Direct inversion in the spectral subspace: A novel method for quantitative and qualitative analysis of chemical mixtures

Abstract

A novel method, called Direct Inversion in the Spectral Subspace (DISS), has been developed for the quantitative (and partly qualitative) analysis of chemical mixtures. The method belongs to the broad group of “supervised classification” methods: its use necessitates the components’ “pure” spectra, either experimental or computed. On the basis of three simple conditions, an elegant, linearized system of equations has been deduced, taking into account a sole restriction via the Lagrange’ multiplier method. This restriction is seemingly redundant but it has been shown that with its use the unknown normalization constant of the components’ descriptive weighted average (CDWA) spectrum can be taken into consideration. The system of linearized equations can be solved repeatedly until convergence. Any kind of spectra can be used; the method does not require the non-negativity of spectral data points. Two versions of the new method have been developed: the normalized and the non-normalized versions regarding the components’ spectra. In ideal cases, the non-normalized version of the DISS method provides a mixture’s accurate composition due to the iteration for getting the correct norm of the CDWA spectrum. Realistically, the normalized version of the DISS method identifies a mixture’s composition within a few molar percentage points accuracy, according to the test results in IR and 1H-NMR spectroscopy. The normalized method functions without any calibration measurements and needs only a control of accuracy; it is hoped that it will be a useful tool for chemical and biochemical analysis as well as for spectral databases. The DISS method is also useful for qualitative analyses in a limited sense: in the case of computed spectra of the components the set of the de facto components determined could be somewhat wider than those existing in the real system.