Simulating the ghost: quantum dynamics of the solvated electron

The hydrated electron, e⁻(aq), has been a subject of intense research for over half a century due to its fundamental importance in various fields, including electrochemistry, photochemistry, high-energy chemistry, and biology (where its non-equilibrium precursor causes radiation damage to DNA). Despite its apparent simplicity as the smallest anion and a basic reducing agent, accurately modeling its behavior is complex. Standard density functional theory (DFT) methods often suffer from delocalization error, hindering accurate radical modeling. While accurate liquid water descriptions are possible with methods like MP2, these are computationally expensive. Previous picosecond-scale MP2-based molecular dynamics (MD) simulations supported the cavity structure, but longer timescales and the inclusion of nuclear quantum effects (NQEs) were computationally prohibitive. This study aims to overcome these limitations by using a machine-learning model to achieve long-timescale quantum dynamics of the hydrated electron with MP2-quality accuracy, enabling a comprehensive characterization of its properties and the impact of NQEs.

Extensive research has focused on understanding the solvated electron, employing various theoretical methods. Density Functional Theory (DFT), while frequently used, suffers from delocalization errors, making accurate modeling of radicals challenging. Higher-level methods like Møller-Plesset perturbation theory (MP2) offer improved accuracy but are computationally expensive. Previous work using MP2-based MD simulations provided insights into the cavity structure of the solvated electron, but these simulations were limited by computational constraints, hindering exploration of longer timescales and the impact of nuclear quantum effects. The use of machine-learning potentials, specifically Behler-Parrinello Neural Networks (BPNNs), has shown potential to address these challenges, offering a balance between accuracy and computational efficiency.

This research employed a machine-learning approach using a Behler-Parrinello Neural Network (BPNN) to create a force field capable of accurately representing the solvated electron's interaction with water. The BPNN was trained using forces and energies calculated at the MP2 level of theory, incorporating data from both classical and quantum MD simulations. This approach implicitly captures the electron's influence on the water structure without explicitly including it in the model, treating it as a "ghost electron". The model was then used in long-timescale (hundreds of picoseconds) simulations to investigate the solvated electron's properties. Nuclear quantum dynamics were modeled using the thermostatted ring-polymer (TRP) MD method. Spin densities were evaluated using DFT with a hybrid functional to analyze the electron's localization. Structural properties were characterized using radial distribution functions (RDFs), and vibrational spectroscopy was analyzed using the vibrational density of states (VDOS). The study included simulations with both H₂O and D₂O to assess isotope effects. Classical MD simulations were also performed for comparison. Absorption spectra were calculated using time-dependent DFT, and quantum alchemical exchange calculations were conducted to probe isotopic segregation. The authors conducted extensive validation and benchmarking to ensure the accuracy and reliability of their machine-learning model.

The machine-learning model accurately reproduced the solvated electron's cavity structure and localization dynamics. Simulations showed that the electron initially exists in a delocalized state before undergoing a pre-solvation stage and finally localizing within a cavity formed by 4-5 water molecules. The inclusion of NQEs revealed a novel transient diffusion mechanism involving the formation of a twin cavity, where the electron shuttles between two adjacent cavities. This mechanism was observed only in quantum simulations of H₂O, not in D₂O. The radial distribution functions indicated a mean e⁻-O coordination number of 4.5, consistent with previous DFT estimates. NQEs broadened the peaks in RDFs and reduced long-range order. The analysis of the hydrogen bond (HB) network showed that cavity formation is associated with an increase in undercoordinated water molecules. The calculated electronic absorption spectrum for single-cavity structures agreed reasonably well with experimental data. The calculated vibrational density of states showed excellent agreement with experimental values for bending modes in both light and heavy water, highlighting the importance of including NQEs. The simulated isotope effects, including the doublet in the bending region for HOD mixtures, aligned with experimental observations. Quantum alchemical exchange calculations supported experimental findings on isotopic segregation.

The study's findings address the long-standing challenges in accurately modeling the solvated electron. The successful application of a machine-learning potential enabled the observation of previously inaccessible dynamics, including the transient twin-cavity diffusion mechanism. The detailed analysis of the structural and spectral properties validated the model's accuracy and provided insights into the electron's interaction with the surrounding water molecules. The inclusion of NQEs significantly improved the agreement between simulations and experimental data, particularly in vibrational spectroscopy. The observed twin-cavity structure highlights the limitations of classical MD simulations in capturing the full complexity of the solvated electron's behavior. The results are significant for advancing our understanding of fundamental chemical processes involving electron transfer and solvation.

This research demonstrates a successful approach to simulating the quantum dynamics of the solvated electron using a machine-learning model. The model accurately reproduces experimental observations, providing insights into the structure, dynamics, and vibrational spectroscopy of this elusive species. The discovery of a novel transient diffusion mechanism involving a twin cavity highlights the importance of considering nuclear quantum effects. Future research could explore the extension of this methodology to other solvated species and investigate the role of the twin-cavity mechanism in various chemical processes.

While the machine-learning model achieved high accuracy, its applicability is limited to the specific system size and conditions it was trained on. The model's performance in regions of configuration space that were poorly sampled during training (e.g., twin-cavity structures) might require further investigation. The absorption spectrum calculations relied on approximate methods (time-dependent DFT), which could introduce uncertainties. The agreement between the simulated and experimental vibrational spectra is not perfect in all regions, which could be due to limitations of the quantum dynamics approximation used.