# Numerical Analysis

## Associated Group Prof. Klaus Neymeyr (University of Rostock)

**Overview**

Modern spectroscopic and computer-based measurement techniques are valuable tools for the analysis of chemical reaction systems and for structure elucidation. Large volumes of data are generated in the case of a high frequency resolution and with a large number of spectra, e.g. if a chemical reaction is monitored.

These data cannot be inspected by classical non-computerized analyzing techniques. Instead mathematical methods and numerical algorithms can be used for a successful analysis.

Figure 1: Left: Sequence of FT-IR spectra of Rhodium-catalyzed hydroformylation ^{[6]}.

Right: Investigated subrange of frequencies.

The research group deals, among others, with the so-called factor analysis of sequences of spectra. If a chemical reaction system is monitored for a number of n times of measuring and on a frequency grid containing k single frequencies, then all spectra can be stored within a k x n matrix. The Lambert-Beer law says that this matrix can be factorized in a matrix containing the pure component spectra and a further matrix which contains the concentration profiles along the times axis of the pure components. Mathematically this is a nonnegative matrix factorization problem. Usually, this problem has no unique solution. Instead, appropriate technqiues are used in order to extract a single and hopefully the chemically correct solution. If this step can be completed successfully, then the mathematical procedure has delivered the complete information on the pure components.

In close cooperation with the research groups of Prof. Armin Börner and Prof. Ralf Ludwig we have succesfully applied newly developed pure component factorization methods for transition metal catalyzed carbonylation reactions.

In this project we have had a close interlocking between the mathematical fundamental work in numerical mathematics and the research work on catalytic systems at the LIKAT.

Figure 2: Results of a pure component factorization for the FT-IR data. The factorization has used a kinetic model and concentration profiles are given in absolute values.

On the one hand the new methods have provided new insight into the chemical reaction systems and on the other hand all the questions arising from the chemical application problem were the starting points for additional mathematical research work.

A comparatively new approach without further assumptions for the matrix factorization problem is to compute the set of all feasible solutions and then to plot an appopriate representation of this set. This results in the so-called Area of Feasible Solutions (AFS). Within a DFG funded project we have developed the FACPACK software which allows to compute the AFS very fast and with simultaneous control of the approximation accuracy [1,2].

The computation of the AFS is often only the first step in a factor analysis. The subsequent adaption to a kinetic model (e.g. Michaelis-Menten kinetics) can be a second step. Within a hybrid approach the computation of the factorization can be coupled to the optimal adaption of the kinetic parameters. The basic procedure has been used multiple times for reaction systems from homogeneous catalysis.

The interdisciplinary project has resulted not only in mathematical-algorithmic developments but also in a deepened understanding of the rhodium- and iridium-catalyzed hydroformylation process. The research group is looking forward to further cooperations on chemometric problems.

*Literatur:*

1. M. Sawall, C. Kubis, D. Selent, A. Börner, K. Neymeyr: A fast polygon inflation algorithm to compute the area of feasible solutions for three-component systems. I: Concepts and applications., *J. Chemometrics* *27(5),* **2013**, 106-116.

2. M. Sawall, K. Neymeyr: A fast polygon inflation algorithm to compute the area of feasible solutions for three component systems. II: Theoretical foundation, inverse polygon inflation and FAC-PACK implementation., *J. **Chemometrics 28(8)*, **2014,** 633-644.

3. M. Sawall, K. Neymeyr: On the area of feasible solutions and its reduction by the complementarity theorem., *Anal. Chim. Acta 828*, **2014,** 17-26.

4. C. Kubis, W. Bauman, E. Barsch, D. Selent, M. Sawall, R. Ludwig, K. Neymeyr, D. Hess, R. Franke, A. Börner: Investigation into the equilibrium of Iridium catalysts for the hydroformylation of olefins by combining in situ high-pressure FTIR- and NMR-spectroscopy., *ACS Catalysis 4*, **2014,** 2097-2108.

5. C. Kubis, M. Sawall, A. Block, K. Neymeyr, R. Ludwig, A. Börner, D. Selent: An Operando FTIR Spectroscopic and Kinetic Study of Carbon Monoxide Pressure Influence on Rhodium Catalyzed Olefin Hydroformylation., *Chemistry - A European Journal 37*, **2014,** 11921-11931.

6. M. Sawall, C. Kubis, R. Franke, D. Hess, D. Selent, A. Börner, K. Neymeyr: How to apply the complementarity and coupling theorems in MCR methods: Practical implementation and application to the Rhodium-catalyzed hydroformylation., *ACS Catalysis 4*,** 2014**, 2836-2843.