Path-integral virial estimator based on the scaling of fluctuation coordinates: Application to quantum clusters with fourth-order propagators

Accurate nuclear quantum statistics on machine-learned classical effective potentials

The contribution of nuclear quantum effects (NQEs) to the properties of various hydrogen-bound systems, including biomolecules, is increasingly recognized. Despite the development of many acceleration techniques, the computational overhead of incorporating NQEs in complex systems is sizable, particularly at low temperatures. In this work, we leverage deep learning and multiscale coarse-graining techniques to mitigate the computational burden of path integral molecular dynamics (PIMD). In particular, we employ a machine-learned potential to accurately represent corrections to classical potentials, thereby significantly reducing the computational cost of simulating NQEs. We validate our approach using four distinct systems: Morse potential, Zundel cation, single water molecule, and bulk water. Our framework allows us to accurately compute position-dependent static properties, as demonstrated by the excellent agreement obtained between the machine-learned potential and computationally intensive PIMD calculations, even in the presence of strong NQEs. This approach opens the way to the development of transferable machine-learned potentials capable of accurately reproducing NQEs in a wide range of molecular systems.

i-PI 3.0: A flexible and efficient framework for advanced atomistic simulations

Atomic-scale simulations have progressed tremendously over the past decade, largely thanks to the availability of machine-learning interatomic potentials. These potentials combine the accuracy of electronic structure calculations with the ability to reach extensive length and time scales. The i-PI package facilitates integrating the latest developments in this field with advanced modeling techniques thanks to a modular software architecture based on inter-process communication through a socket interface. The choice of Python for implementation facilitates rapid prototyping but can add computational overhead. In this new release, we carefully benchmarked and optimized i-PI for several common simulation scenarios, making such overhead negligible when i-PI is used to model systems up to tens of thousands of atoms using widely adopted machine learning interatomic potentials, such as Behler-Parinello, DeePMD, and MACE neural networks. We also present the implementation of several new features, including an efficient algorithm to model bosonic and fermionic exchange, a framework for uncertainty quantification to be used in conjunction with machine-learning potentials, a communication infrastructure that allows for deeper integration with electronic-driven simulations, and an approach to simulate coupled photon-nuclear dynamics in optical or plasmonic cavities.

Two liquid states of distinguishable helium-4: The existence of another non-superfluid frozen by heating

We show that two liquid states can exist in distinguishable helium-4 (4He) obeying Boltzmann statistics by path integral centroid molecular dynamics (CMD) simulations. This is an indication of quantum liquid polyamorphism induced by the nuclear quantum effect. For 0.08-3.3 K and 1-500 bar, we extensively conducted the isothermal-isobaric CMD simulations to explore not only possible states and state diagram but also the state characteristics. The distinguishable 4He below 25 bar does not freeze down to 0.1 K even though it includes no Bosonic exchange effect and, therefore, no Bose condensation. One liquid state, low quantum-dispersion liquid (LQDL), is nearly identical to normal liquid He-I of real 4He. The other is high quantum-dispersion liquid (HQDL) consisting of atoms with longer quantum wavelength. This is another non-superfluid existing below 0.5 K or the temperatures of LQDL. The HQDL is also a low-entropy and fragile liquid to exhibit, unlike conventional liquids, rather gas-like relaxation of velocity autocorrelation function, while there the atoms diffuse without noticeable contribution from quantum tunneling. The LQDL-HQDL transition is not a thermodynamic phase transition but a continuous crossover accompanied by the change in the expansion factor of quantum wavelength. Freezing of HQDL into the low quantum-dispersion amorphous solid occurs by heating from 0.2 to 0.3 K at 40-50 bar, while this P-T condition coincides with the Kim-Chan normal-supersolid phase boundary of real 4He. The obtained state diagram was compared to that of the confined subnano-scale 4He systems, where Bosonic correlation is considerably suppressed.

