In the Laboratory
An Experiment in Electronic Spectroscopy: Information Enhancement Using Second Derivative Analysis B. R. Ramachandran and Arthur M. Halpern Department of Chemistry, Indiana State University, Terre Haute, IN 47809
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distinctly in the case of NH 3) to be almost completely washed out (2). The assignment of the UV absorption of TMA is analogous to that of NH3 ; that is, it corresponds to a transition from the pyramidal ground state (characterized by a double minimum potential) to a planar excited state (characterized by a single minimum potential) and thus it can be logically expected that the TMA spectrum should contain similar vibrational information as does that of NH3 . The logical question arises: Can one accordingly determine the positions of the vibrational features in the TMA spectrum and also estimate the out-of-plane bending frequency of the excitedstate TMA? At first glance, however, it appears that very little or no information can be obtained from the TMA absorption spectrum, since it does not seem to show any vibronic features. Yet upon closer examination1 the spectrum can be seen to possess subtle undulations that are the residual, broadened vibronic features analogous to the ones seen more prominently in the NH 3 spectrum. However, the ripples are not definitive enough to permit unambiguous assignments to be made of their positions (2). Nevertheless, they can be easily transformed in such a way that a surprisingly extensive amount of quantitative spectroscopic information can be retrieved (3). This transformation corresponds to the second derivative (SD) of the “ordinary” (i.e., the zero-derivative, ZD) spectrum of TMA that is shown in Figure 1. In concept, the locations (in wavenumbers) of the maxima of the subtle, broadened features of the ZD spectrum are sought. First we will demonstrate, using synthetic data, how the derivative transformation can, in principle, extract and enhance features that are hidden in a broadened background, such as the ZD spectrum of TMA.
Extinction coefficient of NH 3
Extinction coefficient of TMA
Most undergraduate physical chemistry experiments are so designed that the raw experimental data turn out to be in a ready-to-use form suitable for further analysis and interpretation. This is desirable, since the students can readily analyze the data and derive conclusions about the system being studied. Unfortunately, however, students may get the unrealistic and perhaps erroneous impression that all experimental investigations produce data in a form that can be directly processed for further analysis. Thus it seems worthwhile to include in the laboratory curriculum at least a few experiments in which the raw data contain information in a “latent” form that can be “culled out” only by employing suitable data-manipulation and transformation techniques. In this paper we describe such a physical chemistry or experimental methods experiment in which information about the vibrational states of a molecule can only be obtained by employing a second derivative transformation of the raw spectroscopic data. The electronic absorption spectrum of a molecule contains a considerable amount of information about the excited state that is reached via an electronic transition. In some systems the absorption spectrum shows discrete features that are attributed to the transitions of the molecule from its ground electronic state to various vibrational levels of the excited state. A classic example is provided by the absorption spectrum of NH3 in the 200 nm region (Fig. 1) (1). This absorption corresponds to the promotion of an electron from the nonbonding orbital on nitrogen, n N, to a Rydbergtype orbital, 3sN. About nine discrete bands can be easily identified in the spectrum. These vibrational features correspond to transitions to excited state vibrational levels associated with the out-of-plane bending motion of the NH 3 molecule. The positions of these well-resolved features can be readily determined from the spectrum; furthermore, the frequency of this vibrational mode (out-of-plane bending) can be estimated from the spacing of these features. It is important to know that although NH3 is pyramidal in the electronic ground state, it is planar in the excited state. Thus, the ground state is characterized by a double minimum potential corresponding to the inversion motion; with respect to this degree of freedom, however, a single minimum potential characterizes the excited state. Trimethylamine, TMA, shows an entirely different absorption spectrum, also shown in Figure 1. First, the TMA spectrum shows two absorption bands. A lower-lying absorption, significantly red-shifted relative to the spectrum of NH3, occurs in the region 260–220 nm (38,000–46,000 cm{1) with its maximum lying around 227 nm (44,000 cm {1 ). A second, stronger, absorption overlaps the former at shorter wavelengths; only the shoulder of this absorption is displayed in the TMA spectrum in Figure 1. The longer wavelength absorption is analogous to the one seen in the spectrum of NH3 and is also attributed to the orbital promotion, nN → 3sN. The high-energy transition is assigned as nN → (3p)N . Second, and more importantly from the point of view of this paper, the involvement of the heavy methyl groups in the inversion motion seems to make the vibrational features (seen
Wavenumber (cm{1 )
Figure 1. The vapor phase absorption spectra of (A) NH3 and (B) TMA obtained at 294 K.
