Anal. Chem. 2007, 79, 8281-8292
“Sequential Order” Rules in Generalized Two-Dimensional Correlation Spectroscopy He Huang*
College of Materials Science & Engineering, Hubei University, Wuhan 430062, China
The widely used “sequential order” rules in the generalized two-dimensional (2D) correlation spectroscopy were adopted from the mechanical perturbation-based 2D infrared, where dynamic spectral intensity variation must be a simple sinusoid. 2D correlation analysis is fundamentally a form of parametrization of the integrated or overall relationship between two variable quantities. In generalized 2D correlation spectroscopy, however, the dynamic spectral intensity variations are generally nonperiodic and monotonic, and spectral intensity changes are largely instantaneous. The sequential orders in generalized situations are therefore localized. It is naturally necessary and important to testify whether the analysis result obtained by using the sequential order rules is consistent with the local sequential order of events, which reflects the real sequential order in generalized situations. Unfortunately, this test was not done yet. In this report, the sequential order rules have been tested in the generalized situations using simulated spectra with different local sequential orders and assuming the intensity changes of bands take the exponential forms. It has been found that the sequential order rules correctly identify the local sequential order of two events when spectral intensities of two bands increase or decrease at the same direction but fail when spectral intensities change at different directions. In addition, 2D correlation analysis cannot distinguish the local sequential order from the rate difference of events. A theoretical analysis demonstrates that the synchronous and asynchronous spectra in the generalized 2D correlation spectroscopy may also indicate the linear/nonlinear relationship, in addition to the integrated or overall sequential order of events. The synchronous and asynchronous spectra in the generalized 2D correlation spectroscopy do not necessarily provide the information on the local sequential order or rate difference of events. Generalized two-dimensional (2D) correlation spectroscopy is an extension of the mechanical perturbation-based 2D infrared (IR) method1,2 and was developed to overcome the limitation of * To contact the author. E-mail:
[email protected]. (1) Ozaki, Y., Noda, I., Eds. Two-dimensional Correlation Spectroscopy; American Institute of Physics: Melville, NY, 2000. Noda, I.; Ozaki, Y. Two-dimensional Correlation Spectroscopy: Applications in Vibrational and Optical Spectroscopy; John Wiley & Sons Ltd.: Chichester, UK, 2004. (2) Marcott, C.; Dowrey, A. E.; Noda, I. Anal. Chem. 1994, 66, A1065-A1075. 10.1021/ac0708590 CCC: $37.00 Published on Web 10/06/2007
© 2007 American Chemical Society
the latter where the time-dependent behavior (i.e., waveform) of dynamic spectral intensity variations must be a simple sinusoid in order to effectively employ the original data analysis scheme. It has been claimed that this method is generally applicable to any transient or time-resolved spectra having an arbitrary waveform. More specifically, the spectral variations are not restricted to sinusoidal change at all. Spectral change of any arbitrary waveform can now be analyzed using generalized 2D correlation spectroscopy. Moreover, not only can the time-resolved data be analyzed but also multiple spectra collected under varying conditions, such as temperature, pressure, voltage, concentration, etc., can be analyzed. Furthermore, the extension to various areas of spectroscopy, e.g., UV, near-infrared, Raman and fluorescence spectroscopy, etc., or even to any form of analytical technique (e.g., chromatography, microscopy, and so on) has been found. So-called heterospectral correlation, i.e., the investigation of correlation among bands in two different types of spectroscopy was also introduced. As a result, generalized 2D correlation spectroscopyhasbeenconsideredtobeatremendousimprovement1-3 on the initial mechanical perturbation-based 2D IR method and has been applied to a variety of samples, from simple inorganic and organic compounds to complex biological systems.1-5 The wide application of generalized 2D correlation spectroscopy lies in two major advantages of this technique. One is the enhanced spectral resolution. The other is the so-called “sequential order” rules (sometimes referred to as Noda's rule), applicable to different physical or chemical events.1 The latter has become the more important advantage in generalized 2D correlation spectroscopy. Despite of the generally accepted spectral resolution enhancement, a number of studies,6-18 including our systematic studies in polymers,11-15 have been carried out to investigate the effect (3) Wang, H. C.; Palmer R. A. In Two-dimensional Correlation Spectroscopy; Ozaki, Y., Noda, I.; Eds.; American Institute of Physics: Melville, NY, 2000. (4) Noda, I. Vib. Spectrosc. 2004, 36, 143. (5) Noda, I. J. Mol. Struct. 2006, 799, 2. (6) Greike, A.; Gadaleta, S. J.; Brauner, J. W.; Mendelsohn, R. Biospectroscopy 1996, 2, 341. (7) Czarnecki, M. A. Appl. Spectrosc. 1998, 52, 1583. (8) Czarnecki, M. A. Appl. Spectrosc. 2000, 54, 986. (9) Elmore, D. L.; Dluhy, R. A. Appl. Spectrosc. 2000, 54, 956. (10) Morita, S.; Ozaki, Y. Appl. Spectrosc. 2002, 56, 502. (11) Huang, H. Ph.D. Thesis: Two-dimensional Infrared Correlation Spectroscopy and Its Application to Polymer Materials. The Pennsylvania State University. December 2003. (12) Huang, H.; Malkov, S.; Coleman, M. M.; Painter, P. C. Macromolecules 2003, 36, 8148. (13) Huang, H.; Malkov, S.; Coleman, M. M.; Painter, P. C. Macromolecules 2003, 36, 8156.
