Degeneracy broken

This is the central Jahn-Teller [4,5] result. Three important riders should be noted. First, Fg = 0 for spin-degenerate systems, because F, x F = Fo. This is a particular example of the fact that Kramer s degeneracies, aiising from spin alone can only be broken by magnetic fields, in the presence of which H and T no longer commute. Second, a detailed study of the molecular point groups reveals that all degenerate nonlinear polyatomics, except those with Kramer s... 

The strategy to first use broken symmetry solutions and later restore the correct (spin) density by employing ensembles can be applied successfully to solve many degeneracy related problems. However, in practice it is very rarely used because there are hardly any... 

In Hartree-Fock (HF) theory, the energy of a given orbital is lower when the orbital is occupied than when it is vacant (due to the formal self-energy in the former case), and the degeneracy is broken. Thus, perturbative F1F expressions such as Eq. (2.7) often have wider numerical validity than would be anticipated in naive MO theory. 

The role of electronic structure in Mn and Co site preference and mobility can to some extent be understood through ligand-field theory (LFT). LET qualitatively explains how the degeneracy of the 3d orbitals is broken when a free TM ion is surrounded by coordinating anions. The ligand-field splitting of d orbitals in octahedral and tetrahedral coordination is pictured in Figure 6. ... 

Fig. 8.9. The locus A of double-zero eigenvalue degeneracies of the Hopf bifurcation for cubic autocatalysis with decay. Also shown, as broken curves, are the loci of stationary-state degeneracies, corresponding to the boundaries for isola and mushroom patterns. The curve A lies completely within the parameter regions for multiple stationary states.

Fig. 8.13. (a) The division of the fS0 — K1 parameter region into 11 regions by the various loci of stationary-state and Hopf bifurcation degeneracies. The qualitative forms of the bifurcation diagrams for each region are given in fi)—(xi) in (b), where solid lines represent stable stationary states or limit cycles and broken curves correspond to unstable states or limit cycles, (i) unique solution, no Hopf bifurcation (ii) unique solution, two supercritical Hopf bifurcations (iii) unique solution, one supercritical and one subcritical Hopf (iv) isola, no Hopf points (v) isola with one subcritical Hopf (vi) isola with one supercritical Hopf (vii) mushroom with no Hopf points (viii) mushroom with two supercritical Hopf points (ix) mushroom with one supercritical Hopf (x) mushroom with one subcritical Hopf (xi) mushroom with supercritical and subcritical Hopf bifurcations on separate branches. 

In a hydrogen atom, the orbital energy is determined exclusively by the principal quantum number n—all the different values of / and mi are degenerate. In a multielectron atom, however, this degeneracy is partially broken the energy increases as / increases for the same value of n. 

Nondynamical electron correlation effects are generally important for reaction path calculations, when chemical bonds are broken and new bonds are formed. The multiconfiguration self-consistent field (MCSCF) method provides the appropriate description of these effects [25], In the last decade, the complete active space self-consistent field (CASSCF) method [26] has become the most widely employed MCSCF method. In the CASSCF method, a full configuration interaction (Cl) calculation is performed within a limited orbital space, the so-called active space. Thus all near degeneracy (nondynamical electron correlation) effects and orbital relaxation effects within the active space are treated at the variational level. A full-valence active space CASSCF calculation is expected to yield a qualitatively reliable description of excited-state PE surfaces. For larger systems, however, a full-valence active space CASSCF calculation quickly becomes intractable. 

A curious effect, prone to appear in near degeneracy situations, is the artifactual symmetry breaking of the electronic wave function [27]. This effect happens when the electronic wave function is unable to reflect the nuclear framework symmetry of the molecule. In principle, an approximate electronic wave function will break symmetry due to the lack of some kind of non-dynamical correlation. A typical example of this case is the allyl radical, which has C2v point group symmetry. If one removes the spatial and spin constraints of its ROHF wave function, a lower energy symmetry broken (Cs) solution is obtained. However, if one performs a simple CASSCF or a SCVB [28] calculation in the valence pi space, the symmetry breaking disappears. On the other hand, from the classical VB point of view, the bonding of the allyl radical is represented as a superposition of two resonant structures. 

The essence of the first or second order Jahn-Teller effect is that a high-symmetry geometry generates a real or near degeneracy, which can be broken with stabilization by a symmetry-lowering deformation. Note a... 

Here the subscript 5 stands for symplectic , a Greek word which hterally means entangled . If the system has half-integer spin and all antiunitary symmetries are broken, Kramers degeneracy is lifted and we return to the case Py. 

Fig. 11. Reaction coordinate diagram for ECL system involving rubrene (A) and 9,10-diphenylan-thracene (B). Potential energy curves are presented in the zero-order approximation, without removing the degeneracy at the crossing points of the potential energy curves. Broken lines represent the vibronically excited triplet state.

We have considered several possible explanations for the disappearance of most of the Hg modes. First, the addition of alkali metal atoms to the lattice will introduce symmetry reduction due to A-C o interactions, which will tend to broaden the Hg modes as their degeneracy is broken. However, on the scale of our resolution these effects should be small and, as discussed below, further doping results in a reappear-... 

Figure 2. Fields of the democratic centrifugal force (DCF) on the gyration space at a typical value of (p. The broken lines represent degeneracy between the two gyration radii, — a2. ...

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