Franck-Condon Factors and Their Use in Undergraduate Quantum

In the Classroom

Franck–Condon Factors and Their Use in Undergraduate Quantum Mechanics John C. Wright Department of Chemistry, University of Wisconsin, Madison, WI 53706; [email protected] Theresa Julia Zielinski Department of Chemistry, Medical Technology & Physics, Monmouth University, West Long Branch, NJ 07764

Curriculum reform efforts to expand active-learning methods into upper-level chemistry courses require student exercises and projects that provide an opportunity for students to independently discover the fundamental principles of a subject, obtain deeper insights into how those principles are applied to chemical problems, and experience some of the problems that are representative of the profession. In undergraduate physical chemistry courses in quantum mechanics, the introduction of powerful mathematical software such as Mathcad (1) has allowed undergraduates to avoid much of the mathematical complexity that obscures a fundamental understanding of principles, letting them explore the ways that quantum mechanics describes and interprets interesting chemical problems. This capability has provided new opportunities for teaching undergraduate quantum mechanics (2, 3). For example, it is common to illustrate the expansion of a wave function as a linear combination of eigenfunctions in a complete orthonormal basis set. The sine and cosine eigenfunctions of the particle in a box or the translational wave functions are often used as illustrations of this principle, since the mathematics is accessible at the undergraduate level. In this paper, we describe how symbolic mathematical software makes it possible for undergraduates to expand a wave function in terms of the simple harmonic oscillator wave functions and to explore the relationship between this expansion and the Franck–Condon factors that describe transitions between two electronic states. It also allows students to simulate molecular spectra so they can determine excitedstate geometries. We describe how the implementation of these ideas can serve to introduce active-learning methods into the physical chemistry quantum mechanics course. Examples are provided to illustrate how students can analyze experimental absorption spectra to discover the changes that occur when a molecule is excited to an excited electronic state. A fundamental principle in quantum mechanics is that a wave function, ψ, can be expressed as a linear combination of eigenstates, ψi: ∞

ψ = Σ c i ψi i=1

(1)

where ∞

ci =

{∞

ψψi*dτ

(2)

The expansion is usually illustrated in undergraduate quantum mechanics texts using Fourier series for the basis functions (4 ), but it is also possible to use the wave functions for the simple harmonic oscillator (SHO) because they too form a complete orthonormal basis set. Furthermore, the SHO wave functions have a direct application to molecular spectroscopy

Figure 1. Potential energy for the ground and excited electronic states as a function of an intermolecular coordinate using both a Morse potential and a harmonic potential. Example vibrational wave functions are superimposed on the vibrational energy levels and arrows are drawn to represent absorption emission events. The expanded inset shows the ground state v ′′ = 0 and v ′′ = 1 wave functions and the arrows that represent transitions from the coordinate displacement with maximum probability. Note that there are two positions where the v ′′ = 1 wave function reaches a maximum.

that fosters student learning by showing its significance and practicality. Figure 1 shows the potential energy of the I 2 ground and excited electronic states as a function of a nuclear coordinate displacement from equilibrium (5–8). Both a Morse potential and a simple harmonic potential are shown. The Morse potential uses published parameter values (9). The equilibrium bond distance of the excited electronic state is increased and the vibrational energy is lowered because the bond energy of the excited state is lower than that of the ground state. The excited state displacement is 0.36 Å relative to the ground state. Figure 1 also shows examples of absorption transitions from the ground state v ′′ = 0 and v′′ = 1 vibrational states to the excited electronic state. The expanded view in the inset shows how the transitions originate from the most probable part of the vibrational states. In accordance with the Franck–Condon principle, the transitions are vertical because their time scale is too rapid to allow a change in the molecular geometry. The optical cross section, σ, for an absorption transition can be written as

σv′v′′ =

4πf v′′ ωµ2 3hc

∞ {∞

ψv′ ψv′′*dτ

2

Γ 2 ωo – ω + Γ 2

(3)

where v ′′ and v ′ designate the vibrational state of the ground

JChemEd.chem.wisc.edu • Vol. 76 No. 10 October 1999 • Journal of Chemical Education