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Figure 1. Peak intensities of two bands at 1800 and 1750 cm-1 decreasing at the same rate. The intensity at 1800 cm-1 decreases two steps prior to that at 1750 cm-1.
of variables, such as overlapping, bandwidth change, peak shift and data set used to calculate the 2D plots, on the final results. It has been found that the new features revealed in the 2D plots do not necessarily correspond to real infrared absorption bands. Great care must be taken when applying this technique to various systems. In this report, we are going to cast some further insights on the sequential order rules. Sequential order rules are derived from the signs of a crosspeak in its synchronous and asynchronous spectra. It has been believed that the signs of a cross-peak in generalized 2D correlation spectroscopy could provide useful information on the sequential order of events observed by the spectroscopic technique along the external variable. The sequential order rules are as follows:1 The sign of an asynchronous cross-peak becomes positive if the intensity change at ν1 occurs predominantly before ν2. On the other hand, the peak sign becomes negative if the change at ν1 occurs predominantly after ν2. However, this sign rule is (14) Huang, H.; Malkov, S.; Coleman, M. M; Painter, P. C. J. Phys. Chem. A 2003, 107, 7697. (15) Huang, H.; Malkov, S.; Coleman, M. M.; Painter, P. C. Appl. Spectrosc. 2004, 58, 1074. (16) Lefevre, T.; Arseneault, K.; Pezolet, M. Biopolymers 2004, 73, 705. (17) Wang, Y. W.; Gao, W. Y.; Noda, I.; Yu, Z. W. J. Mol. Struct. 2006, 799, 128. (18) Yu, Z. W.; Wang, Y. W.; Liu, J. Appl. Spectrosc. 2005, 59, 388.
8282 Analytical Chemistry, Vol. 79, No. 21, November 1, 2007
reversed if the synchronous correlation intensity at the same coordinate becomes negative, i.e., Φ (ν1, ν2) < 0. Furthermore, if the intensity of the synchronous cross-peak is zero, the sequential order cannot be determined. The above sequential order rules have been widely used for more than one decade in fields such as small molecules, polymers, and proteins.19-28 It has become a routine way to draw conclusions on the sequential order of events based on the signs of a crosspeak in the 2D plots. The sequential order rules were developed from the mechanical perturbation-based 2D IR, where, as men(19) Robert, P.; Lavenant, L.; Renard. D. Appl. Spectrosc. 2002, 56, 1180. (20) Zhang, J. M.; Duan, Y. X.; Sato, H.; Shen, D. Y.; Yan, S. K.; Noda, I.; Ozaki. Y. J. Phys. Chem. B 2005, 109, 5586. (21) Zhang, J. M.; Duan, Y. X.; Shen, D. Y.;Yan, S. K.; Noda, I.; Ozaki, Y. Macromolecules, 2004, 37, 3292. (22) Wu, P.; Siesler, H. J. Mol. Struct. 2000, 521, 37-47. (23) Wu, P.; Siesler, H. Chem. Phys. Lett. 2003, 374, 74-78. (24) Sasic, S, Amari, T, Ozaki, Y. Anal. Chem. 2001, 73, 5184. Sasic S, Ozaki Y. Anal. Chem. 2001, 73, 2294. (25) Wang, G. F.; Geng, L. Anal. Chem. 2005, 77, 20. Kim, Y. O.; Jung, Y. M.; Kim, S. B.; Park, S. M. Anal. Chem. 2004, 76, 5236. (26) Wu, Y.; Murayama, K.; Ozaki, Y. J. Phys. Chem. B 2001, 105, 6251. (27) Ismoyo, F.; Wang, Y.; Ismail, A. A. Appl. Spectrosc. 2000, 54, 939. Smeller, L.; Heremans, K. Vib. Spectrosc. 1999, 19, 375. (28) Wang, Q.; Sun, S. Q.; Guo, H. B.; Zhou, Q.; Noda, I.; Hu, X. Y. Vib. Spectrosc. 2003, 31, 257-263.
