiliary gauge condition, such as 9/x/Fx = 0, in order to simplify the field equations. In the presence of an externally determined electric current density 4-vector j11, the Maxwell Lagrangian density is 

Because F A V is antisymmetric, the symmetrical derivative ()/l dvFIJ V must vanish. This requires jv to satisfy the equation of continuity, 3vjv = 0, which implies charge conservation in an enclosed volume if net current flow vanishes across its spatial boundary. 

Gauge covariance of the classical theory is due to the invariance of the field tensor F/I V under the local gauge transformation 

From classical field theory we know that the force on a particle of mass m, charge e, and velocity v which is moving in an electromagnetic field is given by 

Now if we know the vector and scalar potentials A and / , equations (3.198) and (3.199) show that the field intensities E and B are uniquely determined. The converse is not true, however, as we now show. Suppose that A and 4 are transformed to A and f according to 

Hence the transformation of A and f to A and p does not change the fields E and B and it therefore follows that A and f are not uniquely determined. The transformation (3.200), (3.201) is known as a gauge transformation and, in general, we require expressions for B and E to be invariant to such a transformation, i.e. to be gauge-invariant. We therefore have some freedom in choosing A and f and two particular choices are common. In the so-called Coulomb gauge we define A such that 

These gauges are equivalent for time-independent problems our previous expressions for A have satisfied (3.205) and we shall continue to use the Coulomb gauge. 

Turning now to our previous expressions for A and p we note that the result given in (3.193) is not satisfactory since V d / 0. The recipe is therefore to make a gauge transformation 

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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... 

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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