2-in-1 Phase Space Sampling for Calculating the Absorption Spectrum of the Hydrated Electron

The investigation of vibrational effects on absorption spectrum calculations often employs Wigner sampling or thermal sampling. While Wigner sampling incorporates zero-point energy, it may not be suitable for flexible systems. Thermal sampling is applicable to anharmonic systems yet treats nuclei classically. The application of generalized smoothed trajectory analysis (GSTA) as a postprocessing method allows for the incorporation of nuclear quantum effects (NQEs), combining the advantages of both sampling methods. We demonstrate this approach in computing the absorption spectrum of a hydrated electron. Theoretical exploration of the hydrated electron and its embryonic forms, such as water cluster anions, poses a significant challenge due to the diffusivity of the excess electron and the continuous motion of water molecules. In many previous studies, the wave nature of atomic nuclei is often neglected, despite the substantial impact of NQEs on thermodynamic and spectroscopic properties, particularly for hydrogen atoms. In our studies, we examine these NQEs for the excess electrons in various water systems. We obtained structures from mixed classical-quantum simulations for water cluster anions and the hydrated electron by incorporating the quantum effects of atomic nuclei with the filtration of the classical trajectories. Absorption spectra were determined at different theoretical levels. Our results indicate significant NQEs, red shift, and broadening of the spectra for hydrated electron systems. This study demonstrates the applicability of GSTA to complex systems, providing insights into NQEs on energetic and structural properties.

1 Supplementary Figures   Figure S2.Fitted and computed spectra for the bulk hydrated electron.Raw data are black, the fitted classical spectrum is blue, the fitted quantized spectrum is red.The integral for the classical spectrum is 0.9408 (0.9432 after fitting), while it is 0.9307 (0.9278 after fitting) for the quantized spectrum.2 Error estimation on the nuclear quantum effects

Nuclear quantum effects on absorption spectra
The absorption spectra is constructed as a sum of the excitation energies X i weighted with their oscillator strengths f i .For each system we used N = 1996 structures taken from the simulations 100 fs apart from each other.We considered 9 excitation energies in the pseudopotential calculations, thus the absorption spectra were determined from n = 17964 data points.The relative weight of the i-th data point can be determined from the oscillator strengths: The mean of an absorption spectrum can be calculated simply as The standard deviation is The confidence intervals of the means are calculated as where α = 0.05 according to the 95% level of significance, t is the Student's distribution.
The confidence interval of the standard deviation is: where χ 2 is the Chi-squared distribution.

Nuclear quantum effects on paired data
Using GSTA we have paired structures, thus we have paired data X cl i , X q i .The mean values calculated as and The difference can be calculated directly on the pairs: The standard deviation is The confidence interval of the difference can be calculated as above, but the confidence interval of the difference can be smaller than that of X cl or X q .
3 Supplementary Tables Table S1.Table S1.The average individual energy levels with 95% confidence intervals for the ground state and the first three excited states of the excess electron.The configurations are collected and averaged from the QCMD trajectories (cl ) and after applying the GSTA procedure (q).Energies are in eV.
Table S2.Table S2.Average decrease of the individual energy levels for the ground state and the first three excited states of the excess electron upon quantization with 95 % confidence intervals.The configurations are collected and averaged from the QCMD trajectories (cl ) and after applying the GSTA procedure (q).Energies are in meV.

Figure S1 .
Figure S1.Fitted and computed classical (raw) spectra for the bulk hydrated electron with the first three fitted subbands.The other eight higher transitions are not shown but included in the full spectra.

Figure S3 .
Figure S3.Fitted spectra for the n = 45 and the n = 200 hydrated electron clusters.The classical spectra are blue, the quantized spectra are red.The first three subbands are also shown by the dashed curves.

Figure S5 .
Figure S5.The absorption spectra of the n = 200 hydrated electron cluster computed from twenty QCMD configurations using the QCMD method (blue) and TD-DFT calculations (red).The top figure shows the computed and normalized spectra to unity, the bottom figure illustrates the shift of the TD-DFT spectrum by 0.5 eV.The spectra are computed from the first four transitions for each configuration.

Table S4 .
Means and standard deviations of the first subband with confidence intervals at level of 95%.

Table S5 .
Means and standard deviations of the second subband with confidence intervals at level of 95%.

Table S6 .
Means and standard deviations of the third subband with confidence intervals at level of 95%.

Table S7 .
Average ground state energies and energy gaps (eV) using QCMD oneelectron calculations for various sets of configurations with compressed and elongated OH lengths (see the main text for more details).