Visualization of Wavefunctions of the Ionized Hydrogen Molecule©
John L. Johnson
Department of Engineering Science and Mechanics
The Pennsylvania State University
University Park, PA 16802
United States

Using Mathcad and the Shooting Method, students can numerically solve the Schrodinger equation for the ionized hydrogen molecule. The solutions to the angular portions of the wavefunctions are the spheroidal harmonics, which students can plot to visualize the shapes of molecular orbitals. These exact solutions are compared with those obtained from the linear combination of atomic orbital (LCAO) method, using hydrogenic orbitals as basis functions. From the products of the angular and radial wavefunctions, students can also plot the electronic charge densities to see the concentration of charge that distinguishes bonding and anti-bonding states. The bonding properties are confirmed by plotting the energies of the wavefunctions as a function of internuclear distance for the eight lowest molecular energy states.

Mathcad plot of the 3 sigma g orbital for the ionized hydrogen molecule as given by the spheroidal harmonic with a degree of 2 and order 0.
Editor's Commentary
Audiences: Upper-Division Undergraduate, Graduate Education / Research
Pedagogies: Computer-Based Learning
Domains: Physical Chemistry
Topics: Mathematics / Symbolic Mathematics, MO Theory, Quantum Chemistry
File NameDescriptionSoftware TypeSoftware Version
IonizedHydrogen.pdf Read-Only Document
IonizedHydrogen.zip JCE ZIP archive contains: HMI.mcd, Oblate spheroidal coordinates.mcd, Prolate spheroidal coordinates.mcd, Spherical coordinates.mcd. Fully interactive with Mathcad 11 or higher.
JCE JCE Subscribers only: name and password or institutional IP number access required.
Johnson, John L. J. Chem. Educ. 2004, 81, 1535.
Comments to: jlj120@psu.edu
©Copyright 2003 Journal of Chemical Education