Ring Polymer Molecular Dynamics Method Research Articles

Temperature effects on the branching dynamics in the model ambimodal (6 + 4)/(4 + 2) intramolecular cycloaddition reaction.

C14H20 (tetradecapentaene) is a simple model system exhibiting post transition-state behavior, wherein both the (6 + 4) and (4 + 2) cycloaddition products are formed from one ambimocal transition state structure. We studied the bifurcation dynamics starting from the two ambimodal transition state structures, the chair-form and boat-form, using the quasi-classical trajectory, classical molecular dynamics, and ring-polymer molecular dynamics methods on the parameter-optimized semiempirical GFN2-xTB potential energy surface. It was found that the calculated branching fractions differ between the chair-form and boat-form due to the different nature in the IRC pathways. We also investigated the effects of temperature on bifurcation dynamics and found that, at higher temperatures, trajectories stay longer in the intermediate region of the potential energy surface.

Rate coefficients for the O + H2 and O + D2 reactions: how well ring polymer molecular dynamics accounts for tunelling.

We present here extensive calculations of the O(3P) + H2 and O(3P) + D2 reaction dynamics spanning the temperature range from 200 K to 2500 K. The calculations have been carried out using fully converged time-independent quantum mechanics (TI QM), quasiclassical trajectories (QCT) and ring polymer molecular dynamics (RPMD) on the two lowest lying adiabatic potential energy surfaces (PESs), 13A' and 13A'', calculated by Zanchet et al. [J. Chem. Phys., 2019, 151, 094307]. TI QM rate coefficients were determined using the cumulative reaction probability formalism on each PES including all of the total angular momenta and the Coriolis coupling and can be considered to be essentially exact within the Born-Oppenheimer approximation. The agreement between the rate coefficients calculated by using QM and RPMD is excellent for the reaction with D2 in almost the whole temperature range. For the reaction with H2, although the agreement is very good above 500 K, the deviations are significant at lower temperatures. In contrast, the QCT calculations largely underestimate the rate coefficients for the two isotopic variants due to their inability to account for tunelling. The differences found in the disagreements between RPMD and QM rate coefficients for the reactions for both the isotopologues are indicative of the ability of the RPMD method to accurately describe systems where tunelling plays a relevant role. Considering that both reactions are dominated by tunelling below 500 K, the present results show that RPMD is a very powerful tool for determining rate coefficients. The present QM rate coefficients calculated on adiabatic PESs slightly underestimate the best global fits of the experimental measurements, which we attribute to the intersystem crossing with the singlet 11A' PES.

Machine learning‐assisted study of correlation between post‐transition‐state bifurcation and initial phase information at the ambimodal transition state

AbstractThe Diels–Alder cycloaddition of cyclopentadiene and nitroethene, the intramolecular cycloaddition between a diene and triene, and the Diels–Alder cycloaddition of 2‐hydroxyacrolein with 1,3‐butadiene involving post‐transition‐state bifurcation (PTSB) were studied. These cycloaddition reactions were investigated using quasi‐classical trajectory (QCT), classical molecular dynamics (MD), ring‐polymer molecular dynamics (RPMD) simulations, and supervised machine‐learning binary classification techniques. Room‐temperature dynamics simulations started from the ambimodal transition state (TS) using the QCT, classical MD, and RPMD methods presented similar dynamics. Binary classification revealed that the initial geometry displacement from the ambimodal TS for the Diels–Alder cycloaddition of cyclopentadiene and nitroethene contributed to the branching dynamics and that the initial momenta for the intramolecular cycloaddition between a diene and triene and the Diels–Alder cycloaddition of 2‐hydroxyacrolein with 1,3‐butadiene played a significant role in the bifurcation dynamics.

Molecular dynamics simulations and machine-learning assisted study of the reaction path bifurcation: Application to the intramolecular Diels–Alder cycloaddition between cyclobutadiene and butadiene

The intramolecular Diels–Alder cycloaddition between cyclobutadiene and butadiene, which involves post-transition-state bifurcation (PTSB), has been investigated using the ring-polymer molecular dynamics (RPMD), classical MD and quasi-classical trajectory (QCT) simulations, and supervised machine-learning (ML) technique. The branching fraction of this reaction strongly depends on the simulation temperature. While QCT, classical MD and RPMD simulations performed at room temperature, starting from the ambimodal transition state (TS), showed the similar dynamics, QCT overestimated the minor branching fraction at low temperature. The supervised machine-learning analysis revealed that the initial coordinates and momenta for low frequency modes at the ambimodal TS contribute to the branching dynamics. The classical MD and RPMD methods follow similar conditions to quantum Boltzmann distributions, whereas the artificially localized phase distribution, considering that the zero-point energy is wedged into the classical particles. As a result, the QCT method can provide inadequate bifurcation dynamics, such as the overestimation of the minor product.

