Dirac equation electric potentials

The most unsatisfactory features of our derivation of the molecular Hamiltonian from the Dirac equation stem from the fact that the Dirac equation is, of course, a single particle equation. Hence all of the inter-electron terms have been introduced by including the effects of other electrons in the magnetic vector and electric scalar potentials. A particularly objectionable aspect is the inclusion of electron spin terms in the magnetic vector potential A, with the use of classical field theory to derive the results. It is therefore of interest to examine an alternative development and in this section we introduce the Breit Hamiltonian [16] as the starting point. We eventually arrive at the same molecular Hamiltonian as before, but the derivation is more satisfactory, although fundamental difficulties are still present. 

In the absence of interactions, electrons are described by the Dirac equation (1928), which rules out the quantum relativistic motion of an electron in static electric and magnetic fields E= yU and B = curl A (where U and A are the scalar and vectorial potentials, respectively) [43-45]. As the electrons involved in a solid structure are characterized by a small velocity with respect to the light celerity c (v/c 10 ) a 1/c-expansion of the Dirac equation may be achieved. More details are given in a paper published by one of us [46]. At the zeroth order, the Pauli equation (1927), in which the electronic spin contribution appears, is retrieved then conferring to this last one a relativistic origin. At first order the spin-orbit interaction arises and is described by the following Hamiltonian... 

Here q is the electric charge of the particle described by the Dirac equation. Particles in a magnetic field are described by the potential matrix... 

Our discussion of the electron-positron interpretation shows that the Dirac equation can have bound states for both attractive and repulsive electric potentials. Consider for example an attractive electrostatic potential well eV(x) < 0. It may support some bound states at energies En in the gap A... 

In the presence of an electric potential V (e.g. from nuclei), the time-independent Dirac equation may be written as in eq. (8.8), where we have again explicitly indicated the electron mass. 

The second, third and foiuth terms inside the first summation in equation (3) are the perturbations introduced into the hamiltonian by the effects of the external fields. The fourth term, describing the electric field perturbation, is linear in the external potential or electric field. The second and third terms give rise to linear and quadratic responses ro a constant, uniform magnetic field. Smaller terms, arising from the Dirac equation, which represent spin-orbit coupling etc. have been omitted. 

A second and more serious consequence of the appearance of the potential in the denominator is that the ZORA Hamiltonian is not invariant to the choice of electric gauge. Adding a constant to the potential should result in the addition of a constant to the energy, which is indeed the case for the Dirac equation. For ZORA, the relation between the ZORA and the Dirac eigenvalue for a one-electron system is given by (18.9). If we add a constant. A, to the Dirac energy in this equation, we get... 

To show the consistency of the various expressions for the electrical force law, consider a point charge, whose density is represented by a Dirac delta function. One can show mathematically that substituting <723(r) for /j(r) in equation (3) implies a potential <72/. The force F on a test charge qi at point r (again, using E = — V0) then yields equation (1), so we have come full circle all the formulations are equivalent. In different situations it may be more convenient to employ one mathematical form than another, as we shall see. 

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