Instantaneous Normal Mode Spectrum Research Articles

Comment on "Explaining the specific heat of liquids based on instantaneous normal modes".

In a recent paper [Phys. Rev. E 104, 014103 (2021)10.1103/PhysRevE.104.014103] Baggioli and Zaccone formulate a theoretical description of the specific heat of liquids by using Debye's expression for the specific heat of solids and inserting a density of states which they claim to represent the instantaneous-normal-mode (INM) spectrum of a liquid. However, the quantum-mechanical procedure of Debye cannot be used for the relaxational excitations of a classical liquid. Furthermore, the authors' formula for the INM spectrum does not represent the known INM spectra of simple liquids, and the derivation of this formula from their model equationof motion is mathematically in error. These and a number of other inconsistencies render their work not very helpful for studying the specific heat of liquids.

Predicting plasticity of amorphous solids from instantaneous normal modes.

We present a mathematical description of amorphous solid deformation and plasticity by extending the concept of instantaneous normal modes (INMs) to deformed systems, which allows us to retain the effect of strain on the vibrational density of states (VDOS). Starting from the nonaffine lattice dynamics (NALD) description of elasticity and viscoelasticity of glasses, we formulate the linear response theory up to large deformations by considering the strain-dependent tangent modulus at finite values of shear strain. The (nonaffine) tangent shear modulus is computed from the VDOS of affinely strained configurations at varying strain values. The affine strain, found analytically on the static (undeformed) snapshot of the glass, leads to configurations that are rich with soft low-energy modes as well as unstable modes (negative eigenvalues) that are otherwise completely "washed out" and lost if one lets the system fully relax after strain. This procedure is consistent with the structure of NALD. The INM spectrum of deformed states allows for the analytical prediction of the stress-strain curve of a model glass. Good parameter-free quantitative agreement is shown between the prediction and simulations of athermal quasistatic shear of a coarse-grained polymer glass.

Modeling the instantaneous normal mode spectra of liquids as that of unstable elastic media

We study the instantaneous normal mode (INM) spectrum of a simulated soft-sphere liquid at different equilibrium temperatures T We find that the spectrum of eigenvalues [Formula: see text] has a sharp maximum near (but not at) [Formula: see text] and decreases monotonically with [Formula: see text] on both the stable and unstable sides of the spectrum. The spectral shape strongly depends on temperature. It is rather asymmetric at low temperatures (close to the dynamical critical temperature) and becomes symmetric at high temperatures. To explain these findings we present a mean-field theory for [Formula: see text], which is based on a heterogeneous elasticity model, in which the local shear moduli exhibit spatial fluctuations, including negative values. We find good agreement between the simulation data and the model calculations, done with the help of the self-consistent Born approximation (SCBA), when we take the variance of the fluctuations to be proportional to the temperature T More importantly, we find an empirical correlation of the positions of the maxima of [Formula: see text] with the low-frequency exponent of the density of the vibrational modes of the glasses obtained by quenching to [Formula: see text] from the temperature T We discuss the present findings in connection to the liquid to glass transformation and its precursor phenomena.

What does the instantaneous normal mode spectrum tell us about dynamical heterogeneity in glass-forming fluids?

We examine the instantaneous normal mode spectrum of model metallic and polymeric glass-forming liquids. We focus on the localized modes in the unstable part of the spectrum [unstable localized (UL) modes] and find that the particles making the dominant contribution to the participation ratio form clusters that grow upon cooling in a fashion similar to the dynamical heterogeneity in glass-forming fluids, i.e., highly mobile (or immobile) particles form clusters that grow upon cooling; however, a comparison of the UL mode clusters to the mobile and immobile particle clusters indicates that they are distinct entities. We also show that the cluster size provides an alternate method to distinguish localized and delocalized modes, offering a significant practical advantage over the finite-size scaling approach. We examine the trajectories of particles contributing most to the UL modes and find that they have a slightly enhanced mobility compared to the average, and we determine a characteristic time quantifying the persistence time of this excess mobility. This time scale is proportional to the structural relaxation time τα of the fluid, consistent with a prediction by Zwanzig [Phys. Rev. 156, 190 (1967)] for the lifetime of collective excitations in cooled liquids. Evidently, these collective excitations serve to facilitate relaxation but do not actually participate in the motion associated with barrier crossing events governing activated transport. They also serve as a possible concrete realization of the "facilitation" clusters postulated in previous modeling of glass-forming liquids.