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In the Classroom

and excited electronic states, respectively, fv ′′ is the fraction of molecules in the ground state, ω is the incident light frequency, ωo is the transition frequency, µ is the electronic transition moment, c is the speed of light, Γ is the transition line width, and ψv is the vibrational wave function for vibrational state v (10). If there are multiple transitions, the crosssections for each are additive. The last factor describes the Lorentzian line shape of an absorption transition. For this work, it is important to recognize the middle factor as the integral that describes the overlap between the two vibrational states—that is, the probability that the molecular states have common coordinate values. The square of the integral is the Franck–Condon factor. Note also that the overlap integral is the same factor that arises when we expand the excited electronic state’s vibrational state, ψv ′, in terms of the ground state’s vibrational states, ψv ′′ (or vice versa). Then

ψv′ =

∞

Σ c v′′v′ψv′′ v′′=1

(4)

where c v′′v′ ≡ v′|v′′ ≡

∞ {∞

ψv′ ψv′′*dτ

The dependence on the overlap arises because the probability of the vertical transition depends on the molecule’s probability of having the molecular coordinate displacements characterized by the two vibrational wave functions.

It is well known that the simple harmonic-oscillator wave function is described by 2

ψn ξ = Nn H n ξ e {ξ /2

(5)

where

Nv =

1 v 2 v! 4 π

(6)

Hn(ξ) are the Hermite polynomials defined by the generating function n

2 n 2 H n ξ = {1 e ξ d n e { ξ dξ

(7)

– Here, ξ is a normalized displacement defined by ξ = √ α x, ⁄ µ is the reduced mass, ω is the vibrational frequency, α = µωo/h, o and x is the displacement (4 ). For I 2, αg is 4.04 × 1022 and αe is 2.36 × 1022 m{2 for the ground and excited states, respectively (8). The first 7 wave functions on the left of Figure 2 and representative Mathcad derivations are shown in Table 1. They represent the vibrational wave functions, ωv, for vibrational quantum numbers of v = 0 through 6. Absorption spectra are formed from transitions between the ground state and the different vibrational levels of the excited electronic state, whereas fluorescence spectra are formed from transitions between the excited electronic state and the different vibrational levels of the ground state. The ground and excited state wave functions can be expressed as

ψg

αg x = Ng Hg

αg x e {

αg x–x og

αe x = Ne He

αe x e {

αe x–x oe

2

/2

and

ψe

2

/2

where xgo and xeo are the ground- and excited-state equilibrium displacements. One can also use normalized displacements to––describe the transition using either––the ground-state, ξ g = √ αg (x – xgo), or excited-state, ξ e = √ αe (x – xeo), normalized displacement. It is convenient to use ξ g for fluorescence transitions from an excited state to the different ground state v′′ levels and ξ e for absorption transitions from the ground state to the different excited state v′ levels. If we define R ≡ ωg /ωe (or equivalently R = αg /αe), then the ground and excited state wave functions can be written as 2

ψ v ′′(ξ g) = Nv ′′H v ′′(ξ g)e { ξ g /2

and

–

–

ψ v ′[(ξ g – ∆g)/√R ] = Nv ′H v ′[(ξ g – ∆g)/√R ]e { (ξ g –∆ g) /2R 2

using ξ g coordinates, and as – – ψ v ′′[√R (ξ e – ∆e)] = Nv ′′H v′′[√R (ξ e – ∆e)]e { R(ξ e +∆ e) /2 2

Figure 2. The left-hand column shows the wave functions for the first 7 vibrational states. The top right-hand graph shows the excited electronic state vibrational wave function directly after it was created by excitation from the ground state v ′′ = 0 mode. The ordinate axis is the excited electronic state’s normalized coordinate displacement, ξ e. The wave functions below it show the linear combination with a limited number of ground-state wave functions. The standard deviation for the fit improves with increasing numbers of basis functions and has values of 0.20, 0.13, 0.074, 0.038, 0.017, 0.0073, and 0.00028, respectively with 1, 2, …, 7 basis functions.

