SPANA for Spectral Data Analyses [Example] Copyright (c) 2010 ykuroda All right reserved.

Contents
1. Example of Equilibrium Analyses
2. Use of Multi-Titration Data
3. Titration by Spectroscopically Active Titrant
4. Analyses of Data Accompanying Dilution during Titration
5. Inverse Titration
6. Reduction of Number of Optimized Parameters (Fixation of Parameter Values)
7. Estimation of Spectra of Intermediates
8. Concentration of Chemical Species during Titration
9. Simulation
10. Wave Separation
11. Analyses of Multicomponent Spectra
12. 3D Ｖｉｅｗ
13. Analyses of NMR Data
14. Analyses of Concentration Data
15. Least Square Analyses for User's Function

1. Example of Equilibrium Analysis

a) Theoretical Model
   The analysis of the simple sequential 1:2 complex formation system, where the solution of A is titrated with B, is taken up as an example. The equilibria are

                             A    +    B    ↔    AB          K1 = [AB]/[A][B]          eq.1
                           AB    +    B    ↔    BAB        K2 = [BAB]/[AB][B]      eq.2
and
                           [A]₀ = [A] + [AB] + [BAB]              eq.3
                           [B]₀ = [B] + [AB] + 2[BAB]            eq.4
Here, [A]₀ is the initial (total) concentration of A and [B]₀ is the total concentration of the titrant, B. If B is spectroscopically transparent, the absorbance of the sample solution, Abs, is expressed as
                           Abs = ε_A[A] + ε_AB[AB] + ε_BAB[BAB]         eq.5
The difference spectra, ΔAbs, is obtained by using eq.3 and 5 as follow
                           ΔAbs = Abs - ε_A[A] ₀ = Δε₁[AB] + Δε₂[BAB]   eq.6
where
                           Δε₁= ε_AB - ε_A   ,    Δε₂= ε_BAB - ε_A
Since [AB] and [BAB] can be evaluated by using eq.1 - 4 with given K1, K2, [A]₀ and [B]₀, the theoretically estimated ΔAbs is obtained from eq.6.
The least square calculation procedure determines the best parameter set, K1, K2, Δε₁, and Δε₂, to minimize the following residual square sum, ss, resulting the best fitting between the theoretical difference spectra, ΔAbs, and the observed ones, ΔAbs_obs.
                           ss = Σ（ΔAbs - ΔAbs_obs)².

b) Analysis of Titration Data

   The titration is usually performed under the condition of [A]₀ = constant during the experiment. According to eq.6, the values of ΔAbs_obs are directly obtained by subtracting the initial spectra,ε_A[A] ₀ ([B]₀ = 0), from the observed titration spectra.
The practical procedures of the analyses of following spectral data set (Nondilute1-2_example.ana) are described below.

First, these 16 spectra are read and displayed on the screen of SPANA ("View-Activate All" menu or "click on Ch.0" and "shift + click on Ch.15") (Fig.1),

Fig.1 Spectra of original data

the display area is adjusted by using the "Format" dialog box ("View-Format" menu) (in this case, the "Auto Format" box is checked) (Fig.2),

Fig.2

to display all spectra on the full screen (Fig.3).

Fig. 3 Expanded original spectra

Then, the concentrations, [B]₀, for each spectrum are inputted in the "Condition ( t )" items in the "Item Change" dialog box ("Channel-Item Edit" menu) (Fig.4),

Fig.4

and the resultant spectra are overwritten on channel 0 -15 by assigning "0-" for "First Channel", where the hyphen, "-", is a keyword for successive writing (the channel names are automatically changed into Ch.0' and so on) (Fig.5).

Fig.5 Setting of concentration

Before making ΔAbs_obs, the 0-base lines of the spectra should be corrected by using the "Spectrum-Base Line-Average Correction" function assigning the area of 300 - 330 nm with shown vertical line-cursor (Fig.7).

Fig.6

The resultant spectra are stored in the Channel 17 - 32 by setting "17-" as the target channels in the "Base Line" dialog box. (Fig.6).

Fig.7 Base-line correction between 300-330 nm

The base-line corrected spectra are displayed by clearing the original spectra ("View-Deactivate All/Cls" menu) and activating (displaying) channel 17-32 by clicking on channel 17 and shift-clicking on channel 32 (selection of all channel between Channel 17 and 32) (Fig.8).

