Least-Squares Fitting of Multidimensional Spectra to Kubo Line-Shape Models

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.

Figure 1

Dephasing parameters by CLS method (blue) and model fitting (red) of simulated 2D IR waiting-time series. (A) Fit parameters from all 100 trials. (B) Fit parameters obtained from an individual trial (chosen at random) with 95% confidence intervals estimated from the covariance of fit. (C) Average of fit parameters over all 100 trials with 95% confidence interval calculated from the standard error of the mean, with the inset showing a zoom-in of the true-value line.

Figure 2

Variance inflation factor (VIF) is a measure of multicollinearity (or ill-conditioning) in least-squares regression. Plots of VIFs for (A) naively fitting all dephasing parameters (with scaling and offset) to a linear absorption spectrum, (B) the CLS method, and (C) model fitting. Larger VIFs reflect higher multicollinearity.

Figure 3

Plots of the cost function and SIGN for modeling with too few (A, B), the correct number (C, D), and too many (E, F) Kubo components. Each trial corresponds to a random sampling of noise and random starting point for p. The dashed line is associated with the stopping criterion.

Figure 4

Select examples of stalling or unusual convergence. (A, B) An example of boundary stalling. (C, D) An example of stalling due to multicollinearity between degenerate Kubo components. (E, F) An example in which one component approaches the true value and the other approaches null (i.e., Δ² = 0), where either stalling or convergence may occur.

Figure 5

Model fitting to low SNR 2D IR waiting-time series. Simulation is representative of a cyanylated cysteine residue in the protein calmodulin, where Δ₂² is static relative to the vibrational lifetime. Examples of (A) transient absorption and (B) 2D spectrum with an SNR of 10:1. (C) Fit parameters from 100 trials. (D) Fit parameters obtained from an individual trial with 95% confidence intervals estimated from covariance of fit. (E) Average of fit parameters over all 100 trials with 95% confidence interval calculated from the standard error of the mean.

Figure 6

CLS of MeSCN in DMSO. (A) Comparison between 2020 data (magenta squares) and 2021 data (black circles). 2021 data are background-subtracted prior to CLS analysis, while 2020 data are not. (B) Comparison between 0–1 (black circles) and 1–2 (blue triangles) CLS for 2021 data.

Figure 7

Plots of (A) cost function and (B) scale invariant gradient norm for one- (orange) and two (blue)-Kubo component models fitted to 2020 data.

Figure 8

Plots of (A, D) 2020 data, (B, E) model fit result, and (C, F) residual for T_w = 0.4 ps and 50 ps. Residual in panel (C) shows a structured response, which is unaccounted for by the model.

Figure 9

(A) Linear absorption spectra as measured by the probe (black, solid), FTIR (blue, dot-dashed), and simulated with parameters obtained by model fitting the 2D IR waiting-time series (red, dashed). (B) Results of the CLS method fitting to the upper 90% (blue, dashed), 80% (green, dashed), and 70% (orange, dashed) of the probe spectrum (black, solid) to obtain the Kubo amplitude and homogeneous dephasing.

Figure 10

Masks used in undersampling the waiting time T_W of MeSCN in DMSO. Each row corresponds to a different sampling mask. Black dots represent the inclusion of a 2D spectrum in fitting for a given waiting time. In all cases, we exclude the first 300 fs of waiting time to avoid spurious nonresonant and time-ordering signals from pulse overlap.

Figure 11

Plots of (A) the cost function C and (B) the scale invariant gradient norm versus fitting iteration for a series of undersampled waiting time T_w. The dashed line in panel (B) is associated with the stopping criterion.

Figure 12

Comparison of dephasing parameters obtained by the CLS method and modeling fitting. (A) Kubo time constants obtained by the CLS method for 2020 and 2021 data. (B, C) Kubo amplitudes and homogeneous dephasing obtained by the CLS method for 2021 data. This is plotted for three different fitting ranges of linear absorption (see Figure 9B) to demonstrate how sensitive these parameters are to the linear absorption spectrum. (D–F) Model fitting of 2021 data as a function of the number of waiting-time points used in fitting. Model fitting to 2020 data also shown for comparison. Error bars are 95% confidence intervals estimated from the covariance of fit.