On the evaluation of turnover frequencies in unicyclic homogeneous catalysis. Experimental, numerical, and statistical aspects

Abstract Transition-metal homogeneous catalysis provides a means for the chemoselective, regioselective and stereoselective synthesis of fine chemicals, and the turnover frequency TOF( x , T , P ) can be used as a measure of the specific activity of unicyclic homogeneous catalytic systems. Given an appropriate in situ spectroscopic method for the experimental measurement of both the organometallic and organic reactant concentrations, various numerical approaches for the evaluation of TOF( x , T , P ) are considered. First, the associated problem of spectrophotometric calibration is investigated, particularly as it regards the use of infrared spectroscopy. Clearly, accurate calibrations are a pre-requisite to the calculation of TOF( x , T , P ), and the propagation of error becomes an important issue. Accordingly, a practical guide to calibration is suggested which takes into account some of the particular difficulties associated with transition-metal systems. Secondly, differencing and quadrature methods are compared against spline fitting techniques for the evaluation of TOF( x , T , P ). Realistic confidence intervals for the resulting estimate of TOF( x , T , P ) are derived for each of the methods and comparisons are drawn on this basis. Attention is given to both the presence of error in the explanatory variable of the resulting regressions as well as the error correlation structures arising from derivative and integral approximations. Asymptotic properties of the estimation schemes are also discussed. Overall, the central difference method was found most suitable for the analysis of the present experimental data due to its ease of use and desirable statistical properties. Typical sources of error and the magnitude of these errors, from the analysis of real in situ infrared spectroscopic measurements of the homogeneous rhodium catalyzed hydroformylation of alkenes to yield aldehydes, are discussed in detail.