Fig.8 Base-line corrected spectra

Then, in order to making the ΔAbs_obs spectra, the conditions for the subtraction are set in the "Data Edit" dialog box ("Spectrum-"-"-Channel" menu), i.e., Ch. "All" for the target spectra, the subtracter, ε_A[A] ₀, channel (Ch."17"), and the channels which store the resulting difference spectra (Ch."34-") (Fig.9),

Fig.9

where "All" is a keyword meaning all active spectra and the resulting spectra are stored in successive Ch.34-49. The resulting difference spectra are shown as below (by changing the display status and format) (Fig.10).

Fig.10 Difference spectra

Then, the sampling points for the least square calculation are collected by the cursor operation ("Analyses-Least Square Analyses-Point Data" menu) (Fig.11),

Fig.11 Data sampling at 415, 450 and 516nm

where the data are sampled at 415nm (single click), 450nm (single click) and 516nm (double click for the final point) . In the cursor operations for data sampling, "single-click" at an arbitrary position is used for successive sampling and "double-click" for the final sampling to show the table of collected data in the "Result" table (Fig.12).*

Fig.12

Click of the "Least Sq." button shows the "Least Square Analyses" dialog box to select the appropriate model for optimization (Fig.13).

Fig.13

In this case, select "Sample : Constant" , and "Sequential 1 : n Complex Formation System - n : 2" in the page of Equilibrium Models 1, and click the "Go" button to show the "Least Square Optimization" dialog box (Fig.14).

Fig.14

The dialog box shows the parameters to be optimized, K1, K2, Δε₁, Δε₂ (shown as Δε*1) at 415nm, Δε₁, Δε₂(Δε*2) at 450nm, Δε₁, Δε₂(Δε*3) at 516nm. Set the concentration A and initial guess values for the parameters and click the "Calc." button to start calculation.

Fig.15

The final optimized parameters are shown in the dialog box together with their standard deviations (sd) and the final residual square sum, ss (Fig.15).
In order to show the results graphically, click the "Plot" button to show the "Graph Format" dialog box (Fig.16).

Fig.16

Check the "Auto Format" box and click "OK" button, then the graph plotting the observed data and the theoretical curves is shown as below (Fig.17).

Fig.17 Plotting graph of [B]₀ vs. observed data and titration curves at 516nm (red), at 450nm (purple) and at 415nm (yellow)

* It should be noted that the reliability of the results is significantly depend on the data extraction and the model selection. For the example shown here, if the data at ~520nm changing monotonously are employed for analyses, the simple 1:1 complex formation model gives good fitting too.

2. Use of Multi-Titration Data
   Data sampling at multi-wave length is effective in obtaining statistically accurate results. Another more effective procedure is use of multi-titration data which are collected from the independent titration experiments. For example, SPANA can analyzes the data collected from the titration experiments at the different [A]₀ all together, which are arranged as data blocks separated by, at least, one blank channel as shown below for the data in "Nodilute1-1-multi.ana" file containing the following data sets (Fig.18).
                       Set 1 : Ch. 15 - 24 ( [A]₀ = 2 x 10^-5 M )
                       Set 2 : Ch. 28 - 38 ( [A]₀ = 1 x 10^-5 M )
                       Set 3 : Ch. 42 - 51 ( [A]₀ = 5 x 10^-6 M )

Fig.18 Spectra of data set 1 (white), 2 (light blue) and 3 (red).

   The different points than the standard procedures for analyses of the single data set is that of calculation setting, i.e., after sampling the data and selecting the theoretical model, set "Number of Data Set = 3" in the "Least Square Optimization" dialog box, and, then, the additional information, the first channel and [A]₀ value for each data set (Fig.19).

