Instantaneous Normal Mode Theory Research Articles

Instantaneous normal mode analysis for OKE reduced spectra of liquid and supercooled water: Contributions of low-density and high-density liquids

In this paper, we investigated the optical Kerr effect (OKE) reduced spectral density (RSD) of liquid and supercooled water at ambient pressure and focused on the contributions of low-density liquid (LDL) and high-density liquid (HDL), which local structures are characterized with tetrahedral order and disorder, respectively. The water systems were generated by performing molecular dynamics (MD) simulations with TIP4P/2005 force fields, and molecules were classified into LDL and HDL with the microscopic structural descriptor [Nature commun. 3 (2014) 3556.]. The OKE RSD was calculated with a model involving the intrinsic polarizabilities of individual molecules and the dipole-induced-dipole mechanism between them. The instantaneous normal mode (INM) analysis was used to dissect the OKE RSD obtained by MD simulations. By considering water as a mixture of LDL and HDL, the OKE RSD in the linearized INM theory is a sum of three contributions, which are resulted from the two partial liquids individually and a cross correlation associated with both of them, in a formulism similar as that for the static structure factor of a binary mixture. By exploring with two separation schemes on the collective polarizability, where they are complemental to each other, the INM analysis provides an insight into the physical causes for the bands in the OKE RSD of liquid and supercooled water and the relation of each band to the local structures of LDL and HDL.

Field-Modulating Modes of Solvents for Describing Electrostatic Intermolecular Vibrational Interactions in Solution

A theoretical method is developed for examining the intermolecular vibrational motions that are responsible for modulating the external electric field at the position of the solute in solution. The derivation is done in the context of the instantaneous normal mode theory by using an algebraic property of the interaction formula, in the same way as in the intensity-carrying mode theory of intramolecular vibrations developed in our previous studies. The vibrational motions derived by the present method are called field-modulating modes (FMMs). These modes play an essential role in electrostatic intermolecular vibrational interactions in solution. As an example of application of the present method, the case of an acetonitrile solution of the nitrate ion (NO 3 - ) is studied. Various characteristics of the FMMs, which are important for vibrational polarization of the solute, are examined in detail. The relation of the present picture to vibrational relaxation theories is also discussed.

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.

Potential energy surfaces and coordinate dependence

For coordinate systems with a nontrivial metric tensor covariant derivatives must be used to obtain properties that are coordinate independent. Applications to instantaneous normal mode theory and bifurcation points are presented as illustrations.

A combined instantaneous normal mode and time correlation function description of the optical Kerr effect and Raman spectroscopy of liquid CS2

The depolarized reduced Raman and corresponding optical Kerr effect (OKE) spectral density of ambient CS2 have been calculated by way of time correlation function (TCF) and instantaneous normal mode (INM) methods and compared with experimental OKE data. When compared in the reduced Raman spectrum form, where the INM spectrum is proportional to the squared polarizability derivative weighted density of states (DOS), the INM results agree nearly quantitatively (at all but the lowest frequencies) with the TCF results. Both are in excellent agreement with experimental measurements. The INM signal has a significant contribution from the imaginary INMs. Within our INM theory of spectroscopy the imaginary INMs contribute like the real modes, at the magnitude of their imaginary frequency. When only the real modes are allowed to contribute, and the spectrum is rescaled to account for the missing degrees of freedom, the results are much poorer, as has been observed previously. When the spectra are compared in their OKE form, the INM spectrum is found to lack the low-frequency spike which is associated with long time scale rotational diffusion, and it is not surprising that an INM theory would not capture such a feature. The results demonstrate that while the OKE and spontaneous depolarized Raman spectrum contain the same information, they clearly highlight different dynamical time scales. At higher frequencies (ω>25 cm−1) the INM OKE results are in excellent agreement with TCF and experimental results. The TCF results capture the low-frequency spike and are in agreement with experiment everywhere within the precision of the present calculations. The molecular contributions to the OKE signal are analyzed using INM methods.

Instantaneous normal mode theory of more complicated correlation functions: Third- and fifth-order optical response

A general procedure is given for calculation of time correlations in liquids, including complex multitime functions, with instantaneous normal modes (INM). The method requires equilibrium averages only and eliminates any need to invoke a fictitious harmonic well. The third and fifth order optical response functions, and the self- and collective-density correlations, are obtained and compared with the results of other methods. INM provides several advantages for evaluating the nonlinear response function required for fifth-order scattering. The time expansions of the response functions, and the role of anharmonicity, are discussed.

