Path Integral Molecular Dynamics Method Research Articles

Location of the Shared Proton in Proton-Bound Dimer Compound of Hydrogen Sulfate and Formate: Path Integral Molecular Dynamics Study.

The structure of the proton-bound dimer compound of hydrogen sulfate and formate has been studied by considering nuclear quantum effects (NQEs) using the path integral molecular dynamics method. This study unveiled the location of the shared proton and answered the following question: "Is the shared proton localized on either an anion or located around the center of two anions?" We have elucidated that the shared proton is distributed in the region beyond the transition state due to the NQEs, even though the shared proton did not completely overcome the transition state for the proton shuttle.

Accurate calculation of tunneling splittings in water clusters using path-integral based methods.

Tunneling splittings observed in molecular rovibrational spectra are significant evidence for tunneling motion of hydrogen nuclei in water clusters. Accurate calculations of the splitting sizes from first principles require a combination of high-quality inter-atomic interactions and rigorous methods to treat the nuclei with quantum mechanics. Many theoretical efforts have been made in recent decades. This Perspective focuses on two path-integral based tunneling splitting methods whose computational cost scales well with the system size, namely, the ring-polymer instanton method and the path-integral molecular dynamics (PIMD) method. From a simple derivation, we show that the former is a semiclassical approximation to the latter, despite that the two methods are derived very differently. Currently, the PIMD method is considered to be an ideal route to rigorously compute the ground-state tunneling splitting, while the instanton method sacrifices some accuracy for a significantly smaller computational cost. An application scenario of such a quantitatively rigorous calculation is to test and calibrate the potential energy surfaces of molecular systems by spectroscopic accuracy. Recent progress in water clusters is reviewed, and the current challenges are discussed.

Nuclear quantum effects in fullerene-fullerene aggregation in water.

We studied the effects of the quantum delocalization in space of the hydrogen atoms of water in the aggregation process of two fullerene molecules. We considered a case using a purely repulsive water-fullerene interaction, as such a situation has shown that water-mediated effects play a key role in the aggregation process. This study becomes feasible, at a reduced computational price, by combining the path integral (PI) molecular dynamics (MD) method with a recently developed open-system MD technique. Specifically, only the mandatory solvation shell of the two fullerene molecules was considered at full quantum resolution, while the rest of the system was represented as a mean-field macroscopic reservoir of particles and energy. Our results showed that the quantum nature of the hydrogen atoms leads to a sizable difference in the curve of the free energy of aggregation; that is, that nuclear quantum effects play a relevant role.

Accurate path-integral molecular dynamics calculation of aluminum with improved empirical ionic potentials

We show that an accurate description of the interplay between the anharmonic ionic interaction and the zero-point fluctuation of ions in crystalline aluminum (Al) can be achieved using the path-integral molecular dynamics (PIMD) method in conjunction with improved empirical modified embedded-atom method (MEAM) potentials. Our results show that the zero-point fluctuation of ions is noticeable at a temperature between 200 K and 250 K, roughly half of the Al Debye temperature of 428 K, with a selective influence on mechanical properties. The effect provides appreciable corrections to the lattice constant $a$, bulk modulus $B$, and elastic constant ${C}_{11}$ at low temperature, but without much influence on elastic constants ${C}_{12}$ and ${C}_{44}$. With a revised MEAM potential which takes into consideration the influence of zero-point fluctuations, the PIMD method has a much improved accuracy in $a, B$, and the $C$'s up to 700 K, when compared with precision experimental measurements. The largest errors of $a$ and $B$ can even be reduced by about an order in percentage below 300 K.

Path integral molecular dynamics simulations for Green's function in a system of identical bosons.

Path integral molecular dynamics (PIMD) has been successfully applied to perform simulations of large bosonic systems in a recent study [Hirshberg et al., Proc. Natl. Acad. Sci. U. S. A. 116, 21445 (2019)]. In this work, we extend PIMD techniques to study Green's function for bosonic systems. We demonstrate that the development of the original PIMD method enables us to calculate Green's function and extract momentum distribution from our simulations. We also apply our method to systems of identical interacting bosons to study Berezinskii-Kosterlitz-Thouless transition around its critical temperature.