Instantaneous normal mode analysis for intermolecular and intramolecular vibrations of water from atomic point of view

By exploiting the instantaneous normal mode (INM) analysis for models of flexible molecules, we investigate intermolecular and intramolecular vibrations of water from the atomic point of view. With two flexible SPC/E models, our investigations include three aspects about their INM spectra, which are separated into the unstable, intermolecular, bending, and stretching bands. First, the O- and H-atom contributions in the four INM bands are calculated and their stable INM spectra are compared with the power spectra of the atomic velocity autocorrelation functions. The unstable and intermolecular bands of the flexible models are also compared with those of the SPC/E model of rigid molecules. Second, we formulate the inverse participation ratio (IPR) of the INMs, respectively, for the O- and H-atom and molecule. With the IPRs, the numbers of the three species participated in the INMs are estimated so that the localization characters of the INMs in each band are studied. Further, by the ratio of the IPR of the H atom to that of the O atom, we explore the number of involved OH bond per molecule participated in the INMs. Third, by classifying simulated molecules into subensembles according to the geometry of their local environments or their H-bond configurations, we examine the local-structure effects on the bending and stretching INM bands. All of our results are verified to be insensible to the definition of H-bond. Our conclusions about the intermolecular and intramolecular vibrations in water are given.

Localization transition of instantaneous normal modes and liquid diffusion

We analyze the structure of instantaneous normal modes (INM) associated with the Hessian matrix of a liquid. Utilizing a scaling theory developed in the framework of Anderson localization studies, we unambiguously identify the crossover point in the INM spectrum between extended and localized modes. We establish a relation between the unstable, delocalized INMs and the liquid diffusion coefficient that appears valid over a large temperature range, covering both Arrhenius and non-Arrhenius regimes of temperature dependence. These results suggest a possible route to theoretically relate dynamics to thermodynamical properties of the liquid via the tomography of the INMs.

Multifractality of instantaneous normal modes at mobility edges

In terms of the multifractal analysis, we investigate the characteristics of the instantaneous normal modes (INMs) at two mobility edges (MEs) of a simple fluid, where the locations of the MEs in the INM spectrum were identified in a previous work [B. J. Huang and T. M. Wu, Phys. Rev. E 79, 041105 (2009)]. The mass exponents and the singularity spectrum of the INMs are obtained by the box-size and system-size scalings under the typical average. The INM eigenvectors at a ME exhibit a multifractal nature and the multifractal INMs at each ME yield the same results in generalized fractal dimensions and singularity spectrum. Our results indicate that the singularity spectrum of the multifractal INMs agrees well with that of the Anderson model at the critical disorder. This good agreement provides numerical evidence for the universal multifractality at the localization-delocalization transition. For the multifractal INMs, the probability density function and the spatial correlation function of the squared vibrational amplitudes are also calculated. The relation between the probability density function and the singularity spectrum is examined numerically, so are the relations between the critical exponents of the spatial correlation function and the mass exponents of the multifractal INMs.

Entropy, diffusivity and the energy landscape of a waterlike fluid

Molecular dynamics simulations and instantaneous normal mode (INM) analysis of a fluid with core-softened pair interactions and waterlike liquid-state anomalies are performed to obtain an understanding of the relationship between thermodynamics, transport properties, and the potential energy landscape. Rosenfeld scaling of diffusivities with the thermodynamic excess and pair correlation entropy is demonstrated for this model. The INM spectra are shown to carry information about the dynamical consequences of the interplay between length scales characteristic of anomalous fluids, such as bimodality of the real and imaginary branches of the frequency distribution. The INM spectral information is used to partition the liquid entropy into two contributions associated with the real and imaginary frequency modes; only the entropy contribution from the imaginary branch captures the nonmonotonic behavior of the excess entropy and diffusivity in the anomalous regime of the fluid.

Local structural effects on low-frequency vibrational spectrum of liquid water: The instantaneous-normal-mode analysis

Currently, the designations for the low-frequency vibrational spectrum of liquid water are still diversified. In this Letter, using the instantaneous normal mode (INM) formalism for subensembles, we have examined the effects of local structure on the low-frequency INM spectrum of liquid water. The classified subensembles of water molecules are characterized by either the geometry of Voronoi polyhedron or the H-bond configuration. With these subensembles, the translational contribution to the INM spectrum is investigated. Providing insights into the local structural dependence of the vibrations between H-bonded water molecules, our results are consistent with the conventional designations for the Raman spectrum of liquid water.