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and

2

ψ v ′(ξ e) = Nv ′H v ′(ξ e)e { ξ e /2

using ξe coordinates. Here, the –– relative displacement of the ground and––excited state is ∆g ≡ √ αg (xeo – xgo) for ξg coordinates – and ∆e ≡ √ αe (xeo – xgo) for ξ e coordinates. Note that ∆e = ∆g/√R. For I2, ∆g= 7.18, ∆e= 5.49, and R = 1.71 (5–8). We will use the ξg coordinates for fluorescence spectra and ξe coordinates for absorption spectra.

Journal of Chemical Education • Vol. 76 No. 10 October 1999 • JChemEd.chem.wisc.edu

In the Classroom

Table 1. Representative Mathcad Derivations for First 7 Wave Functions

n : = 0..40 N n :=

1 n 2 ⋅ n! 4 π n

2 2 n x2 d ⋅ { 1 ⋅ e x ⋅ n e {x 2 dx {1 2 ψ x → 2 ⋅ N 1 ⋅ exp ⋅ x ⋅ exp x 2 ⋅ x ⋅ exp { x 2 2

n := 1

ψ x := N n ⋅ exp {

n := 2

ψ x := N n ⋅ exp {

n := 3

ψ x := N n ⋅ exp {

2 2 n x2 d ⋅ { 1 ⋅ e x ⋅ n e {x 2 dx

ψ x → { N 3 ⋅ exp

{1 2 2 ⋅ x ⋅ exp x 2 ⋅ 12⋅ x ⋅ exp {x – 8 ⋅ x 3 ⋅ exp { x 2 2

n

2 2 x2 n d ⋅ { 1 ⋅ e x ⋅ n e {x 2 dx 2 {1 2 2 2 ψ x → N2 ⋅ exp ⋅ x ⋅ exp x 2 ⋅ { 2 ⋅ exp {x + 4 ⋅ x ⋅ exp { x 2

n

n

2 2 n x2 d ⋅ { 1 ⋅ e x ⋅ n e {x 2 dx {1 2 2 2 2 ψ x → N 4 ⋅ exp ⋅ x ⋅ exp x ⋅ 12⋅ exp {x – 48⋅ x 2 ⋅ exp { x 2 + 16⋅ x 4 ⋅ exp {x 2

n := 4

ψ x := N n ⋅ exp {

n := 5

ψ x := N n ⋅ exp {

n := 6

ψ x := N n ⋅ exp {

n

2 2 n x2 d ⋅ { 1 ⋅ e x ⋅ n e {x 2 dx {1 2 2 2 2 ψ x → { N 5 ⋅ exp ⋅ x ⋅ exp x ⋅ {120⋅ x ⋅ exp {x + 160⋅ x 3 ⋅ exp { x 2 – 32⋅ x 5 ⋅ exp {x 2

ψ x →N 6 ⋅ exp

n

2 2 n x2 d ⋅ { 1 ⋅ e x ⋅ n e {x 2 dx

{1 2 ⋅x ⋅exp x 2 ⋅ {120⋅exp {x 2 +720⋅x 2⋅exp { x 2 –480⋅x 4⋅exp {x 2 +64⋅x 6⋅exp {x 2 2

Consider an absorption spectrum originating from the ground electronic state that has a displacement of ∆e relative to the excited state. For the ground state vibrational wave function written as a linear combination of excited-state wave functions

ψv′′

R ξe + ∆e =

∞

Σ c v′′v′ ψv′ ξe

v′=1

(8)

symbolic mathematics software can be used to determine the expansion coefficients and the Franck–Condon factors: ∞

c v′′v′ =

{∞

ψv′ ψv′′ dτ

(9)