Fig.19

3. Titration by Spectroscopically Active Titrant
   In the spectroscopic titration, the spectroscopically inactive (transparent) titrant is usually chosen, because the spectroscopically active titrant results in complex spectrum transformation due to overlapping of the effects of complex formation and titrant addition. However, if the absorbances of the titrant are so small that the Lambert-Beer's low is applicable during the experiment, the spectroscopic data are correctable for the extra absorbances of the titrant.
In such case, the observed absorbance, Abs, for the 1:2 complexation system is
                          Abs = ε_A[A] + ε_B[B] + ε_AB[AB] + ε_BAB[BAB]         eq.5'
and the difference spectrum is
                          ΔAbs' = Abs - ε_A[A] ₀ -ε_B[B] ₀= Δε₁[AB] + Δε₂[BAB]   eq.6'
where
                          Δε₁= ε_AB - ε_A  - ε_B  ,    Δε₂= ε_BAB - ε_A - 2ε_B
The values of ε_B[B] ₀ are not constant and must be evaluated for each titration data, independently. However, since the careful experiment make it possible to evaluate the ε_B[B] ₀ in SPANA, experimentally or calculably (by, for example, using standard spectrum of a known concentration), the optimization analyses become possible by using eq.6' which has the same form as eq.6.

4. Analyses of Data Accompanying Dilution during Titration
   Usually, the concentration of the titrated sample, [A]₀, must be kept constant during titration to obtain meaningful results. The experimental procedures, however, are sometimes troublesome and wasteful of the samples. If the Lambert-Beer's low is true during the experiment, the spectroscopic data which accompany sample dilution caused by addition of the titrant solution are correctable without losing accuracy.
   For example, the theoretical model of the 1:2 complexation model shown above is modified as follows,

                           [A]₀ = (V/(V + v)) [A]_org = [A] + [AB] + [BAB]              eq.3'
                           [B]₀ = (v/(V + v)) [B]_org = [B] + [AB] + 2[BAB]            eq.4'

where [A]_org, [B]_org , V, and v are the original concentrations of A, B, the initial volume of A, and the added volume of the titrant, B, respectively. Then, the difference spectra are modified,

                  ΔAbs^dil = ((V + v)/V) Abs - ε_A[A]_org
                              = ((V + v)/V) (Δε₁[AB] + Δε₂[BAB] )                  eq.6'

Thus, the difference spectra, ΔAbs^dil, made by subtraction of the initial spectrum, ε_A[A]_org, from the observed titration spectra modified by the factor of (V + v)/V can be analyzed by the least square optimization as the "ΔAbs^dil vs. v" system. SPANA has the special function which modifies the data in this manner by using the "Concentration Correction" dialog box (the "Spectrum-Conc.Correct" menu) (Fig.20).

Fig.20

In this example, the modified spectra, ((V + v)/V) Abs, are calculated for original observed spectra in Ch. 0-14 by using V=3000μL and v=0-300μL, and stored in "Ch.20 -" .
Thus, making ΔAbs^dil, selecting an appropriate "Diluted" model (for example, "1:2 Complexation System-Sequential Eq.-Diluted") in the "Least Square Analyses" dialog box and setting the necessary parameters, [A]_org, [B]_org, and V, required in the "Least Square Optimization" dialog box (Fig.21), the optimization is performed for the "ΔAbs^dil vs. v" system to give the equilibrium constants and difference molar extinction coefficients.

Fig.21

5. Inverse Titration
   For the titration of 1 : n (A : B) multi-complex formation system, B is often chosen as the titrant, because the normal saturation behavior, formation of the 1 : n complex at the infinite concentration of the titrant, is expected. In the case of the combination of spectroscopically transparent A and avtive B, the titration experiment using A as the titrant, however, is also possible, though the saturation at the infinite concentration of the titrant means formation of the 1 : 1 complex. For the analyses of such titration, change the titrant species (A) in the "Least Square Optimization" dialog box (Fig.22).

Fig.22

6. Reduction of Number of Optimized Parameters (Fixation of Parameter Values)
   In the least square calculation optimizing many parameters, there may be the difficulty that the calculation can find no appropriate converging point for the best fitting and/or falls into a local minimum of ss. If some parameters are determinable by other experiments, there is a chance that these difficulties are avoided by fixing the known parameter values.
   Checking the check box beside the parameter display area, the parameter is treated as the constant and removed from optimization calculation (see Fig.22).

7. Estimation of Spectra of Intermediates
   SPANA has the function for estimating the intermediary spectra which are not directly observed. The "Calc." operation with check of the "Spectra estimation" check box in the "Least Square Optimization" dialog box shows the "Estimated Spectra" dialog box (Fig.23) after optimization

Fig.23

For the present sequential 1:2 complex formation system, the estimated difference spectra for AB and BAB are calculated and stored in the assigned channels. The spectra of AB and BAB are obtained by adding the initial spectrum, ε_A[A] ₀, to the resulting difference spectra.