Journal of Chemical Education • Vol. 75 No. 2 February 1998 • JChemEd.chem.wisc.edu
In the Laboratory The Second Derivative (SD) Transformation In Figure 2 is shown a Gaussian curve, hereafter called the “main envelope”, generated using the following parameters: x¯ (the mean of x-values) = 1.0 and σ (the standard deviation in x) = 0.3; the curve is shown for the range of x values between 0.0 and 0.4. We superimposed on this curve two other Gaussian features that are centered at x = 0.15 and 0.25, both having σ values of 0.015, and the intensities at their maxima normalized to correspond to 2% of the intensities of the main envelope at those positions. The resultant of these signals (the main envelope plus the two Gaussian smears), denoted as the ZD curve, also shown in Figure 2, is our working curve. Since the two features are overwhelmed by the strong main envelope, their presence is hardly noticeable in the ZD curve. Can one then accurately extract the information about the locations of the maxima of those features (i.e., x = 0.15 and 0.25) in the ZD curve by derivative analysis? Some degree of clarification of these positions could, in principle, be achieved through the first derivative (FD) of the signal. The maxima of the undulations in a ZD curve would correspond to the zero-crossing points in the FD curve. But two serious complications arise that make this transformation impractical: (i) besides maxima, the minima of the modulations in a ZD curve also become the zero-crossing points in its first FD transform,2 and (ii) the rising envelope of the ZD curve causes the crossing points of the features in the FD curve track a “shifting” rather than an “absolute” zero. As a result of these complications, the FD curve, although laden with more information than the ZD curve, cannot be easily and conveniently used to obtain quantitative assignments. These points are illustrated in Figure 2. In the SD transformation, the maxima of a ZD function become minima (and vice versa), and thus the ambiguity of assigning the positions of the features as mentioned above
x Figure 2. Demonstration of efficacy of the SD analysis in identification of the positions of poorly resolved features. ME: the main Gaussian background (???? , x¯ = 1.0; σ = 0.3). GF: the two Gaussian features ( x¯ = 0.15 and 0.25; σ = 0.015, whose maxima are normalized to 2% of the intensities of ME at x = 0.15 and x = 0.25, respectively) that were used to smear the ME curve. ZD: the working zero derivative curve obtained by combining ME and GF. FD: the first derivative curve of ZD. SD: the second derivative curve of ZD. The GF curve is also shown 50× magnified for clarity.
is removed. Furthermore, the positions of the SD minima are practically unaffected by the presence of a rising background that complicates the result of an FD transformation. Figure 2 includes the SD curve of the synthetic ZD curve as well. As can be seen, the results of the FD and SD transforms, in comparison with the ZD curve, are more revealing. The hardly noticeable features of the ZD curve are extracted and enhanced dramatically in the derivative transforms (especially the SD), and the minima in the SD transform (0.15 and 0.25) correspond to the maxima of the two ripples buried in the ZD curve.3 Thus it is demonstrated that one can, in principle, determine the positions of poorly-resolved vibronic features of the TMA spectrum using the SD transformation technique. Experimental Procedure We used a Varian Cary 5 UV-vis spectrophotometer for data acquisition in our experiments. However, it should be emphasized that any spectrophotometer that can acquire and store data digitally can be used. Some instruments, like the one we used (and the Cary 1 and Cary 3), have the facility for data acquisition directly in the second derivative mode. With such instruments, the SD spectrum is displayed on the monitor (or plotted by the printer/plotter) in real time, and the positions of the minima can be estimated using the cursor. If such a utility is not available in the instrument used, the SD transformation of the raw absorbance– wavelength (i.e., ZD) data can be performed externally by using numerical differentiation methods with a software utility such as EXCEL (Microsoft), Mathcad (Mathsoft), RS/1 (BBN Domain), or PSI Plot (Poly Software International), and the minima can be obtained from the SD data. A brief procedure for obtaining the SD data from the ZD data, using EXCEL, is described in the Appendix. Trimethylamine (Aldrich) was used without further treatment in our experiments. A vacuum rack equipped with an MKS Baratron gauge for measurement of gas pressures was used for filling TMA in a gas cell, which was a 1-cm quartz absorption cell fitted with a PTFE stopcock. A pressure of about 17–25 torr is convenient for the experiment. The choice of appropriate settings