Research: Science and Education
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Bond Length Dependence on Quantum States as Shown by Spectroscopy Kieran F. Lim (
)
School of Biological and Chemical Sciences, Deakin University, Geelong, Victoria 3217, Australia;
[email protected] Theory Changes in molecular energies, ∆Emolec, are conveniently partitioned into electronic, vibrational, and rotational contributions1: ∆ E molec = ∆ Eel + ∆ E vib + ∆E rot
(1)
For linear molecules, the rotational energies are quantized, with second-order Taylor-series expansion,
E rot ( J ) = h B J ( J + 1 ) B =
(2)
h 8π µ r 2
(3)
2
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where h is Planck’s constant, B is the rotational constant, J is the angular momentum quantum number, µ is the reduced mass, and r is the bond length. The energy of a vibration–rotation absorption transition is given by, ∆E = A + h B⬘J ⬘ ( J ⬘ + 1 ) − hB ⬙ J ⬙ ( J ⬙ + 1)
(4)
A = ∆ E vib
(5)
where the single prime (´) and double prime (˝) notation indicates the final (upper) and initial (lower) quantum vibrational state, respectively. The final and initial angular momentum quantum numbers are related by the selection rules : P branch: J ⬘ = J ⬙ − 1
Q brancch: J ⬘ = J ⬙
(6)
R branch: J ⬘ = J ⬙ + 1
Closed-shell diatomic molecules exhibit only the P and R branches, as shown in Figure 1. The intensities of the individual peaks reflect the molecular populations corresponding to the (initial) J ˝ rotational energy levels: I p ( J ⬙ ) ∝ ( 2 J ⬙ + 1) exp
−h B ⬙ J ⬙ ( J ⬙ + 1) kBT
(7)
The energy of a rovibronic absorption transition is also given by eqs 4 and 6, but the energy term A has contribu-
R-branch
Absorbance
Undergraduate students often have the misconception that molecules have fixed, unchanging bond lengths. This misconception is fostered by freshman general chemistry and introductory organic chemistry texts that present molecules as static species with fixed equilibrium bond lengths, re. The concept that a molecule can have vibrational state-dependent bond lengths, r0, r1, r2, … , is not introduced until more advanced courses. This can lead to learning difficulties since students’ conceptions are resistant to change: awareness of the misconception is required for acceptance of the correct (new) concept (1). Most “proofs” of quantum state-dependent bond lengths rely on detailed quantitative analysis of IR spectra (2–8) and hence are not “intuitive”. Similarly, the application of the Franck–Condon principle to UV–vis spectroscopy can demonstrate that average bond lengths in the ground and excited states are different (3–6); however, without a detailed quantitative analysis (2–5, 9) the argument that electronic spectra “prove” an increase in bond length is less convincing. A simple visual (10, 11) explorative exercise that demonstrates the relationship between spectra and bond lengths would minimize the dependence on math analysis. This article discusses how a spreadsheet simulation of linear-molecule spectra can be used to explore the dependence of rotational band spacings and contours on average bond lengths in the initial and final quantum states. This gives students the opportunity to do mathematical “experiments” (10, 11), including testing unphysical scenarios. The use of the spreadsheet, either as a classroom demonstration or as a laboratory exercise, caters to those learners who do not favor a symbolic or mathematical representation, that is, “mathematical intelligence” (12), in their learning. The simulation of hydrogen chloride IR, iodine UV– vis, and nitrogen UV–vis spectra clearly show whether the average bond lengths have increased or decreased on vibrational or electronic excitation. The spreadsheet simulation, rotband.xls, documentation, and an example of possible usage in a laboratory situation are included in the Supplemental Material.W
3100
3000
P-branch
2900
2800
2700
2600
ⴚ1
Wavenumber / cm
Figure 1. Typical student spectrum of hydrogen chloride showing the fundamental vibration–rotation transition. Each peak is split due to contributions from H⫺35Cl and H⫺37Cl. The vibration– rotation intensities are distorted due to saturation and insufficient signal averaging.
