Parametric approximation symmetry

Only for a special class of compound with appropriate planar symmetry is it possible to distinguish between (a) electrons, associated with atomic cores and (7r) electrons delocalized over the molecular surface. The Hiickel approximation is allowed for this limited class only. Since a — 7r separation is nowhere perfect and always somewhat artificial, there is the temptation to extend the Hiickel method also to situations where more pronounced a — ix interaction is expected. It is immediately obvious that a different partitioning would be required for such an extension. The standard HMO partitioning that operates on symmetry grounds, treats only the 7r-electrons quantum mechanically and all a-electrons as part of the classical molecular frame. The alternative is an arbitrary distinction between valence electrons and atomic cores. Schemes have been devised [98, 99] to handle situations where the molecular valence shell consists of either a + n or only a electrons. In either case, the partitioning introduces extra complications. The mathematics of the situation [100] dictates that any abstraction produce disjoint sectors, of which no more than one may be non-classical. In view if the BO approximation already invoked, only the valence sector could be quantum mechanical9. In this case the classical remainder is a set of atomic cores in some unspecified excited state, called the valence state. One complication that arises is that wave functions of the valence electrons depend parametrically on the valence state. 

The presence of the higher order harmonics of the magnetic helix in the incommensurate phase is characteristic for the temperature interval where the Lifshits invariant is comparable with an anisotropy invariant [11], For magnetic systems with a one-parametric thermodynamic potential the propagation vector q is not equal to zero already at the temperature where the system orders, T1 =TP. As an anisotropy invariant is proportional to rj for a crystal with tetragonal symmetry, then it becomes comparable with Lifshits invariant proportional to q t] 2 much below Ti near the transition into a low-temperature commensurate phase. However, in copper metaborate q grows sharply from approximately zero at temperature 7) < Tp (Fig. 7) [5],... 

In addition to approximations for c(1,2 pl), a suitable parametrization of the trial function n(r,m) is required. The position dependence can be described in full detail by a Fourier expansion, the form of which is determined by the choice of crystal symmetry for the solid phase. More simply, a Gaussian distribution of molecular density about the sites of the crystal lattice may be assumed the accuracy of this latter approximation has been verified for... 

In spite of the formal similarities, the parametrization of the PPP and the CNDO Hamiltonians are quite different. In particular, the CNDO model is not restricted to the first-neighbor approximation. For this reason the particle-hole symmetry does not apply for... 

The matrix element in eq. (143) is valid for transitions between two crystal-field levels. Because of the radial integrals, the calculation of the matrix element is very tedious and can in fact only be done if some approximations are made. Axe (1963) treated the quantities Aiaj3(k,X) in eq. (143) as adjustable parameters. In this expression, X is equal to 2, 4 or 6, and k is restricted to values of Ail. The values of q are determined by crystal-field symmetry constraints and lie between 0 and k. This parametrization scheme was used by Axe for the intensity analysis of the fluorescence spectrum of Eu(C2H5S04)3 9H20. Porcher and Caro (1978) introduced the notation Bxki for the intensity parameters ... 

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