for the following data acquisition parameters is important for a successful experiment. (i) Data interval: the data point density should be so chosen such that the data interval is shorter than the spacing between adjacent features. A data interval of 0.25 to 0.30 nm is suitable in this experiment. (ii) Spectral band width: the spectral band width (bandpass) should be narrower than the spacings. An SBW of about 0.25 to 0.30 nm is appropriate in this experiment. (iii) Signal-to-noise ratio: if a “constant S/N acquisition mode” option is available, a high S/N ratio (500–600) can be used, to suppress the effect of any experimental noise. If the instrument is equipped with the linear wavenumber scanning option, it is desirable to use that mode. Furthermore, since it is only the positions of the vibrational features that we seek in this experiment, there is no need for converting the absorbances into molar absorptivity coefficients. We used the following conditions in our experiment: TMA pressure: 18.2 torr Temperature: 21.0 °C Data acquisition parameters: Wavelength range: 200–285 nm Data interval: 0.25 nm SBW: 0.25 nm S/N: 600
JChemEd.chem.wisc.edu • Vol. 75 No. 2 February 1998 • Journal of Chemical Education
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In the Laboratory
The SD spectrum of TMA (Fig. 3) shows two distinct sections, corresponding to the two absorption bands seen in the ZD spectrum (i.e., nN → 3s N and nN → 3pN transitions) and each section reveals a number of oscillations. As mentioned earlier, the minima of these oscillations correspond to the maxima of the undulations present in the ZD spectrum. About 19 and 12 features are revealed in the longer wavelength (> 223 nm) and shorter wavelength (< 223 nm) sections of the SD spectrum, respectively. The wavenumber positions of these minima and the spacing between adjacent features are listed in Table 1. These data show that the spacings of the assigned features are reasonably constant, 378 (σ = 18) cm{1 (for > 223 nm) and 325 (σ = 16) cm {1 (for < 223 nm) (Fig. 4). The regularity of the spacings indicates that the out-of-plane bending potentials of TMA in the excited states are considerably harmonic. The analysis of these data provides an opportunity for the student to learn about vibrational potentials. A harmonic potential can be expressed in the form 2
(2)
Here, n is the vibrational quantum number, h is the Planck’s constant, ν is the vibrational frequency (in s {1), c is the velocity of light (in cm s{1 ), and ν˜ the vibrational frequency (in wavenumbers, cm{1). From eq 2 it can be seen that the harmonic oscillator predicts a constant spacing between vibrational levels, and thus the data obtained from the SD analysis of TMA are consistent with this picture. It is also interesting to note, from the results of the SD analysis, that the vibrational frequencies of the nN → 3sN and nN → 3pN excited states are very similar (378 Å ± 18 and 325 Å ± 16 cm{1). This result is not surprising, since both excited states are planar and the out-of-plane bending motion thus seems to occur with similar frequencies irrespective of the orbital (3sN or 3p N) to which the electron is promoted from the nonbonding nN orbital of the ground state. Summary The second derivative transformation of the absorption spectrum of trimethylamine is extremely effective in enhancing and extracting the information about the vibrational characteristics of the excited state of the molecule, lying obscure in the raw absorption data. Besides being simple, the experiment has many desirable aspects. In addition to reinforcing the knowledge about the vibrational properties of molecules that the student derives from the physical chemistry lecture course, the experiment exposes the student to vacuum handling techniques during the actual run and to various numerical analytical methods during the data analysis.
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Figure 3. The ZD, FD, and ZD spectra of TMA obtained at 294 K.
(1)
where V(x) is the potential energy of the system as a function of the displacement coordinate x, x0 is the equilibrium value of that coordinate corresponding to the minimum of the potential well, and k is a constant characteristic of the system and the vibrational mode involved. Application of the Schrödinger equation to the harmonic potential results in the quantization of vibrational energy levels as shown in eq 2.
E n = n + 1 hν = n + 1 hcν~ 2 2
Wavenumber (cm {1)
Vibrational Spacing (cm{1)
V x = 1 k x – x0 2
Extinction Coefficient (M {1cm{1)
Results and Discussion
Vibrational Feature Number
Figure 4. A plot of the vibrational spacings, ∆˜ν vs. a running integer from the result of the SD analysis (see eq 2). h: Features observed in the low-lying transition ( > 223 nm); e : features assigned in the high-energy transition ( < 223 nm).
