Visualization-Based Approach to Condensed-Phase Line Broadening Using Polyene Chains

Abstract

Condensed-phase spectral line shapes encode the strength and timescale of interactions between molecules and their environments, yet these ideas are often difficult to introduce at the undergraduate level due to their reliance on formal theoretical treatments. We present a visualization-based approach that combines analytic results with numerical simulations to illustrate the physical origins of spectral line broadening in conjugated molecular systems. Using a time-dependent Hückel Hamiltonian, we derive closed-form expressions for coherent electronic motion in finite polyene chains and show how these results provide direct insight into the role of molecular orbital structure in light absorption. Environmental effects are introduced through stochastic fluctuations of the Hamiltonian matrix elements, allowing students to observe how system--environment interactions disrupt coherent motion and produce scattering-like features in electronic trajectories. Real-space animations and simulated absorption spectra provide an intuitive link between microscopic dynamics and measured line shapes. The MATLAB code provided with this work offers an accessible platform for integrating computation and visualization into undergraduate instruction while introducing key concepts in condensed-phase spectroscopy.

Figures (3)

• Figure 1: Coherent electronic dynamics are computed for a 10-membered polyene chain with a tridiagonal Hückel Hamiltonian. (a) Time-dependent expectation value of the carbon-site index computed with diagonal ($\delta\alpha$) and off-diagonal ($\delta\beta$) fluctuations set equal to zero. (b)--(d) Real-space snapshots of the electronic probability distribution evaluated at successive turning points of the trajectory at $t = 0$, 1, and 5 fs, respectively. In the absence of stochastic fluctuations, the electron dynamics are strictly periodic, leading to exact recurrences in the probability distribution.
• Figure 2: Fluctuations induce scattering and decoherence in electronic trajectories along a polyene chain. (a) Time-dependent expectation value of the carbon atom index computed with diagonal fluctuations set to $\delta\alpha = 0.00$ eV and off-diagonal fluctuations $\delta\beta = 0.06$ eV. (b)--(d) Real-space snapshots of the electronic probability distribution evaluated at successive turning points of the trajectory at $t = 0$, 1, and 5 fs, respectively. Repeated scattering events arising from dynamic fluctuations of the electronic couplings disrupt phase coherence, leading to a progressive loss of coherent electronic motion along the chain.
• Figure 3: The electronic trajectory is fitted to Equation \ref{['eq:stretched_damped_cosine']} to obtain the corresponding absorption spectrum. Panels (a)--(b) show results for diagonal disorder with $\delta\alpha = 0.9$ eV, while the off-diagonal fluctuations are held fixed at $\delta\beta = 0.0$ eV. Panels (c)--(d) show the corresponding trajectory and absorption spectrum for off-diagonal disorder with $\delta\beta = 0.1$ eV and $\delta\alpha = 0.0$ eV. Compared to diagonal fluctuations, substantially smaller off-diagonal fluctuations lead to faster damping of the time-domain coherence and broader absorption features, highlighting the enhanced sensitivity of electronic coherence to torsional fluctuations along the polyene chain.