Franck-Condon Pekarian Functions

The Pekarian function provides an analytical formulation of the Franck-Condon principle for vibronic spectra, using the Effective Single-Mode (ESM) approximation.

Key Concept: All vibrational modes are collapsed into a single effective mode with parameters ${\Omega }_{\text{eff}}$ (effective frequency) and $S_{\text{eff}}$ (effective Huang-Rhys factor). This simplification matches the natural output of quantum chemistry ESM calculations.

Relation to multi-mode treatment: Both Pekarian (ESM) and multi-mode approaches implement Franck-Condon physics. Pekarian uses one effective mode (simpler, faster), while multi-mode treats each mode explicitly (more detailed, slower).

Definition

The absorption spectra using Pekarian models:

$$I(\nu )=A\exp (-S)\sum _{n=0}^N\frac{S^n}{n!}\cdot f(\nu -{\nu }_0-n\Omega ,{\sigma }_n)$$

where:

• $A$: Overall amplitude
• $S$: Huang-Rhys factor
• ${\nu }_0$: 0-0 transition frequency
• $\Omega$: Effective vibrational frequency
• ${\sigma }_n=\sqrt{{\sigma }_0^2+n{\delta }^2}$: Progressive broadening
• $f$: Lineshape function (Gaussian or Lorentzian)

Physical Interpretation

The Pekarian function models:

1. Poisson distribution: $\exp (-S)S^n/n!$ weights for $n$ quanta
2. Vibronic progression: peaks at ${\nu }_0+n\Omega$
3. Progressive broadening: linewidth increases with $n$

Parameters

Absorption vs. Fluorescence

Absorption (PFa):

• Progression to higher energy: ${\nu }_0+n\Omega$
• Models ground → excited transitions

Fluorescence (PFf):

• Progression to lower energy: ${\nu }_0-n\Omega$
• Models excited → ground emission

References

• Larina, N. & Khodorkovsky, V. "Pekarian Functions for Vibronic Spectra Analysis," New J. Chem. 49(10), 3937-3945 (2025). DOI: 10.1039/d4nj05537c
• Pekar, S. I., Zh. Eksp. Teor. Fiz. 20, 510 (1950)