The box on page 1370 shows example calculations of cv ′′v ′ using Mathcad. An absorption transition results in the vertical transition (see Fig. 1) that promotes the groundstate wave function to the excited state, so the excited-state wave function, ψe, is identical to the ground-state wave function immediately after the excitation. It is interesting to see how well ψ e can be fit with a limited number of excited state basis functions. The left side of Figure 2 shows the basis functions, ψ v ′, for the first 7 excited-state vibrational wave functions. The upper-right wave function shows the displaced state of the initial excited electronic state, ψe , with ∆g= 1.4 in normalized coordinates. The remaining wave functions on the right show the expansion using different numbers of basis states. It can be seen that 7

terms allow a very acceptable approximation for the initial excited state. In this example, the linear combination is given: ψe ξe + 1.4 = 0.613ψ0 ξe – 0.606ψ1 ξe + 0.425ψ2 ξe – 0.243ψ3 ξe + 0.12ψ4 ξe – 0.053ψ5 ξe + 0.021ψ6 ξe

(10)

where the left side specifies the initially excited wave function and the right side shows the linear combination of excitedstate vibrational basis functions that describe it. Since the excited wave function in eq 10 is a linear combination of stationary states, it will exhibit a time dependence after an initial excitation that is interesting for students to explore. The molecular coordinates do not change appreciably during the short time of the transition to the excited electronic state, and the molecular displacement after the transition is different from the equilibrium displacement in the excited state. Classically one expects that the molecule will oscillate about the equilibrium position after the transition occurs. The time dependence of the wave function, ψ e , that results after the emission is given by

ψe =

∞

Σ c v′′v′ ψv′ e { iE t v′=1 v′

(11)

Equation 11 explicitly contains the time dependence represented by eq 10. The probability, ψ e* ψ e , has time-dependent cross-terms that are responsible for the oscillation around the equilibrium position, and symbolic mathematical software


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In the Classroom

Mathcad Example Calculation of the Franck–Condon Factors for Absorption: Projection of a Ground State Vibrational Wave Function on an Excited Electronic State Surface Constants:

R := 1R := 1 1.4 a := R n := 0..6

N n :=

1

n

2 ⋅ n! ⋅ π 4

Ground State Vibrational Function to be expanded in excited state coordinates:

f x :=

4

2

R ⋅ exp { x + a ⋅ R π 2

Basis Functions

ψ0 x := N 0 exp {

x2 2

ψ1 x := N 1 ⋅ 2 ⋅ x ⋅ exp {

such as Mathcad can animate it. The animation of ψ e* ψ e provides students with an example of a time-dependent wave function. The students personally see the position of the spatial probability maximum oscillate between the extreme displacements without changes in its shape and they realize that the behavior matches their expectations for a classical simple harmonic oscillator. The above discussion focused on an absorption transition, but the same approach can describe an emission. In this case, the ground electronic state’s vibrational wave function is expanded in terms of the excited state vibrational wave functions. The end result is identical to the case of absorption, so there is mirror symmetry in the relative intensities of transitions to the different vibrational states in absorption and emission spectra that is often seen in comparing absorption and fluorescence spectra. It is possible to obtain closed-form expressions for the Franck–Condon factors (10). The expressions for the Franck– Condon factors are 2

2

v′|0 =

x2 2

ψ2 x := N 2 ⋅ 4 ⋅ x 2 – 2 ⋅ exp {

x2 2

x2 ψ3 x := N 3 ⋅ 8 ⋅ x – 12 ⋅ x ⋅ exp { 2

2

v′|1 = e

{∆ g2/2

3

ψ4 x := N 4 ⋅ 16 ⋅ x 4 – 48 ⋅ x 2 + 12 ⋅ exp {

v′|v′′ =

3

ψ6 x := N 6 ⋅ 64 ⋅ x 6 – 480 ⋅ x 4 + 720 ⋅ x 2 – 120 ⋅ exp {

min v′v′′ 2

x 2

The expansion coefficients are: 20

c 0 :=

{ 20

20

c 1 :=

{ 20 20

c 2 :=

{ 20

20

c 3 :=

{ 20 20

c 4 :=

{ 20

20

c 5 :=

{ 20 20

c 6 :=

{ 20

v′–1 ∆ g2/2

v′

(12)

v′! ∆ g2 v′ 1– v′ – 1 ! 2v′

2

(13)

and

x2 2

x2 ψ5 x := N 5 ⋅ 32 ⋅ x – 160 ⋅ x + 120 ⋅ x ⋅ exp { 2 5

e {∆g /2 ∆ g2/2

f x ⋅ ψ0 x dx

f x ⋅ ψ1 x dx

Σ

k=0

1 × 0|0 1 k!

