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Atomic orbitals: 4f (general set)

For any atom, there are seven 4f orbitals. The f-orbitals are unusual in that there are two sets of orbitals in common use. Those shown here are the general set. The other set is known as the cubic set and might be more appropriate to use if the atom is in a cubic environment. Three of the orbitals are common to both sets. These are are the 4fxyz, 4fz3, and 4fz(x2-y2) orbitals.

Follow the links towards the base of the page for information about individual orbitals.

The shape of the seven 4f orbitals (general set). From left to right: (top row) 4fz3, (next to top row) 4fxz2, 4fyz2, (next to bottom row) 4fy(3x2-y2), 4fx(x2-3y2), (bottom row) 4fxyz, and 4fz(x2-y2). For each, the copper zones are where the wave functions have negative values and the gold zones denote positive values.

In the general set of 4f orbitals, there are four distinct shapes, each of which possess a number of planar and conical nodes. The 4f orbitals do not possess any radial nodes.

The 4fxyz and 4fz(x2-y2) (bottom row in the image above) each have eight lobes and are related to each other by a 45° rotation about the z-axis. Each orbital has three nodal planes, which for the 4fxyz are the xy, xz, and yz planes.

The 4fy(3x2-y2) and 4fx(x2-3y2) orbitals (next to bottom row in the image above) are related to each other by a 90° rotation about the z-axis. Each orbital has six lobes separated by three nodal planes lying at 60° to each other.

The 4fxz2 and 4fyz2 orbitals (next to top row in the image above) are related to each other by a 90° rotation about the z-axis. At first sight, they are similar in shape to the 4fy(3x2-y2) and 4fx(x2-3y2) orbitals but they are not. While these orbitals contain six lobes, the nodal planes are not at 60° to each other and two of the six lobes are "bean-shaped". Look closely! In the case of the 4fxz2 orbital these two bean-shaped lobes lie on the x-axis while in the 4fyz2 case they lie along the y-axis. These differences may be appreciated better by examining the electron density plots for the 4fxz2 and 4fx(x2-3y2) orbitals.

The 4fz3 orbital (top row in the image above) has a planar node in the xy plane and two conical nodes orientated along the z-axis.

Links to 4f wave functions

Links to 4f electron density functions

Links to 4f electron "dot-density" functions

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Document served: Tuesday 17th July, 2018