2d Correlation Spectroscopy and Its Application in Vibrational Spectroscopy Using Matlab
نویسندگان
چکیده
Two dimensional correlation spectroscopy is a powerfull tool for spectral analysis. It is able to reveal correlations between spectral changes and to deconvolve overlapping peaks. It is easily applicable in a study of biomolecules. Here presented program was created for easy accessibility of all necessary operations. Presented algorithm was deeply tested on regular spectra as well as on a spectra of vibrational optical activity. Basic mathematics needed for understanding and performing two-dimensional correlation spectroscopy was also presented. 1 Two dimensional correlation spectroscopy Two dimensional correlation spectroscopy (2DCoS) originated in a NMR spectroscopy. There were efforts to use it in different methods of optical spectroscopy since its development. The main reason is that 2DCos has an ability to determine an order of bands and also has superior deconvolution abilities. This effort has run into troubles with a time scale because very short light pulses are needed for excitation. That wasn’t possible until construction of femtosecond lasers and therefore existed an effort for creating a generalised form of 2DCoS. Practical use of so called Generalised 2DCoS was enabled in 1986 when Noda [1] utilised its basic theory. In Generalised 2DCos external impulses (pH, temperature, concentration etc.) are used to stimulate the system instead of short excitation pulses. Excited system is then analysed by one or more spectroscopic techniques. This spectra are then processed by a correlation function into a 2D spectra. This method allows us to study the response of our system on different physical-chemical impulses with use of conventional spectrometers and without need of femtosecond lasers and optical stimulation [2]. It also provides information about order of spectral bands in dependency on external impulse and has very strong deconvoluting abilities. The first part of this spectra (so called synchronous) is representing simultaneous or coincidental changes of measured spectral series. This spectrum is allways symetrical along the diagonal and has peaks on the diagonal. Intensity of these peaks (also called autocorrelation) is representing a strength of the band. The peaks off the diagonal are called cross-peaks and are representing a degree of correlation. When this peak is positive then both peaks are changing in the same direction (both increasing or both decreasing). When negative, peaks are changing in the different way (one is decreasing and the other is increasing). These rules are reversed when these peaks have different signs. Second part of this spectra (asynchronous) is representing sequential or successive changes of measured spectral series. It is always antisymetrical along the diagonal and there are no peaks on the diagonal. When the cross-peak is positive then a band from the first spectra is growing earlier or more intensive then a band from second spectra and vice versa. 2DCoS can be performed in two ways. First of them is called homospectral correlation and there are two identical assemblies of data (spectra) on the input. This method can be used to deconvolve and to determine correlations between bands in the spectra. The second one is called heterospectral correlation and there are two different sets of data on the input. They are obtained by measuring the system with different spectroscopic techniques. This is the most powerfull variant because when we know the explanation of some peaks in one type of spectra and we see that this band is strongly correlated with another band in the second spectra it is most probable that these two bands have the same origin. Figure 1: Synchronous (A) and asynchronous (B) spectra with marked correlation squares and peaks. Some basic properties of correlation spectra are marked on figure 1. We can see diagonal peaks in the synchronous part and a symetrical positive cross-peak which indicates that both bands are changing in the same way. There are no diagonal peaks in the asynchronous part and from negative peak in the lower right corner we can say that band on 70 is changing slower than band on 30 (positive peak in the upper left corner is saying the same). In both cases we can draw so called correlation square which is connecting all peaks belonging to two correlating bands. 2 Computation of 2D spectra First of all we need a set of spectra measured on a system which was induced by some external impulse. Spectral intensities of this set can be expressed as I(ν, t) where ν is a spectral characteristic (wavenumber, wavelength or Raman shift) and t is a parameter of an external impulse. That can be an evolution in time, temperature, pH, concentration etc. Only certain range of t can be measured and therefore a dynamical spectrum is implemented as [3]: ỹ(ν, t) = { y(ν, t)− ỹ(ν) if Tmin ≤ t ≤ Tmax, 0 otherwise (1) where ỹ(ν) is a reference spectrum of the system. An average spectrum is usually picked as a reference spectrum but any reasonable spectrum can be chosen. Correlation spectrum is then defined as: χ(ν1, ν2) = 〈ỹ(ν1, t) · ỹ(ν2, t)〉 (2) where the symbol 〈〉 is a crosscorrelation function defined as: C(t, τ) = 〈Φ(τ)|Ψ(t)〉 (3) This spectra expresses a functional dependency between ν1 and ν2 during the interval of external variable t. We will treat this function as a complex one for further simplifying and divide it into a synchronous and asynchronous part [4]: χ(ν1, ν2) = Φ(ν1, ν2) + iΨ(ν1, ν2) (4) and formal definition of 2D correlation spectra is given by: Φ(ν1, ν2) + iΨ(ν1, ν2) = 1 π(Tmax − Tmin) × ∫ ∞ 0 Ỹ1(ω) · Ỹ2(ω)dω (5) The expression Ỹ1(ω) is a Fourier transformation of spectral intensity variations ỹ1(ν1, t) measured in certain spectral variable ν1 during an external impulse t. It can be written as: Ỹ1(ω) = ∫ ∞ −∞ ỹ(ν1, t)e−iωtdt (6) As it is clear from Eq. 1, this Fourier transformation is related with a strict interval of an external impulse between Tmin and Tmax. Ỹ2(ω) is defined in a similar way but it is an inverse Fourier transform. It is possible, with use of classical timeseries analysis and Wiener-Khintchinsky theorem [5], to express synchronous and asynchronous part: Φ(ν1, ν2) = 1 Tmax − Tmin ∫ Tmax Tmin ỹ(ν1, t) · ỹ(ν2, t)dt (7) Ψ(ν1, ν2) = 1 Tmax − Tmin ∫ Tmax Tmin ỹ(ν1, t) · z̃(ν2, t)dt (8) where z̃(ν2, t) is defined as follows:
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