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Cell[CellGroupData[{
Cell[TextData[{
StyleBox["Introduction to Franck-Condon Factors",
FontWeight->"Bold"],
" \[Copyright]"
}], "Title",
TextAlignment->Center],
Cell["\<\
Theresa Julia Zielinski
Monmouth University Department of Chemistry,
Medical Technology, and Physics
West Long Branch, NJ 07764
tzielins@monmouth.edu\
\>", "Text",
TextAlignment->Center],
Cell["and", "Text",
TextAlignment->Center],
Cell["\<\
George M. Shalhoub
La Salle University
Chemistry Department
Philadelphia, PA 19141
shalhoub@lasalle.edu\
\>", "Text",
TextAlignment->Center],
Cell["\<\
\[Copyright] Copyright 1998 by the Division of Chemical Education, Inc., \
American Chemical Society. All rights reserved. For classroom use by \
teachers, one copy per student in the class may be made free of charge. \
Write to JCE Online, jceonline@chem.wisc.edu, for permission to place a \
document, free of charge, on a class Intranet.\
\>", "Text"],
Cell[TextData[{
StyleBox["Translated from Mathcad to Mathematica by",
FontWeight->"Bold"],
": Laura Rachel Yindra, Journal of Chemical Education, University of \
Wisconsin-Madison, August 2003.\n"
}], "Text"],
Cell[CellGroupData[{
Cell["Introduction ", "Subtitle"],
Cell["\<\
This document and the accompanying file, FranckCondonComputation.nb, may be \
useful to students taking a junior or senior college level course in Quantum \
Chemistry or Spectroscopy. When using these materials you should first work \
through this study guide and then proceed to the accompanying computational \
file. This study guide starts by listing the prerequisite skills and \
knowledge level required for successful and efficient completion of the \
computational document which is designed to lead you to an understanding of \
the role of the Franck-Condon principle in spectroscopy. After reviewing \
harmonic oscillator concepts you proceed here to a qualitative examination \
of the Franck-Condon principle. In the accompanying computational document \
you will practice computing Franck-Condon factors and relating them to the \
electronic spectra. You will be able to see how the Franck-Condon factors \
and other experimentally observed properties contribute to the shape of a \
spectrum. The performance objectives given here identify the essential set \
of concepts you are to learn. A summative mastery type question may be added \
to the end of this document by your instructor. This summative exercise will \
pull together the various concepts you will be studying here.\
\>", "Text"]
}, Open ]],
Cell[CellGroupData[{
Cell["Goal", "Subtitle"],
Cell["\<\
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.\
\>", "Text"]
}, Open ]],
Cell[CellGroupData[{
Cell["Prerequisites", "Subtitle"],
Cell[TextData[{
"1. Moderate skill with ",
StyleBox["Mathematica",
FontSlant->"Italic"],
". \n2. Knowledge of what orthonormal functions are. \n3. Experience \
with harmonic oscillator wave functions. \n4. Some experience with \
Fourier Series expansions of functions."
}], "Text"]
}, Open ]],
Cell[CellGroupData[{
Cell["Performance Objectives", "Subtitle"],
Cell["\<\
At the end of both segments of this lesson you should be able to:
1. compute the Franck-Condon factors associated with a given electronic \
transition;
2. express a ground state vibration wave function as a linear combination \
of excited state vibration wave functions;
3. describe orally and in writing the relationship between Franck-Condon \
factors and intensities of observed electronic transitions;
4. draw diagrams and qualitatively estimate the magnitude of the \
Franck-Condon factors using harmonic oscillator wave functions;
5. predict the spectrum associated with a vibration given the energy of \
the electronic transition and the vibrational frequency of the bond.\
\>", "Text"]
}, Open ]],
Cell[CellGroupData[{
Cell["Tasks for achieving the objectives", "Subtitle"],
Cell["\<\
1. review the harmonic oscillator wave functions and their graphical \
representations;
2. complete the readings in your text assigned by your instructor;
3. complete the activities described in this document;
4. complete the activities described in the FranckCondonComputation.nb.\
\>", "Text"]
}, Open ]],
Cell[CellGroupData[{
Cell["Franck-Condon Principle", "Subtitle"],
Cell["\<\
When a molecule absorbs visible or ultraviolet light, a new excited-state \
electronic structure is substituted for the original electronic structure. \
This occurs on such a short time scale that there is no significant change \
in the position of the nuclei of the molecule. The transition between the \
two states is vertical. If we examine the vibrational wave functions for \
the ground and excited states, the overlap of the wave functions between \
these two states will determine the intensity of the transition. After an \
electronic transition the excited state then adjusts its equilibrium bond \
length. The concept of a vertical transition at constant internuclear \
distance is the Franck-Condon Principle. Mathematically the Franck-Condon \
Principle means that the transition dipole is independent of nuclear \
coordinates. The degree of overlap of the wave functions for the ground \
state and excited state of the oscillator give us the Franck-Condon Factors. \
The Franck-Condon Principle should not be confused with the Born-Oppenheimer \
(BO) Approximation although they seem similar. In the Born-Oppenheimer \
Approximation we consider nuclear motion occurring more slowly than \
