Vibrational Energy Relaxation In Liquids Research Articles

Comprehensive investigation of vibrational relaxation of non-hydrogen-bonded water molecules in liquids

The vibrational dynamics of isolated water molecules dissolved in the nonpolar organic liquids 1,2-dichloroethane (C(2)H(4)Cl(2)) and d-chloroform (CDCl(3)) have been studied using an IR pump-probe experiment with approximately 2 ps time resolution. Analyzing transient, time, and spectrally resolved data in both the OH bending and the OH stretching region, the anharmonic constants of the bending overtone (v=2) and the bend-stretch combination modes were obtained. Based on this knowledge, the relaxation pathways of single water molecules were disentangled comprehensively, proving that the vibrational energy of H(2)O molecules is relaxing following the scheme OH stretch-->OH bend overtone-->OH bend-->ground state. A lifetime of 4.8+/-0.4 ps is determined for the OH bending mode of H(2)O in 1,2-dichloroethane. For H(2)O in CDCl(3) a numerical analysis based on rate equations suggests a bending overtone lifetime of tau(020)=13+/-5 ps. The work also shows that full 2-dimensional (pump-probe) spectral resolution with access to all vibrational modes of a molecule is required for the comprehensive analysis of vibrational energy relaxation in liquids.

Multiphonon vibrational relaxation in liquids: Should it lead to an exponential-gap law?

The profound differences between solids and liquids notwithstanding, high-frequency vibrational energy relaxation in liquids seems to be well described by assuming that the excess energy is being transferred into discrete overtones of some fundamental intermolecular vibrations-precisely the way it is in crystalline solids. In a solid-state context, this kind of analysis can be used to justify the observation that relaxation rates fall off exponentially with the energy being transferred. Liquids, however, have a substantial degree of disorder, causing their relevant intermolecular spectra to have correspondingly diffuse band edges and large bandwidths. It is therefore not at all obvious what should become of this exponential-gap-law phenomenology. We show in this paper how near exponential-gap-law behavior can still be derived for vibrational energy relaxation in liquids. To do so, we take advantage of the simple dynamics that the high-frequency relaxation has when it is launched from an individual instantaneous configuration. Interestingly, the physically relevant region turns out not to be true asymptotic limit of our formalism, but for realistic liquid parameters the behavior in the physical regime differs only slightly from an exact exponential-gap law and is strikingly independent of the details of the intermolecular spectra.

Multiphonon vibrational relaxation in liquids: An exploration of the idea and of the problems it causes for molecular dynamics algorithms

The basic solid-state perspective on energy relaxation—that a solute transfers large amounts of energy to its surroundings by exciting overtones of the solid’s phonons—is sufficiently beguiling that it is tempting to try to apply it to high-frequency vibrational energy relaxation in liquids. We suggest that when the phonon concept is suitably adapted this picture does provide a surprisingly realistic and quantitative portrait of vibrational energy dispersal in solution. Within the nonlinear instantaneous-normal-mode/instantaneous-pair theory of vibrational relaxation, relaxation rates can be formally written as a sum over the contributions of successively higher overtones of fundamental solvent frequencies. However the presence of a significant width to the band of fundamental frequencies in the liquid state means that there could, in principle, be complex interferences between multiple contributing overtones, rendering the overtone picture no more than a formal construct. What we find is that such interferences do not occur. Despite the fact the band shape is log normal—with a relatively long band tail—the relaxation is invariably dominated by a single overtone. This same perspective also helps us understand one of the failings of the common velocity-Verlet molecular dynamics algorithm in predicting high-frequency energy relaxation.

High-frequency vibrational energy relaxation in liquids: The foundations of instantaneous-pair theory and some generalizations

When the relevant frequencies get high enough, vibrational energy relaxation in liquids should, in principle, be governed by instantaneous-pair theory. The basic idea is that in any significantly contributing liquid configuration there is a single critical solvent molecule and that solute relaxation rates are determined by the time evolution of that molecule’s distance from the solute. The theory posits, moreover, that dynamics can always be modeled as a simple one-dimensional, two-body, scattering process with the liquid playing a role only in determining the initial conditions for the scattering. In this article we reformulate this theory so that it can address both polyatomic solutes and molecular solvents and we show that fundamental assumptions and basic approach remain valid even with multiple solute and solvent sites and with long-ranged intermolecular forces. We further show that while the corrections are often not large, it is possible to make systematic improvements by allowing for the multidimensionality of the solute–solvent scattering. We then turn to the instantaneous-normal-mode (INM) interpretation and implementation of the theory. At the lowest level, INM analysis enables us to define the “high frequencies” relevant to the theory as being outside the INM band of the liquid’s intermolecular vibrations and to think of the liquid as generating these frequencies from the overtones of a single INM mode. This kind of analysis predicts a temperature dependence to high-frequency vibrational relaxation remarkably similar to that of solid-state multiphonon models. However, by systematically improving this INM formulation we find that we can also explore the steps a liquid has to take to handle the relaxation of frequencies within its natural band. As the frequency decreases, a liquid evidently needs to invoke more and more of its band to drive the important solvent dynamics. Nonetheless, we continue to find that none of this important dynamics ever seems to involve anything more than the solute’s first solvation shell.

