Nonrelativistic states formalism

An Extended (Sufficiency) Criterion for the Vanishing of the Tensorial Field Observability of Molecular States in a Hamiltonian Formalism An Interpretation Lagrangeans in Phase-Modulus Formalism A. Background to the Nonrelativistic and Relativistic Cases Nonreladvistic Electron... 

A weakly bound state is necessarily nonrelativistic, v Za (see discussion of the electron in the field of a Coulomb center above). Hence, there are two small parameters in a weakly bound state, namely, the fine structure constant a. and nonrelativistic velocity v Za. In the leading approximation weakly bound states are essentially quantum mechanical systems, and do not require quantum field theory for their description. But a nonrelativistic quantum mechanical description does not provide an unambiguous way for calculation of higher order corrections, when recoil and many particle effects become important. On the other hand the Bethe-Salpeter equation provides an explicit quantum field theory framework for discussion of bound states, both weakly and strongly bound. Just due to generality of the Bethe-Salpeter formalism separation of the basic nonrelativistic dynamics for weakly bound states becomes difficult, and systematic extraction of high order corrections over a and V Za becomes prohibitively complicated. 

We also mention that recently a density functional approach to excited states of relativistic systems has been formulated [45], using ensembles of unequally weighted states. This formalism is restricted to the electrostatic limit and the no-sea approximation (see Section 3.2). Moreover, it remains unclear how the spontaneous emission of photons, which is possible in QED in contrast to the standard nonrelativistic many-body theory, is handled for the excited states involved. 

Quite generally, it must be stated that some additional effort is required to develop the RDFT towards the same level of sophistication that has been achieved in the nonrelativistic regime. In particular, all exchange-correlation functionals, which are available so far, are functionals of the density alone. An appropriate extension of the nonrelativistic spin density functional formalism on the basis of either the time reversal invariance or the assembly of current density contributions (which are e.g. accessible within the gradient expansion) is one of the tasks still to be undertaken. 

Let us start with the field-free SO effects. Perturbation by SO coupling mixes some triplet character into the formally closed-shell ground-state wavefunction. Therefore, electronic spin has to be dealt with as a further degree of freedom. This leads to hyperfine interactions between electronic and nuclear spins, in a BP framework expressed as Fermi-contact (FC) and spin-dipolar (SD) terms (in other quasirelativistic frameworks, the hyperfine terms may be contained in a single operator, see e.g. [34,40,39]). Thus, in addition to the first-order and second-order ct at the nonrelativistic level (eqs. 5-7), third-order contributions to nuclear shielding (8) arise, that couple the one- and two-electron SO operators (9) and (10) to the FC and SD Hamiltonians (11) and (12), respectively. Throughout this article, we will follow the notation introduced in [58,61,62], where these spin-orbit shielding contributions were denoted... 

A full relativistic theory for coupling tensors within the polarization propagator approach at the RPA level was presented as a generalization of the nonrelativistic theory. Relativistic calculations using the PP formalism have three requirements, namely (i) all operators representing perturbations must be given in relativistic form (ii) the zeroth-order Hamiltonian must be the Dirac-Coulomb-Breit Hamiltonian, /foBC, or some approximation to it and (iii) the electronic states must be relativistic spin-orbitals within the particle-hole or normal ordered representation. Aucar and Oddershede used the particle-hole Dirac-Coulomb-Breit Hamiltonian in the no-pair approach as a starting point, Eq. (18),... 

The indices i,J denote electrons, whereas X,fi stand for nuclei with charges and Z j. Relativistic, quasirelativistic or nonrelativistic expressions may be inserted into this Hamiltonian for the one- and two-electron operators h and g, respectively. In some cases, e.g. the relativistic all-electron Dirac-CouIomb-(Breit/Gaunt) Hamiltonian, it is necessary to (formally) bracket the Hamiltonian by projection operators onto many-electron (positive energy) states in order to avoid problems connected with unwanted many-electron-positron (negative energy) states. 

We have provided a pedagogical derivation of the traditional, nonrelativistic form of multiple scattering theory based on the optical potential formalism. We have also discussed in detail each of the important advances made over the past ten years in the numerical application of the NR formalism. These include the full-folding calculation of the first-order optical potential, off-shell NN t-matrix contributions, relativistic kinematics and Lorentz boost of the NN t-matrix, electromagnetic effects, medium corrections arising from Pauli blocking and binding potentials in intermediate states, nucleon... 

There is no fundamental change in the concept of correlation between relativistic and nonrelativistic quantum chemistry in both cases, correlation describes the difference between a mean-field description, which forms the reference state for the correlation method, and the exact description. We can also define dynamical and non-dynamical correlation in both cases. There is in fact no formal difference between a nonrelativistic spin-orbital-based formalism and a relativistic spinor-based formalism. Thus we should be able to transfer most of the schemes for post-Hartree-Fock calculations to a relativistic post-Dirac-Hartree-Fock model. Several such schemes have been implemented and applied in a range of calculations. The main technical differences to consider are those arising from having to deal with integrals that are complex, and the need to replace algorithms that exploit the nonrelativistic spin symmetry by schemes that use time-reversal and double-group symmetry. 

Brooks, Johansson and Skriver (1984) investigated the band structure of UC and ThC by nonrelativistic and relativistic (based on the Dirac formalism) LMTO methods. They analysed the electron density changes in the compounds as compared with free atoms, as well as the influence of pressure on the band structure. Crystal pressures as a function of lattice constants (equations of state) were calculated as well as theoretical values of the lattice constants. The calculated trends in the variations of lattice constants and bulk moduli agree well with the available experimental data. Some of the most important results of these calculations are shown in Figs. 2.20 and 2.21. 

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