Symmetry self-consistent

The early Hartley model [2, 3] of a spherical micellar stmcture resulted, in later years, in some considerable debate. The self-consistency (inconsistency) of spherical symmetry witli molecular packing constraints was subsequently noted [4, 5 and 6]. There is now no serious question of tlie tenet tliat unswollen micelles may readily deviate from spherical geometry, and ellipsoidal geometries are now commonly reported. Many micelles are essentially spherical, however, as deduced from many light and neutron scattering studies. Even ellipsoidal objects will appear... 

Mechanical Properties. The hexagonal symmetry of a graphite crystal causes the elastic properties to be transversely isotropic ia the layer plane only five independent constants are necessary to define the complete set. The self-consistent set of elastic constants given ia Table 2 has been measured ia air at room temperature for highly ordered pyrolytic graphite (20). With the exception of these values are expected to be representative of... 

Fig. 6. Self-consistent band structure (48 valence and 5 conduction bands) for the hexagonal II arrangement of nanotubes, calculated along different high-symmetry directions in the Brillouin zone. The Fermi level is positioned at the degeneracy point appearing between K-H, indicating metallic behavior for this tubule array[17. ...

The calculations that have been carried out [56] indicate that the approximations discussed above lead to very good thermodynamic functions overall and a remarkably accurate critical point and coexistence curve. The critical density and temperature predicted by the theory agree with the simulation results to about 0.6%. Of course, dealing with the Yukawa potential allows certain analytical simplifications in implementing this approach. However, a similar approach can be applied to other similar potentials that consist of a hard core with an attractive tail. It should also be pointed out that the idea of using the requirement of self-consistency to yield a closed theory is pertinent not only to the realm of simple fluids, but also has proved to be a powerful tool in the study of a system of spins with continuous symmetry [57,58] and of a site-diluted or random-field Ising model [59,60]. 

Besides the elementary properties of index permutational symmetry considered in eq. (7), and intrinsic point group symmetry of a given tensor accounted for in eqs. (8)-(14), much more powerful group-theoretical tools [6] can be developed to speed up coupled Hartree-Fock (CHF) calculations [7-11] of hyperpolarizabilities, which are nowadays almost routinely periformed in a number of studies dealing with non linear response of molecular systems [12-35], in particular at the self-consistent-field (SCF) level of accuracy. 

It is evident that the approach described so far to derive the electronic structure of lanthanide ions, based on perturbation theory, requires a large number of parameters to be determined. While state-of-the-art ab initio calculation procedures, based on complete active space self consistent field (CASSCF) approach, are reaching an extremely high degree of accuracy [34-37], the CF approach remains widely used, especially in spectroscopic studies. However, for low point symmetry, such as those commonly observed in molecular complexes, the number of CF... 

The obvious disadvantage of this simple LG model is the necessity to cut off the infinite expansion (26) at some order, while no rigorous justification of doing that can be found. In addition, evaluation of the vertex function for all possible zero combinations of the reciprocal wave vectors becomes very awkward for low symmetries. Instead of evaluating the partition function in the saddle point, the minimization of the free energy can be done within the self-consistent field theory (SCFT) [38 -1]. Using the integral representation of the delta functionals, the total partition function, Z [Eq. (22)], can be written as... 

Abstract The hadronic equation of state for a neutron star is discussed with a particular emphasis on the symmetry energy. The results of several microscopic approaches are compared and also a new calculation in terms of the self-consistent Green function method is presented. In addition possible constraints on the symmetry energy coming from empirical information on the neutron skin of finite nuclei are considered. 

The aim of this contribution is to review the present understanding of the hadronic EoS, and the nuclear symmetry energy (SE) in particular. In the past the latter quantity has been computed mostly in terms of the Brueckner-Hartree-Fock (BHF) approach. In order to get an idea about the accuracy of the BHF result we present a recent calculation [1] in terms of the self-consistent... 

Eqs. (9),(11) the condition eq. (12) in the random phase approximation (RPA) formalism leads to a self-consistency relation between the symmetry potential and the Landau parameter F [26] ... 

For small asymmetries, the superconducting state is homogeneous and the order parameter preserves the space symmetries. For most of the systems of interest the number conservation should be implemented by solving equations for the gap function and the densities of species self-consistently. In such a scheme the physical quantities are single valued functions of the asymmetry and temperature, contrary to the double valued results obtained in the non-conserving schemes. 

There exists no uniformity as regards the relations between localized orbitals and molecular symmetry. Consider for example an atomic system consisting of two electrons in an (s) orbital and two electrons in a (2px) orbital, both of which are self-consistent-field orbitals. Since they belong to irreducible representations of the atomic symmetry group, they are in fact the canonical orbitals of this system. Let these two self-consistent-field orbitals be denoted by Cs) and (2p), and let (ft+) and (ft ) denote the two digonal hybrid orbitals defined by... 

Once the "synthesis tree" has been elaborated, we must proceed to the evaluation of the alternative pathways and compare them with possible synthetic schemes in order to optimise the chosen route and make it as self-consistent as possible. However, all synthetic plans must be flexible enough to allow new alternative solutions when things do not happen as anticipated. In this sense. Woodward referred very often to opportunism and of taking advantage of the "surprises" which may occur during the execution of a synthesis. Through the different stages of a synthesis new aspects may evolve and even important discoveries may be made. Such was the case, for instance, in the vitamin B12 synthesis in which the considerations of the stereochemistry of an intermediate, opposite to the one anticipated, led Woodward to the discovery of the principle of conservation of orbital symmetry [29]. 

Vibrational 57) and F NMR 68) spectroscopy were used to establish for CIF3O2 the following structure of symmetry C v, which according to semi-empirical linear combination of atomic orbitals-molecular orbitals (LCAO-MO) self-consistent field (SCF) calculations 239) is most stable ... 

The Hartree-Fock or self-consistent-field approximation is a simplification useful in the treatment of systems containing more than one electron. It is motivated partly by the fact that the results of Hartree-Fock calculations are the most precise that still allow the notion of an orbital, or a state of a single electron. The results of a Hartree-Fock calculation are interpretable in terms of individual probability distributions for each electron, distinguished by characteristic sizes, shapes and symmetry properties. This pictorial analysis of atomic and molecular wave functions makes possible the understanding and prediction of structures, spectra and reactivities. 

J. Paldus, Hartree-Fock Stability and Symmetry Breaking. In R. Garbo and M. Klobukowski (Eds.) Self-Consistent Field Theory and Applications (Elsevier, Amsterdam, 1990), pp. 1-45. 

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