electronic motion because the nuclei have more mass than the electrons. The \
BO approximation states that the rate of change of nuclear wave functions as \
the nuclear positions change is much smaller than the change in electronic \
wave functions. This permits solution of the Schrodinger equation for \
molecules using a fixed set of nuclear coordinates that define the potential \
field in which the electrons move.\
\>", "Text"],
Cell[TextData[{
StyleBox["Exercise 1",
FontWeight->"Bold",
FontVariations->{"Underline"->True}],
" ",
StyleBox[" At the bottom half of a clean piece of paper draw the \
potential energy as a function of bond length for the ground state of a bond \
in a molecule. Sketch in horizontal lines for a few vibrational energy \
levels. Sketch in the wave functions for the vibrational levels into your \
diagram.",
FontWeight->"Bold"]
}], "Text"],
Cell[TextData[{
StyleBox["Exercise 2",
FontWeight->"Bold",
FontVariations->{"Underline"->True}],
" ",
StyleBox[" On the top half of the same sheet draw the potential energy as \
a function of bond length for the first excited electronic state. For this \
exercise consider the case where the excited state bond length is shorter \
than that in the ground state. This is not an unrealistic situation \
because, for some molecules, excitation is from a nonbonding or antibonding \
orbital to another orbital further removed from the nuclei. Such an \
excitation would cause the bond length to shorten in the excited state. \
Consider also how the shape of the excited state curve will differ from that \
of the ground state. Next draw in the horizontal lines for the vibrational \
energy levels of the excited state. Finally, sketch in the wave functions \
for the vibrational levels in your diagram. ",
FontWeight->"Bold"],
" ",
StyleBox[" \n\nNote: The two diagrams should share a common PE axis (y) and \
bond extension axis (x). \n\nHow does the dissociation energy of the \
excited state compare to that of the ground state? \n\nWhere would the \
excited state potential energy diagram lie with respect to the ground state \
potential energy diagram if the transition was from a bonding to an \
antibonding orbital?",
FontWeight->"Bold"]
}], "Text"],
Cell["\<\
Before an electronic transition occurs, most molecules will be in the lowest \
vibrational level (v = 0) of the electronic ground state. Absorption of \
energy during an electronic transition places the molecule into one of the \
vibrational levels of a higher electronic state. Such a transition most \
likely would occur from the most probable (equilibrium) bond length of the \
ground state vibrational energy level, vertically to various vibrational \
energy levels of the excited state.\
\>", "Text"],
Cell[TextData[{
"According to the Franck-Condon Principle, the nuclear coordinates change \
more slowly than the electronic coordinates so that a transition is \
vertical on an energy vs. bond length diagram. ",
StyleBox["Draw such a transition on your diagram.",
FontWeight->"Bold"],
" The vertical transition that is most probable is the one that has an \
excited state vibrational wave function maximum at the same place as the \
ground state wave function maximum. In other words, the probability of a \
transition is measured by the overlap between the vibrational wave functions \
of the ground state and excited states. The greater the overlap the greater \
the probability of the transition. "
}], "Text"],
Cell["\<\
To evaluate these probabilities one merely needs to compute the overlap \
between the vibrational wave functions of the electronic ground state and \
the vibrational wave functions for the excited state. Many of the overlaps \
between the ground state vibrational wave function and the excited state \
vibrational wave functions can be computed. The one with the largest overlap \
is associated with the most intense peak in the spectrum. Other overlaps are \
associated with other less intense peaks in the spectrum.\
\>", "Text"],
Cell[TextData[{
"In order to compute the overlap between the vibrational wave function of \
the ground state and the vibrational wave functions of the excited state, \
the equilibrium extension of the excited state relative to the equilibrium \
extension of the ground state is needed. Usually this is not known for new \
materials. One must make a reasonable guess for the displacement of the \
excited state relative to that of the ground state. This is 'a' in the \
FranckCondonComputation.nb document. To estimate 'a' one must decide if the \
bond length in the excited state is longer or shorter than the length in the \
ground state. Then the overlaps, i.e. computed Franck-Condon factors, are \
tabulated, and compared to the relative intensities of the experimental \
peaks. By varying 'a' a good match between experimental peak intensities and \
Franck-Condon factors would provide the researcher with an estimate of ",
Cell[BoxData[
\(TraditionalForm\`r\_e\)]],
" for the excited state. \n\nFor some molecules, excitation from the \
HOMO to the LUMO will shorten the bond length of the excited state. This \
occurs when the HOMO is nonbonding or slightly antibonding and the LUMO is \
either more diffuse or farther away from the nucleus and thus has less \
impact on the bond length. For other molecules excitation from the HOMO to \
the LUMO will result in a longer bond length for the excited state. This \
occurs in situations where the HOMO is bonding and the LUMO antibonding."
