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Spektri-Sim: Interactive Simulation and Analysis of the Infrared Spectra of Diatomic Molecules Eric D. Glendening* and Jarno M. Kansanaho Department of Chemistry, Indiana State University, Terre Haute, IN 47809; *
[email protected] Analysis of the fundamental band of HCl is a classic spectroscopy experiment in the physical chemistry laboratory (1– 3). Articles focusing on the infrared (IR) spectrum of HCl or other diatomic molecules frequently appear in the Journal, clearly reflecting the popularity of this experiment (4). During the past decade, the effort required to acquire and analyze the HCl spectrum has diminished considerably. IR spectra are typically obtained in a matter of minutes using Fouriertransform (FTIR) spectrometers, and advanced software applications facilitate data analysis. Students in our laboratory encounter the IR spectrum of HCl toward the end of a two-semester physical chemistry sequence. By this time, they have gained considerable experience using both linear and nonlinear regression methods. Analysis of the HCl data represents yet another application of these methods, a straightforward exercise for the students that can be completed without considerable thought. Results for the HCl experiment are excellent, but we frequently find that students fail to recognize the important relationships between the regression parameters (molecular constants) and the spectral features (band origin, rotational splittings, intensity distribution, etc). We have recently developed Spektri-Sim, a Windowsbased software application that offers an appealing alternative to the usual spreadsheet-based analysis of diatomic spectra. The program features a unique, visual approach for evaluating molecular constants in which the student interactively seeks the optimal fit of a simulated spectrum to the experimental spectrum. Using Spektri-Sim forces students to consider the influence that the molecular constants and temperature have on the features of the IR spectrum. Spektri-Sim was developed using LabVIEW 5.1 (5). Other applications of LabVIEW in undergraduate chemistry laboratories have also been reported (6 ). We describe in this paper the theory of diatomic IR spectra (as implemented in Spektri-Sim), the Spektri-Sim user interface, results of typical student data, and the distribution of software. Theory Figure 1 shows the fundamental absorption band of HCl. The lines in this band appear in the infrared region, resulting from rovibrational transitions. Each rovibrational state of the molecule is typically labeled by its vibrational and rotational quantum numbers, v and J. Absorption corresponds to the transition of a molecule from a lower-energy state, designated (v′′, J′′), to a higher-energy state, (v′, J′). For the fundamental band, v′′ = 0 and v′ = 1. The observed (allowed) transitions in closed-shell diatomics like HCl are given by the rotational selection rule, ∆J = ±1. The 11 lines on the right-hand side of Figure 1 arise from P branch transitions for which J′ = J′′ – 1 (i.e., ∆J = ᎑1). The 12 (clearly resolved) lines on the left-hand side are R branch transitions, J′ = J′′ + 1 (∆J = +1). 824
Figure 1. The fundamental absorption band of HCl.
Q branch transitions (∆J = 0) are formally forbidden and therefore are not observed in the spectrum. This accounts for the apparent missing line near the center of the absorption band (the band origin). The energy of a rovibrational state (v, J ) is generally expressed in the form ∼ E(v, J ) = νe(v + 1⁄2) – νe x e(v + 1⁄2)2 + (1) Be J( J + 1) – αe(v + 1⁄2) J( J + 1) – De J 2( J + 1)2 where νe , νe x e , Be , αe, and De are the molecular constants (harmonic vibrational frequency, anharmonicity constant, rotational constant, vibration–rotation coupling constant, and centrifugal distortion constant, respectively). The energy and molecular constants of eq 1 are typically reported in wavenumber (cm᎑1) units. The frequency of a fundamental transition corresponds to the energy difference ∼ν = E∼(1, J′) – E∼(0, J′′) (2) Inserting eq 1 into eq 2 and applying the rotational selection rule, one derives expressions for the frequencies of the R and P branch transitions: ∼ν (J′′) = ν + 2B ( J′′ + 1) – α ( J ′′+ 3)( J′′+ 1) – 4D ( J′′+ 1)3 (3a) R
0
e
e
e
∼ν ( J′′) = ν – 2B J′′ – α J′′( J′′– 2) + 4D J′′ 3 P 0 e e e
(3b)
where the fundamental frequency (band origin), ν0, is given by ν0 = νe – 2νe xe
(4)
If a variable index m (m = J′′ + 1 for R branch transitions, m = ᎑J′′ for P branch transitions) is assigned to each line in the spectrum, then eqs 3 conveniently reduce to a single expression, which is a third-order power series in m. ∼ ν = ν + 2(B – α )m – α m 2 – 4D m 3 (5) m
0
e
e
Journal of Chemical Education • Vol. 78 No. 6 June 2001 • JChemEd.chem.wisc.edu
