Infrared Spectroscopy and Hydrogen-Bond Dynamics of Liquid Water from Centroid Molecular Dynamics with an Ab Initio-Based Force Field

Modeling Infrared Spectroscopy of Nucleic Acids: Integrating Vibrational Non-Condon Effects with Machine Learning Schemes

Vibrational non-Condon effects, which describe how molecular vibrational transitions are influenced by a system's rotational and translational degrees of freedom, are often overlooked in spectroscopy studies of biological macromolecules. In this work, we explore these effects in the modeling of infrared (IR) spectra for nucleic acids in the 1600-1800 cm-1 region. Through electronic structure calculations, we reveal that the transition dipole moments of the C═O and C═C stretching modes in nucleobases are highly sensitive to solvation, hydrogen bonding, and base stacking conditions. To incorporate vibrational non-Condon effects into spectroscopy modeling, we use local electric fields on chromophore atoms as collective coordinates and leverage experimental IR spectra of oligonucleotides to develop deep neural network-based transition dipole strength (TDS) maps for the C═O and C═C chromophores. By integrating molecular dynamics simulations with a mixed quantum/classical treatment of the line shape theory, we apply the TDS maps to calculate the IR spectra of nucleoside 5'-monophosphates, DNA double helices and yeast phenylalanine tRNA. The resulting theoretical spectra show quantitative agreement with experimental measurements. While the predictions for nucleoside 5'-monophosphates are comparable to baseline performance, the TDS maps yield significantly improved IR peak intensities across all oligonucleotides. This theoretical framework effectively bridges atomistic simulations and IR spectroscopy experiments, offering molecular insights into how vibrational non-Condon effects impact the observed spectral features.

Coherence in Chemistry: Foundations and Frontiers

Coherence refers to correlations in waves. Because matter has a wave-particle nature, it is unsurprising that coherence has deep connections with the most contemporary issues in chemistry research (e.g., energy harvesting, femtosecond spectroscopy, molecular qubits and more). But what does the word "coherence" really mean in the context of molecules and other quantum systems? We provide a review of key concepts, definitions, and methodologies, surrounding coherence phenomena in chemistry, and we describe how the terms "coherence" and "quantum coherence" refer to many different phenomena in chemistry. Moreover, we show how these notions are related to the concept of an interference pattern. Coherence phenomena are indeed complex, and ambiguous definitions may spawn confusion. By describing the many definitions and contexts for coherence in the molecular sciences, we aim to enhance understanding and communication in this broad and active area of chemistry.

The heterogeneity of aqueous solutions: the current situation in the context of experiment and theory

The advancement of experimental methods has provided new information about the structure and structural fluctuations of water. Despite the appearance of numerous models, which aim to describe a wide range of thermodynamic and electrical characteristics of water, there is a deficit in systemic understanding of structuring in aqueous solutions. A particular challenge is the fact that even pure water is a heterogeneous, multicomponent system composed of molecular and supramolecular structures. The possibility of the existence of such structures and their nature are of fundamental importance for various fields of science. However, great difficulties arise in modeling relatively large supramolecular structures (e.g. extended hydration shells), where the bonds between molecules are characterized by low energy. Generally, such structures may be non-equilibrium but relatively long-lived. Evidently, the short times of water microstructure exchanges do not mean short lifetimes of macrostructures, just as the instability of individual parts does not mean the instability of the entire structure. To explain this paradox, we review the data from experimental and theoretical research. Today, only some of the experimental results on the lifetime of water structures have been confirmed by modeling, so there is not a complete theoretical picture of the structure of water yet. We propose a new hierarchical water macrostructure model to resolve the issue of the stability of water structures. In this model, the structure of water is presented as consisting of many hierarchically related levels (the stratification model). The stratification mechanism is associated with symmetry breaking at the formation of the next level, even with minimal changes in the properties of the previous level. Such a hierarchical relationship can determine the unique physico-chemical properties of water systems and, in the future, provide a complete description of them.

Fast Quasi-Centroid Molecular Dynamics for Water and Ice

We describe how the fast quasi-centroid molecular dynamics (f-QCMD) method can be applied to condensed-phase systems by approximating the quasi-centroid potential of mean force as a sum of inter- and intramolecular corrections to the classical interaction potential. The corrections are found by using a regularized iterative Boltzmann inversion procedure to recover the inter- and intramolecular quasi-centroid distribution functions obtained from a path integral molecular dynamics simulation. The resulting methodology is found to give good agreement with a previously published QCMD dipole absorption spectrum for liquid water and satisfactory agreement for ice. It also gives good agreement with spectra from a recent implementation of CMD that uses a precomputed elevated temperature potential of mean force. Modern centroid molecular dynamics methods, therefore, appear to be reaching a consensus regarding the impact of nuclear quantum effects on the vibrational spectra of water and ice.