Instantaneous normal mode theory of diffusion and the potential energy landscape: Application to supercooled liquid CS2

The pure translation (TR) imaginary-frequency (or unstable) instantaneous normal modes (INM), which we have proposed as representative of barrier crossing and diffusion, are obtained for seven densities and eight temperatures of supercooled and near-melting liquid CS2 via computer simulation. The self-diffusion constant D, with a range of over two decades, has been determined previously for these 56 states [Li and Keyes, J. Chem. Phys. 111, 328 (1999)], allowing a comprehensive test of the relation of INM to diffusion. INM theory is reviewed and extended. At each density Arrhenius T-dependence is found for the fraction fu of unstable modes, for the product 〈ω〉ufu of the fraction times the averaged unstable frequency, and for D. The T-dependence of D is captured very accurately by fu at higher densities and by 〈ω〉ufu at lower densities. Since the T-dependence of 〈ω〉u is weak at high density, the formula D∝〈ω〉ufu provides a good representation at all densities; it is derived for the case of low-friction barrier crossing. Density-dependent activation energies determined by Arrhenius fits to 〈ω〉ufu are in excellent agreement with those found from D. Thus, activation energies may be obtained with INM, requiring far less computational effort than an accurate simulation of D in supercooled liquids. Im-ω densities of states, 〈ρuTR(ω,T)〉, are fit to the function a(T)ω exp[−(a2(T)ω/T)a3(T)]. The strong T-dependence of D, absent in Lennard-Jones (LJ) liquids, arises from the multiplicative factor a(T); its activation energy is determined by the inflection-point energy on barriers to diffusion. Values of the exponent a3(T) somewhat greater than 2.0 suggest that liquid CS2 is nonfragile in the extended Angell–Kivelson scheme for the available states. A striking contrast is revealed between CS2 and LJ; a3→2 at low-T in CS2 and at high-T in LJ. The INM interpretation is that barrier height fluctuations in CS2 are negligible at low-T but grow with increasing T, while the opposite is true for LJ.

Neighborship structure and dynamics in supercooled liquids

The neighbors of a central atom in the supercooled, unit-density Lennard-Jones liquid are sorted by “neighborship” (first neighbor, second neighbor, etc.), and an analysis of static and dynamical properties is presented. A preliminary model is that neighbors n=1–12 fall in the first shell S1, that n=13,14 are transitional neighbors, and that S2 begins at n=15. S1 is identified as the cage of the central atom, and S1 plus the central atom is considered as a possible cluster; diffusion is proposed to occur via S1→S2 transitions. The radial probability distribution functions, P(n,r), for the nth neighbor are calculated. With decreasing T the shells pull away from each other and from the transitional neighbors, and a mean-field theory of P(n,r) breaks down. It is suggested that such behavior correlates with a dynamical slowing down. Similarly, a diffusive model for the number of original S1 neighbors still in S1 at time t fails for (reduced) T⩽0.80, indicating the onset of collective slow cluster dynamics. Static and dynamic evidence points to T∼0.8 as a temperature below which the liquid becomes more complex. The need to separate fast vibrational dynamics from measures of diffusion is discussed; one atom makes a first passage S1→S2 very quickly. The two-atoms first passage time τ2 is therefore proposed as an approximate single-atom diffusive time. The rate τ2−1 is in excellent agreement with the barrier hopping rate ωh calculated from instantaneous normal mode theory.

Non-Cartesian coordinates for instantaneous normal mode theory of atomic liquids

A hybrid coordinate system of spherical polar coordinates for the mutual nearest-neighbor pairs and Cartesian coordinates for the unpaired atoms is introduced for instantaneous normal mode (INM) analysis of atomic liquids. Densities of states (dos) calculated with the hybrid coordinates in a unit-density, supercooled Lennard-Jones liquid differ from those obtained with Cartesian coordinates, primarily at imaginary frequency. A brief discussion of coordinate dependence is presented, with an analytic treatment of the frequency moments, and it is argued that the hybrid dos are more physically meaningful. INM theory strives to relate Im ω modes to diffusion and barrier crossing, but spurious nondiffusive contributions must be removed. Hybrid coordinates yield substantially fewer Im ω indicating that some nondiffusive modes are simply Cartesian artifacts. Normalized hybrid and Cartesian Re ω dos are nearly identical, as are velocity correlation functions C(t) obtained by treating the Re ω INM as a complete set of harmonic modes. These C(t) are in fair agreement with simulation, but, notably, reach an insufficiently deep negative minimum value at too short a time. A harmonic approximation using the hybrid-translational Re ω dos, in which the hybrid modes are projected onto the center-of-mass translations of the mutual neighbor pairs plus the unpaired atoms, yields much better agreement.

