Notation degenerate orbitals

Consider the electronic configuration of carbon again Is 2s 2pl Remember, there are three different p orbitals in the 2p subshell the p orbital lies on the x-axis the p orbital lies on the y-axis and the p orbital lies on the z-axis. The different p orbitals are degenerate. To obey Hund s rule, these degenerate orbitals must be filled singly before spin pairing occurs. To obey the Pauli exclusion principle, when an orbital is full with two electrons, these electrons must have opposite spins. This is not shown using spectroscopic notation, but is seen when orbital box notation is used. 

The notation n that is used here is almost never strictly correct according to group theory, where it reserved for doubly degenerate orbitals in linear molecules (Chapter 6). It is, however, widely used to refer to loal symmetry the expression n interaction is used when the two orbitals share a common nodal plane and have lateral overlap (3-1). 

Labels refer to the notation of A-orbitals and (A, S)-terms appropriate for linear symmetry (Doon)- This has the same consequencies for d-orbitals as the trigonal symmetry D31,. 2 is orbitally non-degenerate and 77 and A are doubly degenerate. This is also true for if formed from d-electrons, whereas it could split in the general case of Dsh. 

In Table I we give the irreducible representations, in Mulliken s notation, contained in the central and attached orbitals for compounds of other types of symmetry. When 3 or 4 atoms are trigonally or tetragonally attached, we have supposed that the plane of these atoms is a plane of symmetry, as in (N03) or Ni(CN)4 When there is no such symmetry plane, as in NHg, the distinctions between u and g, or between primes and double primes, are to be aholished,9 and the symmetries degenerate to Csv, Civ instead of DSh, D h- When 6 atoms are attached in the scheme Z>3, or 8 in % they are arranged respectively at the corners of a trigonal and a square prism. 

We write creation and annihilation operators for a state 1/1) as a and aA, so that ) = a lO). We use the spin-orbital 2jm symbols of the relevant spin-orbital group G as the metric components to raise and lower indices gAA = (AA) and gAA = (/Li)3. If the group G is the symmetry group of an ion whose levels are split by ligand fields, the relevant irrep A of G (the main label within A) will contain precisely the states in the subshell, the degenerate set of partners. For example, in Ref. [10] G = O and A = f2. In the triple tensor notation X of Judd our notation corresponds to X = x( )k if G is a product spin-space group if spin-orbit interaction is included to couple these spaces, A will be an irrep appearing in the appropriate Kronecker decomposition of x( )k. 

In atomic spectroscopy, the notation S, P, D, F is employed to indicate terms with L values of 0, 1, 2, 3 respectively. The multiplicity is indicated by the number at the top as in 3P, for example. The spectral terms in an octahedral complex can be expressed in terms of only five ligand field quantum numbers which are written as A1( A, E, Tx and T2. The orbital degeneracies of these are 1, 1, 2, 3, 3 respectively. In discussing the absorption spectra of transition metal complexes, one should consider the number of terms into which each term of the free ion splits. The reason for this splitting is of course the differentiation of the degenerate d orbitals into two sets of orbitals. The S and P terms of the free ion are not split but transform as A1 and Tx respectively. The D terms are split into E and Ta while the F terms are split into Aj, Tj and T2 7. As in atomic spectroscopy, the multiplicity of each term is indicated by the number at the top, as for example 3T . 

Figme 1 see text for notation. These configurations are orbitally degenerate. In 5 and 6 this orbital degeneracy is removed by spin-orbit coupling, but it is not removed by this mechanism in 4 (see Section 53.4.S). 

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