Structural evolution of water-in-propylene carbonate mixtures revealed by polarized Raman spectroscopy and molecular dynamics

The liquid structure of systems wherein water is limited in concentration or through geometry is of great interest in various fields such as biology, materials science, and electrochemistry. Here, we present a combined polarized Raman and molecular dynamics investigation of the structural changes that occur as water is added incrementally to propylene carbonate (PC), a polar, aprotic solvent that is important in lithium-ion batteries. Polarized Raman spectra of PC solutions were collected for water mole fractions 0.003 ≤ χwater ≤ 0.296, which encompasses the solubility range of water in PC. The novel approach taken herein provides additional hydrogen bond and solvation characterization of this system that has not been achievable in previous studies. Analysis of the polarized carbonyl Raman band in conjunction with simulations demonstrated that the bulk structure of the solvent remained unperturbed upon the addition of water. Experimental spectra in the O-H stretching region were decomposed through Gaussian fitting into sub-bands and comparison to studies of dilute HOD in D2O. With the aid of simulations, we identified these different bands as water arrangements having different degrees of hydrogen bonding. The observed water structure within PC indicates that water tends to self-aggregate, forming a hydrogen bond network that is distinctly different from the bulk and dependent on concentration. For example, at moderate concentrations, the most likely aggregate structures are chains of water molecules, each with two hydrogen bonds.

Matsubara dynamics approximation for generalized multi-time correlation functions

We introduce a semi-classical approximation for calculating generalized multi-time correlation functions based on Matsubara dynamics, a classical dynamics approach that conserves the quantum Boltzmann distribution. This method is exact for the zero time and harmonic limits and reduces to classical dynamics when only one Matsubara mode is considered (i.e., the centroid). Generalized multi-time correlation functions can be expressed as canonical phase-space integrals, involving classically evolved observables coupled through Poisson brackets in a smooth Matsubara space. Numerical tests on a simple potential show that the Matsubara approximation exhibits better agreement with exact results than classical dynamics, providing a bridge between the purely quantum and classical descriptions of multi-time correlation functions. Despite the phase problem that prevents practical applications of Matsubara dynamics, the reported work provides a benchmark theory for the future development of quantum-Boltzmann-preserving semi-classical approximations for studies of chemical dynamics in condensed phase systems.

Many-Body Effects in Aqueous Systems: Synergies Between Interaction Analysis Techniques and Force Field Development

Interaction analysis techniques, including the many-body expansion (MBE), symmetry-adapted perturbation theory, and energy decomposition analysis, allow for an intuitive understanding of complex molecular interactions. We review these methods by first providing a historical context for the study of many-body interactions and discussing how nonadditivities emerge from Hamiltonians containing strictly pairwise-additive interactions. We then elaborate on the synergy between these interaction analysis techniques and the development of advanced force fields aimed at accurately reproducing the Born-Oppenheimer potential energy surface. In particular, we focus on ab initio-based force fields that aim to explicitly reproduce many-body terms and are fitted to high-level electronic structure results. These force fields generally incorporate many-body effects through (a) parameterization of distributed multipoles, (b) explicit fitting of the MBE, (c) inclusion of many-atom features in a neural network, and (d) coarse-graining of many-body terms into an effective two-body term. We also discuss the emerging use of the MBE to improve the accuracy and speed of ab initio molecular dynamics.

An extensive assessment of the performance of pairwise and many-body interaction potentials in reproducing ab initio benchmark binding energies for water clusters n = 2-25