A path integral molecular dynamics study on the NH4+ rotation in NH4+⋯XH2 (X = Be or Mg) dihydrogen bond systems

The nuclear quantum effects (NQEs) in dihydrogen bond (DHB) complexes, i.e., NH4+⋯BeH2 and NH4+⋯MgH2, have been investigated using multicomponent quantum mechanics (MC_QM) calculations and path integral molecular dynamics (PIMD) simulation. The MC_QM method considers the NQEs, whereas PIMD considers both the NQEs and the thermal effects. The linear C3v structure is maintained in the optimized structures obtained by the static MP2 and MC_MP2 calculations, whereas the average structures obtained by the PIMD simulation are nonlinear. The strong DHB interaction in NH4+⋯MgH2 suppresses the fluctuation in the Hδ+NMg and Hδ-MgN angles, and hence, the NH4+ rotation did not occur in the simulation of NH4+⋯MgH2. The analysis of the radius of gyration revealed that the nuclear quantum fluctuation in the perpendicular direction is suppressed by the formation of the DHB complex, whereas that in the parallel direction is slightly enhanced in both the Hδ+ and Hδ- nuclei.

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.

Time correlation functions for quantum systems: Validating Bayesian approaches for harmonic oscillators and beyond

The quantum harmonic oscillator is the fundamental building block to compute thermal properties of virtually any dielectric crystal at low temperatures in terms of phonons, extended further to cases with anharmonic couplings, or even disordered solids. In general, Path Integral Monte Carlo or Path Integral Molecular Dynamics methods are powerful tools to determine stochastically thermodynamic quantities without systematic bias, not relying on perturbative schemes. Addressing transport properties, for instance calculating thermal conductivity from PIMC, however, is substantially more difficult. Although correlation functions of current operators can be determined by PIMC from analytic continuation on the imaginary time axis, Bayesian methods are usually employed for the numerical inversion back to real time response functions. This task not only strongly relies on the accuracy of the PIMC data but also introduces noticeable dependence on the model used for the inversion. Here, we address both difficulties with care. In particular, we first devise improved estimators for current correlations, which substantially reduce the variance of the PIMC data. Next, we provide a neat statistical approach to the inversion problem, blending into a fresh workflow the classical stochastic maximum entropy method together with recent notions borrowed from statistical learning theory. We test our ideas on a single harmonic oscillator and a collection of oscillators with a continuous distribution of frequencies and provide indications of the performance of our method in the case of a particle in a double well potential. This work establishes solid grounds for an unbiased, fully quantum mechanical calculation of transport properties in solids.

Multi-Level Monte Carlo Path Integral Molecular Dynamics for Thermal Average Calculation in the Nonadiabatic Regime

With the path integral approach, the thermal average in a multi-electronic-state quantum systems can be approximated by the ring polymer representation on an extended configuration space, where the additional degrees of freedom are associated with the surface index of each bead. The primary goal of this work is to propose a more efficient sampling algorithm for the calculation of such thermal averages. We reformulate the extended ring polymer approximation according to the configurations of the surface indexes, and by introducing a proper reference measure, the reformulation is recast as a ratio of two expectations of function expansions. By quantitatively estimating the sub-estimators, and minimizing the total variance of the sampled average, we propose a multi-level Monte Carlo path integral molecular dynamics method (MLMC-PIMD) to achieve an optimal balance of computational cost and accuracy.

Tunneling Splittings in Water Clusters from Path Integral Molecular Dynamics.

We present calculations of tunneling splittings in selected small water clusters, based on a recently developed path integral molecular dynamics (PIMD) method. The ground-rotational-state tunneling motions associated with the largest splittings in the water dimer, trimer, and hexamer are considered, and we show that the PIMD predictions are in very good agreement with benchmark quantum and experimental results. As the tunneling spectra are highly sensitive to both the details of the quantum dynamics and the potential energy surface, our calculations are a validation of the MB-Pol surface as well as the accuracy of PIMD. The favorable scaling of PIMD with system size paves the way for calculations of tunneling splittings in large, nonrigid molecular systems with motions that cannot be treated accurately by other methods, such as the semiclassical instanton.

SCC-DFTB-PIMD Method To Evaluate a Multidimensional Quantum Free-Energy Surface for a Proton-Transfer Reaction.

The self-consistent charge density functional tight binding method was combined with the path-integral molecular dynamics method for the first time to evaluate the two-dimensional free-energy surface including nuclear quantum effects of a proton-transfer reaction in a 2,4-dichlorophenol-trimethylamine complex. A statistically converged two-dimensional quantum free-energy surface was evaluated by the multidimensional blue moon ensemble method. The accuracy was guaranteed by optimizing the repulsive potential between the sp3-hybridized nitrogen and hydrogen atoms in a SCC-DFTB3 parameter set for the system to reproduce high-level quantum chemical calculations. The present study illustrates the usefulness of this new approach to investigate nuclear quantum effects in various realistic proton-transfer reactions.

