Abstract
We report a comprehensive study of the efficacy of least-squares fitting of multidimensional spectra to generalized Kubo line-shape models and introduce a novel least-squares fitting metric, termed the scale invariant gradient norm (SIGN), that enables a highly reliable and versatile algorithm. The precision of dephasing parameters is between 8× and 50× better for nonlinear model fitting compared to that for the centerline-slope (CLS) method, which effectively increases data acquisition efficiency by 1–2 orders of magnitude. Whereas the CLS method requires sequential fitting of both the nonlinear and linear spectra, our model fitting algorithm only requires nonlinear spectra but accurately predicts the linear spectrum. We show an experimental example in which the CLS time constants differ by 60% for independent measurements of the same system, while the Kubo time constants differ by only 10% for model fitting. This suggests that model fitting is a far more robust method of measuring spectral diffusion than the CLS method, which is more susceptible to structured residual signals that are not removable by pure solvent subtraction. Statistical analysis of the CLS method reveals a fundamental oversight in accounting for the propagation of uncertainty by Kubo time constants in the process of fitting to the linear absorption spectrum. A standalone desktop app and source code for the least-squares fitting algorithm are freely available, with example line-shape models and data. We have written the MATLAB source code in a generic framework where users may supply custom line-shape models. Using this application, a standard desktop fits a 12-parameter generalized Kubo model to a 106 data-point spectrum in a few minutes.
Highlights
The Kubo line shape is a common model of spectral diffusion in frequency fluctuation correlation functions (FFCFs) owing to its simple, closed-form expression, and flexibility to describe the limiting cases of homogeneous and inhomogeneous dephasing.[1]
We find that our model fitting routine improves precision over the CLS method by 8−15× on average for Kubo time constants and 8−50× for Kubo amplitudes and homogeneous dephasing, which is due, in part, to a novel figure of merit used in our fitting algorithm that we refer to as the scale invariant gradient norm (SIGN)
We introduce a scale invariant gradient norm (SIGN) capable of identifying, and distinguishing between, algorithmic stalling and convergence at a local or global minimum
Summary
The Kubo line shape is a common model of spectral diffusion in frequency fluctuation correlation functions (FFCFs) owing to its simple, closed-form expression, and flexibility to describe the limiting cases of homogeneous and inhomogeneous dephasing.[1]. There are, several shortcomings of the CLS method, which are characteristics of most of the other common metrics It is unreliable for characterizing relatively fast processes due to the short-time approximation,[13,14] which, following the second-order cumulant expansion, ignores dephasing during coherence times. It requires a second step of fitting to the linear absorbance spectrum to obtain absolute values for Kubo amplitudes and homogeneous dephasing. The CLS method still requires a remarkably high SNR (e.g., ∼100:1) to yield reliable results, which has limited its application.[40]
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