Path Integral Molecular Dynamics Research Articles

Enabling large-scale quantum path integral molecular dynamics simulations through the integration of Dcdftbmd and i-PI codes.

A large-scale quantum chemical calculation program, Dcdftbmd, was integrated with a Python-based advanced atomistic simulation program, i-PI. The implementation of a client-server model enabled hierarchical parallelization with respect to replicas and force evaluations. The established framework demonstrated that quantum path integral molecular dynamics simulations can be executed with high efficiency for systems consisting of a few tens of replicas and containing thousands of atoms. The application of the framework to bulk water systems, with and without an excess proton, demonstrated that nuclear quantum effects are significant for intra- and inter-molecular structural properties, including oxygen-hydrogen bond distance and radial distribution function around the hydrated excess proton.

Ring Polymer Molecular Dynamics with Electronic Transitions.

Full quantum dynamics of molecules and materials is of fundamental importance, which requires a faithful description of simultaneous quantum motions of the electron and nuclei. A new scheme is developed for nonadiabatic simulations of coupled electron-nuclear quantum dynamics with electronic transitions based on the Ehrenfest theorem and ring polymer molecular dynamics. Built upon the isomorphic ring polymer Hamiltonian, time-dependent multistate electronic Schrödinger equations are solved self-consistently with approximate equation of motions for nuclei. Each bead bears a distinct electronic configuration and thus moves on a specific effective potential. This independent-bead approach provides an accurate description of the real-time electronic population and quantum nuclear trajectory, maintaining a good agreement with the exact quantum solution. Implementation of first-principles calculations enables us to simulate photoinduced proton transfer in H_{2}O-H_{2}O^{+} where we find a good agreement with experiment.

Theoretical Calculations of the Thermal Rate Coefficients for the Interstellar NH3+ + H2 → NH4+ + H Reaction on a New Δ-Machine Learning Potential Energy Surface

The NH3+ + H2 → NH4+ + H ion–molecule hydrogen-exchange reaction is important for understanding the mechanisms of ammonia formation in interstellar medium and circumstellar environments. The thermal rate coefficients of this reaction are calculated using the ring polymer molecular dynamics approach, which can account for the nuclear quantum effects, on a newly developed ab initio potential energy surface (PES). The PES was constructed by fitting a large number of ab initio points at the MP2/cc-pVDZ and CCSD(T)/cc-pVTZ levels using a Δ-machine learning approach. The obtained rate coefficients exhibited a minimum at T ∼ 130 K, and this characteristic temperature-dependent behavior qualitatively agreed with the experimental measurements. This study confirms that quantum tunneling plays a significant role in the reaction dynamics along with the existence of the metastable NH3+···H2 pre-reactive complex in the entrance channel of the PES.

Routine Molecular Dynamics Simulations Including Nuclear Quantum Effects: From Force Fields to Machine Learning Potentials.

We report the implementation of a multi-CPU and multi-GPU massively parallel platform dedicated to the explicit inclusion of nuclear quantum effects (NQEs) in the Tinker-HP molecular dynamics (MD) package. The platform, denoted Quantum-HP, exploits two simulation strategies: the Ring-Polymer Molecular Dynamics (RPMD) that provides exact structural properties at the cost of a MD simulation in an extended space of multiple replicas and the adaptive Quantum Thermal Bath (adQTB) that imposes the quantum distribution of energy on a classical system via a generalized Langevin thermostat and provides computationally affordable and accurate (though approximate) NQEs. We discuss some implementation details, efficient numerical schemes, and parallelization strategies and quickly review the GPU acceleration of our code. Our implementation allows an efficient inclusion of NQEs in MD simulations for very large systems, as demonstrated by scaling tests on water boxes with more than 200,000 atoms (simulated using the AMOEBA polarizable force field). We test the compatibility of the approach with Tinker-HP's recently introduced Deep-HP machine learning potentials module by computing water properties using the DeePMD potential with adQTB thermostatting. Finally, we show that the platform is also compatible with the alchemical free energy estimation capabilities of Tinker-HP and fast enough to perform simulations. Therefore, we study how NQEs affect the hydration free energy of small molecules solvated with the recently developed Q-AMOEBA water force field. Overall, the Quantum-HP platform allows users to perform routine quantum MD simulations of large condensed-phase systems and will help to shed new light on the quantum nature of important interactions in biological matter.

Mechanism of Cations Suppressing Proton Diffusion Kinetics for Electrocatalysis.