Figure 2. Peak intensities of two bands centered at 1800 and 1750 cm-1 increasing at the same rate. But the intensity at 1800 cm-1 increases two steps prior to that at 1750 cm-1.
tioned earlier, the time-dependent behavior (i.e., waveform) of dynamic spectral intensity variations must be a simple sinusoid. After extension to the generalized 2D correlation spectroscopy, the above sequential order rules were adopted, as 2D correlation is fundamentally a form of parametrization of the integrated or overall relationship between two variable quantities. It must be noted, however, that, in generalized 2D correlation spectroscopy, the dynamic spectral intensity variations are generally nonperiodic. In other words, spectral intensity changes are largely instantaneous and monotonic. The sequential order of events in generalized situations, such as under temperature or pressure perturbations, would be localized or instantaneous, instead of taking an integrated or overall form (we hereafter use the term “local sequential order” instead of sequential order to indicate the instantaneous or local changes in generalized 2D correlation spectroscopy). This is because, in the real world of spectroscopic studies, the changes that a peak can have under various external, nonperiodic perturbations, in general, can be increasing, decreasing, or no change. When one peak's intensity changes, but another one shows no change during a certain period of time, i.e., one set of intensity changes would lag behind another set of changes, a local sequential order would be created. Such local changes are very common in spectroscopic studies. Naturally, it is necessary and important to testify whether or not the sequential order rules, which are derived from periodic data, are applicable to such
instantaneous or local changes. This test, unfortunately, has not been done so far. We therefore first carried out some simulation studies on the application of sequential order rules in generalized situations. Then a theoretical consideration on the correlation analysis was presented. For the reasons to be mentioned below, we also discussed the rate difference of events at the end. EXPERIMENTAL /SECTION Simulations. At least two bands must be considered while comparing the sequential order of their intensity changes. Spectra were simulated using two Gaussian bands centered at 1800 and 1750 cm-1 with half bandwidth of 20 cm-1 and with no peak shift or bandwidth change. For the sake of convenience, the two bands were assumed to have identical initial intensity, and their peak intensity changes take the exponential form. For each simulation, the series was composed of 11 spectra, with spectrum 1 the starting spectrum. Two-Dimensional Correlation Analysis. A standard twodimensional correlation analysis was carried out; i.e., the average (or mean) spectrum of the spectra in the chosen set is subtracted from each of the simulated spectra to obtain a set of “dynamic” spectra. Synchronous and asynchronous correlation spectra are then calculated from these dynamic spectra using the Hilbert transform suggested by Noda.1 The spectra shown on the top and at the side of the 2D correlation maps are the average spectrum. Analytical Chemistry, Vol. 79, No. 21, November 1, 2007
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Figure 3. Intensity at 1800 cm-1 increases, while that at 1750 cm-1 decreases at the same rate. The intensity at 1800 cm-1 changes two steps earlier than that at 1750 cm-1.
In the 2D correlation plots, shaded regions indicate negative correlation intensities, while unshaded ones indicate positive. The calculation level is 16, except where stated elsewhere. The calculation level is associated with the correlation intensity in the 2D plot. The higher the calculation level, the larger the correlation intensity will be. RESULTS AND DISCUSSION Test on the Sequential Order Rules in Generalized Cases. As mentioned earlier, in spectroscopic studies, the instantaneous or local changes that a peak can have under various external, nonperiodic perturbations, in general, can be increasing, decreasing, or no change. To simplify the discussion, we assume that the intensity changes of two bands take the following three rather simple local sequential orders: (I) The peak intensities of two bands decrease at the same rate, but one changes prior to the other. (II) The peak intensities of two bands increase at the same rate, but one changes prior to the other. (III) One peak's intensity increases, while the other one decreases at the same rate, but one changes prior to the other. Case I. In case I, the intensity of the band centered at 1800 cm-1 decreases first, taking the exponential form of y ) e-0.1x. 8284 Analytical Chemistry, Vol. 79, No. 21, November 1, 2007
After two steps (seconds or any other time unit), the intensity of the band centered at 1750 cm-1 starts to decrease, also taking the exponential form of y ) e-0.1x. So that two bands change their intensities at the same rate but at different times, thereby creating a local sequential order between these two events. The simulated spectra are shown in Figure 1A. The normalized intensity values of the two bands at 1800 and 1750 cm-1 plotted against the perturbation steps are presented in Figure 1B. Panels C and D in Figure 1 are the synchronous and asynchronous spectra calculated using the Hilbert-Noda transform of the dynamic spectra (not shown) obtained by mean-centering the spectra shown in Figure 1A. It should be noted that one can see 11 spectra at 1800 cm-1, but only 9 spectra at 1750 cm-1 (Figure 1A). This is because the second and third spectra at 1750 cm-1 are made completely identical to the starting spectrum 1. In this way, the intensity change at 1700 cm-1 was delayed for two steps, giving rise to a local sequential order of the intensity change. After that, the two bands change their intensities at the same rate, as shown in Figure 1A and B. In the synchronous spectrum (Figure 1C), the two cross-peaks at 1800, 1750 and 1750, 1800 cm-1 are positive, indicating that
Figure 4. Intensity at 1800 cm-1 increases faster, while that at 1750 cm-1 decreases simultaneously, but slower. Note: the calculation level in (C) is 32 in order to show the weak auto peak at 1750, 1750 cm-1. Table 1. Signs of the Cross-Peaks in Cases I-III signs of the cross-peaks events simulated
Syna
Asyna
case
1800
1750
1800b
1750b
1800b
1750b
I II III
Vprior vprior vprior
Vafter vafter Vafter
+ + -
+ + -
+ + +
-
a Syn, synchronous spectrum; Asyn, asynchronous spectrum. b 1800 ) (1800, 1750); 1750 ) (1750, 1800).
