Visualizing Atomic Orbitals

Atomic orbitals are the wavefunctions which are solutions of the Schrödinger equation for the hydrogen atom. The subset of atomic orbitals 1s, 2p, 3d and 4d are plotted in three dimensions to exhibit their characteristic shapes.

The orbitals are drawn by showing their boundary surfaces. In the second view + and - signs are attached to the relevant lobes of the orbitals and colorized accordingly.

This Demonstration shows the basic characteristics for a chosen set of 16 atomic orbitals: the type, the absolute value of quantum number m, the number of lobes/nodes, the Cartesian polynomial form of the wavefunctions, and two 3D views of the probability density (boundary surface: with or without phases). Axes and labels can be displayed as an option via a checkbox.

THINGS TO TRY

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DETAILS

In chemistry orbitals can be classified according to their orientation in a rectangular coordinate system. The set of shapes in the snapshots is given for m=0 and for combinations of m=±1,±2,±3.

The three p-orbitals for a given value of n are described by the values m=0,±1; m=0 gives the (Subscript[p, z]) orbital. The angular functions for m≠0 are complex and depend on ϕ, θ, or both. Pairwise linear combinations of complex spherical harmonics (Subscript[Y, l]^m)(θ,ϕ) yield real functions, which can be plotted as boundary surfaces.

For (Subscript[p, x]) and (Subscript[p, y]), for example, we have

(1/(Sqrt[2]))((Subscript[Y, 1]^1)(θ,ϕ)+(Subscript[Y, 1]^(-1))(θ,ϕ)) and (1/((Sqrt[2])i))((Subscript[Y, 1]^1)(θ,ϕ)-(Subscript[Y, 1]^(-1))(θ,ϕ)).

The function pos inside OrbitalModel is shown at the link Problem with SphericalPlot3D plotting. It is used to attach signs to the positive or negative parts of the radial wavefunction. Then both parts are colored differently.

Alternative representations for the seven 4f orbitals can be written. In this Demonstration, the most commonly used convention was chosen [3].

References

[1] P. Atkins, R. Friedman, Molecular Quantum Mechanics, Oxford: Oxford University Press, 2011.

[2] R. King, "Atomic orbitals, symmetry, and coordination polyhedra," Coordination Chemistry Reviews, 197, 2000 pp. 141–168.

[3] M. Winter. "The Orbitron: a gallery of atomic orbitals and molecular orbitals on the WWW." (Jan 2013) http://winter.group.shef.ac.uk/orbitron/AOs/4f/index-gen.html.

Guenther Gsaller

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