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This equation relates four molecular constants (ν0, Be , αe , and De ) to the frequencies of the fundamental absorption band. Equation 5 can be applied to the R and P branch transitions of any diatomic molecule. The intensity distribution of the absorption band depends on transition probabilities, transition frequencies ∼ν, and the thermal populations of the v = 0 rotational states. Herzberg (7 ) gives expressions of the following form for the relative intensities of the R and P branch lines: IR ∝ ∼ νR ( J′′ + 1)exp[᎑J′′( J′′ + 1)hcBe /kT ]
(6a)
IP ∝ ∼νP J′′exp[᎑J′′( J′′ + 1)hcBe /kT ]
(6b)
Line intensities can be therefore predicted from the rotational constant Be and temperature T. Spektri-Sim User Interface Figure 2 shows the main Spektri-Sim window. The upper left-hand panel displays stick representations of two fundamental absorption bands: an experimental band in green (on the left) and a simulated band in red (on the right). The experimental spectrum shown in this Figure is that of the H35Cl molecule (cf Fig. 1). The student provides the experimental assignments of the spectrum via the Experimental Spectrum window (shown in Fig. 3). The frequencies and intensities of the simulated spectrum are calculated using eqs 5 and 6, based on the current values of the molecular constant and temperature controls. The student can manipulate these controls (sliders) using either the mouse or the keyboard. The simulated spectrum is dynamically updated in response to changes in the control values. The student then performs a visual refinement of the molecular constants, seeking optimal overlap of the simulated and experimental spectra. The quality of the user-refined molecular constants can be quantitatively assessed using the standard deviation of regression (SDR)
SDR =
Σm
νme – νm N –4
h 8π 2cµB e
Figure 3. The experimental spectrum window showing the experimental assignments for the fundamental band of H35Cl.
2 1/ 2
(7)
where the frequencies ∼ νm are those of eq 5 and the frequencies ∼ νme are those of the experimental spectrum; N is the total number of assigned lines in the spectrum. Digital as well as analog displays of the SDR appear in the Spektri-Sim window. Like the simulated spectrum, these displays are dynamically updated as the molecular constant controls are changed. Therefore, the student can use the SDR to assist the optimization of the molecular constants. Several elements of the Spektri-Sim window may be disabled, depending on the data provided by the student or the quality of the refined molecular constants. For example, the equilibrium bond length display in Figure 2 is disabled (“grayed out”). The equilibrium bond length is given by
Re =
Figure 2. The Spektri-Sim window showing the fundamental absorption band of H35Cl in green (on the left) and a simulated band in red (on the right).
1/ 2
(8)
where h is Planck’s constant, c is the speed of light, and µ is the reduced mass. The latter, µ = m1m2/(m1 + m 2), is calculated
Figure 4. The Spektri-Sim window after optimization of the molecular constants for H35Cl.
from the isotopic masses, m1 and m2, of the atoms. The bond length is undefined without these masses and therefore is not displayed until the masses are entered. The Refine button is also disabled in Figure 2. When enabled, clicking this button initiates the numerical (Levenberg– Marquardt) optimization of the molecular constants using
JChemEd.chem.wisc.edu • Vol. 78 No. 6 June 2001 • Journal of Chemical Education
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the control values as initial guesses. Refine remains disabled, however, until the SDR is less than 3 cm᎑1. Spektri-Sim thus requires that the student first interactively refine the molecular constants visually before the software-based optimization is invoked. Figure 4 shows the Spektri-Sim window after numerical refinement of the H35Cl molecular constants. The best-fit constants are reported together with their uncertainties (standard deviations in the last digit displayed). The SDR is 3.16 × 10᎑2 cm᎑1. The controls are updated with the best-fit constants and the spectrum shows the optimal overlap of the experimental and simulated spectra. The student can cycle through several displays, alternately examining the IR spectrum (as shown in Fig. 4), a regression plot based on eq 5, and a residuals plot. Note that the equilibrium bond length (1.2747 Å) is displayed when isotopic masses are entered. Using Spektri-Sim to optimize the molecular constants requires the student to consider the following specific relationships between the molecular constants and the features of the IR spectrum: 1. The fundamental frequency determines the spectral location of the band origin. The vibrational control is used to translate the fundamental band across the display. 2. The rotational constant (rotational control) principally determines the splittings of the spectral lines. 3. The vibration–rotation coupling constant (V–R coupling control) largely accounts for the variation of the rotational splitting across the band. 4. The centrifugal distortion constant (distortion control) only weakly influences the spectrum.