the intensities at 1800 and 1750 cm-1 are changing in the same direction, as simulated. In the asynchronous spectrum, the cross-peak at 1800, 1750 cm-1 is positive, and the cross-peak at 1750, 1800 cm-1 is negative, both suggesting that the intensity change at 1800 cm-1 is before that at 1750 cm-1, according to the above sequential order rules. This conclusion is consistent with the local sequential order of the real intensity changes of two bands simulated and demonstrates that the above integrated sequential order rules correctly identify the local sequential order of intensity changes, when the intensities of two bands decrease.
Case II. In case I, two bands with decreasing intensity were simulated and analyzed. In this case, two bands with increasing intensity will be investigated. In Figure 2A are the spectra simulated with the intensity at 1800 cm-1 increasing two steps prior to that at 1750 cm-1. Again, one can see 11 spectra at 1800 cm-1, but only 9 spectra at 1750 cm-1. The normalized intensity values of the two bands plotted against the perturbation steps are presented in Figure 2B. Panels C and D in Figure 2 also present the 2D plots obtained. In the synchronous spectrum, the two cross-peaks at 1800, 1750 and 1750, 1800 cm-1 are positive, indicating that they are changing in the same direction, here increasing. This is as expected. In the asynchronous spectrum, the cross-peak at 1800, 1750 cm-1 is positive and cross-peak at 1750, 1800 cm-1 is negative, both suggesting that the intensity change at 1800 cm-1 is before that at 1750 cm-1, based on the sequential order rules. This conclusion is also consistent with the real spectra simulated and demonstrates that the sequential order rules also hold true in this case, when the intensities of two bands increase, with one changing prior to the other. Comparing Figures 1 and 2, one can easily find that the 2D plots are identical to each other; the synchronous correlation intensities of the cross-peaks are positive, i.e., Φ (ν1, ν2) > 0, while the cross-peak at 1800, 1750 cm-1 is positive and the cross-peak Analytical Chemistry, Vol. 79, No. 21, November 1, 2007
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Figure 5. Intensity at 1800 cm-1 decreasing simultaneously but faster than that at 1750 cm-1. Table 2. Signs of Cross-Peaks in Cases IV-VI signs of the cross-peaks events simulated
Syna
Asyna
case
1800
1750
1800b
1750b
1800b
1750b
IV V VI
vfaster Vfaster vfaster
Vslower Vslower vslower
+ +
-+ +
+ + -
+
at 1750, 1800 cm-1 is negative in the asynchronous plots. These results indicate that the integrated sequential order rules correctly identifies the local sequential order of the two events when the synchronous cross-peak is positive, i.e., Φ (ν1, ν2) > 0. In order to test the validity of the integrated sequential order rules in generalized situations, the case when the synchronous correlation intensity is negative, i.e., Φ (ν1, ν2) < 0, must also be investigated. This brings us to case III, where the intensities of two peaks change in different directions. Case III. In case III, the intensity of the band centered at 1800 cm-1 increases first, taking the exponential form of y ) e0.1x. After two steps, the intensity of the band centered at 1750 cm-1 starts to decrease, taking the exponential form of y ) e-0.1x. So that two 8286 Analytical Chemistry, Vol. 79, No. 21, November 1, 2007
bands change their intensities at the same rate but with different directions. The simulated spectra are shown in Figure 3A. Once more, one can see 11 spectra at 1800 cm-1, but only 9 spectra at 1750 cm-1. The normalized intensity values of the two bands at 1800 and 1750 cm-1 plotted against the perturbation steps are presented in Figure 3B. It is of note that case III involves two different waveforms; i.e., one is the reciprocal of the other, though they are both exponential functions. Although such a mixed mode response may complicate the analysis, it does have its physical background, for example, in the case of the melting of semicrystalline polymers; the intensity of a crystalline band will decrease, but that of an amorphous band will increase. In fact, as claimed in the literature, spectral change of any arbitrary waveform can be analyzed using generalized 2D correlation spectroscopy. Therefore, response with two different waveforms should, in principle, not be a problem to the discussion here. Panels C and D in Figure 3 are the synchronous and asynchronous spectra calculated. In the synchronous spectrum (Figure 3C), the two cross-peaks at 1800, 1750 and 1750, 1800 cm-1 are negative, indicating that the intensities at 1800 and 1750 cm-1 are changing in different directions, as simulated. In the asynchronous spectrum, the crosspeak at 1800, 1750 cm-1 is positive and the cross-peak at 1750,
Figure 6. Intensity at 1800 cm-1 increasing simultaneously but faster than that at 1750 cm-1.