Proton Collective Quantum Tunneling Induces Anomalous Thermal Conductivity of Ice under Pressure

Proton tunneling is believed to be nonlocal in ice, but its range has been shown to be limited to only a few molecules. Here, we measured the thermal conductivity of ice under pressure up to 50 GPa and found it increases with pressure until 20 GPa but decreases at higher pressures. We attribute this nonmonotonic thermal conductivity to the collective tunneling of protons at high pressures, supported by large-scale quantum molecular dynamics simulations. The collective tunneling loops span several picoseconds in time and are as large as nanometers in space, which match the phonon periods and wavelengths, leading to strong phonon scattering at high pressures. Our results show direct evidence of global quantum motion existing in high-pressure ice and provide a new perspective to understanding the coupling between phonon propagation and atomic tunneling.

Generalized fourth-order decompositions of imaginary time path integral: Implications of the harmonic oscillator

The imaginary time path integral formalism offers a powerful numerical tool for simulating thermodynamic properties of realistic systems. We show that, when second-order and fourth-order decompositions are employed, they share a remarkable unified analytic form for the partition function of the harmonic oscillator. We are then able to obtain the expression of the thermodynamic property and the leading error terms as well. In order to obtain reasonably optimal values of the free parameters in the generalized symmetric fourth-order decomposition scheme, we eliminate the leading error terms to achieve the accuracy of desired order for the thermodynamic property of the harmonic system. Such a strategy leads to an efficient fourth-order decomposition that produces third-order accurate thermodynamic properties for general systems.

Quantum polyamorphism in compressed distinguishable helium-4

We demonstrate that two amorphous solid states can exist in ⁴He consisting of distinguishable Boltzmann atoms under compressed conditions. The isothermal compression of normal or supercritical fluid ⁴He was conducted at 3-25 K using the isobaric-isothermal path integral centroid molecular dynamics simulation. The compression of fluid first produced the low-dispersion amorphous (LDA) state possessing modest extension of atomic necklaces. Further isothermal compression up to the order of 10 kbar to 1 Mbar or an isobaric cooling of LDA induced the transition to the high-dispersion amorphous (HDA) state. The HDA was characterized by long quantum wavelengths of atoms extended over several Angstroms and the promotion of atomic residual diffusion. They were related to the quantum tunneling of atoms bestriding the potential saddle points in this glass. The change in pressure or temperature induced the LDA-HDA transition reversibly with hysteresis, while it resembled the coil-globule transition of classical polymers. The HDA had lower kinetic and higher Gibbs free energies than the LDA at close temperature. The HDA was absent at T ≥ 13 K, while the LDA-HDA transition pressure significantly decreased with lowering temperature. The LDA and HDA correspond to the trapped and tunneling regimes proposed by Markland et al. [J. Chem. Phys. 136, 074511 (2012)], respectively. The same reentrant behavior as they found was observed for the expansion factor of the quantum wavelength as well as for atomic diffusivity.

Nuclear quantum effects in thermal conductivity from centroid molecular dynamics

We show that the centroid molecular dynamics (CMD) method provides a realistic way to calculate the thermal diffusivity a = λ/ρc_V of a quantum mechanical liquid such as para-hydrogen. Once a has been calculated, the thermal conductivity can be obtained from λ = ρc_Va, where ρ is the density of the liquid and c_V is the constant-volume heat capacity. The use of this formula requires an accurate quantum mechanical heat capacity c_V, which can be obtained from a path integral molecular dynamics simulation. The thermal diffusivity can be calculated either from the decay of the equilibrium density fluctuations in the liquid or by using the Green-Kubo relation to calculate the CMD approximation to λ and then dividing this by the corresponding approximation to ρc_V. We show that both approaches give the same results for liquid para-hydrogen and that these results are in good agreement with the experimental measurements of the thermal conductivity over a wide temperature range. In particular, they correctly predict a decrease in the thermal conductivity at low temperatures-an effect that stems from the decrease in the quantum mechanical heat capacity and has eluded previous para-hydrogen simulations. We also show that the method gives equally good agreement with the experimental measurements for the thermal conductivity of normal liquid helium.

