Quantum effects in liquid water and ice: model dependence

This paper explores the influence of choice of potential model on the quantum effects observed in liquid water and ice. This study utilizes standard rigid models and a more formal context for the rigid-body centroid molecular dynamics methodology used to perform the quantum simulations is provided. Quantum and classical molecular dynamics simulations are carried out for liquid water and ice Ih at 298 and 220 K, respectively, with the simple point charge/extended and TIP4P-Ew water models. The results obtained for equilibrium and dynamical properties are compared with those recently reported on TIP4P [L. Hernandez de la Pena and P. G. Kusalik, J. Chem. Phys. 121, 5992 (2004); L. Hernandez de la Pena et al., J. Chem. Phys 123, 144506 (2005)]. For the liquid, an energy shift of about 8% and an average molecular uncertainty of about 11 degrees were found independently of the water model. The self-diffusion coefficient consistently increases by more than 50% when going from the classical to the quantum system and quantum dynamics are found to reproduce the experimental isotopic shifts with the models examined. The ice results compare remarkably well with those previously reported for the TIP4P water model; they confirm that quantum effects are considerable and that the quantum mechanical uncertainty and the energy shifts due to quantization are smaller in ice than in liquid water. The relevance of these findings in the context of the construction of water models is briefly discussed.     

Quantum path integral simulation of isotope effects in the melting temperature of ice Ih

The isotope effect in the melting temperature of ice Ih has been studied by free energy calculations within the path integral formulation of statistical mechanics. Free energy differences between isotopes are related to the dependence of their kinetic energy on the isotope mass. The water simulations were performed by using the q-TIP4P/F model, a point charge empirical potential that includes molecular flexibility and anharmonicity in the OH stretch of the water molecule. The reported melting temperature at ambient pressure of this model (T=251?K) increases by 6.5±0.5 and 8.2±0.5?K upon isotopic substitution of hydrogen by deuterium and tritium, respectively. These temperature shifts are larger than the experimental ones (3.8 and 4.5 K, respectively). In the classical limit, the melting temperature is nearly the same as that for tritiated ice. This unexpected behavior is rationalized by the coupling between intermolecular interactions and molecular flexibility. This coupling makes the kinetic energy of the OH stretching modes larger in the liquid than in the solid phase. However, the opposite behavior is found for intramolecular modes, which display larger kinetic energy in ice than in liquid water.     

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.     

Isotope effects in ice Ih: a path-integral simulation

Ice Ih has been studied by path-integral molecular dynamics simulations, using the effective q-TIP4P/F potential model for flexible water. This has allowed us to analyze finite-temperature quantum effects in this solid phase from 25 to 300 K at ambient pressure. Among these effects we find a negative thermal expansion of ice at low temperatures, which does not appear in classical molecular dynamics simulations. The compressibility derived from volume fluctuations gives results in line with experimental data. We have analyzed isotope effects in ice Ih by considering normal, heavy, and tritiated water. In particular, we studied the effect of changing the isotopic mass of hydrogen on the kinetic energy and atomic delocalization in the crystal as well as on structural properties such as interatomic distances and molar volume. For D(2)O ice Ih at 100 K we obtained a decrease in molar volume and intramolecular O-H distance of 0.6% and 0.4%, respectively, as compared to H(2)O ice.     

On the direct calculation of the free energy of quantization for molecular systems in the condensed phase

Using the path integral formalism or the Feynman-Hibbs approach, various expressions for the free energy of quantization for a molecular system in the condensed phase can be derived. These lead to alternative methods to directly compute quantization free energies from molecular dynamics computer simulations, which were investigated with an eye to their practical use. For a test system of liquid neon, two methods are shown to be most efficient for a direct evaluation of the excess free energy of quantization. One of them makes use of path integral simulations in combination with a single-step free energy perturbation approach and was previously reported in the literature. The other method employs a Feynman-Hibbs effective Hamiltonian together with the thermodynamic integration formalism. However, both methods are found to give less accurate results for the excess free energy of quantization than the estimate obtained from explicit path integral calculations on the excess free energy of the neon liquid in the classical and quantum mechanical limit. Suggestions are made to make both methods more accurate.     