(14)

v′! v′′! v′ – k| 0 0|v′′ – k v′ – k ! v′′ – k !

The derivation of these relationships is typically not part of undergraduate courses and is beyond the scope of this paper. Note that the values do not depend on the sign of the excited state displacement, ∆g. Figure 3 shows the Franck– Condon factors for transitions to the first 7 excited-state vi-

f x ⋅ ψ2 x dx

f x ⋅ ψ3 x dx f x ⋅ ψ4 x dx

f x ⋅ ψ5 x dx f x ⋅ ψ6 x dx

Note that the finite integration limits approximate infinite limits.

i := 0..6 Values for the expansion coefficients, c i , are:

i 0 1 2 3 4 5 6

ci 0.613 {0.606 0.425 {0.243 0.12 {0.053 0.021

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Figure 3. The Franck–Condon factors are plotted for transitions from a v ′′ = 0 state of the ground electronic state to different v ′ states of the excited electronic state as a function of the excited state displacement. The values for emission transitions from the v ′ = 0 excited state to the different v ′′ levels of the ground state are identical.


In the Classroom

brational states from the v′′ = 0 ground state. Students can easily generate these curves from the symbolic mathematical software and extract the following observations. First, for small excited-state equilibrium displacements (∆g < 1.4), the transitions with ∆v = v ′ – v′′ = 0 dominate and transitions to higher vibrational states are much weaker. Second, for large equilibrium displacements, the ∆v = 0 transitions are very weak and the maximum intensities occur for large values of ∆v. Third, students should be able to discuss the transitions in terms of the overlap of excited-state and ground-state wave functions and how these overlaps vary as the excited-state displacement changes. Additional reinforcement of this concept can be obtained by directing students to construct Morse potential functions as shown in Figure 1 and slide the upper curve to the left or the right to see the consequences of changing position on the overlap probabilities (9). These behaviors become even clearer if students also simulate spectra. Assuming Lorentzian line shapes and an absorption spectrum from the ground state, the intensity in a spectrum is given by ∞ c 0v′ I v′′=0 → v′ = A Σ π v′=0

Γv′ 2

ν – νv′ + Γv′2

(15)

where A is a proportionality constant that allows fitting the intensity to experimental data, c is the Franck–Condon factor, ν is the spectral frequency, and νv ′ and Γv′ are the frequency and line width (half-width at half-maximum) of the excited v ′ vibrational state. This function is graphed in Figure 4 for three displacements. With symbolic equation software, students can generate the three curves for themselves. This has several important pedagogical advantages. First, time on task is increased, which can improve learning if the learning environment is supportive. Second, students gain experience in constructing their own knowledge if the pedagogical structure provides the necessary guidance. Third, the active/interactive environment is provided through which more effective learning

Figure 4. Spectral simulations of the absorption transitions from v ′′ = 0 state of the ground electronic state to different v ′ states of the excited electronic state. The spectral simulations assume different values for the excited state displacement. The assumed line width and vibrational energy are 400 and 1425 cm{1, respectively. The cis,trans-octatetraene absorption spectrum is shown in the lower right (11 ).