Converged Colored Noise Path Integral Molecular Dynamics Study of the Zundel Cation Down to Ultralow Temperatures at Coupled Cluster Accuracy

For a long time, performing converged path integral simulations at ultralow but finite temperatures of a few Kelvin has been a nearly impossible task. However, recent developments in advanced colored noise thermostatting schemes for path integral simulations, namely, the Path Integral Generalized Langevin Equation Thermostat (PIGLET) and the Path Integral Quantum Thermal Bath (PIQTB), have been able to greatly reduce the computational cost of these simulations, thus making the ultralow temperature regime accessible in practice. In this work, we investigate the influence of these two thermostatting schemes on the description of hydrogen-bonded systems at temperatures down to a few Kelvin as encountered, for example, in helium nanodroplet isolation or tagging photodissociation spectroscopy experiments. For this purpose, we analyze the prototypical hydrogen bond in the Zundel cation (H₅O₂⁺) as a function of both oxygen-oxygen distance and temperature in order to elucidate how the anisotropic quantum delocalization and, thus, the shape of the shared proton adapts depending on the donor-acceptor distance. The underlying electronic structure of the Zundel cation is described in terms of Behler's Neural Network Potentials of essentially converged Coupled Cluster accuracy, CCSD(T*)-F12a/AVTZ. In addition, the performances of the PIQTB and PIGLET methods for energetic, structural, and quantum delocalization properties are assessed and directly compared. Overall, our results emphasize the validity and practical usefulness of these two modern thermostatting approaches for path integral simulations of hydrogen-bonded systems even at ultralow temperatures.

Non-uniqueness of quantum transition state theory and general dividing surfaces in the path integral space

Despite the fact that quantum mechanical principles do not allow the establishment of an exact quantum analogue of the classical transition state theory (TST), the development of a quantum TST (QTST) with a proper dynamical justification, while recovering the TST in the classical limit, has been a long standing theoretical challenge in chemical physics. One of the most recent efforts of this kind was put forth by Hele and Althorpe (HA) [J. Chem. Phys. 138, 084108 (2013)], which can be specified for any cyclically invariant dividing surface defined in the space of the imaginary time path integral. The present work revisits the issue of the non-uniqueness of QTST and provides a detailed theoretical analysis of HA-QTST for a general class of such path integral dividing surfaces. While we confirm that HA-QTST reproduces the result based on the ring polymer molecular dynamics (RPMD) rate theory for dividing surfaces containing only a quadratic form of low frequency Fourier modes, we find that it produces different results for those containing higher frequency imaginary time paths which accommodate greater quantum fluctuations. This result confirms the assessment made in our previous work [Jang and Voth, J. Chem. Phys. 144, 084110 (2016)] that HA-QTST does not provide a derivation of RPMD-TST in general and points to a new ambiguity of HA-QTST with respect to its justification for general cyclically invariant dividing surfaces defined in the space of imaginary time path integrals. Our analysis also offers new insights into similar path integral based QTST approaches.

Path integrals with higher order actions: Application to realistic chemical systems

Quantum thermodynamic parameters can be determined using path integral Monte Carlo (PIMC) simulations. These simulations, however, become computationally demanding as the quantum nature of the system increases, although their efficiency can be improved by using higher order approximations to the thermal density matrix, specifically the action. Here we compare the standard, primitive approximation to the action (PA) and three higher order approximations, the Takahashi-Imada action (TIA), the Suzuki-Chin action (SCA) and the Chin action (CA). The resulting PIMC methods are applied to two realistic potential energy surfaces, for H₂O and HCN-HNC, both of which are spectroscopically accurate and contain three-body interactions. We further numerically optimise, for each potential, the SCA parameter and the two free parameters in the CA, obtaining more significant improvements in efficiency than seen previously in the literature. For both H₂O and HCN-HNC, accounting for all required potential and force evaluations, the optimised CA formalism is approximately twice as efficient as the TIA formalism and approximately an order of magnitude more efficient than the PA. The optimised SCA formalism shows similar efficiency gains to the CA for HCN-HNC but has similar efficiency to the TIA for H₂O at low temperature. In H₂O and HCN-HNC systems, the optimal value of the a₁ CA parameter is approximately 13, corresponding to an equal weighting of all force terms in the thermal density matrix, and similar to previous studies, the optimal α parameter in the SCA was ∼0.31. Importantly, poor choice of parameter significantly degrades the performance of the SCA and CA methods. In particular, for the CA, setting a₁ = 0 is not efficient: the reduction in convergence efficiency is not offset by the lower number of force evaluations. We also find that the harmonic approximation to the CA parameters, whilst providing a fourth order approximation to the action, is not optimal for these realistic potentials: numerical optimisation leads to better approximate cancellation of the fifth order terms, with deviation between the harmonic and numerically optimised parameters more marked in the more quantum H₂O system. This suggests that numerically optimising the CA or SCA parameters, which can be done at high temperature, will be important in fully realising the efficiency gains of these formalisms for realistic potentials.