1800 cm-1 is negative, both suggesting that the intensity change at 1800 cm-1 is after that at 1750 cm-1, according to the sequential order rules. This conclusion is absolutely right if we are comparing the integrated or overall relationship between these two intensity changes, as also clearly indicated in Figure 4. It is, however, contradictory to the real local sequential order simulated. In fact, the conclusion obtained from the sequential order rules does not make much sense in this case. Discussing the integrated or overall relationship between events such as these two intensity changes is meaningless, because what really matters is the local sequential order of intensity changes of these two bands. This finding demonstrates the breakdown of the above integrated sequential order rules in identifying the local and instantaneous intensity changes, when the intensities of two peaks change in different directions. The signs of the cross-peaks at 1800, 1750 and 1750, 1800 cm-1 in the above three cases are summarized in Table 1. It turns out that the signs of a cross-peak in the 2D plots may correctly provide the information on the local sequential order of events when Φ (ν1, ν2) > 0. The reverse of the sequential order rules seems to be true in identifying the local sequential order of events when Φ (ν1, ν2) < 0. These results lead us to the point that the integrated sequential order rules can only interpret part of the simulated events with certain local sequential orders. It may be concluded that the signs of a cross-peak in its synchronous and asynchronous
spectra do not necessarily indicate the local sequential order of an event. This conclusion can be easily verified by similar simulation studies on events without local sequential orders as discussed in the following section. Events without Local Sequential Order But with Rate Difference. In the real world of spectroscopic studies, the peak intensity changes of two bands may be largely simultaneous, with no local sequential order at all. Similar to the above three cases with local sequential orders, there are also three cases where the peak intensities of two bands change simultaneously: (IV) One peak's intensity increases while the other one decreases simultaneously. (V) The peak intensities of both bands decrease simultaneously. (VI) The peak intensities of both bands increase simultaneously. Then another question arises: what would the 2D correlation spectra look like under these cases? To answer this question, two Gaussian bands, again, were simulated, with their peak intensities changing simultaneously, but at different rates. Case IV. In this case, the peak intensity of the 1800 cm-1 band increases faster, taking the form of y ) e0.2x, while that of 1750 cm-1 decreases simultaneously but slower, taking the form of y ) e-0.1x. In panels A and B in Figure 4 are, respectively, the spectra simulated and the normalized intensity values of the two bands centered at 1800 and 1750 cm-1 plotted against the perturbation Analytical Chemistry, Vol. 79, No. 21, November 1, 2007
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Figure 7. Result obtained by changing the order of data set in Figure 6.
steps. Now, one can see 11 spectra for the band centered at 1750 cm-1 as at 1800 cm-1. The 2D plots are presented in Figure 4C and D. Surprisingly, panels C and D in Figure 4 are identical to panels C and D in Figure 3 in terms of peak signs, though they represent different events. Instead of a prior/after relation in case III, a faster/slower change of intensity at 1800 and 1750 cm-1 was simulated in case IV (Figure 4). This observation clearly indicates that 2D correlation analysis cannot distinguish the local sequential order of events from the rate difference of events, when the intensities of two bands change in different directions, i.e., Φ (ν1, ν2) < 0. It is of note that, if the integrated or overall relationship between the intensity changes of these two bands is compared, a conclusion identical to that in case III (Figure 3) will be obtained; the intensity change at 1800 cm-1 is after that at 1750 cm-1, according to the sequential order rules. This conclusion obviously, again, does not make much sense in terms of revealing the local, instantaneous changes of the band intensities. In generalized cases, it is these local, instantaneous changes that reflect the real situations. Results obtained from the sequential order rules may be misleading. 8288
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Case V. In case V, the peak intensities of two bands centered at 1800 and 1750 cm-1 decrease simultaneously, taking the exponential form of y ) e-0.2x and y ) e-0.1x, respectively. In other words, the intensity of 1800 cm-1 band decreases faster than that of 1750 cm-1. The simulated spectra and the normalized intensity values of the two bands at 1800 and 1750 cm-1 plotted against the perturbation steps are shown in Figure 5A and B. The 2D plots were also presented in Figure 5C and D. Interestingly, Figure 5C and D is identical to Figure 1C and D in terms of peak signs. This result also suggests that the rate difference of events can be easily mistaken as the local sequential order of events, when the peak intensities of two bands decrease, i.e., Φ (ν1, ν2) > 0. (While finalizing this article, we came to know that Czarnecki, in his pioneering work of simulation study on 2D correlation spectroscopy,7 also found that two different physical causes may generate a similar pattern for 2D correlation spectra, based on a brief simulation study on exponentially decaying function similar to here. Unfortunately, such an important conclusion was largely sheltered by the overwhelming effect of factors such as random noise, baseline fluctuations, band position, and widths changes and ignored by the community of 2D correlation spectroscopy).
