# 4*f* atomic orbitals

For any atom, there are seven 4*f* orbitals. The *f*-orbitals are unusual in that there are two sets of orbitals in common use. The first set is known as the *general set, this page*. The second set is the *cubic set, this page* and these might be appropriate to use if the atom is in a cubic environment, for instance. Three of the orbitals are common to both sets. These are are the 4*f*_{xyz}, 4*f*_{z3}, and 4*f*_{z(x2-y2)} orbitals.

The higher *f*-orbitals (
5*f*,
6*f*, and
7*f*) are more complex since they have one or more spherical nodes.

## 4*f* atomic orbitals general set

**The shape of the seven 4 f orbitals (general set).** From left to right: (top row) 4

*f*

_{z3}, (next to top row) 4

*f*

_{yz2}, 4

*f*

_{xz2}, (next to bottom row) 4

*f*

_{xyz}, and 4

*f*

_{z(x2-y2)}, (bottom row) 4

*f*

_{y(3x2-y2)}, 4

*f*

_{x(x2-3y2)}. For each, the green zones are where the wave functions have positive values and the white zones denote negative values.

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

The 4*f*_{z3} orbital (top row in the image above) has a planar node in the *xy* plane and two conical nodes with their exes along the *z*-axis.

The 4*f*_{yz2} and 4*f*_{xz2} 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 4*f*_{y(3x2-y2)} and 4*f*_{x(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".

The 4*f*_{xyz} and 4*f*_{z(x2-y2)} (next to 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 4*f*_{xyz} are the *xy*, *xz*, and *yz* planes.

The 4*f*_{y(3x2-y2)} and 4*f*_{x(x2-3y2)} orbitals (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.

## 4*f* atomic orbitals cubic set

**The shape of the seven 4 f orbitals (cubic set).** From left to right: (top row) 4

*f*

_{y3}, 4

*f*

_{z3}, 4

*f*

_{x3}, (middle row) 4

*f*

_{y(z2-x2)}, 4

*f*

_{z(x2-y2)}, and 4

*f*

_{x(z2-y2)}(bottom row) 4

*f*

_{xyz}. For each, the green zones are where the wave functions have positive values and the white zones denote negative values.

In the cubic set of 4*f* orbitals, there are two distinct shapes, each of which possess a number of planar and conical nodes. None of the 4*f* orbitals possess radial nodes.

The 4*f*_{xyz}, 4*f*_{x(z2-y2)}, 4*f*_{y(z2-x2)}, and 4*f*_{z(x2-y2)} (bottom two rows in the image above) each have eight lobes. The 4*f*_{x(z2-y2)}, 4*f*_{y(z2-x2)}, and 4*f*_{z(x2-y2)} orbitals are related to each other by 45° rotations about the *x*, *y*, and *z*-axis respectively. Each orbital has three nodal planes, which for the 4*f*_{xyz} are the *xy*, *xz*, and *yz* planes.

The 4*f*_{x3}, 4*f*_{y3}, and 4*f*_{z3} orbitals (top row in the image above) has a planar node in the *xy* plane and two conical nodes orientated along the *z*-axis. The other two orbitals are related through 90° rotations.

The Orbitron

^{TM}, a gallery of orbitals on the WWW: https://winter.group.shef.ac.uk/orbitron/

Copyright 2002-2023 Prof. Mark Winter [The University of Sheffield]. All rights reserved.