Proton transfer is crucial for electrocatalysis. Accumulating cations at electrochemical interfaces can alter the proton transfer rate and then tune electrocatalytic performance. However, the mechanism for regulating proton transfer remains ambiguous. Here, we quantify the cation effect on proton diffusion in solution by hydrogen evolution on microelectrodes, revealing the rate can be suppressed by more than 10 times. Different from the prevalent opinions that proton transport is slowed down by modified electric field, we found water structure imposes a more evident effect on kinetics. FTIR test and path integral molecular dynamics simulation indicate that proton prefers to wander within the hydration shell of cations rather than to hop rapidly along water wires. Low connectivity of water networks disrupted by cations corrupts the fast-moving path in bulk water. This study highlights the promising way for regulating proton kinetics via a modified water structure.

Mechanism of Cations Suppressing Proton Diffusion Kinetics for Electrocatalysis

AbstractProton transfer is crucial for electrocatalysis. Accumulating cations at electrochemical interfaces can alter the proton transfer rate and then tune electrocatalytic performance. However, the mechanism for regulating proton transfer remains ambiguous. Here, we quantify the cation effect on proton diffusion in solution by hydrogen evolution on microelectrodes, revealing the rate can be suppressed by more than 10 times. Different from the prevalent opinions that proton transport is slowed down by modified electric field, we found water structure imposes a more evident effect on kinetics. FTIR test and path integral molecular dynamics simulation indicate that proton prefers to wander within the hydration shell of cations rather than to hop rapidly along water wires. Low connectivity of water networks disrupted by cations corrupts the fast‐moving path in bulk water. This study highlights the promising way for regulating proton kinetics via a modified water structure.

Nuclear quantum effects on zeolite proton hopping kinetics explored with machine learning potentials and path integral molecular dynamics

Proton hopping is a key reactive process within zeolite catalysis. However, the accurate determination of its kinetics poses major challenges both for theoreticians and experimentalists. Nuclear quantum effects (NQEs) are known to influence the structure and dynamics of protons, but their rigorous inclusion through the path integral molecular dynamics (PIMD) formalism was so far beyond reach for zeolite catalyzed processes due to the excessive computational cost of evaluating all forces and energies at the Density Functional Theory (DFT) level. Herein, we overcome this limitation by training first a reactive machine learning potential (MLP) that can reproduce with high fidelity the DFT potential energy surface of proton hopping around the first Al coordination sphere in the H-CHA zeolite. The MLP offers an immense computational speedup, enabling us to derive accurate reaction kinetics beyond standard transition state theory for the proton hopping reaction. Overall, more than 0.6 μs of simulation time was needed, which is far beyond reach of any standard DFT approach. NQEs are found to significantly impact the proton hopping kinetics up to ~473 K. Moreover, PIMD simulations with deuterium can be performed without any additional training to compute kinetic isotope effects over a broad range of temperatures.

Ring-polymer Molecular Dynamics Simulation for the Adsorption of H2 on Ice Clusters (H2 O)n (n=8, 10, and 12).

In the interstellar medium, the H2 adsorption and desorption on the solid water ice are crucial for chemical and physical processes. We have recently investigated the probabilities of H2 sticking on the (H2 O)8 ice, which has quadrilateral surfaces. We have extended the previous work using classical MD and ring-polymer molecular dynamics (RPMD) simulations to the larger ice clusters, (H2 O)10 and (H2 O)12 , which have pentagonal and hexagonal surfaces, respectively. The H2 sticking probabilities decreased as the temperature increased for both cluster cases, whereas the cluster-size-independent profiles were observed. It is thought that the size independence of the probabilities is qualitatively understood from the similar binding energies for all the three cluster systems. Furthermore, the RPMD sticking probabilities are smaller than the classical ones because of the reduction in the binding energies owing to nuclear quantum effects, such as vibrational quantization.

Stable Solid Molecular Hydrogen above 900K from a Machine-Learned Potential Trained with Diffusion Quantum MonteCarlo.

We survey the phase diagram of high-pressure molecular hydrogen with path integral molecular dynamics using a machine-learned interatomic potential trained with quantum MonteCarlo forces and energies. Besides the HCP and C2/c-24 phases, we find two new stable phases both with molecular centers in the Fmmm-4 structure, separated by a molecular orientation transition with temperature. The high temperature isotropic Fmmm-4 phase has a reentrant melting line with a maximum at higher temperature (1450K at 150GPa) than previously estimated and crosses the liquid-liquid transition line around 1200K and 200GPa.