From the comparison between cases III and IV, as well as cases I and V, it is easy to find that two different events with rate difference or local sequential order may generate the identical 2D patterns in generalized situations. Before any conclusion can be made, we need to compare the result of case II with its counterpart case VI, where the peak intensities of two bands both increase simultaneously but with different rates. Case VI. In this case, the peak intensities of two bands both increase simultaneously. The intensity of 1800 cm-1 band increases faster (y ) e0.2x) than that of 1750 cm-1(y ) e0.1x), as shown in Figure 6A and B. Also shown in Figure 6C and D are the 2D plots. Contradictory to the other two cases (IV and V), the signs of the asynchronous cross-peaks at 1800, 1750 and 1750, 1800 cm-1 are reversed from those of their counterparts as shown in Figure 2, where one peak's intensity increases earlier than the other (case II), making it an exception to the above observations. This exception may be understandable, however. It is of note that the signs of the cross-peaks at 1800, 1750 and 1750, 1800 cm-1 in the asynchronous plot are also different from those of a similar event (case V, in terms of Φ (ν1, ν2) > 0), where two peaks' intensities decrease simultaneously. This is because changing the order of the data in the calculation matrix will change the signs of the cross-peaks in the asynchronous plot.11,29 This conclusion can be easily approved true by changing the data order in Figure 6A and calculating the 2D plots based on the data obtained thereby as shown in Figure 7. Changing the order of the data in Figure 6A equals creating a new event that is similar to that in case V (Figure 5), where both peaks decrease simultaneously but with different rates. The signs of all cross-peaks in cases IV-VI are listed in Table 2. From the above simulation studies, it can be concluded that 2D correlation analysis cannot distinguish the local sequential order of events from the rate difference of events. Naturally, it is crucial to check out if there does exist any local sequential order before a conclusion regarding the local sequential order of events can be made. In 2D correlation spectroscopic studies, the rate difference of an event can be easily mistaken as the local sequential order of an event. In other words, the signs of a cross-peak in its synchronous and asynchronous spectra do not necessarily indicate the local sequential order of events. In fact, this conclusion has its theoretical foundation. Theoretical Analysis. The formal definition of the generalized 2D correlation equation provided in the original paper is rigorous and concise.30 However, the need to employ Fourier transformation of dynamic spectra makes it somewhat cumbersome to compute actual correlation intensities directly from this definition. And the proposed computational method was not described in sufficient detail to be readily adopted for practical computation. Therefore, a more detailed description of a practical method for computing generalized 2D correlation spectra was (29) Czarnik-Matusewicz, B.; Pilorz, S.; Ashton, L.; Blanch. E. W. J. Mol. Struct. 2006, 799, 253. (30) Noda, I. Appl. Spectrosc. 1993, 47, 1329.
presented by Noda, in which the Hilbert transformation was employed.31,32 For a set of spectral data collected at certain discrete intervals of an external physical variable, such as time, temperature, pressure, etc., the dynamic spectra is defined as
˜(v,t) ) y
{
y(v, t) -yj(v) Tmin e t e Tmax 0 other
(1)
jy(ν) is the reference spectrum, typically the time-average spectrum over a period between Tmin and Tmax:
jy(v) )
1 Tmax - Tmin
∫
Tmax
Tmin
y(v,t) dt
(2)
According to the concept of generalized 2D correlation spectroscopy, variable t does not have to be time. It can be concentration, temperature, pressure, etc. The synchronous 2D correlation intensity is then calculated by
Φ(v1,v2) )
m
1 m-1
∑ y˜ (v )•y (v ) 1
j
2
j
(3)
j)1
This equals to the correlation intensity when the correlation time is zero. It has been claimed, as in the mechanical perturbation based 2D IR, that synchronous spectrum represents the simultaneous/in-phase change or overall similarity of the time-dependent behavior of spectral intensity variations measured at two separate wavenumbers. Similarly, asynchronous 2D correlation intensity can be calculated by
Ψ(v1,v2) )
m
1 m-1
∑ y˜ (v )z˜ (v ) j
1
j
2
(4)
j)1
, with the need for a discrete orthogonal spectrum ˜zj(v2), which can be obtained from the dynamic spectrum y˜k(v2) by using a simple linear transformation operation m
∑ N y˜ (v )
(5)
0 if j ) k 1/π(k - j) otherwise
(6)
˜zj(v2) )
jk k
2
k)1
where
Njk )
{
Because the dynamic spectrum vector ˜zj(v2) is orthogonal to ˜yk(v2), this transformation is taken as shifting the phase of each Fourier component of a function forward or backward by π/2. It is generally believed that an asynchronous spectrum represents the out-of-phase/quadrature change or overall dissimilarity of the (31) Noda, I. Presented at the 2nd International symposium on Advanced Infrared Spectroscopy, Durham, NC, 1996; Paper A-16. (32) Noda, I. Appl. Spectrosc. 2000, 54, 994.
