What are the shapes and designations of the f orbitals?

	Vocabulary
	amplitude angular momentum quantum number f orbital magnetic quantum number orbital principal quantum number rare earth metal spectrum subshell valence shell

The exotic, complex f orbital shapes are rarely shown in textbooks. General (and organic) chemistry traditionally focuses on the lighter elements, but the f orbitals aren't occupied in the ground state until element 58 (cerium). Even for elements beyond cerium, the f orbitals are deeply buried beneath the valence shell and they rarely play an important role in chemical change or bonding. However, the orbital shapes can be useful in interpreting spectra and in understanding the structure of some complexes that involve the rare earth elements. So here they are, if you need them.

The yellow and blue colors designate lobes with positive and negative amplitudes, respectively.

Hydrogen f orbital with n=4,l=3,m=-3 The 4f_{y³ - 3x²y} orbital corresponds to n=4, ell =3, and m=-3. Six lobes point to the corners of a regular hexagon in the xy plane, with one pair of lobes along the x-axis. Three nodal planes pass between the lobes and intersect at the z axis.

The 4f_xyz orbital corresponds to n=4, ell =3, and m=-2. Eight lobes point to the corners of a cube, with four lobes above and four lobes below the xy plane. The x and y axes pass through the centers of four of the cube's faces (between the lobes). The three nodal planes are defined by the x, y, and z axes.

Hydrogen f orbital with n=4,l=3,m=-1 The 4f_{5yz² - yr²} orbital corresponds to n=4, ell =3, and m=-1. Six lobes point to the corners of a regular hexagon in the yz plane, with one pair of lobes along the x-axis. The three nodal planes pass between the lobes and intersect at the y axis.

The 4f_{z³ - 3zr²} orbital corresponds to n=4, ell =3, and m=0. Two lobes point along the z-axis, with two bowl-shaped rings above and below the xy plane. The nodal surfaces are the xy plane and a conical surface passing through the nucleus and between the rings and the lobes.

Hydrogen f orbital with n=4,l=3,m=+1 The 4f_{5xz² - xr²} corresponds to n=4, ell =3, and m=+1. Six lobes point to the corners of a regular hexagon in the xz plane, with one pair of lobes along the y-axis. The three nodal planes pass between the lobes and intersect at the x axis.

The 4f_{zx² - zy²} orbital corresponds to n=4, ell =3, and m=+2. It has the same shape as the 4f_xyz orbital, but the corners of the cube are in the planes defined by the x, y, and z axes and the three nodal planes cut between the lobes and intersect along the z axis.

Hydrogen f orbital with n=4,l=3,m=+3 The 4f_{x³ - 3xy²} orbital corresponds to n=4, ell =3, and m=+3. It is identical to the orbital with m_ ell =-3 except that a lobe lies along the y axis instead of along the x axis.

Resources

Atomic and Molecular Orbitals (Craig Counterman, MIT)

A Java applet that displays scatter plots of atomic, hybridized, and molecular orbitals.
http://wulff.mit.edu/orbs/ (11/24/98)

Atomic Orbitals (R. Burk, Carleton College)
A collection of large Chime models of atomic orbitals, including the simplest s, p, d, and f orbitals. Includes a multiple-choice quiz on atomic orbitals. A Javascript-capable browser and the Chime plugin is required.
http://www.carleton.ca/~rburk/chem100/on_line_tuts/orbitals/atomic.htm (11/06/99)

Atomic orbitals (David William Manthey)
Beautiful computer-generated pictures of atomic orbitals, tabulated by quantum number. You can see the shapes of even exotic g, h, i, j, k, and l orbitals on the Grand Table which includes all atomic orbitals with principal quantum number up to and including 10. Radial nodes are shown using cutaway views for the s orbitals. The rendering software is available for downloading.
http://www.orbitals.com/orb/index.html (6/30/98,3/15/02)

General Chemistry (Ken Wilson, University of California at San Diego)
Slide shows and Quicktime movies for introductory chemistry, including basics of quantum mechanics, atomic orbitals, molecular orbitals, states of matter, equilibrium, and entropy. Be warned that the site uses very large images and animations.
http://www-wilson.ucsd.edu/education/gchem/gchem.html (10/11/98)

Mark's Quantum Mechanics Applets (Mark Sutherland, U. Toronto)
Six applets demonstrating quantum mechanical processes and principles, including two- and three-dimensional visualization of hydrogenic s, p, and d orbitals, the uncertainty principle, scattering from a 1D well, the harmonic oscillator, and the particle in an infinitely deep square well. Some extra features of the demo applets are disabled; fully functional versions are available directly from the author.
http://www3.adnc.com/~topquark/quantum/quantumapplets.html (3/11/99, revised 11/5/99)

Molecular Structure Calculations (Colby College)
This NSF-funded service allows students to compare their Lewis structure and VSEPR predictions with quality molecular orbital-level calculations. The server calculates molecular properties such as bond lengths, angles, atomic charges, dipole moment, bond orders, and molecular orbital energies for small molecules that users can enter using a very simple form. The output (usually) includes a 'best Lewis structure' computed from localized molecular orbitals, as well as a Chime molecular model. A library of over 550 previously computed molecules is available.
http://www.colby.edu/chemistry/webmo/mointro.html (7/16/99)

Author: Fred Senese senese@antoine.frostburg.edu

General Chemistry Online! What are the shapes and designations of the f orbitals?

	The 4f_{y³ - 3x²y} orbital corresponds to n=4, =3, and m=-3. Six lobes point to the corners of a regular hexagon in the xy plane, with one pair of lobes along the x-axis. Three nodal planes pass between the lobes and intersect at the z axis.
	The 4f_xyz orbital corresponds to n=4, =3, and m=-2. Eight lobes point to the corners of a cube, with four lobes above and four lobes below the xy plane. The x and y axes pass through the centers of four of the cube's faces (between the lobes). The three nodal planes are defined by the x, y, and z axes.
	The 4f_{5yz² - yr²} orbital corresponds to n=4, =3, and m=-1. Six lobes point to the corners of a regular hexagon in the yz plane, with one pair of lobes along the x-axis. The three nodal planes pass between the lobes and intersect at the y axis.
	The 4f_{z³ - 3zr²} orbital corresponds to n=4, =3, and m=0. Two lobes point along the z-axis, with two bowl-shaped rings above and below the xy plane. The nodal surfaces are the xy plane and a conical surface passing through the nucleus and between the rings and the lobes.
	The 4f_{5xz² - xr²} corresponds to n=4, =3, and m=+1. Six lobes point to the corners of a regular hexagon in the xz plane, with one pair of lobes along the y-axis. The three nodal planes pass between the lobes and intersect at the x axis.
	The 4f_{zx² - zy²} orbital corresponds to n=4, =3, and m=+2. It has the same shape as the 4f_xyz orbital, but the corners of the cube are in the planes defined by the x, y, and z axes and the three nodal planes cut between the lobes and intersect along the z axis.
	The 4f_{x³ - 3xy²} orbital corresponds to n=4, =3, and m=+3. It is identical to the orbital with m_=-3 except that a lobe lies along the y axis instead of along the x axis.