Minimal Coupling—Relativistically

In the last section, a non-relativistic Hamiltonian for a spin-less particle was derived. However, electrons have spin and in general it would be desirable to use a Hamiltonian operator that fulfills the requirements of special relativity. The so-called Dirac Hamiltonian operator is such a relativistic operator for a single particle in the presence of an electromagnetic field. It can be derived in the same ways as the non-relativistic analogue was obtained in the previous section. 

The Lorentz force in Eq. (2.43) is unchanged in special relativity, because electromagnetism in Maxwell s formulation fulfills the requirements of special relativity. Newton s second law, on the other hand. 

In complete analogy to the non-relativistic case we have to set up a Lagrangian again that, inserted in Lagrange s equations, Eq. (2.49), should yield Newton s second law, Eq. (2.64). The following Lagrangian 

Exercise 2.8 Show that the relativistic Lagrangian in Eq. (2.66) also fulfills the Lagrange equations (2.49). 

The components of the canonical momentum vector are again obtained as partial derivatives of the Lagrangian 

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In relativistic theory, we apply the minimal coupling recipe to the Dirac Hamiltonian... 

It is important to note that minimal coupling requires specification of charge. We are interested in electronic solutions and accordingly choose q = e. The positronic solutions are obtained by charge conjugation. We obtain the relativistic wave equation for the electron in the presence of external electromagnetic fields which we shall write as... 

In order to establish a relativistic hyperfine Hamiltonian operator for a many-electron system one faces the problem of setting up a relativistic many-body Hamiltonian which cannot be written down in closed form. If one considers a one-electron system first one can obtain an exact expression for the hyperfine Hamiltonian starting from the one-electron Dirac equation in minimal coupling to the electromagnetic field ... 

It is immediately apparent that the linearity of the Dirac equation with respect to p imparts a linearity of Hi with respect to A(r). To quantum chemists this result may seem perplexing. The minimal coupling substitution p p+eA(r) is common to both the relativistic theory and the more familiar non-relativistic theory. Consequently, the gauge transformation of potentials and amplitudes according to... 

In Equations 10.1, 10.4, and 10.5, the standard implicit sum on the index (x is assumed. The relativistic Kohn-Sham equations are obtained by minimization of the Equation 10.4 with respect to the orbitals. In contrast to the nonrelativistic case, the variational procedure gives rise to an infinite set of coupled equations (see the summation restrictions in Equation 10.3) that have to be solved in a self-consistent manner ... 

External fields are introduced in the relativistic free-particle operator hy the minimal substitutions (17). One should at this point carefully note that the principle of minimal electromagnetic coupling requires the specification of particle charge. This becomes particularly important for the Dirac equation which describes not only the electron, but also its antiparticle, the positron. We are interested in electrons and therefore choose q = — 1 in atomic units which gives the Hamiltonian... 

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