Table 1. Positions of Vibronic Features and Vibrational Spacings in Absorption Spectrum of TMA Position Spacing Position Spacing Position Spacing (cm{1) (cm{1) (cm{1) (cm{1) (cm{1) (cm{1) 37984 – 41741 367 45436 314 38360 376 42093 352 45783 347 38727 367 42492 399 46106 323 39099 372 42854 362 46455 349 39498 399 43245 391 46768 313 39849 351 43600 355 47077 309 40225 376 43995 395 47398 321 40589 364 44395 400 47726 328 40979 390 44791 396 48026 300 41374 395 45122 331 48361 335 Note: Estimated by the SD analysis (see Fig. 3).
Journal of Chemical Education • Vol. 75 No. 2 February 1998 • JChemEd.chem.wisc.edu
In the Laboratory Acknowledgment
Appendix
Partial funding for the spectrophotometer was provided by the National Science Foundation through the Instrument and Laboratory Improvement Program (USE-9151048).
In concept, we want to obtain the SD data—that is, the {(d2A/d˜ν2), ν˜ } array by employing appropriate data manipulation methods to the raw data (ZD data) which consists of the {A(λ), λ} array, where A is the absorbance, ν˜ is wavenumbers (cm{1), and λ is the wavelength (nm). Since the data are usually equispaced in λ (unless the”linear wavenumber scan mode”, rarely an available option in spectrophotometers, was used), it is convenient to perform a numerical differentiation to obtain dA/dλ first, which can then be transformed, by multiplication by λ 2, into the FD data in the correct form: that is, dA/d˜ν.4 The FD data thus obtained can then be differentiated again, repeating the same procedure, to obtain the d(dA/d˜ν)/dλ and, eventually, the d2A/d˜ν2 (SD) arrays. The basic protocol for obtaining the SD array from ZD array using Microsoft EXCEL is described below. The raw data array, {A(λ), λ}, is read into a spreadsheet. A wavenumber, ν˜ , column is created, using ν˜ = 1.0 × 10 7/λ (λ in nm). The first derivative for the second data point (i.e., the second row), (dA/dλ)2 , is calculated in a new column using the relationship (dA/dλ)2 = (A 3 – A1 )/2∆λ, where A1 and A3 are the absorbance data for the first and third data points, respectively, and ∆λ is the wavelength interval in the data. The derivative array for the rest of the data can now be generated by holding the cursor on the bottom-right corner of the second cell in the derivative column to form a “+” sign and dragging the mouse, with the left button pressed, down the column to the (n–1)th data point, where n is the maximum number of points in the data set. The FD array dA/d˜ν is now calculated by multiplying the derivative column by λ2 . A similar procedure is repeated on the FD column to calculate the SD column. The minima in the SD array can be noted by browsing through the column.
Notes 1. If one examines the absorption spectrum of TMA by sighting tangentially along the envelope, the vibronic ripples are more evident. 2. One can, in principle, distinguish between the two types of zero-crossing points in the FD curve as follows. The maxima of the ZD curve show up as zero-crossing points in the falling portions of the FD curve, whereas the minima of the former become zero-crossing points of the rising portions of the FD curve. 3. It should be pointed out that any noise present in the raw data will also be magnified in the derivative transformation, and thus may interfere with a successful real-life data analysis. This problem could be minimized by using proper statistical techniques that would selectively filter out noise from the raw data leaving the vibronic ripples unaffected. In the experiment described in this paper, however, this effect is not a serious issue if the data acquisition is performed as described in the Experimental Procedure section. 4. dA/ dν˜ = ( dA/ d λ)?(d λ/ dν) ˜ = ( dA/d λ)?λ2, since λ = 1.0×10 7/ ν. ˜ The constant term (1.0×107), being only a scaling factor, can be ignored in the multiplication because we are not interested in the absolute amplitudes of the oscillations in the derivative curves.
Literature Cited 1. Robin, M. B. Higher Excited States of Polyatomic Molecules, Vol. 1; Academic: New York, 1974; pp 82–84, 214–217. 2. Matsumi, Y.; Obi, K. Chem. Phys. 1980, 49, 87. 3. Halpern, A. M.; Ziegler, L. D.; Ondrechen, M. J.; J. Am. Chem. Soc. 1986, 108, 3907.
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