Mechanism for singular behavior in vibrational spectra of topologically disordered systems: Short-range attractions

At low-enough fluid densities, we have found some naive singular behavior, like the van Hove singularities in the phonon spectra of lattices, appearing in the instantaneous normal mode spectra of the Lennard-Jones (LJ) 2n-n fluids, which serve as a prototype of topologically disordered systems. The singular behavior cannot be predicted by the mean-field theory, but interpreted by the perturbed binary modes of some special pairs, called the mutual nearest neighbor pairs, at separations corresponding to the extreme binary frequencies, which are solely determined by the attractive part of the LJ 2n-n pair potential. By reducing the range of attraction in the pair potential under the conditions of the same particle diameter and well depth, the tendency for the appearance of the singular behavior shifts to higher fluid densities. From this study, we conclude that pair potential with a short-range attraction can be a mechanism to produce a counterpart of the van Hove singularity in the vibrational spectra of disordered systems without a reference lattice.

Instantaneous normal mode analysis of orientational motions in liquid water: local structural effects.

We have investigated the effects of local structures on the orientational motions in liquid water in terms of the instantaneous normal mode (INM) analysis. The local structures of a molecule in liquid water are characterized by two different kinds of index: the asphericity parameter of its Voronoi polyhedron and the numbers of the H bonds donated and accepted by the molecule. According to the two kinds of index, the molecules in the simulated water are classified into subensembles, for which the rotational contributions to the INM spectrum are calculated. Our results indicate that by increasing the asphericity, the rotational contribution has a shift toward the high-frequency end in the real spectrum and a decrease in the fraction of the imaginary modes. Furthermore, we find that this shift essentially relies on the number of the donated H bonds of a molecule, but has almost nothing to do with that of the accepted H bonds. The local structural effects resulting from the geometry of water molecule are also discussed.

A combined time correlation function and instantaneous normal mode study of the sum frequency generation spectroscopy of the water/vapor interface

Theoretical approximations to the interface specific sum frequency generation (SFG) spectrum of O–H stretching at the water/vapor interface are constructed using time correlation function (TCF) and instantaneous normal mode (INM) methods. Both approaches lead to a (SSP polarization geometry) signal in excellent agreement with experimental measurements; the SFG spectrum of the entire water spectrum, both intermolecular and intramolecular, is reported. The observation that the INM spectrum is in agreement with the TCF result implies that motional narrowing effects play no role in the interfacial line shapes, in contrast to the O–H stretching dynamics in the bulk that leads to a narrowed line shape. This implies that (SSP) SFG spectroscopy is a probe of structure with dynamics not represented in the signal. The INM approach permits the elucidation of the molecular basis for the observed signal, and the motions responsible for the SFG line shape are well approximated as local O–H stretching modes. The complexity of the broad structured SFG signal is due to O–H stretching motions facing toward the bulk or vacuum environments that are characteristic of the interface. The success of both approaches suggests that theory can play a crucial role in interpreting SFG spectroscopy at more complex interfaces. It is also found that many-body polarization effects account for most of the observed signal intensity.

Instantaneous normal mode analysis of Morse liquids

Instantaneous normal mode (INM) analysis of a set of bulk Morse systems is performed to monitor the changes in the curvature distribution that occur as a result of changing the range and curvature of the Morse potential. The liquids are bound by Morse pair potentials, Vα(r)=ε[e−α(1−(r/re))−1]2−ε, and share a common well-depth, ε, and equilibrium pair distance, re, but possess different values of range parameter, α, which is inversely correlated with the range and softness of the potential. INM analysis is used to index the changes in the curvature distribution of the potential energy surface that take place as the range parameter of the pair potential is varied and is shown to provide considerable insights into the accompanying dynamical changes. For example, the fraction of imaginary frequency modes, and therefore the diffusivity, is expected to rise with increasing temperature and decreasing range. In contrast, the Einstein frequency, which is a measure of the curvature of the effective potential well that traps a tagged particle in the liquid state, shows a nonmonotonic behavior with range. We also consider the behavior of INM spectra of liquids in relation to that of solids and gases. It is shown that INM analysis can be used to monitor the transition of a fluid from a liquidlike regime, dominated by collective rearrangements, to a gaslike regime, dominated by binary collisions. The transition to a collision-dominated regime is promoted by decreasing the range of the pair potential. Key INM spectral features are shown to undergo a discontinuous change on melting. Minimization of instantaneous configurations to the nearest saddle removes the imaginary frequencies in the solid phase but not in the liquid phase.

