THE SHAPES OF f ORBITALS

When the wave equation for a hydrogen-like atom is solved in the most direct way for orbitals with the angular momentum quantum number / = 3, the following results are obtained for the purely angular parts (i.e., omitting all numerical factors)  

The seven functions are grouped into sets having projections of the orbital angular momentum on the z axis of 0, 1, 2, and 3. Each of the functions in the pairs with m equal to 1, 2, and 3 is complex as written above, but by taking linear combinations of each pair, for example, 

True Polynomial Simplified Polynomial Normalizing Factor Angular Function 

A set of functions directly useful in problems with cubic symmetry can be obtained by taking the following linear combinations of those in Table AIV. I. 

O E R 4C3 4C32R 4C33 4C3R 3C2 3 C2R 3 C4 3C43R 3C43 3C4fl 6C2 6C 2R 

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Because of their importance to us, the object of the present section is to give the reader more familiarity with the shapes of f orbitals than that provided in Section 7.3. In that section the f orbitals were used to indicate the way that F terms (arising from d" configurations, n = 2, 3, 7 or 8) split in an octahedral ligand field. The labels that were used to describe these f orbitals were abbreviated, just as the label is an abbreviation for d2z2 2 y2. The complete labels for the f orbitals accurately describe the lobes of the orbitals, their relative phases and their positions. Drawings of the cubic set of f orbitals are repeated in Fig. 11.2, together with their abbreviated and... 

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