Spin degeneracy, with

Finally, we consider the situation at finite temperature. In this case, we will take the temperature-dependent occupation number of the single-particle state k to be k for each spin state. This leads to the assignment of (1 — 2 k) as the Cooper pair occupation number of the same wave-vector. Consequently, each occurrence of a Cooper pair will be accompanied by a factor (1 — 2 k). while the occurrence of an electronic state that is not part of a Cooper pair will be accompanied by a factor 2 k taking into account spin degeneracy. With these considerations, the total energy of the ground state at finite temperature will be given by... 

Magnetic circular dicliroism (MCD) is independent of, and thus complementary to, the natural CD associated with chirality of nuclear stmcture or solvation. Closely related to the Zeeman effect, MCD is most often associated with orbital and spin degeneracies in cliromophores. Chemical applications are thus typically found in systems where a chromophore of high symmetry is present metal complexes, poriihyrins and other aromatics, and haem proteins are... 

Singlet States with Spin Degeneracy Only... 

For four electrons, for example, with only spin degeneracy (the number of occupied orbits equalling the number of electrons), Slater gave the function J(SI i — ipii — iPiii- - I iv) as representing the structure in which orbits a and b are bonded together, and also c and d. Here Pi- Piv... 

An example Six electrons with only spin degeneracy. Placing the six orbits in a ring in the order abcdef, the five structures forming a canonical set are those given in Fig. 2. It is seen that... 

Similarly, cross peak positions in RELAY spectra give valuable information about remote connectivities (spins coupled to common partner but not to each other) and are very useful in resolving chemical shift degeneracies. An AMX spin system with no coupling between A and X shows a cross peak at ... 

For degenerate states a problem arises with the definition of cumulants. We consider here only spin degeneracy. Spatial degeneracy can be discussed on similar lines. For S 0 there are (2S + 1) different Afs-values for one S. The n-particle density matrix p Ms) = of a single one of these states does not... 

In 1949 Coulson and Fisher[15] introduced the idea of nonlocalized orbitals to the VB world. Since that time, suggested schemes have proliferated, all with some connection to the original VB idea. As these ideas developed, the importance of the spin degeneracy problem emerged, and VB methods frequently were described and implemented in this context. We discuss this more fully later. 

With the spin eigenfunctions of Eqs. (4.14) and (4.15) we have an example of the spin degeneracy alluded to in Chapter 2. Unlike the single singlet function we arrived at for two electrons in Section 2.1.1 we now obtain two. Writing out the equations specifically,... 

We make a small digression and note that the spin-degeneracy problem we have alluded to before is evident in Eq. (5.102). It will be observed that / = 1,..., /x in the index of e s pnp these functions are linearly independent since the efj are. There are, thus, fi linearly independent spin eigenfunctions of eigenvalue S(S + 1). Each of these has a full complement of Ms values, of course. In view of Eq. (5.40) the number of spin functions increases rapidly with the number of electrons. Ultimately, however, the dynamics of a system governs if many or few of these are important. 

Above room temperature, the mobile 3 d electrons are well described by a random mixture of Fel" and FeB ions with the mobile electrons diffusing from iron to iron, some being thermally excited to FeA ions, but the motional enthalpy on the B sites is AH < kT. As the temperature is lowered through Tc, the Seebeck coefficient shows the influence of a change in mobile-electron spin degeneracy, and at room temperature the Seebeck coefficient is enhanced by correlated multielectron jumps that provide a mobile electron access to all its nearest neighbors. The electron-hopping time xi, = coi = 10" s... 

The high-temperature transport data has been well rationalized 0 on the basis of a diffusional model with AH = 0.16 eV for all x, which is an agreement with localized Fe " -ion configurations responsible for a cooperative Jahn-Teller distortion below Tn. The ratio (1 - c)/c obtained from Eq. (15) for the Seebeck coefficient, with a spin-degeneracy factor P = 2, gives the factor c(l — c) entering the conductivity expression... 

Note that, for a crystal field with no three- or fourfold symmetry axes (D and 0), the spin degeneracy is completely removed by the spin-spin interaction even in the absence of a magnetic field. This is a characteristic... 

An important problem concerns the nature of the first excited state actually involved in the photodimerization reaction (the lowest excited singlet or triplet states). The simple 77-HMO method does not distinguish between the excited singlet and triplet state. Its results concern the first excited state and cannot be correlated precisely with the different multiplicities. For such a correlation, the more refined approximations of the molecular orbital method, which eliminate spin degeneracy, must be used. 

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