can occur. Students will internalize accurate spectroscopic concepts more deeply by doing the work themselves. The next task for the student is to compare the simulated spectra with experimental spectra, particularly since the comparison allows students to determine the excited-state geometries by finding the displacement of the equilibrium coordinates. The spectra of the polyenes are excellent choices because the transitions involving the skeletal stretching vibrations are clearly seen in the spectra and the excited-state displacements change for different carbon chain lengths. Figure 4 shows simulations of the UV–visible absorption spectrum for cis,trans-octatetraene at three bond-length displacements; the last spectrum shows the experimental spectrum (11). For ∆g= 1.51, Figure 3 shows that the simulation is a good representation of the experimental spectrum. Since the intensity varies as |〈0|v′〉| 2, there is a clear vibrational progression from v′′ = 0 to the v′ = 0→6 states with the maximum intensity occurring for the v ′′ = 0 to v ′ = 1 transition. Students can analyze the simulated spectra by matching the relative intensities of the vibrational features of each displacement to the experimental spectrum and thereby determine the excitedstate displacement associated with the vibrational mode that is measured. The iodine absorption spectrum for the X 1Σg+ → B 3Π0u+ transition is a second example that can provide deeper insights into the information that is available from molecular spectroscopy. The iodine absorption spectrum is easily obtained and is often the basis for an undergraduate laboratory experiment (12–18). The spectrum can be analyzed by calculating the Franck–Condon factors using different values for the excitedstate displacement and approximating the vibrational wave functions as a simple harmonic oscillator. The best value for the displacement is found by finding the best match between the simulated spectrum and the experimental spectrum. An example of the simulated iodine absorption spectrum is shown in Figure 5a. The line positions for Figure 5 were calculated using the published procedures and parameters that describe the vibrational states, including the anharmonicity effects (6–8). The line intensities were calculated using the procedures described in this paper and do not include the anharmonicity effects. It is well known that the anharmonicity causes the vibrational lines to grow closer as v′ increases until they merge into the dissociative continuum that occurs at 20064 cm{1. The absorption in the dissociative continuum is modeled by assuming a single, broad state with the parameters defined in the legend for Figure 5. The Franck–Condon factors were calculated using eqs 12–14. The iodine line-widths and excited-state displacements for the simulation are given in the legend for Figure 5. Note that it is necessary to use a larger excited-state displacement (∆g = 9.6) than the published value of 7.18 (5–8). The difference is caused by the excitedstate anharmonicity and will be discussed shortly. The simulated iodine spectrum in Figure 5a has two vibrational progressions, the first for transitions from v′′ = 0 to the v′ states and the second from v′′ = 1. The experimental spectrum includes these transitions but it also includes a v ′′ = 2 progression that we neglect. The progression based on transitions from v′′ = 0 has one maximum absorption occurring near the v ′′ = 0 → v ′ = 28. The simulation shows the progression based on transitions from v ′′ = 1 has two maxima. The first occurs for the v′′ = 1 → v′ = 22 and the second occurs for the


1371

In the Classroom

v ′′ = 1 → v ′ = 33 transition (which overlaps the v ′′ = 0 → v ′ = 30 transition). If one examines the v ′′ = 1 wave function in the expanded inset of Figure 1, it is clear there are two vibrational coordinates where the probability is a maximum, corresponding to the two turning points for a simple harmonic oscillator. In between, the Franck–Condon factor becomes zero when v′ = ∆ g2/2, as can be seen from eq 13. In the simulation, this point occurs at ≈v ′′ = 1 → v ′ = 26. Interestingly, the observed iodine spectrum does not agree with the simulation because the transitions near the second maximum (v′′ = 1 → v ′ = 33 transition) do not appear. The absence of the second maximum in the progressions based on v ′′ = 1 can be a powerful example for students to discover the importance of simulations in deepening the understanding of molecular structure. The simulations have been based on using simple harmonic-oscillator wave functions for the Franck–Condon factor calculations, and the excitedstate potential is not harmonic for the very highly excited vibrational states characteristic of the iodine molecule absorption spectrum. This becomes especially clear from examining Figure 1, which shows the excited state v ′ = 22, 27, and 33 vibrational wave functions, assuming a harmonic potential and the correct ∆g= 7.18 displacement. These are the final states for the most intense absorption transitions from v ′′ = 0 and v ′′ = 1 in our simulations. They oscillate rapidly, and their overlap with the v′′ = 0 and v′′ = 1 states is small except near the turning points. However, the turning points for these

Figure 5. (a) Spectral simulations of the I2 absorption spectrum using a harmonic potential for the relative intensities. The transitions from the v ′′ = 0 and v ′′ = 1 transitions are indicated. (b) Spectrum is identical except that transitions from v ′′ = 1 to v ′ states higher than v ′ = 29 have been suppressed. These spectra assume the published transition frequencies, a normalized excited state displacement ξ g = 9.6, Lorentzian broadening with a line width of 11 cm{1, a room temperature population of v ′′ = 0 and v ′′ = 1, and a dissociative state at 525 nm with a 2000 cm{1 line width.