Collective Proton Transfers in Cyclic Water–Ammonia Tetramers: A Path Integral Machine-Learning Study

We present results from machine-learning-based path integral molecular dynamics simulations that describe isomerization paths articulated via collective proton transfers along mixed, cyclic tetramers combining water and ammonia at cryogenic conditions. The net result of such isomerizations is a reverse of the chirality of the global hydrogen-bonding architecture along the different cyclic moieties. In monocomponent tetramers, the classical free energy profiles associated with these isomerizations present the usual symmetric double-well characteristics whereas the reactive paths exhibit full concertedness among the different intermolecular transfer processes. Contrastingly, in mixed water/ammonia tetramers, the incorporation of a second component introduces imbalances in the strengths of the different hydrogen bonds leading to a partial loss of concertedness, most notably at the vicinity of the transition state. As such, the highest and lowest degrees of progression are registered along OH···N and O···HN coordinations, respectively. These characteristics lead to polarized transition state scenarios akin to solvent-separated ion-pair configurations. The explicit incorporation of nuclear quantum effects promotes drastic depletions in the activation free energies and modifications in the overall shape of the profiles which include central plateau-like stages, indicating the prevalence of deep tunneling regimes. On the other hand, the quantum treatment of the nuclei partially restores the degree of concertedness among the evolutions of the individual transfers.

Path-integral molecular dynamics predictions of equilibrium H and O isotope fractionations between brucite and water

Precise prediction of hydrogen (H) isotope fractionations among different substances is a long-standing challenge in isotope geochemistry, as it needs treatments beyond the harmonic approximation. Path-integral molecular dynamics (PIMD) simulations have recently been proved to be valid in predicting equilibrium isotope fractionations for light elements. However, the lack of reliable force fields hinders the application of PIMD to the condensed phases. In this work, the deep potential models trained on the first-principles molecular dynamics (FPMD) data are applied to PIMD simulations to determine the D/H and 18O/16O isotope fractionation factors between brucite and water. To quantitatively assess the influence of quantum effects, the D/H and 18O/16O isotope fractionations between brucite and water are also determined under the framework of harmonic approximation.By comparing the results of the PIMD and those of harmonic calculations, we find that H and O atoms in brucite and water are strongly affected by the anharmonic effects. The accuracy of the harmonic isotope fractionations mainly depends on the cancellation percentage of the anharmonic effects on the reduced partition function ratios (RPFRs) of the two substances. The isotope fractionations predicted by PIMD simulations are close to the experimental results at high temperatures. However, at low temperatures, the predicted isotope fractionations are different from those of experiments. The discrepancies are attributed to the approximations in the density functional theory (DFT) functionals used in this work, i.e., the inaccuracy in predicting the proton positions in brucite and water. Moreover, we find that the H isotope fractionations are extremely sensitive to the change of pressure. The direction of H isotope fractionation between brucite and water (at 300 K) will be even inversed as the pressure increases to more than 3 GPa. This strong pressure sensitivity may be a common characteristic of hydrous minerals and water systems. Therefore, the extrapolated H isotope fractionations used in the studies of dehydration of subducting slabs or deep Earth H distribution based on the isotope fractionations under low pressures may need to be rechecked.

Competing Nuclear Quantum Effects and Hydrogen-Bond Jumps in Hydrated Kaolinite.

Recent work has shown that the dynamics of hydrogen bonds in pure clays are affected by nuclear quantum fluctuations, with different effects for the hydrogen bonds holding different layers of the clay together and for those within the same layer. At the clay-water interface there is an even wider range of types of hydrogen bond, suggesting that the quantum effects may be yet more varied. We apply classical and thermostated ring polymer molecular dynamics simulations to show that nuclear quantum effects accelerate hydrogen-bond dynamics to varying degrees. By interpreting the results in terms of the extended jump model of hydrogen-bond switching, we can understand the origins of these effects in terms of changes in the quantum kinetic energy of hydrogen atoms during an exchange. We also show that the extended jump mechanism is applicable not only to the hydrogen bonds involving water, but also those internal to the clay.

Data-Efficient Machine Learning Potentials from Transfer Learning of Periodic Correlated Electronic Structure Methods: Liquid Water at AFQMC, CCSD, and CCSD(T) Accuracy.