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tions from both electronic and vibrational energies: (8)
A = ∆ E el + ∆ E vib
Many teaching-laboratory spectrometers will be unable to distinguish individual rovibronic transitions, but the diatomic vibronic (electronic–vibration) spectral bands will exhibit characteristic asymmetry owing to the unresolved rotational fine structure, as shown in Figure 2. Equations 4, 6, and 7 have been programmed into a spreadsheet, rotband.xls, which is available in the Supplemental Material.W This spreadsheet simulation can be used by students to explore the relationship between molecular parameters such as bond lengths and the appearance of absorption spectra. Alternatively, the spreadsheet simulation can be
Absorbance
v⬙ = 0
v⬙ = 1
520
540
Application to Diatomic IR Spectra Diatomic IR spectra show asymmetric spacings (see Figure 1). Textbooks note that the asymmetric spacings result from different average bond lengths in the initial and final vibrational states, but place most emphasis on the similarity in spacing, for example: ...there is approximately equal spacing between adjacent R-branch lines and between adjacent P-branch lines…the band is not quite symmetrical but shows a convergence in the R branch and a divergence in the P branch…due principally to the inequality of B0 and B1… (5)
The rotational constant, Bv, and the equilibrium bond distance, rv, for the vth vibrational state are related by eq 3.The relative ordering of B0 and B1 (and r0 and r1) is determined from mathematical analysis of the equations or numerical analysis of the spectrum (3–9), which is not always “intuitive”. All the above-mentioned references (3–9) lack a simple visual (10, 11) demonstration of how the vibration–rotation spacings reflect changes in bond length. This visual demonstration is easily done, using a numerical simulation of a spectral transition (through eqs 4, 6, and 7). Students can easily test the three possibilities:
A
500
used by instructors to illustrate the same concepts. At Deakin University, the former approach is used. The following sections discuss how numerical simulations can be used for qualitative interpretation of spectra to show that bond lengths are not fixed, but that their expectation value changes on vibrational and electronic excitation.
560
580
(a) r´ > r˝
600
(b) r´ = r˝
Wavelength / nm
(c) r´ < r˝
Absorbance
B
19000
18800
18600
17200
17000
17800
Wavenumber / cmⴚ1 Figure 2. Typical student spectrum of iodine showing the B 3ΠOu+ ← X 1Σ +g vibronic transitions. (A) The main (v” = 0) progression at λ = 500–560 nm and the hot-band ( v” = 1, 2) progressions at λ > 540 nm. (B) The 526–540 nm (19,000–18,530 cm᎑1) portion of the main progression and the 578–595 nm (17300–16800 cm ᎑1) portion of the hot-band progression on an expanded axis. Note that the horizontal axes use different units for A and B, but the direction of the horizontal axes is the same.
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Figure 3 clearly shows that only an increase in the average bond length with vibrational excitation, scenario A, gives qualitative agreement with the spacings in the observed spectrum (see Figure 1), with R-branch convergence and P-branch divergence.2 Several methods for the analysis of diatomic infrared vibration–rotation spectra have been published in this Journal and elsewhere (8); all of the methods have advantages and disadvantages, with no single “best” method. Regardless of the particular analysis method, some students struggle with the level of math that is required. A secondary use of the rotband.xls spreadsheet is to verify students’ analyses by testing whether the student results can simulate the experimental spectrum. Application to Diatomic UV–Vis Spectra Electronic transitions are usually associated with large changes in molecular geometry. Absorption UV–vis spectra are characterized by transitions from low ν˝ to high ν´, owing to the Franck–Condon principle (5–7). Fluorescence (emission) spectra are normally characterized by transitions from low ν´ to high ν˝ and occur at lower energy (longer wavelength) than the corresponding absorption spectra. This
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Research: Science and Education
5, which shows only one vibronic transition, with an obvious bandhead at ca. 17,965 cm᎑1. The transition is asymmetric, with a “half-width” at half-maximum of approximately 1 cm᎑1 on the higher energy side (i.e., the bandhead itself ), but extending over more than 40 cm᎑1 on the lower energy side. This asymmetry of the vibronic peaks in Figure 2 and Figure 5 must be due to r´ >> r˝.3 Note that oscillations due to the unresolved rovibronic transitions are reproduced qualitatively in the simulated spectrum; there is (also) reasonable quantitative agreement with the oscillations’ spacing at ca. 2 cm᎑1. Transitions between different excited electronic states can either increase or decrease the average bond length, depending on the relative bonding and antibonding characters of
A r⬘ > r ⬙
3100
A r⬘ >> r ⬙
3000
2900
2800
2700
2600
Energy
Absorbance
is visualized in Figure 4A, with an increase in average bond length on excitation (r´ >> r˝). However, Figure 4B, with r´ r˝) is associated with R-branch convergence in IR spectroscopy. In UV–vis spectra, changes in bond length may be very large: the branch “convergence” often results in a reversal of the branch sequence and a resultant band head (e.g., 5, 7 ), as shown in the iodine B 3ΠOu+ ← X 1Σ+g absorption spectrum of Figure
ⴚ1
Wavenumber / cm
Absorbance
B r⬘ = r ⬙
Interatomic Distance (r )
3100
3000
2900
2800
2700
B r⬘ > r”, and (B) r’ r˝.
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ⴚ1
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148
22000
Wavenumber / cm
B simulation
18000
23000
ⴚ1
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Figure 6. Detail of the nitrogen C 3Πu → B 3Π g vibronic emission spectrum. (A) Experimental nitrogen emission spectrum (rovibronic peaks are not fully resolved). The experimental spectrum was provided by Mark Riley (University of Queensland, Australia). (B) Simulated spectrum, assuming some line broadening, with a modified Fortrat diagram (3) indicating the wavenumbers of the individual R-branch and P-branch rovibronic peaks. Asymmetry of the vibronic peaks indicates that the upper vibronic state has a shorter average bond length than the lower vibronic state, r´