}], "Text"],
Cell["\<\
Which of the vibrational levels of the excited state will have the highest \
probability of being achieved? Which will come next? The answers depend on \
1) the extension of the molecule at the instant it experiences an electronic \
transition and 2) the excited state vibrational wave functions at that \
extension relative to the initial state vibrational wave function at the \
instant of the transition. In general we consider the ground state to be at \
its equilibrium bond length.
The intensities of vibrational transitions are governed by the value of the \
overlap integral connecting the two vibrational levels. This is shown in \
equation (1) where the vp and v subscripts identify the ground state and \
excited state wave functions respectively.\
\>", "Text"],
Cell[BoxData[
\(\[Integral]\(\([\[Phi]\_vp]\)\^*\)\ \[Phi]\_v\ \[DifferentialD]\[Tau] =
c\_\(vp, v\)\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \((1)\)\)], "Text",
NumberMarks->False],
Cell["\<\
The integral in equation (1) is called the vibrational overlap integral and \
the integration is over all nuclear coordinates. The square of the overlap \
value is called the Franck-Condon factor. The larger the factor the larger \
the peak observed in the UV-vis spectrum.\
\>", "Text"],
Cell[BoxData[
\(Abs[c\_\(vp, v\)]\^2\)], "Text",
NumberMarks->False],
Cell["\<\
Franck-Condon factors allow one to estimate the relative intensities of \
vibrational transitions.
The sum of Franck-Condon factors for transitions from a given vibrational \
state is equal to 1.0. Verification of this statement is left to the student \
as a master level exercise.
You will compute Franck-Condon factors in the FranckCondonComputation.nb \
document.\
\>", "Text"],
Cell["\<\
Note: This is a good time to review the harmonic oscillator wave functions. \
Where are the maxima as a function of bond extension? What are the \
implications for the atomic positions of the bonded atoms as the vibration \
quantum number increases?\
\>", "Text"],
Cell["\<\
Note: Electronic transitions occur from a vibrational state of the ground \
electronic state to a vibrational state of the excited electronic state. \
There is no strict selection rule on vibrational quantum numbers during an \
electronic transition. There is only Franck-Condon modulation of the \
transition probability.\
\>", "Text"],
Cell[TextData[{
" ",
StyleBox["Exercise 3 \na. From what internuclear distance of the molecule \
in the ground state is the transition to the excited state likely to occur? \
",
FontWeight->"Bold"],
" \n",
StyleBox["b. What excited state vibrational state is most likely to be at \
the end of the transition? Why? How would you quantify your answer? ",
FontWeight->"Bold"],
" \n",
StyleBox["c. Predict the relative intensities of various electronic \
transitions (from a vibrational level in the ground electronic state to a \
vibrational level in the excited electronic state) if ",
FontWeight->"Bold"],
Cell[BoxData[
\(TraditionalForm\`\(\(\ \)\(r\_e\)\)\)]],
" ",
StyleBox[" were the same for each state and sketch them using a \
qualitative bar graph.",
FontWeight->"Bold"],
" ",
StyleBox[" ",
FontWeight->"Bold"]
}], "Text"],
Cell["\<\
Now proceed to the FranckCondonComputation.nb document where you will be \
asked to compute the overlap between a vibrational level in the ground state \
and a vibrational level in the excited state. In the computational document \
you will also be able to check the way the Franck-Condon factors along with \
the full width at half maximum (FWHM) property of a Gaussian function \
determine the shape of a spectrum consisting of several vibronic \
transitions.\
\>", "Text"]
}, Open ]],
Cell[CellGroupData[{
Cell["Mastery Exercise", "Subtitle"],
Cell["\<\
To be provided by the instructor.
You can demonstrate mastery of the concepts by correctly determining the \
relative intensity order of peaks in a given simple experimental electronic \
spectrum. You should also be able to describe your work accurately in writing \
and/or orally.\
\>", "Text"]
}, Open ]],
Cell[CellGroupData[{
Cell["Acknowledgment:", "Subtitle"],
Cell["\<\
TJZ thanks John Wright of the Chemistry Department of the University of \
Wisconsin - Madison for discussions that initiated the development of this \
document and the accompanying FranckCondonComputation.mcd document. TJZ \
also acknowledges that partial support was provided by the National Science \
Foundation's Division of Undergraduate Education through grant DUE #9354473. \
Additional partial support was provided by the New Traditions project at the \
University of Wisconsin - Madison through the National Science \
Foundation's Division of Undergraduate Education grant DUE #9455928.\
\>", "Text"]
}, Open ]]
}, Open ]]
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