ℏ2 expansion of the transmission probability through a barrier.

Ninety years ago, Wigner derived the leading order expansion term in ℏ2 for the tunneling rate through a symmetric barrier. His derivation included two contributions: one came from the parabolic barrier, but a second term involved the fourth-order derivative of the potential at the barrier top. He left us with a challenge, which is answered in this paper, to derive the same but for an asymmetric barrier. A crucial element of the derivation is obtaining the ℏ2 expansion term for the projection operator, which appears in the flux-side expression for the rate. It is also reassuring that an analytical calculation of semiclassical transition state theory (TST) reproduces the anharmonic corrections to the leading order of ℏ2. The efficacy of the resulting expression is demonstrated for an Eckart barrier, leading to the conclusion that especially when considering heavy atom tunneling, one should use the expansion derived in this paper, rather than the parabolic barrier approximation. The rate expression derived here reveals how the classical TST limit is approached as a function of ℏ and, thus, provides critical insights to understand the validity of popular approximate theories, such as the classical Wigner, centroid molecular dynamics, and ring polymer molecular dynamics methods.

New Stable and Fast Ring-Polymer Molecular Dynamics for Calculating Bimolecular Rate Coefficients with an Example of OH + CH4

The accurate and efficient calculation of the rate coefficients of chemical reactions is a key issue in the research of chemical dynamics. In this work, by applying the dimension-free ultrastable Cayley propagator, the thermal rate coefficients of a prototypic high dimensional chemical reaction OH + CH4 → H2O + CH3 in the temperature range of 200 to 1500 K are investigated with ring polymer molecular dynamics (RPMD) on a highly accurate full-dimensional potential energy surface. Kinetic isotope effects (KIEs) for three isotopologues of the title reaction are also studied. The results demonstrate excellent agreement with experimental data, even in the deep tunneling region. Especially, the Cayley propagator shows a high calculation efficiency with little loss of accuracy. The present results confirmed the applicability of the RPMD method, particularly the speed-up using a Cayley propagator, in theoretical calculations of bimolecular reaction rates.

Clustering, collision, and relaxation dynamics in pure and doped helium nanoclusters: Density- vs particle-based approaches.

The clustering, collision, and relaxation dynamics of pristine and doped helium nanodroplets is theoretically investigated in cases of pickup and clustering of heliophilic argon, collision of heliophobic cesium atoms, and coalescence of two droplets brought into contact by their mutual long-range van der Waals interaction. Three approaches are used and compared with each other. The He time-dependent density functional theory method considers the droplet as a continuous medium and accounts for its superfluid character. The ring-polymer molecular dynamics method uses a path-integral description of nuclear motion and incorporates zero-point delocalization while bosonic exchange effects are ignored. Finally, the zero-point averaged dynamics approach is a mixed quantum-classical method in which quantum delocalization is described by attaching a frozen wavefunction to each He atom, equivalent to classical dynamics with effective interaction potentials. All three methods predict that the growth of argon clusters is significantly hindered by the helium host droplet due to the impeding shell structure around the dopants and kinematic effects freezing the growing cluster in metastable configurations. The effects of superfluidity are qualitatively manifested by different collision dynamics of the heliophilic atom at high velocities, as well as quadrupole oscillations that are not seen with particle-based methods, for droplets experiencing a collision with cesium atoms or merging with each other.

Nuclear quantum effects on the dynamics and glass behavior of a monatomic liquid with two liquid states.

We perform path integral molecular dynamics (PIMD) simulations of a monatomic liquid that exhibits a liquid-liquid phase transition and liquid-liquid critical point. PIMD simulations are performed using different values of Planck's constant h, allowing us to study the behavior of the liquid as nuclear quantum effects (NQE, i.e., atoms delocalization) are introduced, from the classical liquid (h = 0) to increasingly quantum liquids (h > 0). By combining the PIMD simulations with the ring-polymer molecular dynamics method, we also explore the dynamics of the classical and quantum liquids. We find that (i) the glass transition temperature of the low-density liquid (LDL) is anomalous, i.e., Tg LDL(P) decreases upon compression. Instead, (ii) the glass transition temperature of the high-density liquid (HDL) is normal, i.e., Tg HDL(P) increases upon compression. (iii) NQE shift both Tg LDL(P) and Tg HDL(P) toward lower temperatures, but NQE are more pronounced on HDL. We also study the glass behavior of the ring-polymer systems associated with the quantum liquids studied (via the path-integral formulation of statistical mechanics). There are two glass states in all the systems studied, low-density amorphous ice (LDA) and high-density amorphous ice (HDA), which are the glass counterparts of LDL and HDL. In all cases, the pressure-induced LDA-HDA transformation is sharp, reminiscent of a first-order phase transition. In the low-quantum regime, the LDA-HDA transformation is reversible, with identical LDA forms before compression and after decompression. However, in the high-quantum regime, the atoms become more delocalized in the final LDA than in the initial LDA, raising questions on the reversibility of the LDA-HDA transformation.