Kinetic isotope effects and how to describe them

We review several methods for computing kinetic isotope effects in chemical reactions including semiclassical and quantum instanton theory. These methods describe both the quantization of vibrational modes as well as tunneling and are applied to the ⋅H + H2 and ⋅H + CH4 reactions. The absolute rate constants computed with the semiclassical instanton method both using on-the-fly electronic structure calculations and fitted potential-energy surfaces are also compared directly with exact quantum dynamics results. The error inherent in the instanton approximation is found to be relatively small and similar in magnitude to that introduced by using fitted surfaces. The kinetic isotope effect computed by the quantum instanton is even more accurate, and although it is computationally more expensive, the efficiency can be improved by path-integral acceleration techniques. We also test a simple approach for designing potential-energy surfaces for the example of proton transfer in malonaldehyde. The tunneling splittings are computed, and although they are found to deviate from experimental results, the ratio of the splitting to that of an isotopically substituted form is in much better agreement. We discuss the strengths and limitations of the potential-energy surface and based on our findings suggest ways in which it can be improved.

Modeling quantum nuclei with perturbed path integral molecular dynamics

Here we combine perturbation theory with the Feynman–Kac imaginary-time path integral approach to quantum mechanics for modeling quantum nuclear effects.

The quantum nature of nuclear motions plays a vital role in the structure, stability, and thermodynamics of molecules and materials. The standard approach to model nuclear quantum fluctuations in chemical and biological systems is to use path-integral molecular dynamics. Unfortunately, conventional path-integral simulations can have an exceedingly large computational cost due to the need to employ an excessive number of coupled classical subsystems (beads) for quantitative accuracy. Here, we combine perturbation theory with the Feynman–Kac imaginary-time path integral approach to quantum mechanics and derive an improved non-empirical partition function and estimators to calculate converged quantum observables. Our perturbed path-integral (PPI) method requires the same ingredients as the conventional approach, but increases the accuracy and efficiency of path integral simulations by an order of magnitude. Results are presented for the thermodynamics of fundamental model systems, an empirical water model containing 256 water molecules within periodic boundary conditions, and ab initio simulations of nitrogen and benzene molecules. For all of these examples, PPI simulations with 4 to 8 classical beads recover the nuclear quantum contribution to the total energy and heat capacity at room temperature within a 3% accuracy, paving the way toward seamless modeling of nuclear quantum effects in realistic molecules and materials.

EXPONENTIAL PROPAGATORS (INTEGRATORS) FOR THE TIME-DEPENDENT SCHRÖDINGER EQUATION

The time-dependent Schrödinger Equation (TDSE) is a parabolic partial differential equation (PDE) comparable to a diffusion equation but with imaginary time. Due to its first order time derivative, exponential integrators or propagators are natural methods to describe evolution in time of the TDSE, both for time-independent and time-dependent potentials. Two splitting methods based on Fer and/or Magnus expansions allow for developing unitary factorizations of exponentials with different accuracies in the time step △t. The unitary factorization of exponentials to high order accuracy depends on commutators of kinetic energy operators with potentials. Fourth-order accuracy propagators can involve negative or complex time steps, or real time steps only but with gradients of potentials, i.e. forces. Extending the propagators of TDSE's to imaginary time allows to also apply these methods to classical many-body dynamics, and quantum statistical mechanics of molecular systems.

