Spin degeneracy

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... 

Table 12-4 gives the characters, basis functions, and the case a, b, or c to which the irreducible representation of Hu and belong for the point T. The degeneracy in and is the usual Kramers spin degeneracy, which is removed in cases (2) and (4) because of the absence of 6 in these symmetry groups. 

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... 

In an octahedral crystal field, for example, these electron densities acquire different energies in exactly the same way as do those of the J-orbital densities. We find, therefore, that a free-ion D term splits into T2, and Eg terms in an octahedral environment. The symbols T2, and Eg have the same meanings as t2g and eg, discussed in Section 3.2, except that we use upper-case letters to indicate that, like their parent free-ion D term, they are generally many-electron wavefunctions. Of course we must remember that a term is properly described by both orbital- and spin-quantum numbers. So we more properly conclude that a free-ion term splits into -I- T 2gin octahedral symmetry. Notice that the crystal-field splitting has no effect upon the spin-degeneracy. This is because the crystal field is defined completely by its ordinary (x, y, z) spatial functionality the crystal field has no spin properties. 

Fig. 5. Relation of activation parameters for Fe[HB(pz)3]2. The quantity — TASl is partitioned into — TASieg corresponding to spin degeneracy of A, state and — corresponding to...

Here, the factor 2 accounts for spin degeneracy, and the sum is over the occupied single-particle states. In terms of y/i, the energy functional given by Eq. (6) can be written as ... 

In this expression p is a mass parameter associated to the electronic fields, i.e. it is a parameter that fixes the time scale of the response of the classical electronic fields to a perturbation. The factor 2 in front of the classical kinetic energy term is for spin degeneracy. The functional f [ i , ] plays the role of potential energy in the extended parameter space of nuclear and electronic degrees of freedom. It is given by. 

The configurational entropy evaluated above omits all mention of electron spin. Because there are two spin directions, a spin degeneracy of 2, these must be included in the possible configurations. When this term is included, the Heikes equation becomes the Chaikin-Beni equation ... 

In general, the spin degeneracy can be specified by a parameter (3, to give... 

If cations in the material can take two valence states, as occurs in many transition-metal compounds, Ti3+/Ti4+ or Co3+/Co4+, for example, then the value of the spin degeneracy, (3, must take into account these configurations and can take values greater than 2. 

The spin degeneracy of the term is not affected by the crystal field since 3C is assumed not to contain spin terms. 

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. 

When there are only two electrons the analysis is much simplified. Even quite elementary textbooks discuss two-electron systems. The simplicity is a consequence of the general nature of what is called the spin-degeneracy problem, which we describe in Chapters 4 and 5. For now we write the total solution for the ESE 4 (1, 2), where the labels 1 and 2 refer to the coordinates (space and spin) of the two electrons. Since the ESE has no reference at all to spin, 4 (1, 2) may be factored into separate spatial and spin functions. For two electrons one has the familiar result that the spin functions are of either the singlet or triplet type. 

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. 

The above treatment of a three-electron case shows how to generate quartet (spin states are named in terms of their spin degeneracies 2S+1) and doublet states for a configuration of the form... 

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