Path Integrals for Nonadiabatic Dynamics: Multistate Ring Polymer Molecular Dynamics.

This review focuses on a recent class of path-integral-based methods for the simulation of nonadiabatic dynamics in the condensed phase using only classical molecular dynamics trajectories in an extended phase space. Specifically, a semiclassical mapping protocol is used to derive an exact, continuous, Cartesian variable path-integral representation for the canonical partition function of a system in which multiple electronic states are coupled to nuclear degrees of freedom. Building on this exact statistical foundation, multistate ring polymer molecular dynamics methods are developed for the approximate calculation of real-time thermal correlation functions. The remarkable promise of these multistate ring polymer methods, their successful applications, and their limitations are discussed in detail.

Quantum versus classical unimolecular fragmentation rate constants and activation energies at finite temperature from direct dynamics simulations.

In the present work, we investigate how nuclear quantum effects modify the temperature dependent rate constants and, consequently, the activation energies in unimolecular reactions. In the reactions under study, nuclear quantum effects mainly stem from the presence of a large zero point energy. Thus, we investigate the behavior of methods compatible with direct dynamics simulations, the quantum thermal bath (QTB) and ring polymer molecular dynamics (RPMD). To this end, we first compare them with quantum reaction theory for a model Morse potential before extending this comparison to molecular models. Our results show that, in particular in the temperature range comparable with or lower than the zero point energy of the system, the RPMD method is able to correctly capture nuclear quantum effects on rate constants and activation energies. On the other hand, although the QTB provides a good description of equilibrium properties including zero-point energy effects, it largely overestimates the rate constants. The origin of the different behaviours is in the different distance distributions provided by the two methods and in particular how they differently describe the tails of such distributions. The comparison with transition state theory shows that RPMD can be used to study fragmentation of complex systems for which it may be difficult to determine the multiple reaction pathways and associated transition states.

On-the-Fly Ring-Polymer Molecular Dynamics Calculations of the Dissociative Photodetachment Process of the Oxalate Anion.

The dissociative photodetachment dynamics of the oxalate anion, C2O4H− + hν → CO2 + HOCO + e−, were theoretically studied using the on-the-fly path-integral and ring-polymer molecular dynamics methods, which can account for nuclear quantum effects at the density-functional theory level in order to compare with the recent experimental study using photoelectron–photofragment coincidence spectroscopy. To reduce computational time, the force acting on each bead of ring-polymer was approximately calculated from the first and second derivatives of the potential energy at the centroid position of the nuclei beads. We find that the calculated photoelectron spectrum qualitatively reproduces the experimental spectrum and that nuclear quantum effects are playing a role in determining spectral widths. The calculated coincidence spectrum is found to reasonably reproduce the experimental spectrum, indicating that a relatively large energy is partitioned into the relative kinetic energy between the CO2 and HOCO fragments. This is because photodetachment of the parent anion leads to Franck–Condon transition to the repulsive region of the neutral potential energy surface. We also find that the dissociation dynamics are slightly different between the two isomers of the C2O4H− anion with closed- and open-form structures.

Isotopic separation of helium through graphyne membranes: a ring polymer molecular dynamics study.

Microscopic-level understanding of the separation mechanism for two-dimensional (2D) membranes is an active area of research due to potential implications of this class of membranes for various technological processes. Helium (He) purification from the natural resources is of particular interest due to the shortfall in its production. In this work, we applied the ring polymer molecular dynamics (RPMD) method to graphdiyne (Gr2) and graphtriyne (Gr3) 2D membranes having variable pore sizes for the separation of He isotopes, and compare for the first time with rigorous quantum calculations. We found that the transmission rate through Gr3 is many orders of magnitude greater than Gr2. The selectivity of either isotope at low temperatures is a consequence of a delicate balance between the zero-point energy effect and tunneling of 4He and 3He. In particular, a remarkable tunneling effect is reported on the Gr2 membrane at 10 K, leading to a much larger permeation of the lighter species as compared to the heavier isotope. RPMD provides an efficient approach for studying the separation of He isotopes, taking into account quantum effects of light nuclei motions at low temperatures, which classical methods fail to capture.

