Intrinsic degeneration

Wc shall deal also with the ease in which such relations exist only for certain values of the, ik the mechanical system is then non-dcgcncratc the particular motions in question, for which (ti>)=0, we shall call accidentally degenerate, whilst the motions of a degenerato system [( tv)=0 identically] are spokon of as intrinsically degenerate. 

We consider next the case of intrinsic degeneration, i.e. we assume that a relation... 

The method of approximation described in 41 can also fail when the unperturbed system is not intrinsically degenerate, if there exist relations of the form... 

If for the moment we disregard the presence of intrinsic degenerations, but assume the existence of several accidental degenerations, we can choose the angle and action variables wk0, Jk° (A=l, 2. . . /) of the unperturbed system so that the v° s (a=l, 2. . . s) differ from zero and are incommensurable, while vf)°(p=s + l. . . /) vanish for the particular values which the Jj.° s have in the case of the unperturbed motion. We assume therefore that an (/—s)-fold accidental degeneration exists. 

The unperturbed system consists therefore of two circular orbits of the same size. In addition to the double limiting degeneration due to the circular orbits, we have also a double intrinsic degeneration, arising from the fact that the planes of the two orbits are... 

Perturbations of a Non-degenerate System 42. Application to the Non-harmonic Oscillator 43. Perturbations of an Intrinsically Degenerate System 44. An Example of Accidental Degeneration. 45. Phase Relations in the Case of Bohr Atoms and Molecules... 

Electrons and most other fiindamental particles have two distinct spin wavefunctions that are degenerate in the absence of an external magnetic field. Associated with these are two abstract states which are eigenfiinctions of the intrinsic spin angular momentum operator S... 

Most of the AIMD simulations described in the literature have assumed that Newtonian dynamics was sufficient for the nuclei. While this is often justified, there are important cases where the quantum mechanical nature of the nuclei is crucial for even a qualitative understanding. For example, tunneling is intrinsically quantum mechanical and can be important in chemistry involving proton transfer. A second area where nuclei must be described quantum mechanically is when the BOA breaks down, as is always the case when multiple coupled electronic states participate in chemistry. In particular, photochemical processes are often dominated by conical intersections [14,15], where two electronic states are exactly degenerate and the BOA fails. In this chapter, we discuss our recent development of the ab initio multiple spawning (AIMS) method which solves the elecronic and nuclear Schrodinger equations simultaneously this makes AIMD approaches applicable for problems where quantum mechanical effects of both electrons and nuclei are important. We present an overview of what has been achieved, and make a special effort to point out areas where further improvements can be made. Theoretical aspects of the AIMS method are... 

Perry, V. H., Brown, M. C., Lunn, E. R., Tree, P., and Gordon, S. Evidence that very slow Wallerian degeneration in C57BL/ Ola mice is an intrinsic property of the peripheral nerve. Eur. J. Neurosci. 2 802-808,1990. 

In 1937 Jahn and Teller applied group-theoretical methods to derive a remarkable theorem nonlinear molecules in orbitally degenerate states are intrinsically unstable with respect to distortions that lower the symmetry and remove the orbital degeneracy.37 Although Jahn-Teller theory can predict neither the degree of distortion nor the final symmetry, it is widely applied in transition-metal chemistry to rationalize observed distortions from an expected high-symmetry structure.38 In this section we briefly illustrate the application of Jahn-Teller theory and describe how a localized-bond viewpoint can provide a complementary alternative picture of transition-metal coordination geometries. 

Degenerate semiconductors can be intrinsic or extrinsic semiconductors, but in these materials the band gap is similar to or less than the thermal energy. In such cases the number of charge carriers in each band becomes very high, as does the electronic conductivity. The compounds are said to show quasi-metallic behavior. 

Figure 4. Plot of intrinsic barrier to degenerate exchange (intrinsic barrier for X + CHSX) vs. methyl cation affinity for selected nucleophiles.

Intrinsic Nucleophilicity vs. Leaving Group Ability. As pointed out earlier, our definition of intrinsic nucleophilicity in terms of barriers to degenerate exchange reactions necessarily implies that nucleophilicity and leaving group ability are equivalent. This can be seen in either of two ways. First,... 

Vitamin Bn deficiency Deficiency, although rare, results in two serious problems megaloblastic anaemia (which is identical to that caused by folate deficiency) and a specific neuropathy called Bi2-associated neuropathy or cobalamin-deficiency-associated neuropathy (previously called, subacute combined degeneration of the cord). A normal healthy adult can survive more than a decade without dietary vitamin B12 without any signs of deficiency since it is synthesised by microorganisms in the colon and then absorbed. However, pernicious anaemia develops fairly rapidly in patients who have a defective vitamin B12 absorption system due to a lack of intrinsic factor. It results in death in 3 days. Minot and Murphy discovered that giving patients liver, which contains the intrinsic factor, and which is lightly cooked to avoid denaturation, cured the anaemia. For this discovery they were awarded the Nobel Prize in Medicine in 1934. 

A frequent cause of vitamin B12 deficiency is atrophic gastritis leading to a lack of intrinsic factor. Besides megaloblastic anemia, damage to mucosal linings and degeneration of myelin sheaths with neurological sequelae will occur (pernicious anemia). 

A detailed analysis of the resulting curves yields that, for the degenerate exchange reaction (N = X), the intrinsic barrier can be written as (50), where 0 < r < 1 is the slope parameter, and P the avoided crossing constant of the two curves. 

Compounds with a narrow HOMO-LUMO gap (Figure 5.5d) are kinetically reactive and subject to dimerization (e.g., cyclopentadiene) or reaction with Lewis acids or bases. Polyenes are the dominant organic examples of this group. The difficulty in isolation of cyclobutadiene lies not with any intrinsic instability of the molecule but with the self-reactivity which arises from an extremely narrow HOMO-LUMO gap. A second class of compounds also falls in this category, coordinatively unsaturated transition metal complexes. In transition metals, the atomic n d orbital set may be partially occupied and/or nearly degenerate with the partially occupied n + 1 spn set. Such a configuration permits exceptional reactivity, even toward C—H and C—C bonds. These systems are treated separately in Chapter 13. 

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