Capturing anharmonic effects in single vibronic level fluorescence spectra using local harmonic Hagedorn wavepacket dynamics

TL;DR

The paper develops a local harmonic Hagedorn wavepacket framework for single vibronic level (SVL) fluorescence spectra to partially account for anharmonicity through the local harmonic potential $V_{ ext{LHA}}(q;q_t)$. By preserving the exact TDSE solutions of Hagedorn functions and keeping the expansion coefficients $c_K$ constant, SVL spectra for all initial vibrational levels can be obtained from a single Gaussian trajectory, enabling efficient on-the-fly ab initio calculations. Benchmarking against exact quantum results in a two-dimensional Morse-type system shows the local-harmonic approach outperforms global-harmonic models, particularly for higher excitations, while on-the-fly calculations for anthracene and difluorocarbene demonstrate practical applicability and system-dependent anharmonic effects. The method provides a practical diagnostic for harmonic-model adequacy and lays groundwork for extensions to initial-surface anharmonicity and rovibrational coupling in more complex spectra.

Abstract

Hagedorn wavepacket dynamics yields exact single vibronic level (SVL) fluorescence spectra in global harmonic models. To partially describe the effects of anharmonicity, important in the spectra of real molecules, we describe a combination of the Hagedorn wavepacket approach to SVL spectroscopy with the local harmonic approximation. In a proof-of-principle study [Phys. Rev. A 111, L010801 (2025)], we successfully demonstrated the utility of this method by computing the SVL spectra of difluorocarbene, a floppy molecule with moderately anharmonic potential. Here, we describe the theory in detail and analyse the method more thoroughly. To assess the accuracy of the method independently of electronic structure errors, we use a two-dimensional Morse-type potential for which exact quantum benchmarks are available, and show that the local harmonic approach yields more accurate results than global harmonic approximations, especially for the emission spectra from higher initial vibrational levels. Next, we compare the global and local harmonic SVL spectra of anthracene, where the more expensive local harmonic corrections turn out to be less important as long as the correct global harmonic model is used. We also present additional local harmonic results for difluorocarbene, where treating anharmonicity is essential for accurate evaluation of the spectra. Yet, we also show that the structure of the difluorocarbene spectra can be explained qualitatively (but not quantitatively) with a reduced-dimensional harmonic model, for which the spectral intensities can be evaluated analytically.

Figures (6)

• Figure 1: SVL spectra computed with Hagedorn wavepackets (red dashed lines) propagated using the local, adiabatic, or vertical harmonic approximation of a two-dimensional coupled Morse potential are compared to exact quantum spectra (black solid lines); the initial vibrational excitation is indicated at the top.
• Figure 2: Differences ($\sigma_{\text{Hagedorn}} - \sigma_{\text{exact}}$) between the SVL spectra of a two-dimensional coupled Morse potential computed with Hagedorn wavepacket dynamics and the corresponding exact spectra. See the caption of Figure \ref{['fig:2d']} for more details.
• Figure 3: Comparison of SVL spectra computed using local (black solid lines), adiabatic (red dashed lines) and vertical (blue dotted lines) harmonic approximations from levels $0^0$, $\overline{11}^1$, $\overline{11}^2$, and $12^1$ of anthracene. The wavenumbers are not empirically scaled.
• Figure 4: Comparison of the experimental (black solid line) SVL fluorescence spectra from $2^1$ and $2^3$ levels of CF$_2$ from Ref. King_Stephenson:1979 with computed (red dotted line) spectra using the local, adiabatic, and vertical harmonic Hagedorn wavepacket dynamics.
• Figure 5: Comparison between SVL spectral intensities computed with the analytical formula [Equation (\ref{['eq:1d_analytical']}), black sticks] and with the Hagedorn approach (red solid lines) in a reduced one-dimensional displaced harmonic oscillator model based on CF$_2$ for emission from excited bending-mode levels with initial vibrational quantum numbers $v^\prime = 0, 1, 2, 3$; the final vibrational levels $v$ are indicated along the top axis. Transitions marked by *, **, and *** are analyzed in Figure \ref{['fig:cf2_fc']}.