State‐selective cross sections from ring polymer molecular dynamics

AbstractUnderstanding the influence of different forms of energy (eg, translational, vibrational, rotational) on chemical reactions is a key goal and great challenge in physical chemistry. Very recently, we proposed a new approach to obtain state‐selective cross sections that approximately include quantum effects such as zero‐point energy and tunneling. The method is a combination of the widely used quasiclassical trajectory approach (QCT) and the ring polymer molecular dynamics method and thus is numerically very efficient and easily employed. Here, we present a detailed description of the method and exhaustive tests of its accuracy and applicability. The robustness of the approach is tested, as well as the convergence with the number of beads. The approach is then applied to several prototypical X + H2(ν = 0, 1), X = Mu, H, D, F, Cl reactions over a wide range of collision energies. Good agreement with rigorous quantum dynamics simulations is found for most cases. Encouraging improvement over QCT results is found for particular cases, while only a small increase in numerical cost is required.

A Machine Learning Approach for Rate Constants. II. Clustering, Training, and Predictions for the O(3P) + HCl → OH + Cl Reaction.

Following up on our recent paper, which reported a machine learning approach to train on and predict thermal rate constants over a large temperature range, we present new results by using clustering and new Gaussian process regression on each cluster. Each cluster is defined by the magnitude of the correction to the Eckart transmission coefficient. Instead of the usual protocol of training and testing, which is a challenge for present small database of exact rate constants, training is done on the full data set for each cluster. Testing is done by inputing hundreds of random values of the descriptors (within reasonable bounds). The new training strategy is applied to predict the rate constants of the O(3P) + HCl reaction on the 3A' and 3A″ potential energy surfaces. This reaction was recently focused on as a "stress test" for the ring polymer molecular dynamics method. Finally, this reaction is added to the databases and training is done with this addition. The freely available database and new Python software that evaluates the correction to the Eckart transmission coefficient for any reaction are briefly described.

VTST and RPMD kinetics study of the nine-body X + C2H6 (X ≡ H, Cl, F) reactions based on analytical potential energy surfaces.

Thermal rate constants of nine-atom hydrogen abstraction reactions, X + C2H6 → HX + C2H5 (X ≡ H, Cl, F) with qualitatively different reaction paths, have been investigated using two kinetics approaches - variational transition state theory with multidimensional tunnelling (VTST/MT) and ring polymer molecular dynamics (RPMD) - and full dimensional analytical potential energy surfaces. For the H + C2H6 reaction, which proceeds through a noticeable barrier height of 11.62 kcal mol-1, kinetics approaches showed excellent agreement between them (with differences less than 30%) and with the experiment (with differences less than 60%) in the wide temperature range of 200-2000 K. For X = Cl and F, however, the situation is very different. The barrier height is either low or very low, 2.44 and 0.23 kcal mol-1, respectively, and the presence of van der Waals complexes in the entrance channel leads to a very flat topography and, consequently, imposes theoretical challenges. For the Cl(2P) reaction, VTST/MT underestimates the experimental rate constants (with differences less than 86%), and RPMD demonstrates better agreement (with differences less than 47%), although the temperature dependence is opposite to the experiment at low temperatures. Finally, for the F(2P) reaction, available experimental information shows discrepancies, both in the absolute values of the rate constants and also in the temperature dependence. Unfortunately, kinetics theories did not resolve this discrepancy. Different possible causes of these theory/experiment discrepancies were analyzed.

Rate coefficients of roaming reaction H+MgH using ring polymer molecular dynamics

The ring-polymer molecular dynamics (RPMD) was used to calculate the thermal rate coefficients of the multi-channel roaming reaction H+MgH→Mg+H2. Two reaction channels, tight and roaming, are explicitly considered. This is a pioneering attempt of exerting RPMD method to multi-channel reactions. With the help of a newly developed optimization-interpolation protocol for preparing the initial structures and adaptive protocol for choosing the force constants, we have successfully obtained the thermal rate coefficients. The results are consistent with those from other theoretical methods, such as variational transition state theory and quantum dynamics. Especially, RPMD results exhibit negative temperature dependence, which is similar to the results from variational transition state theory but different from the ones from ground state quantum dynamics calculations.

Quantum Effects on the D + H3+ → H2D+ + H Deuteration Reaction and Isotopic Variants.