8. Concentration of Chemical Species during Titration
    The "Plot" operation with check of the "Conc. Sim." box shows the concentration of the chemical species existing in the system during titration graphically. In the case of the present sequential 1 : 2 complex formation system, the graph of [A], [B], [AB], and [BAB] vs. [B]₀ are prepared (Fig.24).

Fig.24 Concentration variations of [A] (white), [AB] (yellow) and [BAB] (purple).
[B] is not shown graphically because of its too high concentration. 

9. Simulation
The "Plot" operation in the "Least Square Optimization" dialog box makes the necessary graph by using the numerical values set in the parameter text boxes^.* Therefore, setting the appropriate numerical vlues there manually, the "Plot" operation makes the graph simulated by using these parameters, which can be compared with the optimised one. (Fig..25)

　　　　　　　　　　
　　　　　　　　　　　　　　Fig.25

* The simulation mode can be also invoked directly from the "Analysis - Process Simulation" menu.

10. Wave Separation
  SPANA separates the overlapped spectra by using Gauss, Lorentz or Voigt functions. For example, the procedures in which the spectrum in the "Overlap.ana" file is separated by Gauss functions is shown below (Fig.26).

Fig.26 Sample spectrum for wave separation

  At first, the Gaussians expected in the spectrum are roughly defined by using the "+" cursor (the "Analysis-Wave Separation-Manual-Gauss" menu). Left click of this cursor leaves a bright point on the screen and the Gaussian (dark green Gaussian in Fig.24) is generated every these 3 bright points (the top and two points showing the half width at half maximum of Gaussian are usually easy to define). The selection whether the generated Gaussian should be used for separation is assigned on the "Wave Set" dialog box, i.e., the "Cancel" button for discard, or the "Set" for the adoption which continues the operation of another definition of Gaussian, or the "OK" which means that the Gaussian is adopted as the final Gaussian component for the calculation (Fig.27).

Fig.27 Generation of Gaussian component

 After defining all Gaussians (in the present case, 3 Gaussians are defined), the wave length range where the optimization is applied is indicated with the vertical cursor shown on the screen (in this case, 375- 451 nm) and the channels which store the resulting Gaussian components are assigned as usual (Fig.28).

Fig.28 Setting of simulation range

 After completion of calculation, the Gaussian components (Gaussian 1-3) and final simulated spectrum are stored in the suggested channels , in this case, Ch. 2-4 and Ch.5, respectively (Fig.29).。

Fig.29 The spectra of original (white), simulated (red), and elemental Gaussians (dark blue, dark green and dark red)

Note : In the default status, the calculation of the wave separation is performed by using the unit of the wave number, Kcm^-1 (alteration is possible). The numerical data for the final Gaussian components are shown in the "Edit" dialog box ("Channel-Channel Edit" menu) as height of top (h), peak position (v Kcm^-1), and half width at half height (w Kcm^-1) as shown in Fig 30.

Fig.30 Numerical data for Gaussian in Ch. 2.

11. Analyses of Multicomponent Spectra
  If the standard individual spectra of the components contained in a sample are available, SPANA estimates their compositions in its observed spectrum. The example is given in the "regression.ana" file which contains 4 standard spectra (Ch. 0-4) and 4 sample spectra (Ch. 10-13) as shown in Fig.31 and 32.

Fig.31 Sample data for regression analysis of multicomponent spectra (standard spectra).

The regression analyses of SPANA estimates the contents of the standard spectra in the sample spectra as the form of
          Sample X = a･Standard A + b･Standard B + c･Standard C + d･Standard D    eq.7

Fig.32 Sample data for regression analysis of multicomponent spectra (sample spectra)

In order to set the conditions for analyses, open the "Regression Analyses" dialog box (the "Analysis-Regression" menu) and set the sample channel (here, "All"), storage channel for the resultant simulated spectra (here, "16-") and the channels of the standard spectra used for the analysis (here, Ch. 0, 1, 2, 3) up to 10 (Fig.33).