Path integral simulations of quantum Lennard-Jones solids

Path integral simulations are used to study the thermodynamic and structural properties of quantum Lennard-Jones solids as a function of the degree of quantum delocalization or the de Boer parameter. Simulations in the isothermal–isobaric ensemble are performed using a Fourier path integral Monte Carlo technique. Among the more striking trends in the properties of quantum solids which emerge from this study is the strong dependence of the number density on the degree of delocalization, rather than the temperature. The large lattice expansions, under NPT conditions, associated with quantum solids, are necessarily accompanied by significant decreases in the binding energies. The kinetic energies per particle indicate that even a semiclassical Lennard-Jones solid is far from the classical equipartition regime at temperatures as high as 70% of the melting temperature. The Lindemann index, the bond orientational order parameters and the structure factors are used to monitor the degree of solidlike order. The Lindemann index increases sharply with the de Boer parameter and is accompanied by a substantial decrease in the average coordination number. This local disorder in the solid phase has a significant effect on the second-order bond-orientational order parameters, but leaves the third-order rotational invariants unchanged. The intensity of structure factors dies out very rapidly as the magnitude of the wave vector increases. Trends in lattice rigidities and the deviation of typical instantaneous configurations from the equilibrium lattice geometry as a function of quantum effects are estimated from the instantaneous normal mode spectra. It is also shown that configurational properties of the quantum path centroids, when compared with observable values based on averaging over instantaneous configurations, provide insights into the relation between local disorder and the relative magnitude of thermal and quantum fluctuations.

Comparison of inherent, instantaneous, and saddle configurations of the bulk Lennard-Jones system

The configurational energies, order parameters and normal mode spectra associated with inherent structure, inherent saddle, and instantaneous configurations of the bulk Lennard-Jones system are compared. Instantaneous structures are generated by sampling configurations from an isothermalisobaric ensemble Monte Carlo simulation. Local minimization of the potential, starting from a given instantaneous configuration is used to determine the corresponding inherent structure. The inherent saddles are obtained by local minimization on a pseudo-potential surface defined in terms of the square magnitude of the potential gradient. In the solid phase, no stationary points of order greater than zero are sampled and minimizations of both the potential, as well as of the pseudo-potential, always lead to the same global minimum energy crystalline configuration. The energies of instantaneous configurations of the solid show a clear negative correlation with the second-order bond orientational parameters. The instantaneous normal mode spectrum of the solid close to melting has a fairly prominent imaginary branch and is sufficiently smoothed out by local disorder that it qualitatively resembles the liquid phase INM spectrum. In the liquid phase, the inherent, saddle, and instantaneous structures form distinct sets of configurations. The thermal averages of the saddle energies and force constants lie between that of the instantaneous and inherent structures. The temperature dependence of the mean saddle energy and force constant is essentially parallel to that of the corresponding instantaneous quantities. The fraction of imaginary modes for the saddle configurations is approximately half that of the instantaneous configurations. The most striking similarity between the instantaneous and saddle configurations is the linear relationship between the index density and the configurational energy. The most notable difference between the two sets of configurations is the reduction to zero of the fraction of imaginary modes of the saddle configurations on freezing, making the saddle normal mode spectra qualitatively different in the liquid and solid phases.

Inelastic effects in electron tunneling through water layers

Calculations of tunneling matrix elements associated with electron transfer through molecular environments are usually done for given frozen nuclear configurations, with the underlying assumption being that nuclear motions are slow relative to the time scale of a tunneling event. This paper examines this issue for the case of electron tunneling through water. The motivation for this study is a recent calculation [Peskin et al., J. Chem. Phys. 111, 7558 (1999)] that indicates that electron tunneling through water may be enhanced by tunneling resonances in the range of ∼1 eV below the vacuum barrier, and finds that the lifetimes of such resonances are in the 10 fs range, same order as OH stretch periods. Our calculation is based on the absorbing-boundaryconditions-Green’s-function (ABCGF) method and proceeds in two steps. First we consider the effect of a single symmetric OH-stretch mode on electron tunneling in an otherwise frozen water environment, and establish that the inelastic tunneling probability is small enough to justify an approach based on perturbation theory limited to single phonon transitions. Next we note that on the short time scale of a tunneling event, even under resonance conditions, water nuclear dynamics may be represented in the instantaneous normal modes picture. We generalize the ABCGF method to take into account low order inelastic scattering from a continuum of such harmonic normal modes. We find that near resonance the total inelastic transmission probability is of the same order as the elastic one, and may lead to an additional ∼20–40% enhancement of the overall transmission in the range of up to 1 eV below the vacuum barrier. The absolute energy exchange is small, of the order of 1% of the incident electron energy. Surprisingly, we find that the main contribution to the inelastic transmission is associated with energy transfer into the rotational–librational range of the water instantaneous normal mode spectrum.