We assess the performance of 7 pairwise additive (TIP3P, TIP4P, TIP4P-ice, TIP5P, OPC, SPC, SPC/E) and 8 families of many-body potentials (q-AQUA, HIPPO, AMOEBA, EFP, TTM, WHBB, MB-pol, MB-UCB) in reproducing high-level ab initio benchmark values, CCSD(T) or MP2 at the complete basis set (CBS) limit for the binding energy and the many-body expansion (MBE) of water clusters n = 2-11, 16-17, 20, 25. By including a large range of cluster sizes having dissimilar hydrogen bonding networks, we obtain an understanding of how these potentials perform for different hydrogen bonding arrangements that are mostly outside of their parameterization range. While it is appropriate to compare the results of ab initio based many-body potentials directly to the electronic binding energies (D_e's), the pairwise additive ones are compared to the enthalpies at T = 298 K, ΔH(298 K), as the latter class of force fields are parametrized to reproduce enthalpies (implicitly accounting for zero-point energy corrections) rather than binding energies. We find that all pairwise additive potentials considered overestimate the reference ΔH values for the n = 2-25 clusters by >13%. For the water dimer (n = 2) in particular, the errors are in the range 83-119% for the pairwise additive potentials studied since these are based on an effective rather than the true 2-body interaction specifically designed as a means of partially accounting for the missing many-body terms. This stronger 2-body interaction is achieved by an enhanced monomer dipole moment that mimics its increase from the gas phase monomer to the condensed phase value. Indeed, for cluster sizes n ≥ 4 the percent deviations become slightly smaller (albeit all exceeding 13%). In contrast, we find that the many-body potentials perform more accurately in reproducing the electronic binding energies (D_e's) throughout the entire cluster range (n = 2-25), all reproducing the ab initio benchmark binding energies within ±7% of the respective CBS values. We further assess the ability of a subset of the many-body potentials (MB-UCB, q-AQUA, MB-pol, and TTM2.1-F) to also reproduce the magnitude of the ab initio many-body energy terms for water cluster sizes n = 7, 10, 16 and 17. The potentials show an overall good agreement with the available benchmark values. However, we identify characteristic differences upon comparing the many-body terms at both the ab initio-optimized geometries and the respective potential-optimized geometries to the reference ab initio values. Additionally, by applying this analysis to a wide range of cluster sizes, trends in the MBE of the potentials with increasing cluster size can be identified. Finally, in an attempt to draw a parallel between the pairwise additive and many-body potentials, we report the analysis of the individual molecular dipole moments for water clusters with 1 to ∼4 solvation shells with the TTM2.1-F potential. We find that the internally solvated water molecules have in general a larger molecular dipole moment ranging from 2.6-3.0 D. This justifies the use of an enhanced, with respect to the gas-phase value, molecular dipole moment for the pairwise additive potentials, which is intended to fold in the many body terms into an effective (enhanced) pairwise interaction through the choice of the charges. These results have important implications for the development of future generations of efficient, transferable, and highly accurate classical interaction potentials in both the pairwise additive and many-body categories.

Coarse-Graining of Imaginary Time Feynman Path Integrals: Inclusion of Intramolecular Interactions and Bottom-up Force-Matching

Feynman's imaginary time path integral formalism of quantum statistical mechanics and the corresponding quantum-classical isomorphism provide a tangible way of incorporating nuclear quantum effect (NQE) in the simulation of condensed matter systems using well-developed classical simulation techniques. Our previous work has presented the many-body coarse-graining of path integral (CG-PI) theory that builds an isomorphism between the quantum partition function of N distinguishable particles and the classical partition function of 2N pseudoparticles. In this present work, we develop a generalized version of the many-body CG-PI theory that incorporates many-body interactions in the force field. Based on the new derivation, we provide a numerical CG-PI (n-CG-PI) modeling strategy parametrized from the underlying path integral molecular dynamics (PIMD) trajectories using force matching and Boltzmann inversion. The n-CG-PI models for two liquid systems are shown to capture well both the intramolecular and intermolecular structural correlations of the reference PIMD simulations. The generalized derivation of the many-body CG-PI theory and the n-CG-PI model presented in this work extend the scope of the CG-PI formalism by generalizing the previously limited theory to incorporate force fields of realistic molecular systems.

Experimental Determination of Relationship between Intramolecular O-D Bond Length and Its Stretching Vibrational Frequency of D2O Molecule in the Liquid State

Experimental evidence has been obtained for the structure-spectra relationship of hydrogen bonds in aqueous solutions. Intramolecular O-D distance, r_OD, has been determined by the least-squares fitting analysis of the neutron interference term in the high-Q region observed for pure D₂O and concentrated aqueous solutions. The average O-D stretching frequency, ν_OD, has been obtained from the position of the center of gravity of the observed ATR-IR O-D stretching band. The linear relationship between r_OD and ν_OD has been confirmed in the liquid state. The slope of dν_OD/dr_OD is evaluated to be -21 000 ± 1000 cm^-1 Å^-1.