Fig.33

Click of the "Analyze" button start the regression analysis calculation and, then, shows the content coefficients of the standard spectra, a, b, c and d in eq.7, for the sample spectra in the "Result" Table (Fig.34).

Fig.34

The resultant simulated spectra are stored in the indicated channels (here, Ch. 16-19) as "Composed Ch. X". These composed spectra are shown on the screen as similarly as usual spectrum by "left-click" operation and, furthermore, the component spectra are also shown by "right-click" operation as shown in Fig.35.

Fig.35 Simulated spectrum (dark red, Ch.18), component spectra (dark blue) and Sample C spectrum (purple, Ch.12, almost overlapping).

The content coefficients, a, b, c and d, are also recorded in the "Edit" dialog box ( "Channel-Channel Edit" menu) as shown in Fig.36.

Fig.36

12. 3Ｄ Ｖｉｅｗ
　　The "View-3D View" menu opens the new windows for the 3 dimensional expression of the collected data which has three orthogonal axes of wave length (x), spectral strength (y), and condition (t or z) variables. The spectra are shown in the x-y-z 3 dimensional box of which tilt angle and the size are set in the bottom part of the window. Click of the "Show" button shows the simple spectra in the box (Fig.37).

　　　　Fig. 37　Surface-None,　Color　Separation-Channel

　　Other types of 3D graphs such as the spectral surface with mesh representation, the contour map, and the spectral screens may be shown by setting the necessary parameters in the "Option" dialog box (Fig.38 and 39).

Fig.38

Fig.39 Left: Surface-Mesh, Color Separation-Height , Right: Surface-Screen,

13. Analyses of NMR Data

a) Titration
For chemical exchange-equilibrium which is rapid on the NMR time-scale,　
　　　　　　　　　AB_i-1 + B <-> AB_i 　　　　　　i = 1 - n
the individual observed chemical shift δ_obs is the mole-fraction-weighted average of the shifts δ for the corresponding nuclei in chemical species AB_i existing in the system.　
　　　　　　　　　δ_obs = ΣP_ABi δ_ABi
Here,
　　　　　　　　　P_ABi = [AB_i] / [A]_Total ,　　[A]_Total = Σ[AB_i] ,　　 i = 0 - n.
then, the change of the chemical shift, Δδ_obs, is
　　　　　　　　Δδ_obs = δ_obs - δ_A = Σ(Δδ_ABi/[A]_Total)[AB_i] ,　　　Δδ_ABi = δ_ＡBi - δ_Ａ, 　　i = 1- n
Since, regarding Δδ_X/[A]_Total as Δε_X , the relationship between the observed values and the concentrations is the same as that for the electronic spectrum, the optimization models of SPANA are also applicable for this type of NMR titration data.　

The analysis is performed for the chemical shift data which are set in the "Result" table by using the numerical key-board or by reading the data file of the ".lsq" format.
Following is the example of analysis for the data of the attached sample file, "NMR1-2.lsq", containing two chemical-shift change data (y1 and y2) vs. ligand concentration (t) for a 1:2 complex formation system (Fig.40).

Fig.40

The optimization by using the 1:2 sequential complex formation model results in the following parameters (Fig.41),

Fig.41

K1 = 5.7 x 10³, K2 = 1.0 x 10³, Δδ1_AB = 4.1 (Δε1 x [A]_Total), Δδ1_AB2 = 0.98 (Δε*1 x [A]_Total), Δδ2_AB = 1.0 (Δε2 x [A]_Total), Δδ2_AB2 = 2.0 (Δε*2 x [A]_Total) (Fig.41)

b) NMR Line Shape Fitting
　The ¹H-NMR signal shape, I(ν), under the equilibrium, A <-> A', is theoretically determined as the function containing the following 7 parameters,³⁾
　 ν_A : chemical shift of A , ν_A' : chemical shift of A' , P_A : mole-fraction of A , k_A : rate constant for A -> A'
　　T_2A : spin-spin relxation time of A , T_2A' : spin-spin relaxation time of A' , C : normalization factor　
The NMR line shape fitting method of SPANA (Fig. 42) optimizes these 7 parameters to minimize the residual square sum between the theoretical and observed signal shapes.

