TCE

The Chemical Educator

ISSN: 1430-4171 (electronic version)

Table of Contents

Abstract Volume 28 (2023) pp 100-104

Understanding the Quantum Mechanical Variational Method through Coding

Amanda Morgenstern*^,†, Charles Morgenstern, Jessica Bergherm^‡, and Sally Meyer*^,‡

†Department of Chemistry and Biochemistry, University of Colorado Colorado Springs, 1420 Austin Bluffs Pkwy, Colorado Springs, CO 80918, amorgens@uccs.edu; ‡Department of Chemistry and Biochemistry, Colorado College, 14 E. Cache La Poudre,Colorado Springs, CO 80903, smeyer@coloradocollege.edu
Received August 9, 2022. Accepted April 26, 2023.

Published: 8 July 2023

Abstract. This article presents a project for physical chemistry or computational chemistry courses. There are four computational experiments within the project, all focused on applying the variational method to the simple molecular system, H . The molecular ion H has an exact quantum mechanical solution to the Schrödinger equation within the Born Oppenheimer approximation, which makes it an ideal system to study. This project has students write code to approximately solve the H system using the variational method and has students compare their results to the exact solution as well as results from Hartree-Fock calculations. Students learn many important concepts from classical and quantum mechanics in this project including 1) setting up the Schrödinger equation, 2) using the variational method, 3) applying the Born Oppenheimer approximation, 4) using atomic units, 5) applying a coordinate transformation, 6) selecting a basis set, and 7) creating a potential energy surface. These concepts are presented at a level that allows students with a variety of backgrounds to understand and carry out the experiments. Furthermore, the experiments are written to be (mostly) self-contained, such that instructors can choose to have students perform individual experiments or perform all four experiments followed by writing a full project report. Three of the experiments use Wolfram Mathematica software (or can be performed with similar open-source packages), and one experiment uses the free online WebMO interface to access Gaussian software. The more general learning objectives of these experiments involve 1) breaking a complex problem into smaller pieces, 2) using assumptions to simplify a complex problem, 3) testing a method on a problem whose exact solution is known, 4) writing and debugging code to solve a chemical problem, and 5) implementing and understanding numerical methods. This project has been tested at two different institutions, Colorado College and the University of Colorado Colorado Springs.

Key Words: Computers in Chemistry; physical chemistry; computational chemistry; quantum mechanics; variational method; coding; hydrogen molecular ion

(*) Corresponding author. (E-mail: smeyer@coloradocollege.edu)

Article in PDF format(170 KB) HTML format

Supporting Materials:

© The Chemical Educator 1996-2025