Quasi-harmonic approximation of thermodynamic properties of ice Ih, II, and III

Several thermodynamic properties of ice Ih, II, and III are studied by a quasi-harmonic approximation and compared to results of quantum path integral and classical simulations. This approximation allows to obtain thermodynamic information at a fraction of the computational cost of standard simulation methods, and at the same time permits studying quantum effects related to zero-point vibrations of the atoms. Specifically, we have studied the crystal volume, bulk modulus, kinetic energy, enthalpy, and heat capacity of the three ice phases as a function of temperature and pressure. The flexible q-TIP4P/F model of water was employed for this study, although the results concerning the capability of the quasi-harmonic approximation are expected to be valid independently of the employed water model. The quasi-harmonic approximation reproduces with reasonable accuracy the results of quantum and classical simulations showing an improved agreement at low temperatures (T< 100 K). This agreement does not deteriorate as a function of pressure as long as it is not too close to the limit of mechanical stability of the ice phases.     

Ring-polymer Molecular Dynamics Simulation for the Adsorption of H2 on Ice Clusters (H2O)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 (H₂O)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, (H₂O)₁₀ and (H₂O)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.     

Impacts of quantization on the properties of liquid water

The results of classical and quantum simulations of liquid water over a wide range of temperatures are compared to probe the impact of quantization on the properties of liquid water. We show that, when treated quantum mechanically, water molecules have an enhanced probability of accessing nontetrahedral coordination in the local three-dimensional structure. We discuss how this enhanced probability, also called "effective tunneling", is related to the dynamics of the hydrogen-bond breaking and molecular diffusion in the liquid. We explore in detail how local molecular environments affect the manifestation of quantum effects and identify a previously unreported and apparently unique behavior of the quantum mechanical uncertainty of the water molecule as a function of temperature. The nonmonotonic behavior of the quantum mechanical uncertainty with temperature is shown to be due to the notable strength of the water-water interaction in the condensed phase and becomes further evidence of the importance of the water structure in the properties of this ubiquitous liquid.     

On the Debye-Waller factor of hexagonal ice: a computer simulation study

We investigate by molecular dynamics (MD) simulations the temperature dependence of the Debye-Waller (DW) factor of hexagonal ice with 25 different proton-disordered configurations. Each initial configuration is composed of 288 water molecules with no net dipole moment. The intermolecular interaction of water is described by TIP4P potential. Each production run of the simulation is 15 ns or longer. We observe a change in slope of the DW factor around 200 K, which cannot be explained within the framework of either classical or quantum harmonic approximation. Configurations generated by MD simulations are subjected to the steepest descent energy minimization. Analysis of the local energy minimum structures reveals that water molecules above 200 K jump to other lattice sites via some local energy minimum structures which contain some water molecules sitting on the locations other than the lattice sites. As time evolves, these defect molecules move back and forth to the lattice sites yielding defect-free structures. Those motions are responsible for the unusual increase in the DW factor at high temperatures. In making a transition from an energy-minimum structure to another one, a small number of water molecules are involved in a highly cooperative fashion. The larger DW factor at higher temperature arises from jump-like motions of water molecules among these locally stable configurations which may or may not be a family of the proton-disordered ice forms satisfying the "ice rule".     

Classical and quantum gibbs free energies and phase behavior of water using simulation and cell theory

