Degeneracy, orbital

There is a degeneracy factor of two associated with a n orbital compared with the nondegeneracy of a (7 orbital, so that it might be expected that the integrated intensity of the second band system would be twice that of each of the other two. However, although the second band system is the most intense, other factors affect the relative intensities so that they are only an approximate guide to orbital degeneracies. 

Nevertheless, crystal fields cannot be completely ignored. The intensities of a number of bands ( hypersensitive bands) show a distinct dependence on the actual ligands which are coordinated. Also, in the same way that crystal fields lift some of the orbital degeneracy (2L -)- 1) of the terms of d" ions, so they lift some of the 2J -)- 1 degeneracy of the sates of P ions, though in this case only by the order of 100cm This produces fine structure in some bands of Ln spectra. 

BalKSl Balasubramanian, K. Symmetry simplifications of spare types in configuration int by orbital degeneracy. Internat. J. Quantum Chem. 20 (1981) 1255-1271. 

Consider the orbital angular momentum of a free-ion term. Here L = 3 and the orbital degeneracy is 7. Application of Van Vleck s formula (5.8) predicts an effective magnetic moment. 

Gardner 45) has observed the spectrum of Cl atoms adsorbed on a silica-gel surface at 77°K. The experimental results indicate that the orbital degeneracy of the 3p atomic orbital has been removed as a result of the electrostatic interaction with the surface. From the occupancy of the atomic orbitals one would predict that gx >011 — 2.00 and indeed the experimental g values are = 2.012 with g = 2.003. The hyperfine coupling indicates that the impaired electron is highly localized in the 3p orbitals. 

In 1937 Jahn and Teller applied group-theoretical methods to derive a remarkable theorem nonlinear molecules in orbitally degenerate states are intrinsically unstable with respect to distortions that lower the symmetry and remove the orbital degeneracy.37 Although Jahn-Teller theory can predict neither the degree of distortion nor the final symmetry, it is widely applied in transition-metal chemistry to rationalize observed distortions from an expected high-symmetry structure.38 In this section we briefly illustrate the application of Jahn-Teller theory and describe how a localized-bond viewpoint can provide a complementary alternative picture of transition-metal coordination geometries. 

So far, we have discussed the crystalline field acting on the ion A due to an octahedral environment of six B ligand ions. In many optically ion activated crystals, such as Ti +rAlaOj, the local symmetry of the active ion A is slightly distorted from the perfect octahedral symmetry Oh symmetry). This distortion can be considered as a perturbation of the main octahedral field. In general, this perturbation lifts the orbital degeneracy of the tag and eg levels and then produces additional structure in the tag eg absorption/emission bands. 

Too often the Jahn-Teller principle is presented in a reversed fashion—i.e., the distortion of a coordination polyhedron eliminates orbital degeneracy. This formulation renders the principle ineffective, because the distortion is at first sight attributed to external forces, whereas in fact it is the result of the intrinsic instability of the symmetric condition. 

Under the action of the electric field of the 0 environment, the state of the ( ) 3d configuration will split up into two states. The orbital degeneracy of a D state is 2 X 2 -f- 1 = 5. From the above discussion each of the resulting states must belong to one of the irreducible representations of Oh given in Table I. The state W corresponds to an irreducible representation of the group of symmetry operations of a sphere, i.e., the full rotation group R(3). 

Fia. 24. Energy level diagram for 3d configuraUon quartet terms under the action of crystal fields of various symmetries. Orbital degeneracy of a state is indicated in parentheses. 

Jahn-Teller distortions of octahedra are expected to be much smaller than in the cases mentioned above, if the orbital degeneracy occurs in the non-bonding d e-orbitals. Indeed no structures of fluorides are known, with the possible exception of FeF2 (24,130), in which distortions without doubt are effected by this degeneracy of rfs-orbitals. Weinstock and Claassen [322, 323) reported evidence of d5mamic Jahn-Teller effects on the vibrational properties of hexafluoride molecules, however. 

Jahn, H. A., and E. Teller Stability of polyatomic molecules in degenerate electronic states. I. Orbital degeneracy. Proc. Roy. Soc. A161, 220 (1937). 

Typical for cyclic jr—systems are d enerate orbitals degeneracy can disappear by do— nor-or acceptor-perturbation ... 

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