Five-fold degenerate orbitals

In the third row K uses a doubly occupied 4s band of bonding orbitals, and Ca uses two s-p hybrid orbitals. In Sc the 3d-electron is comparable in energy to the 4s, and the d orbitals form a five-fold degenerate band. The free atoms of the first transition series all have high spin d-electrons in accordance with Hund s rule. We expect similar behavior in the solid state, so that only 2 1/2 electrons will fill the bonding d levels without double occupancy. This is the case for Sc, Ti and V. 

The colours of the transition metal ions and their complexes are, in fadt, due to electronic transitions involving the d electrons. The d shell in an atom or ion is five-fold degenerate, i.e. there are five d orbitals which are equal in energy in the absence of an external magnetic or electric field. An electron... 

The total number of states for the three levels P, and is 5 + 3 + 1 = 9. When spin-orbit interaction was neglected, we had a P term that consisted of (2L + 1)(25 + 1) = 3(3) = 9 equal-energy states. With spin-orbit interaction, the 9-fold-degenerate term splits into three closely spaced levels P2 with five states, P with three states, and Pq with one state. The total number of states is the same for the term P as for the three levels arising from this term. 

Now consider the consequence on these orbital types of the quantum number mi. For an s orbital, / = 0 and W/ can only equal 0. This means that for any value of n, there is only one s orbital it is said to be singly degenerate. For ap orbital, / = 1, and there are three possible W/ values + 1, 0, 1. This means that there are three p orbitals for a given value of n when n>2 the set of p orbitals is said to be triply or three-fold degenerate. For a d orbital, 1 = 2, and there are five possible values of mp +2, +1,0, — 1, 2, meaning that for a given value of n (n > 3),... 

A set of /-orbitals is 7-fold degenerate and there is more than one way to represent them. You will encotmter both the general and cubic sets of /-orbitals. The cubic set is commonly used and is readily related to tetrahedral, octahedral and cubic ligand fields. The cubic set comprises the/ 3, f f fxyi fz x — y ) f— x ) fx(z —y ) atomic orbitals. Figure 27.1 shows representations of the/ 3 and orbitals and indicates how the remaining five atomic orbitals are related to them. In Fig. 27.1b, each of the eight lobes of the fxyz orbital points towards one comer of a cube. Each / orbital contains three nodal planes. 

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