Brief Remarks on Relativistic DFT

Frequently, noncollinearity is due to spin-orbit coupling. Although spin-orbit terms can be added as a perturbation to the equations of SDFT, a complete description requires a relativistic formulation. A generalization of DFT that does account for spin-orbit coupling and other relativistic effects is rdft. ° Here the fundamental variable is the relativistic four-component current and the Kohn-Sham equation is now of the form of the single-particle Dirac equation, instead of the Schrodinger equation. 

In practice, one typically employs so-called relativistic SDFT (R-SDFT), in which a Gordon decomposition of the four-current is performed to separate orbital and spin degrees of freedom, and only the spin magnetization is maintained as a fundamental variable. This reduced RDFT is not Lorentz-invariant (which is not essential in solid-state physics and quantum chemistry, where a preferred frame is provided by the laboratory) and does not account for orbital magnetism (except, possibly, induced by spin-orbit coupling, which can be treated as a perturbation). Relativistic CDFT (R-CDFT) can overcome both limitations, but is less frequently used since it is more complicated to implement, and less is known about approximate four-current functionals. 

Magnetic Fields Coupling to Spins and Currents CDFT 

There are at least four conceptually distinct ways in which orbital magnetism can appear in a physical system. One is the presence of external magnetic fields B(r), whose vector potential A(r) enters the Hamiltonian via the usual minimal substitution in the kinetic energy 

This substitution is, formally, easy to perform in the many-body and the Kohn-Sham Hamiltonians of SDFT, but the presence of the vector potential complicates the task of solving these equations. Maintaining gauge invariance is not trivial in approximate calculations. Moreover, in extended systems the vector potential breaks translational invariance, so Bloch s theorem cannot be used anymore.Still, couplings of external vector potentials become relevant in many situations, and ways to deal with them have been developed, e.g. for the calculation of nuclear magnetic shielding tensors and spin-spin coupling constants. 

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