Obtaining the atomistic structure and dynamics of disordered condensed-phase systems from first-principles remains one of the forefront challenges of chemical theory. Here we exploit recent advances in periodic electronic structure and provide a data-efficient approach to obtain machine-learned condensed-phase potential energy surfaces using AFQMC, CCSD, and CCSD(T) from a very small number (≤200) of energies by leveraging a transfer learning scheme starting from lower-tier electronic structure methods. We demonstrate the effectiveness of this approach for liquid water by performing both classical and path integral molecular dynamics simulations on these machine-learned potential energy surfaces. By doing this, we uncover the interplay of dynamical electron correlation and nuclear quantum effects across the entire liquid range of water while providing a general strategy for efficiently utilizing periodic correlated electronic structure methods to explore disordered condensed-phase systems.

Non-adiabatic ring polymer molecular dynamics in the phase space of the SU(N) Lie group.

We derive the non-adiabatic ring polymer molecular dynamics (RPMD) approach in the phase space of the SU(N) Lie Group. This method, which we refer to as the spin mapping non-adiabatic RPMD (SM-NRPMD), is based on the spin-mapping formalism for the electronic degrees of freedom (DOFs) and ring polymer path-integral description for the nuclear DOFs. Using the Stratonovich-Weyl transform for the electronic DOFs and the Wigner transform for the nuclear DOFs, we derived an exact expression of the Kubo-transformed time-correlation function (TCF). We further derive the spin mapping non-adiabatic Matsubara dynamics using the Matsubara approximation that removes the high frequency nuclear normal modes in the TCF and derive the SM-NRPMD approach from the non-adiabatic Matsubara dynamics by discarding the imaginary part of the Liouvillian. The SM-NRPMD method has numerical advantages compared to the original NRPMD method based on the Meyer-Miller-Stock-Thoss (MMST) mapping formalism due to a more natural mapping using the SU(N) Lie Group that preserves the symmetry of the original system. We numerically compute the Kubo-transformed position auto-correlation function and electronic population correlation function for three-state model systems. The numerical results demonstrate the accuracy of the SM-NRPMD method, which outperforms the original MMST-based NRPMD. We envision that the SM-NRPMD method will be a powerful approach to simulate electronic non-adiabatic dynamics and nuclear quantum effects accurately.

Direct Elucidation of the Vibrationally Averaged Structure of Benzene: A Path Integral Molecular Dynamics Study.

Path integral molecular dynamics (PIMD) simulations for C6H6, C6D6, and C6T6 have been carried out to directly estimate the distribution of projected C-H(D,T) bond lengths onto the principal axis plane. The average values of raw C-H(D,T) bond lengths obtained from PIMD simulations are in the order of ⟨RC-H⟩ > ⟨RC-D⟩ > ⟨RC-T⟩ due to the anharmonicity of the potential energy curve. However, the projected C-H(D,T) bond lengths are almost the same as those reported by Hirano et al. [J. Mol. Struct. 2021, 1243, 130537]. Our PIMD simulations directly and strongly support the explanation by Hirano et al. for the experimental observations that almost the same projected C-H(D) bond lengths are found for C6H6 and C6D6. The PIMD simulations also predicted the same projected bond lengths for C6T6 as those of C6H(D)6. In addition to the previous local mode analysis, the present PIMD simulations predicted, for benzene isotopologues, that the vibrationally averaged structure is planar but non-flat.

Accurate global machine learning force fields for molecules with hundreds of atoms.

Global machine learning force fields, with the capacity to capture collective interactions in molecular systems, now scale up to a few dozen atoms due to considerable growth of model complexity with system size. For larger molecules, locality assumptions are introduced, with the consequence that nonlocal interactions are not described. Here, we develop an exact iterative approach to train global symmetric gradient domain machine learning (sGDML) force fields (FFs) for several hundred atoms, without resorting to any potentially uncontrolled approximations. All atomic degrees of freedom remain correlated in the global sGDML FF, allowing the accurate description of complex molecules and materials that present phenomena with far-reaching characteristic correlation lengths. We assess the accuracy and efficiency of sGDML on a newly developed MD22 benchmark dataset containing molecules from 42 to 370 atoms. The robustness of our approach is demonstrated in nanosecond path-integral molecular dynamics simulations for supramolecular complexes in the MD22 dataset.

Unraveling the Nature of Hydrogen Bonds of "Proton Sponges" Based on Car-Parrinello and Metadynamics Approaches.