The title reaction and its isotopic variants are studied using quasi-classical trajectory (QCT) (without taking into account corrections to account for the possible zero point energy breakdown) and ring polymer molecular dynamics (RPMD) methods with a full dimensional and accurate potential energy surface which presents an exchange barrier of approximately 0.144 eV. The QCT rate constant increases when the temperature decreases from 1500 to 10 K. On the contrary, the RPMD rate constant decreases with decreasing temperature, in semiquantitative agreement with recent experimental results. The present RPMD results are in between the thermal and translational experimental rate constants, extracted from the measured data to eliminate the initial vibrational excitation of H3+, obtained in an arc discharge. The difference between the present RPMD results and experimental values is attributed to the possible existence of non thermal vibrational excitation of H3+, not completely removed by the semiempirical model used for the analysis of the experimental results. Also, it is found that, below 200 K, the RPMD trajectories are trapped, forming long-lived collision complexes, with lifetimes longer than 1 ns. These collision complexes can fragment by either redissociating back to reactants or react to products, in the two cases tunneling through the centrifugal and reaction barriers, respectively. The contribution of the formation of the complex to the total deuteration rate should be calculated with more accurate quantum methods, as has been found recently for reactions of larger systems, and the present four atoms system is a good candidate to benchmark the adequacy of RPMD method at temperatures below 100 K.

Ring-polymer molecular dynamics studies of thermal rate coefficients for reaction F + H2O → HF + OH

The prototype tetra-atomic reaction F + H2O → HF + OH plays a significant role in both atmospheric and astronomical chemistry. In this work, thermal rate coefficients of this reaction are determined with the ring polymer molecular dynamics (RPMD) method on a full-dimensional potential energy surface (PES). This PES is the most accurate one for the title reaction, as demonstrated by the correct barrier height and reaction energy, compared to the benchmark calculations by the focal point analysis and the high accuracy extrapolated ab initio thermochemistry methods. The RPMD rate coefficients are in excellent agreement with those calculated by the semiclassical transition state theory and a two-dimensional master equation technique, and some experimental measurements. As has been found in many RPMD applications, quantum effects, including tunneling and zero-point energy effects, can be efficiently and effectively captured by the RPMD method. In addition, the convergence of the results with respect to the number of beads is rapid, which is also consistent with previous RPMD applications.

Quantum effects on dislocation motion from ring-polymer molecular dynamics

Quantum motion of atoms known as zero-point vibration was recently proposed to explain a long-standing discrepancy between theoretically computed and experimentally measured low-temperature plastic strength of iron and possibly other metals with high atomic masses. This finding challenges the traditional notion that quantum motion of atoms is relatively unimportant in solids comprised of heavy atoms. Here we report quantum dynamic simulations of quantum effects on dislocation motion within the exact formalism of Ring-Polymer Molecular Dynamics (RPMD). To extend the reach of quantum atomistic simulations to length and time scales relevant for extended defects in materials, we implemented RPMD in the open-source code LAMMPS thus making the RPMD method widely available to the community. We use our RPMD/LAMMPS approach for direct calculations of dislocation mobility and its effects on the yield strength of α-iron. Our simulation results establish that quantum effects are noticeable at temperatures below 50 K but account for only a modest (≈13% at T = 0 K) overall reduction in the Peierls barrier, at variance with the factor of two reduction predicted earlier based on the more approximate framework of harmonic transition state theory. Our results confirm that zero-point vibrations provide ample additional agitation for atomic motion that increases with decreasing temperature, however its enhancing effect on dislocation mobility is largely offset by an increase in the effective atom size, an effect known as quantum dispersion that has not been accounted for in the previous calculations.

Nuclear Quantum Effects in H+ and OH– Diffusion along Confined Water Wires

The diffusion of protons and hydroxide ions along water wires provides an efficient mechanism for charge transport that is exploited by biological membrane channels and shows promise for technological applications such as fuel cells. However, what is lacking for a better control and design of these systems is a thorough theoretical understanding of the diffusion process at the atomic scale. Here we focus on two aspects of this process that are often disregarded because of their high computational cost: the use of first-principles potential energy surfaces and the treatment of the nuclei as quantum particles. We consider proton and hydroxide ions in finite water wires using density functional theory augmented with an apolar cylindrical confining potential. We employ machine learning techniques to identify the charged species, thus obtaining an agnostic definition that takes explicitly into account the delocalization of the charge in the Grotthus-like mechanism. We include nuclear quantum effects (NQEs) through the thermostated ring polymer molecular dynamics method and model finite system size effects by considering Langevin dynamics on the potential of mean force of the charged species, allowing us to extract the same "universal" diffusion coefficient from simulations with different wire sizes. In the classical case, diffusion coefficients depend significantly on the potential energy surface, in particular on how dispersion forces modulate water-water distances. NQEs, however, make the diffusion less sensitive to the underlying potential and geometry of the wire.