Expansion, relativistic

Relativistic density functional theory can be used for all electron calculations. Relativistic DFT can be formulated using the Pauli formula or the zero-order regular approximation (ZORA). ZORA calculations include only the zero-order term in a power series expansion of the Dirac equation. ZORA is generally regarded as the superior method. The Pauli method is known to be unreliable for very heavy elements, such as actinides. 

This potential is referred to in electromagnetism texts as the retarded potential. It gives a clue as to why a complete relativistic treatment of the many-body problem has never been given. A theory due to Darwin and Breit suggests that the Hamiltonian can indeed be written as a sum of nuclear-nuclear repulsions, electron-nuclear attractions and electron-electron repulsions. But these terms are only the leading terms in an infinite expansion. 

An expansion in powers of 1 /c is a standard approach for deriving relativistic correction terms. Taking into account electron (s) and nuclear spins (1), and indicating explicitly an external electric potential by means of the field (F = —V0, or —— dAjdt if time dependent), an expansion up to order 1/c of the Dirac Hamiltonian including the... 

In this paper, we present another application of the semi-relativistic expansion by evaluating the relativistic corrections to the energy up to 1/c and l/c". This gives us explicit correction terms to the usual calculation of anisotropy energy in magnetic systems. 

The semi-relativistic expansion is obtained through the identity (1 — T) = in... 

The calculation of the magnetic anisotropy of non-cubic materials requires an expansion up to 1 /c . Except in the case of fully relativistic calculations, the expansion is never carried out consistently and only the spin-orbit perturbation is calculated to second order (or to infinite order), without taking account of the other terms of the expansion. In this section, we shall follow Gesztesy et al. (1984) and Grigore et al. (1989) to calculate the terms H3 and H. Hz will be found zero and H4 will give us terms that must be added to the second order spin-orbit calculation to obtain a consistent semi-relativistic expansion. 

The purpose of this paper was to explain the new semi-relativistic expansion and to show how it can be used to carry our explicit calculations in physical cases. The results are simple. 

Ag+ with 1.2 A ) [99]. Spin-orbit coupling is neglected in our analysis because the results shown in Table 4.2 are from scalar relativistic Douglas-Kroll calculations. Because of the additional shell expansion of the 5ds/2 orbital due to spin-orbit coupling, we expect a further increase of the polarizability of Au. Table 4.3 also... 

Baerends, E.J., Schwarz, W.H.E., Schwerdtfeger, P. and Snijders, J.G. (1990) Relativistic atomic orbital contractions and expansions magnitudes and explanations. Journal of Physics B-Atomic Molecular and Optical Physics, 23, 3225-3240. 

Kutzelnigg, W. (1989) Perturbation theory of relativistic corrections 1. The non-relativistic limit of the Dirac equation and a direct perturbation expansion. Zeitschrifi fur Physik D, 11, 15-28. 

Wang, F. and Li, L. (2002) A singularity excluded approximate expansion scheme in relativistic density functional theory. Theoretical Chemistry Accounts, 108, 53-60. 

Our approach is based on a systematic semiclassical study of the Dirac equation. After separating particles and anti-particles to arbitrary powers in h, a semiclassical expansion of the quantum dynamics in the Heisenberg picture is developed. To leading order this method produces classical spin-orbit dynamics for particles and anti-particles, respectively, that coincide with the findings of Rubinow and Keller Hamiltonian relativistic (anti-) particles drive a spin precession along their trajectories. A modification of that method leads to a semiclassical equivalent of the Foldy-Wouthuysen transformation resulting in relativistic quantum Hamiltonians with spin-orbit coupling. 

At this point, it is appropriate to present a brief discussion on the origin of the FC operator (d function) in the two-component form (Pauli form) of the molecular relativistic Hamiltonian. Many textbooks adopt the point of view that the FC is a relativistic effect, which must be derived from the Dirac equation [50,51]. In other textbooks or review articles it is stressed that the FC is not a relativistic effect and that it can be derived from classical electrodynamics [52,53] disregarding the origin of the gyromagnetic factor g—2. In some textbooks both derivations are presented [54]. The relativistic derivations suffer from the inherent drawbacks in the Pauli expansion, in particular that the Pauli Hamiltonian can only be used in the context of the first-order perturbation theory. Moreover, the origin of the FC term appears to be different depending on whether one uses the ESC method or FW transformation. 

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