Degenerate levels

To extend the above kinetie model to this more general ease in whieh degenerate levels oeeur, one uses the number of moleeules in eaeh level (N and Nf for the two levels in the above example) as the time dependent variables. The kinetie equations then governing their time evolution ean be obtained by summing the state-to-state equations over all states in eaeh level... 

The pattern of two half-filled degenerate levels persists for larger rings containing 4n 71 electrons. In contrast, all 4 -F 2 systems are predicted to have all electrons paired in bonding MOs with net stabilization relative to isolated double bonds. This pattern provides... 

These rules follow directly from the quantum-mechanical theory of perturbations and the resolution of the secular equations for the orbital interaction problem. The (small) interaction between orbitals of significantly different energ is the familiar second order type interaction, where the interaction energy is small relative to the difference between EA and EB. The (large) interaction between orbitals of same energy is the familiar first order type interaction between degenerate or nearly degenerate levels. 

Thus, only one level is present for 7=0, but three levels with the same energy occur when 7=1, five at 7=2, and so on. The degenerate levels are distinguished by a quantum number K given by... 

The simplest example is that of a doubly degenerate level, for which... 

When spin-orbit coupling is introduced the symmetry states in the double group CJ are found from the direct products of the orbital and spin components. Linear combinations of the C"V eigenfunctions are then taken which transform correctly in C when spin is explicitly included, and the space-spin combinations are formed according to Ballhausen (39) so as to be diagonal under the rotation operation Cf. For an odd-electron system the Kramers doublets transform as e ( /2)a, n =1, 3, 5,... whilst for even electron systems the degenerate levels transform as e na, n = 1, 2, 3,. For d1 systems the first term in H naturally vanishes and the orbital functions are at once invested with spin to construct the C functions. 

Note For the triplet states the functions are given only for the Ms = 1 spin component. For all degenerate levels the a components (as given by the vector coupling coefficients of Table 2) are listed first, and these transform as +1 under oxz and the b components as - 1 under the same operation. 

Another difference between nucleons and electrons is that nucleons pair whenever possible. Thus, even if a particular energy level can hold more than two particles, two particles will pair when they are present. Thus, for two particles in degenerate levels, we show two particles as II rather than II. As a result of this preference for pairing, nuclei with even numbers of protons and neutrons have all paired particles. This results in nuclei that are more stable than those which have unpaired particles. The least stable nuclei are those in which both the number of neutrons and the number of protons is odd. This difference in stability manifests itself in the number of stable nuclei of each type. Table 1.3 shows the numbers of stable nuclei that occur. The data show that there does not seem to be any appreciable difference in stability when the number of protons or neutrons is even while the other is odd (the even-odd and odd-even cases). The number of nuclides that have odd Z and odd N (so-called odd-odd nuclides) is very small, which indicates that there is an inherent instability in such an arrangement. The most common stable nucleus which is of the odd-odd type is 147N. 

To understand how electrons with spin interact, it is useful to examine a system consisting of two electrons, such as the helium atom. Let this two-electron system be described by the wave function, in space coordinates, (r), If the electrons are interchanged the wave function will in general be different, ( ), but since the electrons are identical (ignoring spin) the energy of the system will not be affected. The wave functions therefore belong to degenerate levels. 

Figure 6. Formfactor K(r) and nearest neighbour spacing distribution P(s) for (a) the pi s form the regular representation of the cyclic group Z24 (b) pi s represent the symmetric group. S, (c) a random set of pi s without symmetries. The dashed curve in (b) labeled red. Poisson corresponds to a distribution of degenerate levels being Poisson distributed otherwise.

Diagonalization of the rotation-vibration interaction produces splittings of the individual (degenerate) levels, as shown schematically in Figure 4.22. It is interesting to note that the matrix elements of Eq. (4.124a) can be approximately written as... 

One of the objections to these experiments is that they give no real measure of the extent to which the SiMes group conjugates. The system of degenerate levels is so finely balanced that it takes very little to tip it one way or the other. Indeed there is evidence that the effect is small. In... 

---