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time-dependent behavior of spectral intensity variations measured at two separate wavenumbers. In matrix notation, the above dynamic spectra can be written as
[ ]
˜y(v,t1) y˜(v,t2) y˜(v) ) ‚‚‚ y˜(v,tm)
Table 3. Intensity Changes in Cases I-III Fitted into Exponential Forms case
1800 cm-1
1750 cm-1
I II III
y ) 55.26e-0.1x y ) 40.94e0.2x y ) 45.24e0.1x
y ) 60.85e-0.09x y ) 33.76e0.17x y ) 60.85e-0.09x
(7) Table 4. Signs of Cross-Peaks in All Six Cases signs of the cross-peaks
Synchronous and asynchronous spectra can be calculated as follows:
1 y˜(v1)T˜y(v2) Φ(v1,v2) ) m-1 Ψ(v1,v2) )
1 y˜(v1)TΝy˜(v2) m-1
[ ]
(8) (9)
where N is the Hilbert-Noda transformation matrix, given by
0
-1 1 1 N) π -2 1 3 ‚‚‚
1
1 2
0
1
-1 0
1 3 1 2
‚‚‚
1
‚‚‚
‚‚‚
(10)
1 - 1 0 ‚‚‚ 2 ‚‚‚ ‚‚‚ ‚‚‚ ‚‚‚ -
It is clear that the Hilbert-Noda transform method makes calculation of 2D maps easy and practical, and this is the common practice of calculating generalized 2D plots. Nevertheless, such a correlation analysis may not work as well as expected in general. From the calculation methodology presented above, it can be seen that generalized 2D spectroscopy is derived from discrete/ nonperiodic data.32 However, the theory and terminology of the original mechanical perturbation-based 2D spectroscopy are automatically transferred to the generalized methodology. It is this transformation that has led to various misinterpretations of results. It is worth pointing out that the discrete form of a synchronous spectrum is represented by the inner product of two dynamic spectrum vectors with the time-average spectrum as a reference. Simply speaking, a synchronous spectrum is the average product of two dynamic spectrum vectors. This is in fact the classical method in engineering applications to determine the degree of linear dependence11,12,33-35 of two variables. The nature of an asynchronous spectrum is similar to the synchronous spectrum, i.e., nonlinear dependence of two variables. Therefore, synchronous and asynchronous spectra in generalized 2D correlation spectroscopy may also represent the linear and nonlinear relationship, in addition to the well-known integrated sequential order of events. In other words, synchronous and asynchronous spectra (33) Bendat, J. S.; Piersol, A. G. Engineering Applications of Correlation and Spectral Analysis; 2nd ed.; John Wiley and Sons: New York, 1993. (34) Isaksson, T.; Katsumoto, Y.; Ozaki, Y.; Noda, I. Appl. Spectrosc. 2002, 56, 1289. (35) Lefevre, T.; Pezolet, M. J. Phys. Chem. A 2003, 107, 6366.
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events simulated
Syna
Asyna
1800b 1750b 1800b 1750b
case
1800
1750
I II III IV V VI
Vprior vprior vprior vfaster Vfaster vfaster
Vafter vafter Vafter Vslower Vslower vslower
+ + + +
+ + + +
+ + + + + -
+
rate difference 1800
1750
faster faster faster faster faster faster
slower slower slower slower slower slower
a Syn, synchronous spectrum; Asyn, asynchronous spectrum. b 1800 ) (1800, 1750); 1750 ) (1750, 1800).
do not necessarily provide the information regarding the simultaneous/in-phase or out-of-phase/quadrature change of events. In conclusion, the signs of a cross-peak in the generalized 2D correlation spectroscopy do not necessarily provide the information on the sequential order of events. The above conclusion can be further verified by a theoretical analysis36 on the 2D correlation spectroscopy based on exponential decay dynamic spectra, similar to case V discussed in this report. Consider the time-dependent spectral intensities with exponential decay dynamic behavior at wavenumbers ν1 and ν2. The dynamic spectral intensities are given by
˜(ν1,t) ) A(ν1)e-k(ν1)t y
(11)
˜(ν2,t) ) A(ν2)e-k(ν2)t y
(12)
where A(ν1) and A(ν2) are intensity coefficients and k(ν1) and k(ν2) are the characteristic rate constants of the decay process at wavenumbers ν1 and ν2. The synchronous and asynchronous correlation intensities as well as the phase angle of the correlated spectral intensities are given by
1 1 T k(ν1) + k(ν2)
(13)
1 lnk(ν2) - lnk(ν1) T π(k(ν1) + k(ν2))
(14)
Φ(v1,v2) )
Ψ(v1,v2) )
Θ(v1,v2) ) tan-1
lnk(ν2) - lnk(ν1) π
(15)
Ψ(v1,v2) will be zero only if the rate constants for the decay processes at the two wavenumbers are the same. In other words, (36) Ekgasit, S.; Ishida, H. Appl. Spectrosc. 1995, 49, 1243.
