Gauge term Hamiltonian

The basic variables of a density functional scheme are derived from the structure of the coupling of the electrons to the external fields. However, there are two possible ways to express the Hamiltonian (122) in terms of density and current operators, which differ by the treatment of the gauge term. Using n and j one obtains... 

This contradiction is resolved by noticing that the order 1/c has not been treated consistently in the weakly relativistic expansion which leads to (122). In fact, consistent neglect of all terms of the order 1 /c in the Hamiltonian (128), i.e. of the gauge term, allows a proof of an existence theorem with the variables n and j, at the price of loosing gauge invariance. In other words For any weakly relativistic Hamiltonian one has to choose between consistency in 1 /c and gauge invariance. Only a fully relativistic approach combines both properties. 

Flocke et al (FSK) view the Hubbard model in this problem as essentially the Hiickel model, but supplemented by the penalty term (Hubbard U) associated with double occupancy of an atomic orbital. In its simple form all C - C bonds are characterized by a common hopping parameter, t say, and all C atoms have the common electron-electron repulsion energy U already mentioned above. As PKS stress, the Hubbard Hamiltonian constitutes the simplest gauge-invariant Hamiltonian taking into account electron correlation. The extent of electron correlation is readily varied through the ratio U/t a continuous variation from the (uncorrelated) Hiickel approximation U = 0,t 7 0) to a fully correlated limit t = O U 0) being available. 

The gauge term, which is the difference between the Gaunt interaction and the Breit interaction, produces a spin-free operator that can be interpreted as an orbit-orbit interaction. Thus, both the Gaunt interaction and the gauge term of the Breit interaction give rise to spin-free contributions to the modified Dirac operator. We will use the developments of this section in chapter 17 to derive the Breit-Pauli Hamiltonian. 

Gauge Term Contributions from the Breit Interaction to the Breit-Pauli Hamiltonian... 

The reduction of the remaining contributions from the gauge term is extremely tedious, and in fact gives a zero contribution to the operator to 0(c ). Thus, there is no spin-dependent contribution from the gauge term for the relativistic correction to the electron-electron interaction to the Breit-Pauli Hamiltonian. 

Asymptotic Condition.—In Section 11.1, we exhibited the equivalence of the formulation of quantum electrodynamics in the Coulomb and Lorentz gauges in so far as observable quantities were concerned (t.e., scattering amplitudes). We also noted that both of these formulations, when based on a hamiltonian not containing mass renormalization counter terms, suffered from the difficulty that the... 

In the present section we shall make this difficulfy apparent in a somewhat different way by showing that it is not possible to satisfy the asymptotic condition when the theory is formulated in terms of an unsubtracted hamiltonian of the form jltAll(x) — JS0JV. We shall work in the Lorentz gauge, where the relativistic invariance of the theory is more obvious. 

We have introduced a gauge fixing term, with the limit e —> 0 being taken after the calculations are carried out.In Eq.(5) A stands for the transverse gauge field and (j) is defined as ip = 4>exp(i y). To obtain the free energy density, Veff = T jV (effective potential), we introduce a shifted field 4> — 4> + and split the Hamiltonian into two parts ... 

The two parts of this formula are derived from the same QED Feynman diagram for interaction of two electrons in the Coulomb gauge. The first term is the Coulomb potential and the second part, the Breit interaction, represents the mutual energy of the electron currents on the assumption that the virtual photon responsible for the interaction has a wavelength long compared with system dimensions. The DCB hamiltonian reduces to the complete standard Breit-Pauli Hamiltonian [9, 21.1], including all the relativistic and spin-dependent correction terms, when the electrons move nonrelativistically. 

In Section I, it was argued that 0(3) electrodynamics on the classical level emerges from a vacuum configuration that can be described with an 0(3) symmetry gauge group. On the QED level, this concept is developed by considering higher-order terms in the Hamiltonian... 

Given this approximation, we can transform the Hamiltonian of Eq. (1.44) from the velocity gauge to the so-called length gauge in which the matter-radiation interaction term contains only the dot product of the dipole moment and the electric field. In order to do so we choose x [Eq. (1.5)] as... 

As was shown in [13], to include the relativistic recoil corrections in calculations of the energy levels, we must add to the standard Hamiltonian of the electron-positron field interacting with the quantized electromagnetic field and with the Coulomb field of the nucleus Vc an additional term. In the Coulomb gauge, this term is given by... 

The first term of (3.289) represents a translational Stark effect. A molecule with a permanent dipole moment experiences a moving magnetic field as an electric field and hence shows an interaction the term could equally well be interpreted as a Zeeman effect. The second term represents the nuclear rotation and vibration Zeeman interactions we shall deal with this more fully below. The fourth term gives the interaction of the field with the orbital motion of the electrons and its small polarisation correction. The other terms are probably not important but are retained to preserve the gauge invariance of the Hamiltonian. For an ionic species (q 0) we have the additional translational term... 

It is important to notice, however, that consistent neglect of all terms of the order 1/c (which has not been treated consistently in the weakly relativistic expansion) in the Hamiltonian allows a proof of a HK-theorem on the basis of the variables n and/. In other words Only a fully relativistic approach combines consistency in 1/c with gauge invariance. It remains to be investigated explicitly, whether inclusion of all relevant terms to order 1/c allows to reinstate the physical current j x) as basic variable also in this order as one would expect from the fully relativistic theory. 

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