Theory-Guided Fitting of UV/Visible Spectra

Overview

Theory-guided fitting uses quantum chemistry calculations as initial guesses for fitting experimental UV/visible spectra. This approach combines the strengths of both theoretical predictions (molecular properties like vibronic coupling and frequencies) and experimental measurements (accurate peak positions and intensities).

The method is particularly powerful for multi-component spectra with overlapping electronic transitions, where pure theory may miss environmental effects and manual fitting may miss physical constraints.

Complete Workflow

Step 1: Quantum Chemistry Calculations

• Ground state: optimize geometry, compute Hessian
• Excited states: compute gradients and energies for bright states
• Output: normal modes $\{{\omega }_k\}$, gradients $\{g_k\}$, excitation energies $E_{00}$ (approximation to ${\nu }_0$)

Step 2: Linear Coupling Model Analysis

• Compute mode-specific Huang-Rhys factors from gradients (see Linear Coupling Model)
• Collapse to effective single mode: $S_{\text{eff}}$, ${\Omega }_{\text{eff}}$
• Estimate broadening parameters: ${\sigma }_0$, $\delta$ (see Estimation of Pekarian Parameters)

Step 3: Load Experimental Data

• Read experimental UV/Vis spectrum (wavenumber vs. intensity)
• Normalize intensity to [0,1] range
• Optionally smooth for peak detection

Step 4: Initial Parameter Guess

• Use theory for molecular properties: $S$, $\Omega$, ${\sigma }_0$, $\delta$ (shapes and vibronic coupling)
• Use experiment for positions: ${\nu }_0$ from automatic peak detection
• Use experiment for amplitudes: $A$ from relative peak heights
• Set physical bounds (typically ±50% of theory values)

Step 5: Constrained Optimization

• Run minimization with boundaries with the L-BFGS-B algorithm (scipy.optimize.minimize)
• Enforce energy ordering constraint: ${\nu }_{0,1}<{\nu }_{0,2}<\cdots <{\nu }_{0,n}$ (prevents band swapping)
• Monitor convergence (typically 10-50 iterations with good initial guess)

Step 6: Validation and Interpretation

• Check fit quality: $R^2>0.95$, residuals $<2$-$3\%$ of max intensity
• Verify no systematic trends in residuals
• Compare fitted vs. theoretical parameters

Fitting Options

Function Types

• Lorentzian/Gaussian/Voigt: For broad, unresolved electronic bands
• Pekarian: For vibronically structured bands (see Pekarian Function)
• Mixed: Different function types for different peaks simultaneously

Constraint Options

• Energy ordering: Prevents band swapping during optimization (${\nu }_{0,j}<{\nu }_{0,j+1}$)
• Physical bounds: Limits parameters to reasonable ranges based on theory (±50% typical)
• Fixed parameters: Lock specific parameters if well-known from theory or prior fits

References

• Larina, N. & Khodorkovsky, V. "Pekarian Functions for Vibronic Spectra Analysis," New J. Chem. 49(10), 3937-3945 (2025). DOI: 10.1039/d4nj05537c
• Santoro, F., Lami, A., Improta, R., Bloino, J. & Barone, V. "Effective method for the computation of optical spectra of large molecules at finite temperature including the Duschinsky and Herzberg-Teller effect: The Qx band of porphyrin as a case study," J. Chem. Phys. 128, 224311 (2008). DOI: 10.1063/1.2929846