Figure 8. Normalized intensity values of two bands at 1800 and 1750 cm-1 plotted against the perturbation steps.
any two decay processes having different rates will assume nonzero values of asynchronous correlation intensity, no matter if local sequential order exists or not, as demonstrated in previous sections (Figures 1 and 5). For two simultaneous decay processes having different rates (as in case V), one may still calculate the Θ(v1,v2) value, the so-called phase angle, from eq 15. This value, however, does not make any physical sense regarding the local sequential order of the events. For two decay processes having the same rates but with a local sequential order (as case I), their “real” decay rates are actually different (more on this below), leading to the nonzero values of asynchronous correlation intensity. These findings clearly demonstrate that the signs of a cross-peak in the generalized 2D correlation spectroscopy do not necessarily provide the information on the local sequential order of events. They may also provide the information on the rate difference of events. In fact, in the original paper of the generalized 2D correlation method published in 1993 by Noda,30 a similar interpretation in terms of rate difference was adopted to the exponentially decaying spectra. Then, a question can be asked; what do the signs of a crosspeak really imply in generalized 2D correlation spectroscopy? This question brings us to the next section. Local Sequential Order or Rate Difference? Based on the above simulation results and theoretical discussions, and considering the discrete nonperiodic characteristic of data points used in real calculation of 2D correlation spectroscopy, one may speculate that the signs of a cross-peak may provide the information on the rate difference rather than the local sequential order of events in a more general sense. That is to say, the result in Figure 1, 2, or 3 will also indicate a faster intensity change at 1800 cm-1, but a slower change at 1750 cm-1, which can be easily verified by fitting the intensity data at 1750 and 1800 cm-1 in Figures 1, 2, and 3 into exponential forms. The results are compiled together in Table 3. It is clear that, though the rates of intensity changes at the two wavenumbers were created equal in each case, the existence of a local sequential order between these two events changed the final rates of intensity changes. Table 4 summarizes the signs of cross-peaks in all the six cases. From case I to V, a positive/negative asynchronous crosspeak suggests that the intensity change at ν1 occurs faster/slower
than at ν2, no matter the synchronous correlation intensity at the same coordinate is positive or negative. The above results may look pretty good at the first sight. It cannot explain, however, why the signs of the asynchronous peaks in case VI are different from those in cases I-V. It also cannot explain why the signs of asynchronous peaks in cases II and VI are different, though they represent a quite similar event; i.e., the intensity change at 1800 cm-1 is faster than that at 1750 cm-1. These results suggest that the signs of a cross-peak do not necessarily provide the information on the rate difference of events either. More specifically, 2D correlation can't distinguish the localized differences in the specific rate of signal changes. This statement can be more clearly verified by the following straightforward simulation. We can compare the following two different cases, as shown in Figure 8A and B. In Figure 8A, the rate of change of the intensity at 1800 cm-1 is slower than that at 1750 cm-1 during the observation period. On the other hand, in Figure 8 B, the rate of change of the intensity at 1800 cm-1 is faster than that at 1750 cm-1 during the same observation period. If 2D correlation is sensitive to the difference in the rate of intensity changes, we should expect to see the difference in the 2D correlation spectra between these two cases. In fact, the cross-peak signs are identical to each other for both cases (not shown). Obviously, it is also not safe to draw conclusions on the relative rate difference of events based on the signs of the cross-peaks. In summary, the synchronous and asynchronous spectra in the generalized 2D correlation spectroscopy do not necessarily provide the information on the local sequential order or the rate difference of events. Further research work is needed to tackle the local sequential order or the rate difference of events investigated by using the generalized 2D correlation spectroscopy. CONCLUSIONS (1) Simulation studies demonstrate that the popularly employed sequential order rules can only partially interpret the events with different local sequential orders in generalized situations. (2) 2D correlation analysis cannot distinguish the local sequential order of events from rate difference of events. (3) A theoretical analysis shows that the synchronous and asynchronous spectra in the generalized 2D correlation spectroscopy do not necessarily provide the information on the local sequential order of events. They may Analytical Chemistry, Vol. 79, No. 21, November 1, 2007
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also represent the linear/nonlinear relationship between two events. (4) The signs of a cross-peak in 2D plots also do not necessarily provide the information on the rate difference of two events. ACKNOWLEDGMENT This material is based upon work supported by the National Natural Science Foundation of China under grants 20674016 and
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20774024. The critical and detailed comments from Dr. Isao Noda, the founder of two-dimensional correlation spectroscopy, are gratefully acknowledged. Received for review April 26, 2007. Accepted September 1, 2007. AC0708590