SPATIALLY-DISCRETIZED HIGH-TEMPERATURE APPROXIMATIONS AND THEIR O(N) IMPLEMENTATION ON A GRID

We consider the problem of performing imaginary-time propagation of wavefunctions on a grid. We demonstrate that spatially-continuous high-temperature approximations can be discretized in such a way that their convergence order is preserved. Requirements of minimal computational work and reutilization of data then uniquely determine the optimal grid, quadrature technique, and propagation method. It is shown that the optimal propagation technique is O(N), with respect to the grid size. The grid technique is utilized to compare the Monte Carlo efficiency of the Trotter–Suzuki approximation against a recently introduced fourth-order high-temperature approximation, while circumventing the issue of statistical noise, which usually prevents such comparisons from being carried out. We document the appearance of a systematic bias in the Monte Carlo estimators that involve temperature differentiation of the density matrix, bias that is due to the dependence of the eigenvalues on the inverse temperature. This bias is negotiated more successfully by the short-time approximations having higher convergence order, which leads to non-trivial computational savings.

The inefficiency of re-weighted sampling and the curse of system size in high-order path integration

Computing averages over a target probability density by statistical re-weighting of a set of samples with a different distribution is a strategy which is commonly adopted in fields as diverse as atomistic simulation and finance. Here we present a very general analysis of the accuracy and efficiency of this approach, highlighting some of its weaknesses. We then give an example of how our results can be used, specifically to assess the feasibility of high-order path integral methods. We demonstrate that the most promising of these techniques—which is based on re-weighted sampling—is bound to fail as the size of the system is increased, because of the exponential growth of the statistical uncertainty in the re-weighted average.

An accurate and simple quantum model for liquid water

The path-integral molecular dynamics and centroid molecular dynamics methods have been applied to investigate the behavior of liquid water at ambient conditions starting from a recently developed simple point charge/flexible (SPC/Fw) model. Several quantum structural, thermodynamic, and dynamical properties have been computed and compared to the corresponding classical values, as well as to the available experimental data. The path-integral molecular dynamics simulations show that the inclusion of quantum effects results in a less structured liquid with a reduced amount of hydrogen bonding in comparison to its classical analog. The nuclear quantization also leads to a smaller dielectric constant and a larger diffusion coefficient relative to the corresponding classical values. Collective and single molecule time correlation functions show a faster decay than their classical counterparts. Good agreement with the experimental measurements in the low-frequency region is obtained for the quantum infrared spectrum, which also shows a higher intensity and a redshift relative to its classical analog. A modification of the original parametrization of the SPC/Fw model is suggested and tested in order to construct an accurate quantum model, called q-SPC/Fw, for liquid water. The quantum results for several thermodynamic and dynamical properties computed with the new model are shown to be in a significantly better agreement with the experimental data. Finally, a force-matching approach was applied to the q-SPC/Fw model to derive an effective quantum force field for liquid water in which the effects due to the nuclear quantization are explicitly distinguished from those due to the underlying molecular interactions. Thermodynamic and dynamical properties computed using standard classical simulations with this effective quantum potential are found in excellent agreement with those obtained from significantly more computationally demanding full centroid molecular dynamics simulations. The present results suggest that the inclusion of nuclear quantum effects into an empirical model for water enhances the ability of such model to faithfully represent experimental data, presumably through an increased ability of the model itself to capture realistic physical effects.

Path-integral virial estimator for reaction-rate calculation based on the quantum instanton approximation

The quantum instanton approximation is a type of quantum transition-state theory that calculates the chemical reaction rate using the reactive flux correlation function and its low-order derivatives at time zero. Here we present several path-integral estimators for the latter quantities, which characterize the initial decay profile of the flux correlation function. As with the internal energy or heat-capacity calculation, different estimators yield different variances (and therefore different convergence properties) in a Monte Carlo calculation. Here we obtain a virial (-type) estimator by using a coordinate scaling procedure rather than integration by parts, which allows more computational benefits. We also consider two different methods for treating the flux operator, i.e., local-path and global-path approaches, in which the latter achieves a smaller variance at the cost of using second-order potential derivatives. Numerical tests are performed for a one-dimensional Eckart barrier and a model proton transfer reaction in a polar solvent, which illustrates the reduced variance of the virial estimator over the corresponding thermodynamic estimator.