Segregation on the nanoscale coupled to liquid water polyamorphism in supercooled aqueous ionic-liquid solution

The most intriguing hypothesis explaining many water anomalies is a metastable liquid-liquid phase transition (LLPT) at high pressure and low temperatures, experimentally hidden by homogeneous nucleation. Recent infrared spectroscopic experiments showed that upon addition of hydrazinium trifluoroacetate to water, the supercooled ionic solution undergoes a sharp, reversible LLPT at ambient pressure, possible offspring of that in pure water. Here, we calculate the temperature-dependent signature of the OH-stretching band, reporting on the low/high density phase of water, in neat water and in the same experimentally investigated ionic solution. The comparison between the infrared signature of the pure liquid and that of the ionic solution can be achieved only computationally, providing insight into the nature of the experimentally observed phase transition and allowing us to investigate the effects of ionic compounds on the high to low density supercooled liquid water transition. We show that the experimentally observed crossover behavior in the ionic solution can be reproduced only if the phase transition between the low- and high-density liquid states of water is coupled to a mixing-unmixing transition between the water component and the ions: at low temperatures, water and ions are separated and the water component is a low density liquid. At high temperatures, water and ions get mixed and the water component is a high-density liquid. The separation at low temperatures into ion-rich and ion-poor regions allows unveiling the polyamorphic nature of liquid water, leading to a crossover behavior resembling that observed in supercooled neat water under high pressure.

Machine learning phase space quantum dynamics approaches

Derived from phase space expressions of the quantum Liouville theorem, equilibrium continuity dynamics is a category of trajectory-based phase space dynamics methods, which satisfies the two critical fundamental criteria: conservation of the quantum Boltzmann distribution for the thermal equilibrium system and being exact for any thermal correlation functions (even of nonlinear operators) in the classical and harmonic limits. The effective force and effective mass matrix are important elements in the equations of motion of equilibrium continuity dynamics, where only the zeroth term of an exact series expansion of the phase space propagator is involved. We introduce a machine learning approach for fitting these elements in quantum phase space, leading to a much more efficient integration of the equations of motion. Proof-of-concept applications to realistic molecules demonstrate that machine learning phase space dynamics approaches are possible as well as competent in producing reasonably accurate results with a modest computation effort.

Role of Intermolecular Charge Fluxes in the Hydrogen-Bond-Induced Frequency Shifts of the OH Stretching Mode of Water

The relation between the vibrational properties and the electrostatic situations of the vibrating functional group is useful to predict vibrational spectroscopic features based on, for example, classical molecular dynamics of liquids or biomolecular systems, but to pursue its generality or the extent of applicability, it is required to understand the mechanisms giving rise to it. Here such an analysis is carried out for the OH stretching mode of water. By examining the correlations among various (structural, vibrational, and electrostatic) properties and by analyzing the spatial characteristics of the behavior of electrons occurring upon the vibration, it is shown that the dependence of the vibrational frequency and the dipole derivative of the OH stretching mode on the electric field is not of purely electrostatic origin, and the delocalized electronic motions occurring with this mode, called intermolecular charge fluxes, related to both the dipole first and second derivatives play important roles.

Nuclear Quantum Effects from the Analysis of Smoothed Trajectories: Pilot Study for Water

Nuclear quantum effects have significant contributions to thermodynamic quantities and structural properties; furthermore, very expensive methods are necessary for their accurate computation. In most calculations, these effects, for instance, zero-point energies, are simply neglected or only taken into account within the quantum harmonic oscillator approximation. Herein, we present a new method, Generalized Smoothed Trajectory Analysis, to determine nuclear quantum effects from molecular dynamics simulations. The broad applicability is demonstrated with the examples of a harmonic oscillator and different states of water. Ab initio molecular dynamics simulations have been performed for ideal gas up to the temperature of 5000 K. Classical molecular dynamics have been carried out for hexagonal ice, liquid water, and vapor at atmospheric pressure. With respect to the experimental heat capacity, our method outperforms previous calculations in the literature in a wide temperature range at lower computational cost than other alternatives. Dynamic and structural nuclear quantum effects of water are also discussed.

Dephasing and Decoherence in Vibrational and Electronic Line Shapes

Absorption and emission line shapes of vibrational and electronic transitions in liquids are broadened by interactions with the "bath" (in this case, the rotational and translational degrees of freedom of all the molecules in the liquid). If these degrees of freedom are treated classically, the broadening process is often known as dephasing. If, on the other hand, the bath degrees of freedom are instead treated quantum mechanically, there is additional broadening due to what is known in the chemical-physics literature as decoherence. The question addressed in this paper is the relative importance of decoherence (bath quantum effects) and dephasing. We present general developments of this subject for absorption and emission line shapes, discover several new relationships connecting classical and quantum treatments of the bath, and also consider the Stokes shift (difference in peak frequencies in absorption and emission). We next draw some general conclusions by considering a model system whose transition-frequency time-correlation function has only one bath time scale. We then consider a realistic system of the vibrational OH stretch transition of dilute HOD in liquid D₂O at room temperature. For this system, we conclude that bath quantum effects are not very important, except for the Stokes shift. More generally, we argue that this is the case for many vibrational and most electronic transitions in room-temperature liquids.