A method to calculate the classical and quantum free energy of a liquid from a computer simulation by using cell theory [J. Chem. Phys. 2007, 126, 064504] is tested for liquid water and ice Ih against experiment as a function of temperature. This fast and efficient method reproduces reasonably well the experimental values of entropy, enthalpy, and free energy of a liquid across the supercooled, stable, and superheated range of temperatures considered. There are small differences between classical and quantum results of water at 298 K, necessitating a small correction term to reproduce water's enthalpy of vaporisation. Only at higher temperatures is entropy underestimated by up to 9 J K(-1) mol(-1) as verified by thermodynamic integration calculations. Satisfactory agreement for ice, however, is only obtained by using the quantum formulation. Even then, at higher temperatures, the entropies exceed experiment by up to 15 J K(-1) mol(-1). Further insight into the quantum nature of water is provided by inspecting the temperature dependence of the frequencies. The harmonic approximation is further supported by the harmonic force and torque distributions and the very similar entropies obtained from the force and torque variances. All these results suggest that the single molecule harmonic oscillator approximation for water, although not exact, provides a rapid, insightful, and useful means to evaluate the thermodynamic properties of water from a computer simulation in a way that can account for the quantization of water's energy levels.     

Calibration of force-field dependency in free energy landscapes of peptide conformations by quantum chemical calculations

The free energy landscapes of peptide conformations were calibrated by ab initio quantum chemical calculations, after the enhanced conformational diversity search using the multicanonical molecular dynamics simulations. Three different potentials of mean force for an isolated dipeptide were individually obtained by the multicanonical molecular dynamics simulations using the conventional force fields, AMBER parm94, AMBER parm96, and CHARMm22. Each potential of mean force was then calibrated based upon the umbrella sampling algorithm from the adiabatic energy map that was calculated separately by the ab initio molecular orbital method, and all of the calibrated potentials of mean force coincided well. The calibration method was also applied to the simulations of a peptide dimer in explicit water models, and it was shown that the calibrated free energy landscapes did not depend on the force field used in the classical simulations, as far as the conformational space was sampled well. The current calibration method fuses the classical free energy calculation with the quantum chemical calculation, and it should generally make simulations for biomolecular systems much more reliable when combining with enhanced conformational sampling.     

A classical polarizable model for simulations of water and ice

We develop a classical rigid polarizable model of water for molecular simulations of water and ice. The model uses the Rowlinson five-site geometry: oxygen bearing the Lennard-Jones interaction and linearly polarizable point dipole, two positively charged hydrogens, and two massless negative charges placed symmetrically off oxygen so that the experimental dipole moment is reproduced. The target properties are the densities of water and ice Ih, diffusivity, enthalpies of fusion and vaporization, and the ice Ih melting point. The surface tension at lower temperatures is by 7% underestimated whereas the dielectric constant by 6% overestimated. Diffusivity and viscosity worsen at higher temperatures, although the Stokes radius is overestimated only by 2-7%. The ice Ih melting temperature is 260 K and the temperature of maximum density is 269 K. Rescaling the charges by a factor of 1.01 and Lennard-Jones energy by 1.0201 improves the melting point and energy-related quantities but shifts the agreement of kinetic properties to higher temperatures. For the model we propose abbreviation POL4D.     

Quantized time correlation function approach to nonadiabatic decay rates in condensed phase: application to solvated electrons in water and methanol

A new, alternative form of the golden rule formula defining the nonadiabatic transition rate between two quantum states in condensed phase is presented. The formula involves the quantum time correlation function of the energy gap, of the nonadiabatic coupling, and their cross terms. Those quantities can be inferred from their classical counterparts, determined via molecular dynamics simulations. The formalism is applied to the problem of the nonadiabatic p-->s relaxation of an equilibrated p-electron in water and methanol. We find that, in both solvents, the relaxation is induced by the coupling to the vibrational modes and the quantum effects modify the rate by a factor of 2-10 depending on the quantization procedure applied. The resulting p-state lifetime for a hypothetical equilibrium excited state appears extremely short, in the sub-100 fs regime. Although this result is in contrast with all previous theoretical predictions, we also illustrate that the lifetimes computed here are very sensitive to the simulated electronic quantum gap and to the strongly correlated nonadiabatic coupling.     

