An effective bath state approach to model infrared spectroscopy and intramolecular dynamics in complex molecules

TL;DR

The study advances the effective bath state (EBS) method for intra-molecular vibrational dynamics by incorporating polynomial couplings in the bath and enabling finite-temperature infrared spectroscopy. By coarse-graining the bath into a ladder of effective energy states and rigorously treating system–bath couplings, the approach captures non-Markovian dynamics and recurrences that arise from a finite environment. Benchmarking on a 10-mode model demonstrates accurate IR spectra and IVR behavior at multiple temperatures, and a detailed application to phenylacetylene shows good agreement with experiments while highlighting the importance of resonant pathways and higher-order bath couplings. Overall, the work provides a scalable framework to study vibronic dynamics and IR responses in complex molecules, with potential extensions to emission spectroscopy and higher-dimensional effective baths.

Abstract

When a molecule contains more than a few atoms, its full-dimensional dynamics becomes untractable, especially when introducing temperature effects. In such a case, it can be interesting to focus only on a few degrees of freedom and to model the rest of the molecule as a finite-dimensional bath. In this prospect, we extend the effective bath state (EBS) method that we had first developed and benchmarked in [J. Chem. Phys. \textbf{160}, 044107 (2024)] to describe the spectroscopy and intramolecular dynamics of complex isolated molecules. The EBS method is a system-bath approach based on the coarse-graining of the bath into a reduced set of effective energy states. It allows for a significant reduction of the bath dimension and makes finite-temperature calculations more accessible. In order to treat a realistic molecule, the method is extended to include polynomial couplings in the bath coordinates. The ability of the method to model temperature-resolved infrared spectra and to follow population transfers between the vibrational modes of the molecule is first tested on a 10-mode model system. The extended method is then applied to the realistic case of phenylacetylene.

Figures (8)

• Figure 1: Representation of the coarse-graining procedure of the bath for a model system containing four modes (mode 1 is the system). Left: Energy scale where the first three energy grains are indicated. Center: energy levels of the three harmonic bath modes with their respective frequencies $\omega_k$ and their maximum number of quanta $N_k$. Right: Transformation of the bath modes into a single ladder of effective energy states $\ket{m}$ (in blue). Individual microstates are represented in black inside the EES that contains them, with a label indicating the corresponding set of quantum numbers $\left(n_2,n_3,n_4\right)$. The number $\rho(m)\Delta E$ of microstates in each EES is also indicated. One specific bath state with one quantum of energy in mode 2 and one in mode 4 is highlighted in green. The position of the associated microstate $\left(1,0,1\right)$ is represented in green inside the effective state ladder. As shown in grey, its position is obtained by summing the energies of the individual bath modes. The coarse-graining procedure allows the basis set size to drop from $N_2\times N_3\times N_4$ microstates to a much smaller number $M$ of effective states.
• Figure 2: Illustration of possible transitions involving two bath modes $j$ and $k$, with frequencies $\omega_j > \omega_k$. Left: a combination band where modes $j$ and $k$ both gain one quantum of energy, leading to a transition energy $\hbar\omega_j+\hbar\omega_k$. Right: a difference band where mode $j$ gains one quantum of energy while mode $k$ loses one, leading to a transition energy $\hbar\omega_j-\hbar\omega_k$.
• Figure 3: Absorption spectra of the 10-mode model system in the spectral region of the mode of interest, obtained at 0, 300 and 600 K. The EBS and full-dimensional results are superimposed in red and black, respectively. Both spectra have been re-normalized to have the same maximal intensity. The system transitions $v \to v'$ are denoted as (S)$_{v,v'}$. The transitions $n_k \to n'_k$ of a given bath mode $k$ are denoted as (B$_k$)$_{n_k,n_k'}$. When there are several bath hot bands involved in a feature, it is denoted as (B)$_{\rm h.b.}$. In the upper panel, the mode of interest harmonic ($\omega_h$) and fundamental ($\omega_{01}$) frequencies are indicated by vertical dashed lines.
• Figure 4: Time-evolution of the 10-mode model system starting from $\ket{v = 1, m = 0}$. (a) Evolution of the population in the vibrational states of the mode of interest $i_0 =3$. (b) Evolution of the population in the effective bath states, labeled by their energy. The bath contains all the modes except $i_0$. All the bath states gaining at least 1% population are labeled. In both panels, the full-dimensional results are shown as thick brown lines.
• Figure 5: Time-evolution of the population in (a) the first excited state of the mode of interest, and (b) its second excited state, for bath temperatures ranging from 100 K to 600 K. In panel (a), the inset emphasizes the half-life times obtained at different temperatures, i.e., the time at which the population in $v = 1$ drops below 50%.