Additionally, the student will discover how temperature (Temperature control) influences the intensity distribution of the simulated spectrum. The distribution broadens with increasing temperature as states of higher J ′′ value become thermally populated. At very low temperatures, just a few lines appear in the spectrum, since only states of low J value are significantly populated. Results Spectra for HCl and CO were acquired on a Midac– 2000 FT-IR spectrometer using the Grams/32 software interface. Results from the Spektri-Sim analysis of the resulting data are shown in Table 1, together with literature values (8). The values in parentheses are the estimated uncertainties (one standard deviation in the last digit reported). The calculated molecular constants and equilibrium bond lengths are in excellent agreement with the literature values. Distribution Spektri-Sim is available, free-of-charge, by contacting the corresponding author or, preferably, by downloading compressed copies of the installation diskettes from http://carbon.indstate. edu/spektri-sim. The distribution includes the LabVIEW 5.1 Run-Time Engine, documentation (with a tutorial) in PDF
826
Table 1. Molecular Constants and Equilibrium Bond Lengths of H3 5 Cl and 1 2 C1 6 O Constant/ cm᎑1
Equilibrium Bond Length/Å a H Cl 35
12
C1 6 O
Spektri-Sim
Ref 7
Spektri-Sim
Ref 7
ν0 b
2885.57(1)
2885.64
2142.94(2)
2143.29
Be αe
10.592(2)
10.5909
1.932(1)
1.9313
0.3035(2)
0.3019
0.01751(7)
0.01748
De
0.00052(1)
0.00053
0.000006(1)
0.0000064
Re
1.2747
1.2746
1.1282
1.2181
aThe values in parentheses represent one standard deviation uncertainty in the last reported digit. bSee eq 4.
format, and sample data for HCl, HBr, and CO. Spektri-Sim has been tested (and performs well) in the Windows 95/98 environment using 133-MHz Pentium or faster processors. Acknowledgment We acknowledge Arthur Halpern for his encouragement and valuable comments during the development of SpektriSim and the preparation of this manuscript. Literature Cited 1. Halpern, A. M. Experimental Physical Chemistry: A Laboratory Textbook, 2nd ed.; Prentice-Hall: Upper Saddle River, NJ, 1997; pp 567–587. 2. Shoemaker, D. P.; Garland, C. W.; Nibler, J. W. Experiments in Physical Chemistry, 6th ed.; McGraw-Hill: New York, 1996; pp 397–404. 3. Daniels, F; Williams, J. W.; Bender, P.; Alberty, R. A.; Cornwell, C. D.; Harriman, J. E. Experimental Physical Chemistry, 7th ed.; McGraw-Hill: New York, 1970; pp 247–256. 4. See, for example: Zielinski, T. J. J. Chem. Educ. 2000, 77, 668–670. Schwenz, R. W.; Polik, W. F. J. Chem. Educ. 1999, 76, 1302–1307. Iannone, M. J. Chem. Educ. 1998, 75, 1188– 1189. Schor, H. H. R.; Teixeira, E. L. J. Chem. Educ. 1994, 71, 771–774. Rieck, D. F.; Kundell, F. A.; Clements, P. J. J. Chem. Educ. 1989, 66, 682. Prais, M. G. J. Chem. Educ. 1986, 63, 747–752. Ford, T. A. J. Chem. Educ. 1979, 56, 57–58. 5. LabVIEW 5.1; National Instruments Corp., 6504 Bridge Point Parkway, Austin, TX 78730-5039. 6. Henderson, G. J. Chem. Educ. 1999, 76, 868–870. Gostowski, R. J. Chem. Educ. 1996, 73, 1103–1107. Drew, S. M. J. Chem. Educ. 1996, 73, 1107–1111. Muyskens, M. A.; Glass, S. V.; Wietsma, T. W.; Gray, T. M. J. Chem. Educ. 1996, 73, 1112–1114. Ogren, P. J.; Jones, T. P. J. Chem. Educ. 1996, 73, 1115–1116. 7. Herzberg, G. Molecular Spectra and Molecular Structure I. Spectra of Diatomic Molecules, 2nd ed.; Van Nostrand: Princeton, NJ, 1950. 8. Literature data for the 0–1 bands of HCl and DCl can be found in Rank, D. H.; Eastman, D. P.; Rao, B. S.; Wiggins, T. A. J. Opt. Soc. Am. 1953, 52, 37.
Journal of Chemical Education • Vol. 78 No. 6 June 2001 • JChemEd.chem.wisc.edu