Instantaneous normal mode analysis of hydrated electron solvation dynamics

The instantaneous normal mode (INM) method is implemented in the context of mixed quantum-classical molecular dynamics (MD) simulations and applied to the analysis of the short-time solvation dynamics of the hydrated electron. Numerically suitable equations for computing the solvent dynamical matrix (Hessian) for both ground and excited adiabatic electronic states are derived using analytical derivative methods of quantum chemistry. Standard diagonalization of the Hessian leads to the sets of eigenfrequencies and eigenvectors that underlie the INM theory. Comparison of the hydrated electron and pure water INM spectra and the corresponding mode participation ratios shows that the quantum solute enhances the participation of collective low-frequency unstable modes (imaginary frequencies) at the expenses of stable ones. Distinct differential INM spectra, involving distinct solvent configurational averages, are introduced to describe the changes experienced by the solvent INMs upon the vertical excitation of the electron. The overall picture is that the INMs associated with lower frequency translational and rotational motions, as well as fast librational reorientations are markedly affected by the photoexcitation, as opposed to the localized internal vibrations of the individual water molecules. The INM solvation response for the upward transition calculated from the real modes agrees with the response obtained directly from the energy gap time correlation up to approximately 100 fs. The agreement extends over much longer times for downward transitions. The INM analysis of the solvation responses following vertical upward and downward transitions reveals that diffusive translational and librational motions are both important mechanisms for the early stages (≲50 fs) of the solvent response, with the latter dominating the first half of this time period. It is also shown that the short-time solvent relaxation involves the combined motion of molecules from the first and second hydration shells. In addition, the linearized INM solvation response calculated for D2O indicates a significant (∼36%) solvent isotope effect in the first 25 fs of the response, where the decay is Gaussian. These results are compared with previous studies of the hydrated electron solvation dynamics.

Instantaneous normal modes analysis of amorphous and supercooled silica

The dynamics of a model for amorphous and supercooled silica (SiO2), a strong glass former, is studied using instantaneous normal mode (INM) analysis. The INM spectra at a variety of temperatures are calculated via molecular dynamics simulation. At temperatures below the glass transition temperature, the dominant contribution to the soft highly anharmonic modes comprising the imaginary frequency region of the INM spectrum are found to correspond to coupled rotations of SiO4 tetrahedral units, consistent with interpretations of neutron scattering experiments [B. B. Buchenau, H. M. Zhou, and N. Nucker, Phys. Rev. Lett. 60, 1318 (1988)] and with previous normal mode analysis of simulation results at T=0 K [S. N. Taraskin and S. R. Elliot, Phys. Rev. B 56, 8623 (1997)].

Role of unstable directions in the equilibrium and aging dynamics of supercooled liquids

The connectivity of the potential energy landscape in supercooled atomic liquids is investigated through a calculation of the instantaneous normal modes spectrum and a detailed analysis of the unstable directions in configuration space. We confirm the hypothesis that the mode-coupling critical temperature is the T at which the dynamics crosses over from free to activated exploration of configuration space. We also observe changes in the local connectivity of configuration space sampled during aging, following a temperature jump from a liquid to a glassy state.

Microscopic morphology and energy surface landscape in a supercooled soft-sphere system

We carry out a systematic study on the microscopic morphology and its relationship with the signatures of the traces swept over the potential energy surface in a soft-sphere fluid system interacting via a truncated Lennard–Jones potential. The state of the model system is tuned by changing its temperature obtained in a molecular dynamics simulation, ranging from the equilibrium fluid phase to the deeply quenched supercooled states. The density and the range of interaction are chosen so that the instantaneously barely isolated centers are present. We analyze the microscopic structural origin of the features in the instantaneous-normal-mode (INM) spectrum and search for evidence of state transformation. It is found that the presence of drastic changes in the INM spectra upon entering the deep supercooled regime is accompanied by the reduction in the spatial connectivity. We develop a method to characterize the spatial configuration in an attempt to address the origin of the temperature-dependent changes in the spectra.