Capturing intrinsic site-dependent spectral signatures and lifetimes of isolated OH oscillators in extended water networks

The extremely broad infrared spectrum of water in the OH stretching region is a manifestation of how profoundly a water molecule is distorted when embedded in its extended hydrogen-bonding network. Many effects contribute to this breadth in solution at room temperature, which raises the question as to what the spectrum of a single OH oscillator would be in the absence of thermal fluctuations and coupling to nearby OH groups. We report the intrinsic spectral responses of isolated OH oscillators embedded in two cold (~20 K), hydrogen-bonded water cages adopted by the Cs⁺·(HDO)(D₂O)₁₉ and D₃O⁺·(HDO)(D₂O)₁₉ clusters. Most OH oscillators yield single, isolated features that occur with linewidths that increase approximately linearly with their redshifts. Oscillators near 3,400 cm^-1, however, occur with a second feature, which indicates that OH stretch excitation of these molecules drives low-frequency, phonon-type motions of the cage. The excited state lifetimes inferred from the broadening are considered in the context of fluctuations in the local electric fields that are available even at low temperature.

Multi-time formulation of Matsubara dynamics

Matsubara dynamics has recently emerged as the most general form of a quantum-Boltzmann-conserving classical dynamics theory for the calculation of single-time correlation functions. Here, we present a generalization of Matsubara dynamics for the evaluation of multitime correlation functions. We first show that the Matsubara approximation can also be used to approximate the two-time symmetrized double Kubo transformed correlation function. By a straightforward extension of these ideas to the multitime realm, a multitime Matsubara dynamics approximation can be obtained for the multitime fully symmetrized Kubo transformed correlation function. Although not a practical method, due to the presence of a phase-term, this multitime formulation of Matsubara dynamics represents a benchmark theory for future development of Boltzmann preserving semiclassical approximations to general higher order multitime correlation functions. It also reveals a connection between imaginary time-ordering in the path integral and the classical dynamics of multitime correlation functions.

Introductory lecture: advances in ion spectroscopy: from astrophysics to biology

This introduction provides a historical context for the development of ion spectroscopy over the past half century by following the evolution of experimental methods to the present state-of-the-art. Rather than attempt a comprehensive review, we focus on how early work on small ions, carried out with fluorescence, direct absorption, and photoelectron spectroscopy, evolved into powerful technologies that can now address complex chemical problems ranging from catalysis to biophysics. One of these developments is the incorporation of cooling and temperature control to enable the general application of "messenger tagging" vibrational spectroscopy, first carried out using ionized supersonic jets and then with buffer gas cooling in radiofrequency ion traps. Some key advances in the application of time-resolved pump-probe techniques to follow ultrafast dynamics are also discussed, as are significant benchmarks in the refinement of ion mobility to allow spectroscopic investigation of large biopolymers with well-defined shapes. We close with a few remarks on challenges and opportunities to explore molecular level mechanics that drive macroscopic behavior.

Beyond Badger's Rule: The Origins and Generality of the Structure-Spectra Relationship of Aqueous Hydrogen Bonds

The structure of hydrogen bonded networks is intimately intertwined with their dynamics. Despite the incredibly wide range of hydrogen bond strengths encountered in water clusters, ion-water clusters, and liquid water, we demonstrate that the previously reported correlation between the change in the equilibrium bond length of the hydrogen bonded OH covalent bond and the corresponding shift in its harmonic frequency in water clusters is much more broadly applicable. Surprisingly, this correlation describes the ratios for both the equilibrium OH bond length/harmonic frequency and the vibrationally averaged bond length/anharmonic frequency in water, hydronium water, and halide water clusters. Consideration of harmonic and anaharmonic data leads to a correlation of -19 ± 1 cm^-1/0.001 Å. The fundamental nature of this correlation is further confirmed through the analysis of ab initio Molecular Dynamics (AIMD) trajectories for liquid water. We demonstrate that this simple correlation for both harmonic and anharmonic systems can be modeled by the response of an OH bond to an external field. Treating the OH bond as a Morse oscillator, we develop analytic expressions, which relate the ratio of the shift in the vibrational frequency of the hydrogen-bonded OH bond to the shift in OH bond length, to parameters in the Morse potential and the ratio of the first and second derivatives of the field-dependent projection of the dipole moment of water onto the hydrogen-bonded OH bond. Based on our analysis, we develop a protocol for reconstructing the AIMD spectra of liquid water from the sampled distribution of the OH bond lengths. Our findings elucidate the origins of the relationship between the molecular structure of the fleeting hydrogen-bonded network and the ensuing dynamics, which can be probed by vibrational spectroscopy.

