N-Mode Quantized Anharmonic Vibronic Hamiltonians for Matrix Product State Dynamics

TL;DR

The work addresses the challenge of modeling photochemical vibronic dynamics with strong anharmonicity and complex nonadiabatic couplings. It introduces an n-mode quantized vibronic Hamiltonian in a second-quantized form and couples it to tangent-space TD-DMRG, enabling accurate time evolution of high-dimensional vibronic systems. Applied to maleimide, the study demonstrates S0 → S4 excitation dynamics using six active modes, producing an absorption spectrum in good agreement with experiment and validating convergence behavior with respect to the bond dimension $m$ and local basis size $N_{max}$ (e.g., propagation up to 400 fs with $m=75$). This framework offers a scalable, controllable approach to simulating complex photochemical dynamics and can be extended to larger systems and finite-temperature conditions.

Abstract

Theoretical predictions of photochemical processes are essential for interpreting and understanding spectral features. Reliable quantum dynamics calculations of vibronic systems require precise modeling of anharmonic effects in the potential energy surfaces and off-diagonal nonadiabatic coupling terms. In this work, we present the n-mode quantization of all vibronic Hamiltonian terms comprised of general high-dimensional model representations. This results in a second-quantized framework for accurate vibronic calculations employing the density matrix renormalization group algorithm. We demonstrate the accuracy and reliability of this approach by calculating the excited state quantum dynamics of maleimide. We analyze convergence and the choice of parameters of the underlying time-dependent density matrix renormalization group algorithm for the n-mode vibronic Hamiltonian, demonstrating that it enables accurate calculations of complex photochemical dynamics.

Figures (8)

• Figure 1: Cuts through the $S_3$ and $S_4$ potential energy surfaces along the selected vibrational mode coordinates. Vibrational mode indices are assigned according to the magnitude of the corresponding harmonic frequencies. Arrows attached to the molecular structures indicate the displacement of the atoms in each of the selected normal mode coordinate. Atom color code: gray -- carbon, white -- hydrogen, red -- oxygen, blue -- nitrogen.
• Figure 2: Comparison of the experimental gas-phase absorption spectrum of maleimide taken from Ref. lehr2020role (top) and the TD-DMRG spectrum obtained in this work (with a time step of $0.5$ fs, a total propagation time of $800$ fs, and a bond dimension of 75). Top: Two experimental curves are shown. One represents the raw data and the other shows the spectrum with its envelope subtracted to allow for a better comparison with the calculated spectrum. All fundamental transitions as well as some overtones are labeled according to Ref. lehr2020role. Dotted vertical lines connect the experimental values of these transitions with the calculated results.
• Figure 3: Non-truncated and truncated bond dimensions of the MPS over the first 100 microiterations of the TD-DMRG calculation. The data was obtained by a calculation employing a 0.5 fs time step and a maximum bond dimension of $75$.
• Figure 4: Real and imaginary parts of the autocorrelation function obtained by a TD-DMRG calculation of maleimide upon photoexcitation onto the $S_4$ surface with different values for the maximum bond dimension. The top panel shows the real and the bottom panel the imaginary part of the autocorrelation function. The results were obtained by employing a time step of $0.5$ fs and propagating the wave function for $400$ fs.
• Figure 5: Absorption spectrum of the maleimide molecule upon a Franck-Condon excitation to the $S_4$ electronic surface for different values of the maximum bond dimension m. All calculations were conducted with a time step of 0.5 fs for a total propagation time of $400$ fs.