# Two dimensional correlation spectroscopy applications in vibratioal and optical spectroscopy - Isao N.

ISBN 0-471-62391-1

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2.2 GENERALIZED TWO-DIMENSIONAL CORRELATION

2.2.1 DYNAMIC SPECTRUM

Let us consider a perturbation-induced variation of a spectral intensity y(v,t) observed during a fixed interval of some external variable t between Tmm and Tmax. While this external variable t in many cases is the conventional chronological time, it can also be any other reasonable measure of a physical quantity, such as temperature, pressure, concentration, voltage, etc., depending on the type of experiment. The variable v can be any appropriate spectral index used in the field of spectroscopy, including Raman shift, wavenumber or wavelength in IR, NIR and UV-visible studies, scattering angles of X-ray or neutron beam, etc.

The dynamic spectrum y(v,t) of a system induced by the application of an external perturbation is formally defined as

| y(y> t) y(v) for Tmjn <t< Tmax (21)

y ( ’ ) _ | 0 otherwise ( ' )

where y(v) is the reference spectrum of the system. While selection of a proper reference spectrum is not strictly specified, in most cases, it is customary to set y(v) to be the stationary or averaged spectrum given by

1 p Tmax

y(v) = vf, _ T I y(v,t)dt (2.2)

T max T min J Tmin

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Principle of Two-dimensional Correlation Spectroscopy

In some applications, however, it is possible to select a different type of reference spectrum by choosing a spectrum observed at some fixed reference point t = rref, i.e., y{v) = y{v, rref). For example, a reference point can be chosen as the original or ground state of the system, sometimes well before the application of the perturbation (Tref ^ —co). It can also be picked at the beginning (Tref = Tmin) or the end (Tref = Tmax) of the course of spectral measurement period, or even well after the full relaxation of the perturbation effect (Tref ^ +o). The reference spectrum could also be set simply equal to zero; in that case, the dynamic spectrum is identical to the observed variation of the spectral intensity. Each selection of the reference spectrum has its own merit for the specific type of 2D correlation analysis. Without any prior knowledge about the specific physical origin of the dynamic spectrum, the reference spectrum defined by Equation (2.2) probably provides the most robust and preferred form to be used for the correlation analysis.

2.2.2 TWO-DIMENSIONAL CORRELATION CONCEPT

The fundamental concept governing 2D correlation spectroscopy is a quantitative comparison of the patterns of spectral intensity variations along the external variable t observed at two different spectral variables, vi and v2, over some finite observation interval between Tmin and Tmax. The 2D correlation spectrum can be expressed as

X(vi, V2) = (y(vi, t) ? y (V2, t')} (2.3)

The intensity of 2D correlation spectrum X(v1, v2) represents the quantitative measure of a comparative similarity or dissimilarity of spectral intensity variations y(v,t) measured at two different spectral variables, v1 and v2, during a fixed interval. The symbol (} denotes for a cross-correlation function designed to compare the dependence patterns of two chosen quantities on t. The correlation function generically defined by Equation (2.3) is calculated between the spectral intensity variations measured at two independently chosen spectral variables, v1 and v2, which gives the basic two-dimensional nature of this particular correlation analysis.

In order to simplify the mathematical manipulation, we treat X(v1, v2) as a complex number function

X(V1, V2) = 0(V1, V2) + i^(V1, V2) (2.4)

comprising two orthogonal (i.e., real and imaginary) components, known respectively as the synchronous and asynchronous 2D correlation intensities. The synchronous 2D correlation intensity 0(v1, v2) represents the overall similarity or coincidental trends between two separate intensity variations measured

Generalized Two-dimensional Correlation

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at different spectral variables, as the value of t is scanned from Tmin to Tmax. The asynchronous 2D correlation intensity ^(v1; v2), on the other hand, may be regarded as a measure of dissimilarity or, more strictly speaking, out-of-phase character of the spectral intensity variations.

The terminology, such as the synchronous or asynchronous spectrum, was adopted for purely historical reasons. Because earlier conceptual development of perturbation-based 2D correlation analysis had relied heavily on the framework of statistical time-series analysis, the variable t associated with the external perturbation was originally assumed to be the chronological time.5 6 With the generalized scheme of 2D correlation depicted in Figure 2.1, the variable t can be any reasonable physical quantity, such as temperature, pressure, concentration, and so on. However, in order to avoid the unnecessary coinage of awkward terms, such as the synthermal or asynbaric spectrum, traditional terms like synchronous and asynchronous spectrum will be consistently used to refer to the real and imaginary components of the complex 2D correlation spectrum.

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