Optimized Monte Carlo sampling in forward-backward semiclassical dynamics

Forward-backward semiclassical dynamics (FBSD) provides a rigorous and powerful methodology for calculating time correlation functions in condensed phase systems characterized by substantial quantum mechanical effects associated with zero-point motion, quantum dispersion, or identical particle exchange symmetries. The efficiency of these simulations arises from the use of classical trajectories to capture all dynamical information. However, full quantization of the density operator makes these calculations rather expensive compared to fully classical molecular dynamics simulations. This article discusses the convergence properties of various correlation functions and introduces an optimal Monte Carlo sampling scheme that leads to a significant reduction of statistical error. A simple and efficient procedure for normalizing the FBSD results is also discussed. Illustrative examples on model systems are presented.     

Long-range electrostatic interactions in hybrid quantum and molecular mechanical dynamics using a lattice summation approach

A method based on a lattice summation technique for treating long-range electrostatic interactions in hybrid quantum mechanics/molecular mechanics simulations is presented in this article. The quantum subsystem is studied at the semiempirical level, whereas the solvent is described by a two-body potential of molecular mechanics. Molecular dynamics simulations of a (quantum) chloride ion in (classical) water have been performed to test this technique. It is observed that the application of the lattice summations to solvent-solvent interactions as well as on solute-solvent ones has a significant effect on solvation energy and diffusion coefficient. Moreover, two schemes for the computation of the long-range contribution to the electrostatic interaction energy are investigated. The first one replaces the exact charge distribution of the quantum solute by a Mulliken charge distribution. The long-range electrostatic interactions are then calculated for this charge distribution that interacts with the solvent molecule charges. The second one is more accurate and involves a modified Fock operator containing long-range electron-charge interactions. It is shown here that both schemes lead to similar results, the method using Mulliken charges for the evaluation of long-range interactions being, however, much more computationally efficient.     

Direct assessment of quantum nuclear effects on hydrogen bond strength by constrained-centroid ab initio path integral molecular dynamics

The impact of quantum nuclear effects on hydrogen (H-) bond strength has been inferred in earlier work from bond lengths obtained from path integral molecular dynamics (PIMD) simulations. To obtain a direct quantitative assessment of such effects, we use constrained-centroid PIMD simulations to calculate the free energy changes upon breaking the H-bonds in dimers of HF and water. Comparing ab initio simulations performed using PIMD and classical nucleus molecular dynamics (MD), we find smaller dissociation free energies with the PIMD method. Specifically, at 50 K, the H-bond in (HF)(2) is about 30% weaker when quantum nuclear effects are included, while that in (H(2)O)(2) is about 15% weaker. In a complementary set of simulations, we compare unconstrained PIMD and classical nucleus MD simulations to assess the influence of quantum nuclei on the structures of these systems. We find increased heavy atom distances, indicating weakening of the H-bond consistent with that observed by direct calculation of the free energies of dissociation.     

Molecular dynamics simulations of the Ag+ or Na+ cation with an excess electron in bulk water

The properties of an excess electron interacting with a monovalent cation in bulk water are studied by molecular dynamics simulations. Sodium and silver cations are chosen as prototypical cases because of their very different redox properties. In both cases, mixed quantum classical molecular dynamics simulations reproduce the experimental UV-Vis spectra. In the case of silver, we observe a highly polarized neutral atom, corresponding to a dipolar excitonic state. For sodium a contact cation/electron pair is observed. Free energy curves along the cation electron coordinate are calculated using quantum Umbrella Sampling technique. The relative stability of the different chemical species is discussed.     

A powerful computational crystallography method to study ice polymorphism

Classical molecular dynamics (MD) simulations are employed as a tool to investigate structural properties of ice crystals under several temperature and pressure conditions. All ice crystal phases are analyzed by means of a computational protocol based on a clustering approach following standard MD simulations. The MD simulations are performed by using a recently published classical interaction potential for oxygen and hydrogen in bulk water, derived from neutron scattering data, able to successfully describe complex phenomena such as proton hopping and bond formation/breaking. The present study demonstrates the ability of the interaction potential model to well describe most ice structures found in the phase diagram of water and to estimate the relative stability of 16 known phases through a cluster analysis of simulated powder diagrams of polymorphs obtained from MD simulations. The proposed computational protocol is suited for automated crystal structure identification.