Introduction to Franck-Condon Factors©
Theresa Julia Zielinski
Department of Chemistry, Medical Technology, and Physics
Monmouth University
West Long Branch, NJ 07764-1898
United States
mail to: tzielins@monmouth.edu

George M. Shalhoub
Department of Chemistry
La Salle University
Philadelphia, PA 19141-1199
United States
mail to: shalhoub@lasalle.edu

Background Document
The goal of this document is to provide students with an introduction to Franck-Condon Factors and the relationship of these factors to vibronic spectroscopy. The document contains a very brief introduction to Franck-Condon factors through a sequence of guided inquiry type exercises. Specifically, students are asked to use potential energy diagrams for a diatomic molecule to examine a transition in the diatomic molecule from a ground electronic state to an excited electronic state including consideration of the vibrational levels of each state. The overlap of vibrational wave functions is used to introduce Franck-Condon factors. All of the exercises in this document are done with pencil and paper as preparation for more detailed work to be done in the companion computational document FrankCondonComputation, The Franck-Condon Factors.

Computation Document
The goal of this document is to introduce some of the mathematical models that are used to describe vibronic spectra (vibration-electronic spectra) of molecules. After completion of a study of the material presented in this document and the exercises distributed throughout the document, students are expected to be able to describe the relationship between Franck-Condon factors and vibronic spectra, explain the key factors determining the magnitude of the Franck-Condon factors, compute Franck-Condon factors, and construct a vibronic spectrum for a molecule from a knowledge of the wavelength of the electronic transition and the number and the relative intensity of the vibronic peaks. Along the way students are introduced to the idea of using a complete orthonormal set of functions, in this case the harmonic oscillator functions, as a linear combination to represent another function. The final focus of the document is the simulation of spectra in terms of the Franck-Condon factors. The document contains many student exercises and it is highly annotated for the novice Mathcad user. Faculty using this document can easily modify the document to increase student interaction with the mathematical content.

These documents build on the concepts learned in the "Exploring Orthonormal Functions." and "Introductory Explorations of the Fourier Series." documents that are found at this Web site. In this document the first seven harmonic oscillator states for the excited state of a molecule are used to write the linear combination expression for the ground state lowest vibration wave function. The overlap between the ground state function and the excited state is computed and leads directly to the Franck-Condon factors. The sum of the squares of the deviations is evaluated for different linear combinations and the best linear combination chosen by the student. The last activity is the simulation of a sample spectrum using the computed Franck-Condon factors. Students can see the effect of the full width at half maximum on the shape of the Gaussian functions that lead to the computed spectrum.
Editor's Commentary
Audiences: Upper-Division Undergraduate
Pedagogies: Computer-Based Learning
Domains: Physical Chemistry
Topics: Mathematics / Symbolic Mathematics, Quantum Chemistry, Spectroscopy, UV-Vis Spectroscopy
File NameDescriptionSoftware TypeSoftware Version
FranckCondonBackground.mcd JCE Mathcad Computational Document Mathcad
FranckCondonBackground.nb Mathematica Computational Document added August 2003 Mathematica
FranckCondonBackground.pdf Read Only Document for Mathcad
FranckCondonComputation.pdf Read Only Document for Mathcad
FranckCondonBackground[nb].pdf Read Only Document for Mathematica
FranckCondonComputation[nb].pdf Read Only Document for Mathematica
FranckCondonComputation.nb Mathematica Computational Document Mathematica
FranckCondonComputation.mcd JCE Mathcad Computational Document Mathcad
JCE JCE Subscribers only: name and password or institutional IP number access required.
Zielinski, T. J.; Shalhoub, G. M. J. Chem. Educ. 1998 75 1192.
Zielinski, T. J.; Shalhoub, G. M. J. Chem. Educ. 1998 75 1191.
Comments to: Theresa Julia Zielinski at tzielins@monmouth.edu and George M. Shalhoub at shalhoub@lasalle.edu.
©Copyright 1998 Journal of Chemical Education