Quantum dynamics of charged particles in the Wigner formulation of quantum mechanics

We are going to study the kinetic properties of dense hydrogen plasma by a new quantum dynamics method in the Wigner representation of quantum mechanics. This method combines the Feynman and Wigner formulation of quantum mechanics and uses for calculation the direct path integral Monte Carlo (PIMC) and molecular dynamics methods. We are going to solve the Wigner–Liouville equation for dense degenerate hydrogen with the initial condition sampled by the PIMC method from equilibrium plasma state. We hope to do calculations of the hydrogen plasma properties under extreme conditions.

Nuclear quantum effects in molecular dynamics simulations

To take into account nuclear quantum effects on the dynamics of atoms, the path integral molecular dynamics (PIMD) method used since 1980s is based on the formalism developed by R. P. Feynman. However, the huge computation time required for the PIMD reduces its range of applicability. Another drawback is the requirement of additional techniques to access time correlation functions (ring polymer MD or centroid MD). We developed an alternative technique based on a quantum thermal bath (QTB) which reduces the computation time by a factor of ∼20. The QTB approach consists in a classical Langevin dynamics in which the white noise random force is replaced by a Gaussian random force having the power spectral density given by the quantum fluctuation-dissipation theorem. The method has yielded satisfactory results for weakly anharmonic systems: the quantum harmonic oscillator, the heat capacity of a MgO crystal, and isotope effects in 7LiH and 7LiD. Unfortunately, the QTB is subject to the problem of zero-point energy leakage (ZPEL) in highly anharmonic systems, which is inherent in the use of classical mechanics. Indeed, a part of the energy of the high-frequency modes is transferred to the low-frequency modes leading to a wrong energy distribution. We have shown that in order to reduce or even eliminate ZPEL, it is sufficient to increase the value of the frictional coefficient. Another way to solve the ZPEL problem is to combine the QTB and PIMD techniques. It requires the modification of the power spectral density of the random force within the QTB. This combination can also be seen as a way to speed up the PIMD.

Accurate calculation of equilibrium reduced density matrix for the system-bath model: A multilayer multiconfiguration time-dependent Hartree approach and its comparison to a multi-electronic-state path integral molecular dynamics approach

An efficient and accurate method for computing the equilibrium reduced density matrix is presented for treating open quantum systems characterized by the system-bath model. The method employs the multilayer multiconfiguration time-dependent Hartree theory for imaginary time propagation and an importance sampling procedure for calculating the quantum mechanical trace. The method is applied to the spin-boson Hamiltonian, which leads to accurate results in agreement with those produced by the multi-electronic-state path integral molecular dynamics method.

Theoretical calculations of photoelectron spectrum of (Au–CO2)− anion

Photoelectron spectra of the (Au–CO2)− anion complex have been calculated using both quantum path-integral and classical molecular dynamics methods to interpret the experimentally measured spectrum, showing a double-peak feature corresponding to two different anion configurations of Au·CO2− and Au−·CO2. The anionic potential energy surfaces have been developed on the basis of density-functional theory calculation results obtained with several different levels. We found that the calculated photoelectron spectra significantly depend on the relative energies of these two isomers and the isomerization barrier heights. Nuclear quantum effects are playing important roles for obtaining reliable spectra through vibrational quantization. The present calculations also suggest that the photoelectron spectrum intensity around the overlap region of the two anion peaks can provide a fingerprint of the Au·CO2− ↔ Au−·CO2 reaction.

Car–Parrinello and Path Integral Molecular Dynamics Study of the Proton Transfer in the Intramolecular Hydrogen Bonds in the Ketohydrazone–Azoenol System

