Weakly relativistic expansion

It is important to notice, however, that consistent neglect of all terms of the order 1/c (which has not been treated consistently in the weakly relativistic expansion) in the Hamiltonian allows a proof of a HK-theorem on the basis of the variables n and/. In other words Only a fully relativistic approach combines consistency in 1/c with gauge invariance. It remains to be investigated explicitly, whether inclusion of all relevant terms to order 1/c allows to reinstate the physical current j x) as basic variable also in this order as one would expect from the fully relativistic theory. 

This contradiction is resolved by noticing that the order 1/c has not been treated consistently in the weakly relativistic expansion which leads to (122). In fact, consistent neglect of all terms of the order 1 /c in the Hamiltonian (128), i.e. of the gauge term, allows a proof of an existence theorem with the variables n and j, at the price of loosing gauge invariance. In other words For any weakly relativistic Hamiltonian one has to choose between consistency in 1 /c and gauge invariance. Only a fully relativistic approach combines both properties. 

This problem can be illustrated by estimating, for a weak relativistic atom, the first correction to the kinetic energy. Using the usual expansion... 

The weakly relativistic limit of the Hamiltonian (2.20) for fermions in external electric and magnetic fields can be derived with standard techniques, either by direct expansion or by a low order Foldy-Wouthuysen transformation. One obtains... 

In the nonrelativistic context current-density functional theory is based on the nonrelativistic limits of the paramagnetic current (87) and/or the magnetization density (89) [128,129]. In the relativistic situation, however, a density functional approach relying on jp or m can only be considered an approximation, as long as the external magnetic field does not vanish. In order to clarify the relation between these two points of view the weakly relativistic limit of RDFT has to be analyzed. The weakly relativistic limit of the Hamiltonian (23) can be derived either by a direct expansion in 1/c or by a low order Foldy-Wouthuysen transformation,... 

The transverse correction factor can be decomposed into a magnetic and a retardation contribution with opposite signs. The magnetic contribution dominates at higher densities. The expansion of in the weakly relativistic... 

The origin of the nonvauishing Joule-Thomson effect is the effective repulsive (Fermions) and attractive (Bosons) potential exerted on the gas molecules, which arises from the different ways in which quantum states can be occupied in sy.stems obeying Fermi-Dirac and Boso-Einstein statistics, respectively [17]. In other words, the effective fields are a consequence of whether Pauli s antisymmetry principle, which is relativistic in nature [207], is applicable. Thus, a weakly degenerate Fermi gas will always heat up ((5 < 0), whereas a weakly degenerate Bose gas will cool down (5 > 0) during a Joule-Thomson expansion. These conclusions remain valid even if the ideal quantum gas is treated relativistically, which is required to understand... 

In order to prepare the discussion of the relativistic generalization of the HK-theorem in Section 3 we finally consider the renormalization procedure for inhomogeneous systems. As the underlying renormalization program of vacuum QED is formulated within a perturbative framework (see Appendix B) we assume the perturbing potential to be sufficiently weak to allow a power series expansion of all relevant quantities with respect to V. In particular, this allows an explicit derivation of the counterterms required for the field theoretical version of the KS equations, i.e. for the four current and kinetic energy of noninteracting particles. 

The behavior of relativistic wave functions at the Coulomb singularities of the Hamiltonian have been studied [84]. The nuclear attraction potentials don t cause any problem. There are weak singularities of the type r with p slightly smaller than 0, as they are familiar for the H-like ions. The limits r —> 0 and oo commute, and the Kato cusp conditions [85] arise in the nrl. For the coalescence of two electrons the two limits do not commute. An expansion in powers of c is possible to the lowest orders and leads to results consistent with those reported above. 

The four nonrelativistic operators of Eq. (4a) arise from an expansion of the lepton wave functions, while those of Eq. (4b) occur in the hadronic weak current, connecting the large and small components of nucleonic wave functions, and are relativistic. Matrix elements of different rank contribute incoherently to the decay rate. The rank R has the selection rule... 

We have developed a new relativistic treatment in the QMC technique using the ZORA Hamiltonian. We derived a novel relativistic local energy using the ZORA Hamiltonian and tested its availability in the VMC calculation. In addition, we proposed a relativistic electron-nucleus cusp correction scheme for the relativistic ZORA-QMC method. The correction scheme was a relativistic extension of the MO correction method where the 1 s MO was replaced by a correction function satisfying the cusp condition. The cusp condition for the ZORA wave function in electron-nucleus collisions was derived by the expansion of the ZORA local energy and the condition required the weak divergence of the orbital itself. The proposed relativistic correction function is the same as the NR correction function of Ma et al. in the NR... 

In most cases, however, the relativistic effects are rather weak and may be separated into spin-orbit coupling effects and scalar effects. The latter lead to compression and/or expansion of electron shells and can rather accurately be treated by modifying the one-electron part of the non-relativistic many-electron Hamiltonian. With this scalar-relativistic Hamiltonian the (modified) energies and wave functions are computed and subsequently an effective spin-orbit part is added to the Hamiltonian. The effects of the spin-orbit term on the energies and wave functions are commonly estimated using second-order perturbation theory. More information for the interested reader can be found in excellent textbooks on relativistic quantum chemistry [2, 3]. 

The cluster energies are obtained from the solution of the non-relativistic SchrOdinger equation for each system. The expansion of the trial many-electron wavefunction delineates the level of theory (description of electron correlation), whereas the description of the constituent one-electron orbitals is associated with the choice of the orbital basis set. A recent review [54] outlines a path, which is based on hierarchical approaches in this double expansion in order to ensure convergence of both the correlation and basis set problems. It also describes the application of these hierarchical approaches to various chemical systems that are associated with very diverse bonding characteristics, such as covalent bonds, hydrogen bonds and weakly bound clusters. 

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