Impact of nuclear quantum effects on the structural inhomogeneity of liquid water

The 2D Raman-terahertz (THz) response of liquid water is studied in dependence of temperature and isotope substitution ([Formula: see text]O, [Formula: see text]O, and [Formula: see text]O). In either case, a very short-lived (i.e., between 75 and 95 fs) echo is observed that reports on the inhomogeneity of the low-frequency intermolecular modes and hence, on the heterogeneity of the hydrogen bond networks of water. The echo lifetime slows down by about 20% when cooling the liquid from room temperature to the freezing point. Furthermore, the echo lifetime of [Formula: see text]O is [Formula: see text] slower than that of [Formula: see text]O, and both can be mapped on each other by introducing an effective temperature shift of [Formula: see text] K. In contrast, the temperature-dependent echo lifetimes of [Formula: see text]O and [Formula: see text]O are the same within error. [Formula: see text]O and [Formula: see text]O have identical masses, yet [Formula: see text]O is much closer to [Formula: see text]O in terms of nuclear quantum effects. It is, therefore, concluded that the echo is a measure of the structural inhomogeneity of liquid water induced by nuclear quantum effects.

Hydrogen bonding from a valence bond theory perspective: the role of covalency

A valence bond theory based method has been developed to decompose hydrogen bond energies into contributions from geometry, electrostatics, polarization and charge transfer. This decomposition method has been carried out for F-HFH, F-HOH2, F-HNH3, HO-HOH2, HO-HNH3, and H2N-HNH3. Localized valence bond self-consistent field (L-VBSCF) and localized breathing orbital valence bond (L-BOVB) calculations were performed at the PBEPBE/aug-cc-pVDZ optimized geometries. It is shown that inclusion of valence bond structures that explicitly include charge transfer account for at least 32% (likely over half) of the hydrogen bond energy of all systems studied, indicating the dominant role of covalency. This is in agreement with calculated bond lengths, geometry deformation energies, and polarization energies. Electrostatic effects were found to play only a minor role in contrast to some widely held ideas regarding the nature of hydrogen bonding.

Inclusion of nuclear quantum effects for simulations of nonlinear spectroscopy

The computation and interpretation of nonlinear vibrational spectroscopy is of vital importance for understanding a wide range of dynamical processes in molecular systems. Here, we introduce an approach to evaluate multi-time response functions in terms of multi-time double symmetrized Kubo transformed thermal correlation functions. Furthermore, we introduce a multi-time extension of ring polymer molecular dynamics to evaluate these Kubo transforms. Benchmark calculations show that the approximations are useful for short times even for nonlinear operators, providing a consistent improvement over classical simulations of multi-time correlation functions. The introduced methodology thus provides a practical way of including nuclear quantum effects in multi-time response functions of non-linear optical spectroscopy.

Approximating Matsubara dynamics using the planetary model: Tests on liquid water and ice

Matsubara dynamics is the quantum-Boltzmann-conserving classical dynamics which remains when real-time coherences are taken out of the exact quantum Liouvillian [T. J. H. Hele et al., J. Chem. Phys. 142, 134103 (2015)]; because of a phase-term, it cannot be used as a practical method without further approximation. Recently, Smith et al. [J. Chem. Phys. 142, 244112 (2015)] developed a "planetary" model dynamics which conserves the Feynman-Kleinert (FK) approximation to the quantum-Boltzmann distribution. Here, we show that for moderately anharmonic potentials, the planetary dynamics gives a good approximation to Matsubara trajectories on the FK potential surface by decoupling the centroid trajectory from the locally harmonic Matsubara fluctuations, which reduce to a single phase-less fluctuation particle (the "planet"). We also show that the FK effective frequency can be approximated by a direct integral over these fluctuations, obviating the need to solve iterative equations. This modification, together with use of thermostatted ring-polymer molecular dynamics, allows us to test the planetary model on water (gas-phase, liquid, and ice) using the q-TIP4P/F potential surface. The "planetary" fluctuations give a poor approximation to the rotational/librational bands in the infrared spectrum, but a good approximation to the bend and stretch bands, where the fluctuation lineshape is found to be motionally narrowed by the vibrations of the centroid.