An instantaneous normal mode theory of condensed phase absorption: the vibrational spectrum of condensed CS 2 from boiling to freezing

An instantaneous normal mode (INM) theory of condensed phase absorption is applied to determine the infrared spectrum of condensed phase carbon disulfide (CS 2). State points with pressures from 100 kPa (near boiling) to 1.16 GPa (near freezing) are considered, and excellent agreement with time correlation function (TCF) and experimental results is demonstrated. The INM theory includes an approximate treatment of the imaginary INM absorption, and reproduces the Fourier transform of the full system dipole–dipole TCF. The INM formalism also allows for more flexibility in calculating the spectrum, and is far less computationally demanding.

Normal mode dynamics in simple liquids

A dynamical generalization of the instantaneous normal mode (INM) theory of liquid state dynamics is presented. Due to anharmonicities the eigenvalues and eigenvectors of the Hessian matrix change with time. Therefore, regular INM theory gives a description of molecular dynamics valid only for short times. Starting out from the classical equations of motion the velocity correlation function is expressed in terms of a series of propagation matrices. These are calculated by diagonalizing the Hessian matrix at configurations equidistant in time along a short piece of trajectory. Correlation functions calculated by this normal mode propagation (NMP) for a representative selection of atomic systems agree quantitatively with results from molecular-dynamics simulation.

An instantaneous normal mode theory of condensed phase absorption: The collision-induced absorption spectra of liquid CO2

An instantaneous normal mode (INM) theory of condensed phase absorption spectra is presented. The present theory approximately includes contributions to the spectra from the imaginary frequency modes. The line-shapes of the absorptions are analyzed by way a motional narrowing parameter, Γ=Δτc. It is found that the entire intramolecular vibrational spectrum is motionally narrowed, and line-widths determined in this manner are in good agreement with intramolecular spectra obtained from time correlation function results. The low frequency, intermolecular vibrational spectrum is found to be in the motionally narrowed limit only below 50 cm−1. The frequency region from 50–300 cm−1 is intermediate between the slow and fast modulation limit. The far infrared (IR), collision-induced spectrum of dense liquid CO2 is calculated using the present theory and found to be in near quantitative agreement with experimental results.

Instantaneous normal mode prediction for cation and anion diffusion in ionic melts

Instantaneous normal mode (INM) analysis was undertaken for several ionic melts: NaCl at six distinct thermodynamic states, and for a particular state of liquid LiCl, LiF, KF, KI, NaI, ZnCl2, and CuCl. In this Communication, we show that, in most cases, the ratio between the diffusion constants for cations (Dca) and anions (Dan) is predicted from the average frequency of the real (“stable” 〈ωs〉) and imaginary (“unstable” 〈ωu〉) frequency modes of the projection of the total density of states on cations and anions, respectively. The proposed relationship, Dca/Dan=(mca−1〈ωu〉ca〈ωs〉ca−2)⋅(man−1〈ωu〉an〈ωs〉an−2)−1, where mi is the mass of a particular species, is suggested by Keyes’ INM theory for diffusion [J. Chem. Phys. 101, 5081 (1994)], with the further assumption that the parameters which are related to the topology of the multidimensional potential surface are equal for cations and anions. The above equation is shown to be valid for the simple melts NaCl, LiCl, LiF, KF, KI, and NaI, but to fail for the network forming melt ZnCl2 and for CuCl, which shows fast ionic diffusion characteristics.

Far-infrared absorption spectra of water, ammonia, and chloroform calculated from instantaneous normal mode theory

Instantaneous normal mode (INM) theory was used to calculate absolute far-infrared absorption spectra of water, ammonia, and chloroform. Three procedures for weighting the INM density of states to yield absorption intensities were tested against spectra based on dipole time correlation functions generated from molecular dynamics (MD) simulations. Weighting method I, which utilizes only the rotational character of a mode in determining its contribution to absorption, performed slightly better than method II, a more exact treatment which incorporates the extent to which a mode is IR-active. Method III, which includes the contributions of induced dipoles, was successful in describing the influence of induced dipoles on the far-infrared spectra of the model liquids. The contribution to absorption of unstable modes with imaginary frequencies was found to be significant at low frequencies, and was treated by a simple approximation. Agreement between INM theory and the MD analysis was quite good for chloroform and ammonia, but less so for water. Agreement with experimental data in the range of 4–100 cm−1 was generally poor, but no worse than that for the MD calculations, primarily reflecting the simple intermolecular potentials used rather than the computational method itself.

