Spin degeneracy three-electron

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... 

We next consider an acceptor center near the valence band. Here the states can sometimes be described in terms of a deep s-like state (not important in the problem), and three p-like states (p+ = px + ip, P- = px — ip, and pz). Suppose that these p-like states are degenerate (or nearly so, within a few kT), so that their six electrons (three spin-up, three spin-down) can be considered equivalent. Consider the acceptor formed by Cd on a Ga site in GaAs. This site has seven electrons, two in the deep s-like state and five in the p-like states. Therefore, the unoccupied state has degeneracy gA0 = 6 /5 l = 6, and the occupied state (after accepting one electron) has gA1 = 6 /6 = 1. Thus, in Eq. (B41), the degeneracy factor is gAJgA0 = b... 

The situation becomes slightly complicated for systems with more than two electrons due to the spin degeneracy. In the case of three electrons, for example, one can construct doublet (S = ) and quartet (S = ) spin... 

Let us remark that in crystals consisting of aromatic molecules, to which the theory of Sternlicht and McConnell (26) was applied, the excited triplet states are not three-fold degenerate even when an external magnetic field is absent. Due to the dipole spin-spin interaction between electrons the degeneracy is totally or partially removed, depending on the symmetry of the excited state wavefunction. By a phenomenological description of this splitting the so-called Spin-Hamiltonian is usually applied... 

Let us now restrict our discussion to the singlet states of molecules with spin degeneracy only. For a system involving 2n electrons and 2n stable orbitals (such as the Is orbitals in 2n hydrogen atoms), there are (2n) /2nn different ways in which valence bonds can be drawn between the orbitals in pairs. Thus for the case of four orbitals a, b, c, and d the bonds can be drawn in three ways, namely,... 

The existence of numerous isomers illustrates the relative ease of electron transfer between a-orbitals (which dominate in the bonding in planar structures) and Planar structures are the most stable for n < 5, three-dimensional structures for > 5. There is a transition at n = 6 to ground states with minimum spin degeneracy, so that it is essential to incorporate spin in the calculations of lighter clusters. The structural variety is consistent with the metallic nature of the elements The valence sp-shells in the atoms are less than half-filled, and the separation in energy between the highest occupied and lowest unoccupied orbitals is usually small. 

CF+ complexes are d complexes with tjg orbitals occupied by the maximum number of three electrons of the same spin. The ground state is a quartet state (2S -b 1 = 4). The transition t2g Cg is in the visible region and gives a strong green color to the system. Cr + is a strong field case, but the spin is maximum due to the degeneracy. 

Although an octahedron has a much lower symmetry than a sphere it would be reasonable to expect that many-electron wavefunctions would be handled similarly. This is so—symbols such as Eg and A g, like t g, g and aig orbitals, imply, respectively, triple, double and single orbital degeneracy. In each case they are associated with a spin degeneracy which, in each of these three examples, is identical to the spatial degeneracy. However the two vary independently and so symbols such as T g, Eg and A g are perfectly reasonable. 

Table 15-1 Some increased-valence structures expressed in terms of their component three-electron bonds , n = number of electrons = number of unpaired elections D = spin degeneracy...

In general, at least three anchors are required as the basis for the loop, since the motion around a point requires two independent coordinates. However, symmetry sometimes requires a greater number of anchors. A well-known case is the Jahn-Teller degeneracy of perfect pentagons, heptagons, and so on, which will be covered in Section V. Another special case arises when the electronic wave function of one of the anchors is an out-of-phase combination of two spin-paired structures. One of the vibrational modes of the stable molecule in this anchor serves as the out-of-phase coordinate, and the loop is constructed of only two anchors (see Fig. 12). 

The left superscript indicates that the arrangements are all spin triplets. The letter T refers to the three-fold degeneracy just discussed and it is in upper case because the symbol pertains to a many-electron (here two) wavefunction (we use lower-case letters for one-electron wavefunctions or orbitals, remember). The subscript g means the wavefunctions are even under inversion through the centre of symmetry possessed by the octahedron (since each d orbital is of g symmetry, so also is any product of them), and the right subscript 1 describes other symmetry properties we need not discuss here. More will be said about such term symbols in the next two sections. 

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