Solution of the Kohn-Sham-Dirac Equations

The enormous progress in the calculation of solid state properties in the past decades has been pushed by the development of a nrunber of distinct band structure schemes like KKR-, ASW-, LMTO-, LCAO-, PP- and (L)APW-methods which differ essentially in their representation of basis functions. For all of the mentioned methods there exist by now full potential codes which also incorporate relativistic effects in one way or another. 

The Kohn-Sham-Dirac equation (27) introduced in the last section is the basis of most relativistic electronic structure calculations in solid state theory. There are certain aspects which make the numerical solution of this four-component equation more involved than its non-relativistic coimterpart The Hamiltonian of the Kohn-Sham-Dirac equation is, unlike its Schrodinger equivalent and unlike the field-theoretical Hamiltonian (7) with the properly chosen normal order, not bounded below. In the limit of free, non-interacting particles the solutions of the Kohn-Sham-Dirac equation are plane waves with energies e(k) = cVk -I- c, where positive energies correspond to electrons and states with negative energy can be interpreted as positrons. For numerical procedures, which preferably use variational techniques to find electronic solutions, this property of the Dirac operator causes a severe problem, which can be circumvented by certain techniques like the application of a squared Dirac operator or a projection onto the properly chosen electronic states according to their above definition after Eq. (19). 

Since the Dirac operator couples the spin and spatial degrees of freedom, a separation into spin up and down states is no longer possible and the dimension of the eigenvalue problem is doubled in comparison to the 

With a few exceptions practical implementations of Eq. (27) treat the exchange and correlation (XC) field in a collinear approximation with Bxc = Bxc z aligned along the (arbitrary) i-axis. Non-collinear structures play an important role for the groxond state of some magnetic systems as well as for excited state properties like magnon dispersions. A discussion of the technical details of non-collinear schemes and their applications is beyond the scope of this text. Reviews on this subject can be found in Refs. [23] and [24]. 

Despite the great variety of calculational schemes employed, relativistic band structure codes have by now achieved a high level of accuracy. While for example the calculated lattice constant of fcc-Th in early publications covered a broad range of values (Fig. 1), a number of state-of-the-art relativistic full potential methods give reliable values very close to each other, about 2.5 percent below the experimental lattice constant (which is the systematic error of the LDA functional used in the calculations). Moreover, the most accurate schemes coincide in their total energies within a few mHartree per atom, a level of accuracy almost comparable to non-relativistic band structure schemes. 

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