Instantaneous normal mode theory of quantum time correlation functions: Raman spectrum of liquid CS2

An instantaneous normal mode (INM) theory is presented for quantum time correlation functions. It is argued that the INM formalism for classical correlation functions is particularly amenable to quantum correction. The intermolecular (Rayleigh) and allowed vibrational Raman spectra of liquid CS2 are calculated as an illustration. The Applequist–Quicksall polarizability model is employed, yielding the correct values for both the molecular polarizability and its derivatives with respect to the normal coordinates. Agreement with experiment is reasonable for the intermolecular Raman and for some aspects of the allowed Raman, but not for the linewidth. A brief discussion is given regarding the future developments which will be needed for an accurate INM theory of vibrational line shapes.

An instantaneous normal mode analysis of solvation: Methyl iodide in high pressure gases

An instantaneous normal mode (INM) analysis of the short-time solvation dynamics of the B-state (200 nm) Rydberg excitation of methyl iodide in high pressures of Ar (ρ*=0.08, 0.3, and 0.8) is presented. Solute–solvent interaction potentials for this system have been determined by previous absorption and resonance scattering studies. The B-state transition energy correlation function (ECF), also known as the solvation correlation function, calculated by the linear coupling INM theory is in good agreement with the ECF given by molecular dynamics simulation at short times (≤150 fs) that are well beyond the so-called inertial regime (≤100 fs). The shape and peak frequency of the solvation spectra are relatively constant over the wide range of bath densities considered here in contrast to the INM total density of states. This is attributed to the relative density independence of the first peak in the solute–solvent pair distribution function. Similarly, the ECFs are also only modestly dependent on solvent density. A cancellation of the density dependence of the solvation spectrum area and the second moment of the absorption spectrum line shape, and the nearly constant solvation spectrum shape, accounts for the relatively weak density dependence of the ECF decay. A computationally fast, semianalytical method for calculating the weighted density of states incorporating both two- and three-body correlations is shown to be in reasonable agreement with the total INM weighted density of states. A participation ratio analysis of the eigenvectors contributing to the solvation spectrum reveals that single solvent–solute interactions are responsible for the solvation response of the ρ*=0.08 and 0.3 solutions. More collective, totally symmetric solvent motions involving just a few solvent particles, in addition to single solvent interactions, contribute to the solvation response at the liquidlike density of ρ*=0.8. The effects of solvent–solvent repulsions on the shape of the solvation spectrum at this density are also evident by this INM analysis and, in part, account for the modest increase in ECF decay rate at the highest density studied here.

Normal mode theory of two step relaxation in liquids: Polarizability dynamics in CS2

An instantaneous normal mode (INM) theory is given for relaxation in liquids by a fast β process followed by a slow α process. The β process is harmonic dynamics in the wells of the N-body potential, while the α process is structural relaxation coincident with barrier crossing to a neighbor well. The theory introduces a new parameter, the ‘‘harmonic fraction’’ denoted FH, which is the fraction of the mean-square fluctuations of a dynamical variable capable of being relaxed by the harmonic β process. Theory and computer simulation are compared for the polarizability correlation function, PC(t), and the polarizability time derivative correlation function, DPC(t), in a model of CS2 including internal degrees of freedom. Agreement is good, with the INM theory clearly showing the ‘‘signature’’ time dependence of a correlation function undergoing αβ relaxation in a low temperature liquid; there are no adjustable parameters in the theory. The polarizability is calculated in the ‘‘point atomic polarizability approximation’’ (PAPA) which is sensitive to molecular vibrations, so a preliminary classical INM treatment of Raman scattering is obtained. The PAPA overestimates the derivative of the polarizability with respect to the internal coordinates, and in reality the vibrations behave quantum mechanically, so the Raman intensities are inaccurate, but otherwise a plausible description is obtained for several features of the spectrum. It is explained how an improved PAPA will be combined with a quantum INM theory in future Raman calculations.

The formulation of quantum statistical mechanics based on the Feynman path centroid density. V. Quantum instantaneous normal mode theory of liquids

The concept of instantaneous normal modes in liquids is extended into the quantum regime using the Feynman path centroid perspective in quantum statistical mechanics. To accomplish this goal, the variational quadratic approximation for the effective centroid potential is recast in a general multidimensional phase space form. In the context of the effective quadratic approximation, the velocity autocorrelation functions of liquids can then be predicted based on a set of instantaneous quantum normal modes. Representative applications are presented for quantum Lennard-Jones liquids and a quantum particle solvated in a classical fluid. The quantum effective phonon spectrum leads to some revealing observations and interpretations for these systems.