Three-fold degeneracy

Two other examples will sufhce. Methane physisorbs on NaCl(lOO) and an early study showed that the symmetrical, IR-inactive v mode could now be observed [97]. In more recent work, polarized FTIR rehection spectroscopy was used to determine that on being adsorbed, the three-fold degeneracies of the vs and v modes were partially removed [98]. This hnding allowed consideration of possible adsorbate-adsorbent geometries one was that of a tripod with three of the methane hydrogens on the surface. The systems were at between 4 and 40 K so that the equilibrium pressure was very low, about 10 atm. 

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. 

As stated earlier, in the state of thermal equilibrium at room temperature, dihydrogen (H2) contains 25.1% parahydrogen (nuclear singlet state) and 74.9% orthohydrogen (nuclear triplet state) [19]. This behavior reflects the three-fold degeneracy of the triplet state and the almost equal population of the energy levels, as demanded by the Maxwell-Boltzmann distribution. At lower temperatures, different ratios prevail (Fig. 12.5) due to the different symmetry of the singlet and the triplet state [19]. 

The presence of a Ti3+ ion in a distorted octahedral site would also yield a zero electronic entropy term. This results from removal of the three-fold degeneracy of t2g orbitals in the low-symmetry environment. Other effects of electronic entropy on thermodynamic properties of transition metal-bearing minerals are discussed in chapter 7 ( 7.4). 

The many-electron states of an atom in a crystal field or a molecule can obviously not be labelled by the IRs of SO(3), since the Hamilton operator, the angular moment operator and therefore also the many-electron wave functions transform according to the IRs of a less symmetric point group. The lower symmetry may also remove the degeneracies of the LS terms. For example, the ground term of a boron atom becomes T u in an octahedral crystal field so that the three fold degeneracy is retained, but splits into two LS terms of and symmetry when the crystal field symmetry is lowered to C v ... 

The first study on accidental three-state conical intersections was done for the CH cation by Katriel and Davidson. In a tetrahedral geometry, the ground state of CH is a T2 state. Therefore, it is triply degenerate as required by symmetry. Only one degree of freedom exists that will preserve Tj symmetry, and the dimensionality of the seam is one because all the requirements for degeneracy are satisfied by symmetry. The authors found additional three-fold degeneracies in this system even when the tetrahedral symmetry was broken. If no symmetry is present, the cation has 9 degrees of freedom and the dimensionality of the seam becomes 9 — 5 = 4. 

Now take the case for an octahedral vanadium(iii) ion. For d, the ground term is Tig. The spatial degeneracy of a 7 term is three-fold and we describe this with Leff = 1. Using (5.10) we find eff = VlO. So for this Tig term, the crystal field has quenched some, but not all, of the angular momentum of the parent free ion F term. 

Table 3 illustrates the correlation between the O, C4, and Ca symmetries. The C, structure can account for the data, qualitatively, as follows. The highest frequency line is derived from vj of [MFg]" and its shift to higher frequency may simply reflect the departure of the [MFg] species from octahedral symmetry, as discussed recently by Murrell. Derivatives of vj, Vg, and vj also appear (perhaps with small shifts), Vj becoming Raman active in the reduced symmetry. The small splitting of Vj and vg for the platinum compound is also consistent with the postulate, but not with a Cg structure, since any symmetry which retains a three-fold axis of the octahedron will not lift the degeneracy of the Vg (Fj) mode. Note abo that the appearance of a vj derivative (ungerade) is inconsistent with symmetrical approach of two [XeF] cations on either side of the octahedron. The Vg modes in these compounds s m to be split more than in simpler [MFg] salts. This... 

One simple extension would be to vary the orientation to cover the possibility that adsorption occurs with a low-symmetry axis pointing towards the surface, e.g. a C-C single bond or individual C atom could be prone to the surface. However, the most serious omission is, perhaps, the treatment of the LUMO and HOMO as if they retain their three- and flve-fold degeneracies in the adsorbed environment. This is certainly not something that would be expected for Ceo molecules adsorbed onto a surface. 

As a second example consider the case of a quartz crystal, with the space group Ds. One of the representations of the point group >3 has dimension two. Thus, if the vector ko is parallel to the three-fold symmetry axis when the point group Gk0 coincides with the point group >3, a double degeneracy of excitonic terms is, in general, possible. 

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... 

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