A Global Approach to Accurate and Automatic Quantitative Analysis of NMR Spectra by Complex Least-Squares Curve Fitting

Abstract

The performance of quantitative analysis of 1D NMR spectra depends greatly on the choice of the NMR signal model. Complex least-squares analysis is well suited for optimizing the quantitative determination of spectra containing a limited number of signals (<30) obtained under satisfactory conditions of signal-to-noise ratio (>20). From a general point of view it is concluded, on the basis of mathematical considerations and numerical simulations, that, in the absence of truncation of the free-induction decay, complex least-squares curve fitting either in the time or in the frequency domain and linear-prediction methods are in fact nearly equivalent and give identical results. However, in the situation considered, complex least-squares analysis in the frequency domain is more flexible since it enables the quality of convergence to be appraised at every resonance position. An efficient data-processing strategy has been developed which makes use of an approximate conjugate-gradient algorithm. All spectral parameters (frequency, damping factors, amplitudes, phases, initial delay associated with intensity, and phase parameters of a baseline correction) are simultaneously managed in an integrated approach which is fully automatable. The behavior of the error as a function of the signal-to-noise ratio is theoretically estimated, and the influence of apodization is discussed. The least-squares curve fitting is theoretically proved to be the most accurate approach for quantitative analysis of 1D NMR data acquired with reasonable signal-to-noise ratio. The method enables complex spectral residuals to be sorted out. These residuals, which can be cumulated thanks to the possibility of correcting for frequency shifts and phase errors, extract systematic components, such as isotopic satellite lines, and characterize the shape and the intensity of the spectral distortion with respect to the Lorentzian model. This distortion is shown to be nearly independent of the chemical species, of the nature of the molecular site, and of the type of nucleus, but to vary significantly with the spectrometer. The potential of the method for accurate fast and fully automatic quantitative analysis is illustrated in the case of natural-abundance deuterium NMR spectra such as those of linalool and ethyl mandelate.