This article reviews the ring-polymer molecular dynamics model for condensed-phase quantum dynamics. This model, which involves classical evolution in an extended ring-polymer phase space, provides a practical approach to approximating the effects of quantum fluctuations on the dynamics of condensed-phase systems. The review covers the theory, implementation, applications, and limitations of the approximation.

Numerous studies have identified large quantum mechanical effects in the dynamics of liquid water. In this paper, we suggest that these effects may have been overestimated due to the use of rigid water models and flexible models in which the intramolecular interactions were described using simple harmonic functions. To demonstrate this, we introduce a new simple point charge model for liquid water, q-TIP4P/F, in which the O–H stretches are described by Morse-type functions. We have parametrized this model to give the correct liquid structure, diffusion coefficient, and infrared absorption frequencies in quantum (path integral-based) simulations. The model also reproduces the experimental temperature variation of the liquid density and affords reasonable agreement with the experimental melting temperature of hexagonal ice at atmospheric pressure. By comparing classical and quantum simulations of the liquid, we find that quantum mechanical fluctuations increase the rates of translational diffusion and orientational relaxation in our model by a factor of around 1.15. This effect is much smaller than that observed in all previous simulations of empirical water models, which have found a quantum effect of at least 1.4 regardless of the quantum simulation method or the water model employed. The small quantum effect in our model is a result of two competing phenomena. Intermolecular zero point energy and tunneling effects destabilize the hydrogen-bonding network, leading to a less viscous liquid with a larger diffusion coefficient. However, this is offset by intramolecular zero point motion, which changes the average water monomer geometry resulting in a larger dipole moment, stronger intermolecular interactions, and a slower diffusion. We end by suggesting, on the basis of simulations of other potential energy models, that the small quantum effect we find in the diffusion coefficient is associated with the ability of our model to produce a single broad O–H stretching band in the infrared absorption spectrum.

The ring polymer molecular dynamics (RPMD) and partially adiabatic centroid molecular dynamics (PA-CMD) methods are compared and contrasted in an application to the infrared absorption spectrum of a recently parametrized flexible, polarizable, Thole-type potential energy model for liquid water. Both methods predict very similar spectra in the low-frequency librational and intramolecular bending region at wavenumbers below 2500 cm−1. However, the RPMD spectrum is contaminated in the high-frequency O–H stretching region by contributions from the internal vibrational modes of the ring polymer. This problem is avoided in the PA-CMD method, which adjusts the elements of the Parrinello–Rahman mass matrix so as to shift the frequencies of these vibrational modes beyond the spectral range of interest. PA-CMD does not require any more computational effort than RPMD and it is clearly the better of the two methods for simulating vibrational spectra.

The approximate quantum mechanical ring polymer molecular dynamics (RPMD) and linearized semiclassical initial value representation (LSC-IVR) methods are compared and contrasted in a study of the dynamics of the flexible q-TIP4P/F water model at room temperature. For this water model, a RPMD simulation gives a diffusion coefficient that is only a few percent larger than the classical diffusion coefficient, whereas a LSC-IVR simulation gives a diffusion coefficient that is three times larger. We attribute this discrepancy to the unphysical leakage of initially quantized zero point energy (ZPE) from the intramolecular to the intermolecular modes of the liquid as the LSC-IVR simulation progresses. In spite of this problem, which is avoided by construction in RPMD, the LSC-IVR may still provide a useful approximation to certain short-time dynamical properties which are not so strongly affected by the ZPE leakage. We illustrate this with an application to the liquid water dipole absorption spectrum, for which the RPMD approximation breaks down at frequencies in the O–H stretching region owing to contamination from the internal modes of the ring polymer. The LSC-IVR does not suffer from this difficulty and it appears to provide quite a promising way to calculate condensed phase vibrational spectra.

Automatically generating chemical reaction pathways is a significant computational challenge, particularly in the case where a given chemical system can exhibit multiple reactants and products, as well as multiple pathways connecting these. Here, we outline a computational approach to allow automated sampling of chemical reaction pathways, including sampling of different chemical species at the reaction end-points. The key features of this scheme are (i) introduction of a Hamiltonian which describes a reaction “string” connecting reactant and products, (ii) definition of reactant and product species as chemical connectivity graphs, and (iii) development of a scheme for updating the chemical graphs associated with the reaction end-points. By performing molecular dynamics sampling of the Hamiltonian describing the complete reaction pathway, we are able to sample multiple different paths in configuration space between given chemical products; by periodically modifying the connectivity graphs describing the chemical identities of the end-points we are also able to sample the allowed chemical space of the system. Overall, this scheme therefore provides a route to automated generation of a “roadmap” describing chemical reactivity. This approach is first applied to model dissociation pathways in formaldehyde, H2CO, as described by a parameterised potential energy surface (PES). A second application to the HCo(CO)3 catalyzed hydroformylation of ethene (oxo process), using density functional tight-binding to model the PES, demonstrates that our graph-based approach is capable of sampling the intermediate paths in the commonly accepted catalytic mechanism, as well as several secondary reactions. Further algorithmic improvements are suggested which will pave the way for treating complex multi-step reaction processes in a more efficient manner.

We have used the ring polymer molecular dynamics method to study the diffusion of muonium, hydrogen, and deuterium atoms in liquid water and hexagonal ice over a wide temperature range (8–361 K). Quantum effects are found to dramatically reduce the diffusion of muonium in water relative to that predicted by classical simulation. This leads to a simple explanation for the lack of any significant isotope effect in the observed diffusion coefficients of these species in the room temperature liquid. Our results indicate that the mechanism of the diffusion in liquid water is similar to the intercavity hopping mechanism observed in ice, supplemented by the diffusion of the cavities in the liquid. Within the same model, we have also been able to simulate the observed crossover in the c-axis diffusion coefficients of hydrogen and deuterium in hexagonal ice. Finally, we have been able to obtain good agreement with experimental data on the diffusion of muonium in hexagonal ice at 8 K, where the process is entirely quantum mechanical.