Estimation of Pekarian Function Parameters for ESM Method

Overview

The Pekarian function for vibronic spectra in the ESM approximation requires five fundamental parameters:

$$I(\nu )=A\exp (-S_{\text{eff}})\sum _{n=0}^{n_{\text{max}}}\frac{S_{\text{eff}}^n}{n!}\cdot G(\nu -{\nu }_0-n{\Omega }_{\text{eff}},{\sigma }_n)$$

with ${\sigma }_n=\sqrt{{\sigma }_0^2+n{\delta }^2}$ for progressive broadening.

Part I: Electronic and Vibronic Parameters from Quantum Chemistry

The first three parameters are obtained from quantum chemistry calculations combined with the LCM approximation.

I.A: Zero-Zero Transition Energy (${\nu }_0$)

Definition: The 0-0 transition frequency ${\nu }_0$ is the energy difference between the ground electronic state (v=0) and excited electronic state (v=0).

$${\nu }_0\approx E_{\text{exc}}^{\text{vert}}+\text{ZPE corrections}$$

Calculation from quantum chemistry: The vertical excitation energy $E_{00}$ from DFTB+ serves as a computational approximation to ${\nu }_0$. In practice, $E_{00}\approx {\nu }_0$ but may differ due to environmental effects (solvent, temperature, geometry relaxation).

I.B: Effective Huang-Rhys Factor ($S_{\text{eff}}$)

Definition: Total vibronic coupling strength summed over all active modes.

$$S_{\text{eff}}=\sum _{k\in \text{active}}{S}_k$$

where mode-specific Huang-Rhys factors are:

$$S_k=\frac{g_k^2}{2{\omega }_k^4}$$

with $g_k$ = excited state gradient projection onto normal mode $k$, and ${\omega }_k$ = ground state frequency for mode $k$.

Calculation steps:

1. Ground state optimization → Hessian → normal modes $\{L_k\}$, frequencies $\{{\omega }_k\}$
2. Excited state gradient ${\mathbf{g}}_{\text{exc}}$ at ground state geometry
3. Project gradient: $g_k={\mathbf{g}}_{\text{exc}}^T\cdot {\mathbf{L}}_k$
4. Calculate: $S_k=g_k^2/(2{\omega }_k^4)$
5. Sum: $S_{\text{eff}}={\sum }_k{S}_k$

Physical meaning: Controls vibronic progression intensity distribution (Poisson): $I_n=\exp (-S_{\text{eff}})S_{\text{eff}}^n/n!$

Typical values: Rigid aromatics: 0.3-0.8; Flexible aromatics: 0.8-1.5; Charge-transfer states: 1.5-3.0

I.C: Effective Vibrational Frequency (${\Omega }_{\text{eff}}$)

Definition: Intensity-weighted mean frequency of all active modes.

$${\Omega }_{\text{eff}}=\frac{{\sum }_{k\in \text{active}}{S}_k{\omega }_k}{S_{\text{eff}}}=\frac{{\lambda }_{\text{eff}}}{S_{\text{eff}}}$$

where ${\lambda }_{\text{eff}}={\sum }_k{S}_k{\omega }_k$ is the total reorganization energy.

Physical meaning: Determines spacing between vibronic peaks. Strongly coupled modes contribute more.

Typical values: Aromatic hydrocarbons: 1200-1600 cm⁻¹; Conjugated systems: 1400-1600 cm⁻¹

Summary: Parameters from Quantum Chemistry

Workflow:

1. Ground state: Optimize geometry, compute Hessian → $\{{\omega }_k,{\mathbf{L}}_k\}$
2. Excited state: Compute gradient ${\mathbf{g}}_{\text{exc}}$, energy $E_{00}$ (approximation to ${\nu }_0$)
3. LCM: Project gradient → $\{S_k\}$, sum → $S_{\text{eff}}$, weighted average → ${\Omega }_{\text{eff}}$

Part II: σ₀ Estimation (Initial Linewidth)

Physical meaning: The parameter ${\sigma }_0$ represents the linewidth of the 0-0 transition, including inhomogeneous broadening, homogeneous broadening, and ESM approximation error.

Implemented Method: Dispersion

$${\sigma }_0=\sqrt{\frac{{\sum }_k{S}_k({\omega }_k-{\Omega }_{\text{eff}}{)}^2}{S_{\text{eff}}}}$$

Calculates the weighted standard deviation of vibrational modes around the effective frequency. Directly measures how much the actual mode distribution deviates from the single effective mode approximation.

When to use: Default method. Provides physically meaningful measure of ESM quality. Gives reasonable values for typical organic molecules (50-300 cm⁻¹).

Typical Values and Guidelines

• Rigid systems (anthracene, rubrene): 50-150 cm⁻¹
• Flexible systems (polymers, solution): 200-400 cm⁻¹
• Disordered systems (amorphous films): 300-600 cm⁻¹

Sanity checks:

• Usually dispersion < cumulant
• Should be comparable to ${\Omega }_{\text{eff}}/10$
• If ${\sigma }_0>0.5\times {\Omega }_{\text{eff}}$, consider multi-mode method instead

Part III: δ Estimation (Progressive Broadening)

Physical meaning: The parameter $\delta$ represents progressive broadening with increasing vibrational quantum number: ${\sigma }_n=\sqrt{{\sigma }_0^2+n{\delta }^2}$. Arises from anharmonicity, mode coupling, and lifetime effects.

Implemented Method: Linear scaling

$$\delta =0.1\times {\Omega }_{\text{eff}}$$

Captures empirical observation that progressive broadening scales with ~10% of effective frequency, corresponding to typical anharmonicity in organic molecules.

Accuracy: ~15% average error for typical organic molecules. Range: 0.05-0.14 × ${\Omega }_{\text{eff}}$

Typical Values

For ${\Omega }_{\text{eff}}\approx 1000$-$1500$ cm⁻¹:

• $\delta$ typically: 50-200 cm⁻¹
• Corresponds to 5-15% of ${\Omega }_{\text{eff}}$
• Gives ${\sigma }_n/{\sigma }_0\approx 1.5$-$2.5$ at $n=5$

Empirical relationship: $\delta \approx 0.3{\sigma }_0$ (typical)

Summary

Complete workflow:

1. Run quantum chemistry calculation (ground state Hessian + excited state gradient)
2. Calculate ${\nu }_0$ (from $E_{00}$), $S_{\text{eff}}$, ${\Omega }_{\text{eff}}$ from LCM
3. Auto-calculate ${\sigma }_0$ (dispersion) and $\delta$ (simple)
4. Validate against experimental data and refine by fitting if needed

Note: Parameters in this document are starting estimates from theory. Always validate and refine with experimental fitting when available. See Theory-Guided Fitting for full workflow integration.