Quantum theory of multiscale coarse-graining

Coarse-grained (CG) models serve as a powerful tool to simulate molecular systems at much longer temporal and spatial scales. Previously, CG models and methods have been built upon classical statistical mechanics. The present paper develops a theory and numerical methodology for coarse-graining in quantum statistical mechanics, by generalizing the multiscale coarse-graining (MS-CG) method to quantum Boltzmann statistics. A rigorous derivation of the sufficient thermodynamic consistency condition is first presented via imaginary time Feynman path integrals. It identifies the optimal choice of CG action functional and effective quantum CG (qCG) force field to generate a quantum MS-CG (qMS-CG) description of the equilibrium system that is consistent with the quantum fine-grained model projected onto the CG variables. A variational principle then provides a class of algorithms for optimally approximating the qMS-CG force fields. Specifically, a variational method based on force matching, which was also adopted in the classical MS-CG theory, is generalized to quantum Boltzmann statistics. The qMS-CG numerical algorithms and practical issues in implementing this variational minimization procedure are also discussed. Then, two numerical examples are presented to demonstrate the method. Finally, as an alternative strategy, a quasi-classical approximation for the thermal density matrix expressed in the CG variables is derived. This approach provides an interesting physical picture for coarse-graining in quantum Boltzmann statistical mechanics in which the consistency with the quantum particle delocalization is obviously manifest, and it opens up an avenue for using path integral centroid-based effective classical force fields in a coarse-graining methodology.

First-Principles Simulations of Liquid Water Using a Dielectric-Dependent Hybrid Functional

We carried out first-principles simulations of liquid water under ambient conditions using a dielectric-dependent hybrid functional, where the fraction of exact exchange is set equal to the inverse of the high-frequency dielectric constant of the liquid. We found excellent agreement with experiment for the oxygen-oxygen partial correlation function at the experimental equilibrium density and 311 ± 3 K. Other structural and dynamical properties, such as the diffusion coefficient, molecular dipole moments, and vibrational spectra, are also in good agreement with experiment. Our results, together with previous findings on electronic properties of the liquid with the same functional, show that the dielectric-dependent hybrid functional accurately describes both the structural and electronic properties of liquid water.

Nuclear quantum effects on the vibrational dynamics of liquid water

Based on quantum-mechanical path-integral molecular dynamics simulations, the impact of nuclear quantum effects on the vibrational and hydrogen bond dynamics in liquid water is investigated. The instantaneous fluctuations in the frequencies of the O-H stretch modes are calculated using the wavelet method of time-series analysis, while the time scales of the vibrational spectral diffusion are determined from frequency-time correlation functions, joint probability distributions, and the slope of three-pulse photon echo. We find that the inclusion of nuclear quantum effects leads not only to a redshift of the vibrational frequency distribution by around 130 cm^-1 but also to an acceleration of the vibrational dynamics by as much as 30%. In addition, quantum fluctuations also entail a significantly faster decay of correlation in the initial diffusive regime, which is in agreement with recent vibrational echo experiments.

Formulation of state projected centroid molecular dynamics: Microcanonical ensemble and connection to the Wigner distribution

A derivation of quantum statistical mechanics based on the concept of a Feynman path centroid is presented for the case of generalized density operators using the projected density operator formalism of Blinov and Roy [J. Chem. Phys. 115, 7822-7831 (2001)]. The resulting centroid densities, centroid symbols, and centroid correlation functions are formulated and analyzed in the context of the canonical equilibrium picture of Jang and Voth [J. Chem. Phys. 111, 2357-2370 (1999)]. The case where the density operator projects onto a particular energy eigenstate of the system is discussed, and it is shown that one can extract microcanonical dynamical information from double Kubo transformed correlation functions. It is also shown that the proposed projection operator approach can be used to formally connect the centroid and Wigner phase-space distributions in the zero reciprocal temperature β limit. A Centroid Molecular Dynamics (CMD) approximation to the state-projected exact quantum dynamics is proposed and proven to be exact in the harmonic limit. The state projected CMD method is also tested numerically for a quartic oscillator and a double-well potential and found to be more accurate than canonical CMD. In the case of a ground state projection, this method can resolve tunnelling splittings of the double well problem in the higher barrier regime where canonical CMD fails. Finally, the state-projected CMD framework is cast in a path integral form.