Car-Parrinello (CPMD) and path integral molecular dynamics (PIMD) simulations were carried out for 1-(phenylazo)-2-naphthol (I) and 1-(4-F-phenylazo)-2-naphthol (II) (Sudan I) in vacuo and in the solid state at 298 K. The fast proton transfer (FPT) and tautomerism in the ketohydrazone-azoenol systems have been analyzed on the basis of CPMD and PIMD methods level. The two-dimensional free-energy landscape of reaction coordinate δ-parameter and RN···O distances shows the NH tautomer to be more favorable in the gas phase as well as in the solid state according to the CP and PI results, respectively. The hydrogen between the nitrogen and the oxygen atoms adopts a starkly asymmetrical position in the double potential well. The molecular geometry and energy barrier for the intramolecular proton transference were calculated, and the value found suggested a strong hydrogen bond with low barrier for FPT mechanism. These studies and the two-dimensional average index of π-delocalization ⟨λ⟩ landscape of time evolutions of RN1···O1 and RC1═O1 distances for both the crystals indicate that the hydrogen bonds in the crystals of 1-(phenylazo)-2-naphthol (I) and 1-(4-F-phenylazo)-2-naphthol (II) have characteristic properties for the type of bonding model: resonance-assisted hydrogen bonds and low-barrier hydrogen bonds, without the existence of equilibrium in the two tautomers. The infrared spectrum has been calculated, and a comparative vibrational analysis has been performed. The CPMD vibrational results appear to qualitatively agree with the experimental ones.

Spatially resolved proton momentum distributions in KDP from first-principles.

The ferroelectric to paraelectric (PE) phase transition of KH2PO4 (KDP) is investigated as a stringent test of the first-principles, normal modes framework proposed for calculating anharmonic quantum nuclear motion. Accurate spatially resolved momentum distribution functions (MDFs) are directly calculated from the nuclear wavefunction, overcoming the limitations of path-integral molecular dynamics methods. They indicate coherent, correlated tunneling of protons across hydrogen bonds in the PE phase in agreement with neutron Compton scattering data and reproduces the key features of the experimental MDF. It further highlights the role of Slater's lateral configurations in the PE phase. The analysis in terms of normal modes demonstrates the importance of collective, correlated proton motion and underlines the value of the employed framework in interpreting experimental data. This also makes the framework very attractive for application to deuterated KDP to further elucidate the nature of the PE transition and to systems exhibiting strong quantum nuclear effects in general.

Quantum structural fluctuation in para-hydrogen clusters revealed by the variational path integral method.

In this paper, the ground state of para-hydrogen clusters for size regime N ≤ 40 has been studied by our variational path integral molecular dynamics method. Long molecular dynamics calculations have been performed to accurately evaluate ground state properties. The chemical potential of the hydrogen molecule is found to have a zigzag size dependence, indicating the magic number stability for the clusters of the size N = 13, 26, 29, 34, and 39. One-body density of the hydrogen molecule is demonstrated to have a structured profile, not a melted one. The observed magic number stability is examined using the inherent structure analysis. We also have developed a novel method combining our variational path integral hybrid Monte Carlo method with the replica exchange technique. We introduce replicas of the original system bridging from the structured to the melted cluster, which is realized by scaling the potential energy of the system. Using the enhanced sampling method, the clusters are demonstrated to have the structured density profile in the ground state.

An Analysis of Quantum Effect on the Proton Conduction in BaZrO3 Membrane

In this paper, the nuclear quantum effect of the proton on its conduction in BaZrO3 membrane was addressed by using path integral molecular dynamics method. The quantum effect on two conduction paths that are translational and reorientaional motion determined by means of nudged elastic band method were analyzed. The results of the potential barrier along the minimum energy paths show significant changes because of the quantum effect of proton. However, it was found that the tendency of the quantum effect on the conduction paths are different with each path. On the translational path, the potential barrier for the proton conduction decreases with an increase of the quantum effect. In contrast, on the reorientational motion, the potential barrier for the proton conduction increases with an increase of the quantum effect.

Quantum Thermal Bath for Path Integral Molecular Dynamics Simulation.

The quantum thermal bath (QTB) method has been recently developed to account for the quantum nature of the nuclei by using standard molecular dynamics (MD) simulation. QTB-MD is an efficient but approximate method when dealing with strongly anharmonic systems, while path integral molecular dynamics (PIMD) gives exact results but in a huge amount of computation time. The QTB and PIMD methods have been combined in order to improve the PIMD convergence or correct the failures of the QTB-MD technique. Therefore, a new power spectral density of the random force within the QTB has been developed. A modified centroid-virial estimator of the kinetic energy, especially adapted to QTB-PIMD, has also been proposed. The method is applied to selected systems: a one-dimensional double-well system, a ferroelectric phase transition, and the position distribution of an hydrogen atom in a fuel cell material. The advantage of the QTB-PIMD method is its ability to give exact results with a more reasonable computation time for strongly anharmonic systems.