The nature of intra- and intermolecular non-covalent interactions was studied in four naphthalene derivatives commonly referred to as "proton sponges". Special attention was paid to an intramolecular hydrogen bond present in the protonated form of the compounds. The unsubstituted "proton sponge" served as a reference structure to study the substituent influence on the hydrogen bond (HB) properties. We selected three compounds substituted by methoxy, amino, and nitro groups. The presence of the substituents either retained the parent symmetry or rendered the compounds asymmetric. In order to reveal the non-covalent interaction properties, the Hirshfeld surface (HS) was computed for the crystal structures of the studied compounds. Next, quantum-chemical simulations were performed in vacuo and in the crystalline phase. Car-Parrinello molecular dynamics (CPMD), Path Integral Molecular Dynamics (PIMD), and metadynamics were employed to investigate the time-evolution changes of metric parameters and free energy profile in both phases. Additionally, for selected snapshots obtained from the CPMD trajectories, non-covalent interactions and electronic structure were studied. Quantum theory of atoms in molecules (QTAIM) and the Density Overlap Regions Indicator (DORI) were applied for this purpose. It was found based on Hirshfeld surfaces that, besides intramolecular hydrogen bonds, other non-covalent interactions are present and have a strong impact on the crystal structure organization. The CPMD results obtained in both phases showed frequent proton transfer phenomena. The proton was strongly delocalized in the applied time-scale and temperature, especially in the PIMD framework. The use of metadynamics allowed for tracing the free energy profiles and confirming that the hydrogen bonds present in "proton sponges" are Low-Barrier Hydrogen Bonds (LBHBs). The electronic and topological analysis quantitatively described the temperature dependence and time-evolution changes of the electronic structure. The covalency of the hydrogen bonds was estimated based on QTAIM analysis. It was found that strong hydrogen bonds show greater covalency, which is additionally determined by the proton position in the hydrogen bridge.

Incorporating Nuclear Quantum Effects in Molecular Dynamics with a Constrained Minimized Energy Surface

The accurate incorporation of nuclear quantum effects in large-scale molecular dynamics (MD) simulations remains a significant challenge. Recently, we combined constrained nuclear-electronic orbital (CNEO) theory with classical MD and obtained a new approach (CNEO-MD) that can accurately and efficiently incorporate nuclear quantum effects into classical simulations. In this Letter, we provide the theoretical foundation for CNEO-MD by developing an alternative formulation of the equations of motion for MD. In this new formulation, the expectation values of quantum nuclear positions evolve classically on an effective energy surface that is obtained from a constrained energy minimization procedure when solving for the quantum nuclear wave function, thus enabling the incorporation of nuclear quantum effects in classical MD simulations. For comparison with other existing approaches, we examined a series of model systems and found that this new MD approach is significantly more accurate than the conventional way of performing classical MD and generally outperforms centroid MD and ring-polymer MD in describing vibrations in these model systems.

Dynamics study of the post-transition-state-bifurcation process of the (HCOOH)H+ → CO + H3O+/HCO+ + H2O dissociation: application of machine-learning techniques.

The process of protonated formic acid dissociating from the transition state was studied using ring-polymer molecular dynamics (RPMD), classical MD, and quasi-classical trajectory (QCT) simulations. Temperature had a strong influence on the branching fractions for the HCO+ + H2O and CO + H3O+ dissociation channels. The RPMD and classical MD simulations showed similar behavior, but the QCT dynamics were significantly different owing to the excess energies in the quasi-classical trajectories. Machine-learning analysis identified several key features in the phase information of the vibrational motions at the transition state. We found that the initial configuration and momentum of a hydrogen atom connected to a carbon atom and the shrinking coordinate of the CO bond at the transition state play a role in the dynamics of HCO+ + H2O production.

First-principles surface reaction rates by ring polymer molecular dynamics and neural network potential: role of anharmonicity and lattice motion.

Elementary gas-surface processes are essential steps in heterogeneous catalysis. A predictive understanding of catalytic mechanisms remains challenging due largely to difficulties in accurately characterizing the kinetics of such steps. Experimentally, thermal rates for elementary surface reactions can now be measured using a novel velocity imaging technique, providing a stringent testing ground for ab initio rate theories. Here, we propose to combine ring polymer molecular dynamics (RPMD) rate theory with state-of-the-art first-principles-determined neural network potential to calculate surface reaction rates. Taking NO desorption from Pd(111) as an example, we show that the harmonic approximation and the neglect of lattice motion in the commonly-used transition state theory overestimates and underestimates the entropy change during the desorption process, respectively, leading to opposite errors in rate coefficient predictions and artificial error cancellations. Including anharmonicity and lattice motion, our results reveal a generally neglected surface entropy change due to significant local structural change during desorption and obtain the right answer for the right reasons. Although quantum effects are found to be less important in this system, the proposed approach establishes a more reliable theoretical benchmark for accurately predicting the kinetics of elementary gas-surface processes.