Perturbation Approach for Computing Infrared Spectra of the Local Mode of Probe Molecules

Linear and two-dimensional infrared (IR) spectroscopy of site-specific probe molecules provides an opportunity to gain a molecular-level understanding of the local hydrogen-bonding network, conformational dynamics, and long-range electrostatic interactions in condensed-phase and biological systems. A challenge in computation is to determine the time-dependent vibrational frequencies that incorporate explicitly both nuclear quantum effects of vibrational motions and an electronic structural representation of the potential energy surface. In this paper, a nuclear quantum vibrational perturbation (QVP) method is described for efficiently determining the instantaneous vibrational frequency of a chromophore in molecular dynamics simulations. Computational efficiency is achieved through the use of (a) discrete variable representation of the vibrational wave functions, (b) a perturbation theory to evaluate the vibrational energy shifts due to solvent dynamic fluctuations, and (c) a combined QM/MM potential for the systems. It was found that first-order perturbation is sufficiently accurate, enabling time-dependent vibrational frequencies to be obtained on the fly in molecular dynamics. The QVP method is illustrated in the mode-specific linear and 2D-IR spectra of the H-Cl stretching frequency in the HCl-water clusters and the carbonyl stretching vibration of acetone in aqueous solution. To further reduce computational cost, a hybrid strategy was proposed, and it was found that the computed vibrational spectral peak position and line shape are in agreement with experimental results. In addition, it was found that anharmonicity is significant in the H-Cl stretching mode, and hydrogen-bonding interactions further enhance anharmonic effects. The present QVP method complements other computational approaches, including path integral-based molecular dynamics, and represents a major improvement over the electrostatics-based spectroscopic mapping procedures.

Torque and atomic forces for Cartesian tensor atomic multipoles with an application to crystal unit cell optimization

New equations for torque and atomic force are derived for use in flexible molecule force fields with atomic multipoles. The expressions are based on Cartesian tensors with arbitrary multipole rank. The standard method for rotating Cartesian tensor multipoles and calculating torque is to first represent the tensor with n indexes and 3(n) redundant components. In this work, new expressions for directly rotating the unique (n + 1)(n + 2)/2 Cartesian tensor multipole components Θpqr are given by introducing Cartesian tensor rotation matrix elements X(R). A polynomial expression and a recursion relation for X(R) are derived. For comparison, the analogous rotation matrix for spherical tensor multipoles are the Wigner functions D(R). The expressions for X(R) are used to derive simple equations for torque and atomic force. The torque and atomic force equations are applied to the geometry optimization of small molecule crystal unit cells. In addition, a discussion of computational efficiency as a function of increasing multipole rank is given for Cartesian tensors. © 2016 Wiley Periodicals, Inc.

Modeling Molecular Interactions in Water: From Pairwise to Many-Body Potential Energy Functions

Almost 50 years have passed from the first computer simulations of water, and a large number of molecular models have been proposed since then to elucidate the unique behavior of water across different phases. In this article, we review the recent progress in the development of analytical potential energy functions that aim at correctly representing many-body effects. Starting from the many-body expansion of the interaction energy, specific focus is on different classes of potential energy functions built upon a hierarchy of approximations and on their ability to accurately reproduce reference data obtained from state-of-the-art electronic structure calculations and experimental measurements. We show that most recent potential energy functions, which include explicit short-range representations of two-body and three-body effects along with a physically correct description of many-body effects at all distances, predict the properties of water from the gas to the condensed phase with unprecedented accuracy, thus opening the door to the long-sought "universal model" capable of describing the behavior of water under different conditions and in different environments.

Experimentally probing the libration of interfacial water: the rotational potential of water is stiffer at the air/water interface than in bulk liquid

Most properties of liquid water are determined by its hydrogen-bond network. Because forming an aqueous interface requires termination of this network, one might expect the molecular level properties of interfacial water to markedly differ from water in bulk. Intriguingly, much prior experimental and theoretical work has found that, from the perspective of their time-averaged structure and picosecond structural dynamics, hydrogen-bonded OH groups at an air/water interface behave the same as hydrogen-bonded OH groups in bulk liquid water. Here we report the first experimental observation of interfacial water's libration (i.e. frustrated rotation) using the laser-based technique vibrational sum frequency spectroscopy. We find this mode has a frequency of 834 cm(-1), ≈165 cm(-1) higher than in bulk liquid water at the same temperature and similar to bulk ice. Because libration frequency is proportional to the stiffness of water's rotational potential, this increase suggests that one effect of terminating bulk water's hydrogen bonding network at the air/water interface is retarding rotation of water around intact hydrogen bonds. Because in bulk liquid water the libration plays a key role in stabilizing reaction intermediates and dissipating excess vibrational energy, we expect the ability to probe this mode in interfacial water to open new perspectives on the kinetics of heterogeneous reactions at aqueous interfaces.