Relativistic mean field approximations

A description of nuclear matter as an ideal mixture of protons and neutrons, possibly in (5 equilibrium with electrons and neutrinos, is not sufficient to give a realistic description of dense matter. The account of the interaction between the nucleons can be performed in different ways. For instance we have effective nucleon-nucleon interactions, which reproduce empirical two-nucleon data, e.g. the PARIS and the BONN potential. On the other hand we have effective interactions like the Skyrme interaction, which are able to reproduce nuclear data within the mean-field approximation. The most advanced description is given by the Walecka model, which is based on a relativistic Lagrangian and models the nucleon-nucleon interactions by coupling to effective meson fields. Within the relativistic mean-field approximation, quasi-particles are introduced, which can be parameterized by a self-energy shift and an effective mass. 

If we understand FM or magnetic properties of quark matter more deeply, we must proceeds to a self-consistent approach, like Hartree-Fock theory, beyond the previous perturbative argument. In ref. [11] we have described how the axial-vector mean field (AV) and the tensor one appear as a consequence of the Fierz transformation within the relativistic mean-field theory for nuclear matter, which is one of the nonperturbative frameworks in many-body theories and corresponds to the Hatree-Fock approximation. We also demonstrated... 

Aspects of the relativistic theory of quantum electrodynamics are first reviewed in the context of the electronic structure theory of atoms and molecules. The finite basis set parametrization of this theory is then discussed, and the formulation of the Dirac-Hartree-Fock-Breit procedure presented with additional detail provided which is specific to the treatment of atoms or molecules. Issues concerned with the implementation of relativistic mean-field methods are outlined, including the computational strategies adopted in the BERTHA code. Extensions of the formalism are presented to include open-shell cases, and the accommodation of some electron correlation effects within the multi-configurational Dirac-Hartree-Fock approximation. We conclude with a survey of representative applications of the relativistic self-consistent field method to be found in the literature. 

Dirac s relativistic theory for the motion of electrons in molecules was introduced in the preceding chapters. The appearance of positron solutions and the four-component form of the wave function looks problematic at first sight but in practice it turns out that the real challenge is, like in non-relativistic electronic structure theory, the description of the correlation between the motion of electrons. The mean-field approximation that is made in the Dirac-Hartree-Fock (DHF) approach provides a good first step, but gives bond energies and structures that are often too inaccurate for chemical purposes. 

Dirac-Coulomb theory within the mean field approximation (see Chapter 8) is routinely applied to molecules and allows us to estimate the relativistic effects even for large molecules. In the computer era. this means that there are computer programs available that allow anybody to perform relativistic calculations. 

If an atom is considered in the Bom-Oppenheimer approximation, the problem is even simpler, and the electronic equation also holds we can then take, e.g., J = 0. People still try to compute correlated wave functions (i.e., beyond the mean field approximation see Chapter 10) for heavier atoms. Besides, relativistic effects (see Chapter 3) play increasingly important roles for such atoms. Starting with magnesium, they are larger than the correlation corrections. Fortunately, the relativistic corrections for atoms are largest for the inner electronic shells, which are least important for chemists. 

The Dirac equation represents an approximation- and refers to a single particle. What happens with larger systems Nobody knows, but the first idea is to construct the total Hamiltonian as a sum of the Dirac Hamiltonians for individual particles plus their Coulombic interaction (the Dirac-Coulomb apjmmmation). This is practised routinely nowadays for atoms and molecules. Most often we use the mean-field approximation (see Chapter 8) with the modification that each of the one-electron functions represents a four-component bispinor. Another approach is extreme pragmatic, maybe too pragmatic we perform the non-relativistic calculations with a pseudopotential that mimics what is supposed to happen in a relativistic case. 

Mass-polarization, 54 Mass-velocity correction, in relativistic methods, 209, 211 Matrix element, 55, 103 McLean-Chandler basis sets, 160 Mean field approximation, 64 Metal coordination compounds, force field, 36 Metropolis sampling, in Monte Carlo techniques, 376... 

The relativistic mean meson field (R.MF) theory formulated by Teller and others [8, 9, 10] and by Walecka [11] is quite successful in both infinite nuclear matter and finite nuclei[12, 13, 14]. In the RMF model, only positive-energy baryonic states are considered to study the properties of ordinary nuclei. This is the so-called no-sea-approximation . However, an interesting feature of the RMF theory is the existence of bound negative-energy baryonic states. This happens because the interaction with the vector field generated by the baryon-... 

There is only one subtle point with regard to the no-pair approximation that deserves some attention. In the non-relativistic case the Fock space formalism without truncation of the T operators gives just an alternative parametrization of the foil Cl wave function. In the relativistic case the situation is more complex because the states of interest may contain a different number of electrons than the reference state. This means that the no-pair approximation is less appropriate as it is based on a mean-field potential due to a different number of electrons. Formally this problem might be tackled by lifting the no-pair restriction but it will be very hard to turn the resulting complicated formalism into an efficient algorithm. The corrections would probably be small since the difference in potential mainly affects the valence region where the potential is small relative to the rest mass term anyway. 

The new term in Ho approximates the actual interaction of an electron with the other electrons with a mean field U r), chosen to do so as accurately as possible. Applying standard Rayleigh-Schrddinger perturbation theory to Vc then gives the MBPT expansion. It is also possible to generalize to the relativistic case by introducing the instantaneous Breit interaction. 

Many attempts have been undertaken to rewrite the one-electron Dirac equation — of hydrogen-like atoms and also the mean-field SCF type derived in chapter 8 and in matrix form in chapter 10 — to obtain a form that is most suitable for numerical computations. Historically, the transformation and elimination techniques first emerged from such endeavors and were only later studied from a formal point of view as an essential part of the complete picture of relativistic many-electron theory. For instance, the DKH theory was first developed as an efficient low-order approximation to the Dirac equa-... 

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