Fig.42

The results of analysis for the attached sample data, nmrfit-spana.lsq, by the "NMR Line Shape Fitting" model is shown below (Fig.43).

Fig.43

The "Plot" command shows the graphics of the theoretical-observed data fitting (Fig.44).

Fig.44

14. Analyses of Concentration Data
　Since the absorbance of the electronic spectrum is linearly proportional to the concentration of the corresponding chemical species under the Lambert-Beer Law conditions, specifying 1.0 for the proportionality constant, the absorbance term means the concentration.　Thus, the data for the concentration of individual chemical species in the system can be analyzed by SPANA using this relationship.
For example, the analysis of the successive reaction, A -> B -> C, using concentration data is shown below.
The attached data, "Series.lsq", contains following two data sets of the concentrations of A, B and C (y1, y2, y3) vs. time (t) observed for different initial concentrations [A]₀ (Fig.44),
　　　　　{ Ch.1-5 : t = 0.1-1.5 , [A]₀ = 101.53 } and { Ch.6-13 : t = 0.15-2.1 , [A]₀ = 200.1 }:.

Fig.44

In order to analyze these data in concentration mode, following parameter settings are applied to the "Least Square Analyses - Miscellaneous Model - Successive Reaction" dialog box (Fig.45).
Since y1 means [A] which is related to the absorbance term according to the equation of
　　　　　　Abs_obs = ε₁[A] + ε*₁[B] + ε**₁[C],
the values of <ε₁, ε*₁, ε**₁> for y1 are fixed to <1, 0, 0>. 　In the same way, those for y2 an y3 are fixed to <0, 1, 0> and <0, 0, 1>, respectively. The check-boxes for all these parameters are checked to remove from optimization calculation.
Setting two initial concentrations of A and the initial guess values for k1 and k2, the optimization gives the results shown in Fig.45.

Fig.45

The "Plot" command shows the fitting graph of Fig.46.

Fig.46

15. Least Square analyses for User's Function (Ver.5 Only) ⁴)
　　Although the procedures of the least square analyses for the user's function may be different according to the user's situation, the outlines of the analyses are described below, as an example, by using the sample data, "titrn283-323.ana" (for the program of the analyses, see the "Analyses" page).
The file, titrn283-323.ana, contains the following temperature dependent spectroscopic titration data.

The different spectra of this data are shown in Fig. 47.

Fig.47

Collecting necessary data, click of the "External" button in the "Miscellaneous Models" page (Fig.40) of "Least Square Analyses" dialog box shows the "External Least Square Program" dialog box (Fig.48).

Fig.48

If the data contain two or more data sets, set the number of data set and the top channel of each set, select the user's program "titrn_t-dep.exe", for example, and click the "Do Program" button. The program read the "spana_lsq.dta" file and open new MS-DOS WINDOW. According to request of the program, set the necessary data such as the number of parameters, p[i], and constants, c[i], and initial guess of p[i], then the results of calculation are shown as Fig.49

Fig.49

After calculation, this program may generate the data for graphic simulation on user's request, which are used for theoretical simulation curves by clicking the "Plot" button in Fig.41to show the simulation graphs Fig.50.

Fig.50

References.
 1) For example of the nonlinear least square calculation,
    a) S. L. S Jacoby, J. S. Kowalik, J. T. Pizzo, "Interative Methods for Nonlinear Optimization Problems", Prentice Hall, Inc., N.J, 1972.
    b) T. Nakagawa and Y. Koyanagi, "Analyses of Experimental Data by Least Square Method (Japanese)" (UP Applied Mathematics Series 7), University of Tokyo Press, Tokyo, 1982.
 2) For example of regression analyses and wave separation,
    a) D. J. Leggett, "Numerical Analysis of Multicomponent Spectra", Analytical Chemistry, vol.49, 276, 1977
    b) R. D. B. Frasher, E. Suzuki, "Resolution of Overlapping Absorption Bands by Least Squares Procedures", Analytical Chemistry, vol. 38, 1770, 1966.
    c) S. Minami, Ed., "Treatment of Wave Data in Scientific Measurement (Japanese)", CQ Press Inc. (Japan), Tokyo, 1986 and references therein.
3) J. Sandstroem, "Dynamic NMR Spectroscopy.", Academic Press, 1982
4) For details of this section, contact to my e-mail address.