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states do not occur at the same coordinate displacement as the maximum probability for the v ′′ = 0 and 1 states. The excited state equilibrium displacement must be increased by 0.12 Å in order for these states to reach maximum intensity. This additional displacement accounts for our observation that ∆g = 9.6 gave the best agreement in our simulations instead of the expected ∆g = 7.18 (5–8). The difference is caused by the strong anharmonicity of the excited state. In particular, the repulsive part of the molecular potential increases more rapidly than a harmonic potential as the two iodine atoms are brought together more closely during an energetic vibration. This rapid increase also accounts for the absence of absorption transitions from v ′′ = 1 to the v ′ states near v ′ = 33. Referring to Figure 1, it is clear that the transition from the left side of the v ′′ = 1 vibrational state will have the largest overlap with v ′ states that are above the dissociation limit for the excited state. Thus, these vibrational states cannot exist and their absorption features cannot appear. The absence of the second progression from the v ′′ = 1 state can be simulated in two ways. The best way is to use better potentials for determining the wave functions in the excited electronic state. These choices include the Morse potential or RKR potentials (5, 7, 8). This approach is probably beyond the scope of an undergraduate course. The second is to simply set the Franck–Condon factors for the second progression equal to zero. The resulting simulation is shown in Figure 5b. One can see that the simulation provides a reasonable approximation of the observed spectrum except for the differences that are caused by the excited state’s anharmonic potential. There are other examples that can be used to analyze spectra and determine the excited state displacements. Ahmed (19) discussed the example of the unrelaxed Bi2 fluorescence spectrum where the transitions originate from many excited v ′ vibrational states of the excited electronic state. The low stretching frequency of Bi2 allows resolved vibrational spectra from 6 different initial states to 17 different final states, so the full range of spectra can be observed in one example. All the expected progressions are observed in the spectra, providing a classic example of the application of Franck–Condon factors in describing the experimental spectra. The transitions from v ′ > 0 have the multiple maxima that were not observed for I 2. The exercises with the Franck–Condon factors described here were used in physical chemistry courses at the University of Wisconsin Madison, Niagara University, and Monmouth University. These courses emphasize active learning. It is important in such a course to provide challenging problems for students to use to develop their thinking skills (20, 21). In our implementation of these exercises, students worked in cooperative groups or pairs that allowed them to share ideas as they sought to understand and complete the project. Active learning cooperative groups work best when they involve problems that are too difficult for individuals to complete. The many steps and complex concepts of this project provide just such a difficulty level. The implementation of the Franck–Condon project described here started with a clear explanation of the goal and performance objectives. In this case, the goals are to provide an example of how a complete set of basis functions can be used to represent other functions and how that representation is related to the interpretation of molecular spectra. Students


In the Classroom

were expected to reach the following performance objectives: 1. Express an arbitrary wave function as a linear combination of functions from a complete orthonormal set of functions. 2. Compute Franck–Condon factors using the basis set functions and the trial function. 3. Explain the significance of the magnitude of spectral peaks in terms of Franck–Condon factors. 4. Demonstrate how the overlap of functions of the initial state and final state of an electronic transition are dependent on the relative geometries of the states and the overlap of the wave functions for these states. 5. Use simulated spectra to extract the excited-state displacement for the transition.

It is also important for the students to succeed in the project if meaningful learning is to occur. Success requires presenting a clear and logical pathway that directs the thinking process through the discovery process so the students are led to an understanding of the Franck–Condon factors and their relationship to molecular spectroscopy. The background materials must focus the students’ attention on interpreting the results of their work in terms of physical principles. In addition, the course structure needs to provide ample time for the students to work through the process themselves. Finally, the material can challenge students at all skill levels with no fiscal constraints to limit which institutions can afford to provide challenges. Acknowledgment This work was supported by the National Science Foundation under grant DUE-9455928.

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