Vibrational cooling of a highly excited anharmonic oscillator: Evidence for strong vibration–rotation coupling during relaxation

The model of a diatomic molecule in a monoatomic solvent bath is considered to directly elucidate the role of vibration–rotation coupling for vibrational energy relaxation in liquids. Capacities of solute–solvent interactions as well as of Coriolis' and centrifugal forces are used as a means for displaying different energy exchange channels and manifesting the strength of vibration–rotation coupling. The ensemble averaged energy exchange between different degrees of freedom during specific solute vibrational phases is investigated. As an aside, we show that in the reduced density regime 0.5 to 0.75 the scaling of the frequency dependent frictional response with solvent density is practically the same for all effective vibrational frequencies explored by the anharmonic oscillator during energy relaxation.

A theory of vibrational energy relaxation in liquids

A microscopic statistical mechanical theory of the vibrational energy relaxation of a diatomic solute in an atomic solvent is presented. The diatomic is treated as a breathing Lennard-Jones sphere. The relaxation rate is obtained from the Fourier transform of the force–force time-correlation function. The latter is expanded in powers of time (up to t4), and expressions for the expansion coefficients are derived using equilibrium statistical mechanics. These coefficients are used to determine the parameters of an analytic ansatz for this correlation function, which can be evaluated at all times (and thus can be Fourier transformed). The resulting theory for the time-correlation function is compared to numerical results from a molecular dynamics simulation. Theoretical results for the vibrational relaxation rate are compared to experiments on I2 in Xe over a wide range of densities and temperatures.

Theory of vibrational energy relaxation in liquids: Vibrational–translational–rotational energy tranfer

The concepts underlying a theoretical treatment of the vibrational energy relaxation (VER) time T1 of a solute normal mode in a molecular solvent are summarized, and results for T1, valid for VER processes mediated by vibrational–translational–rotational energy transfer, obtained from this treatment are presented. These results are based on the formula T1=βTR−1(ωl), where βTR(ω) is the translational–rotational branch of the friction kernel of the normal mode and where ωl is its liquid phase frequency. βTR(ω) is evaluated as the cosine transform of the autocorrelation function 〈ℱ̃(t)ℱ̃〉0 of the fluctuating generalized force exerted by the solvent on the solute normal mode coordinate conditional that this coordinate is fixed at its equilibrium value and that all solvent molecules are constrained to have their equilibrium geometries. The Gaussian model is utilized to evaluate 〈ℱ̃F(t)ℱ̃〉0 and molecular level expressions for ωl and for the Gaussian model parameters are presented for the infinitely dilute diatomic solution. The expressions involve site density integrals over the coordinates of a single solvent atomic site and over the coordinates of a pair of solvent atomic sites located on the same molecule. The results permit the evaluation of T1 in terms of the atomic masses and gas phase bondlengths of the solute and the solvent molecules, the solute gas phase vibrational frequency, the solute–solvent site–site interaction potentials, and specified equilibrium site–site pair correlation functions of the liquid solution.

Theory of vibrational energy relaxation in liquids: Vibrational–vibrational energy transfer

A theoretical treatment of the vibrational–vibrational (VV) contribution to the vibrational energy relaxation time T1 of a solute normal mode in a molecular solvent, which extends a previous treatment [S. A. Adelman, R. H. Stote, and R. Muralidhar, J. Chem. Phys. 99, 1320 (1993), henceforth called Paper I] of the vibrational–translational–rotational (VTR) contribution to T1, is outlined and expressions for this VV contribution, valid for the infinitely dilute diatomic solution, are presented. The treatment is based on the formula T1=β−1(ωl), where β(ω) is the friction kernel of the relaxing solute mode and where ωl is its liquid phase frequency. β(ω) is evaluated as the cosine transform of the autocorrelation function 〈ℱ̃(t)ℱĩ〉0v of the fluctuating generalized force exerted by the vibrating solvent on the solute normal mode coordinate conditional that this coordinate is fixed at its equilibrium value. 〈ℱ̃(t)ℱ̃〉0v is expressed as a superposition of the rigid solvent autocorrelation function 〈ℱ̃(t)ℱ̃〉0 and a correction which accounts for solvent vibrational motion. For diatomic solvents one has 〈ℱ̃(t)ℱ̃〉0v= 〈ℱ̃(t)ℱ̃〉0+NSMD(t) cos ωDt F(ΩD), where NS=number of solvent molecules, MD(t) is the vibrational force gradient autocorrelation function, ωD and ΩD are solvent molecule liquid phase frequencies, and F(Ω)=1/2ℏΩ−1 coth[ℏΩ/2kBT]. The Gaussian model is assumed for 〈ℱ̃(t)ℱ̃〉0 and MD(t) yielding β(ω) as a superposition of a Gaussian centered at ω=0 which mediates VTR processes and a Gaussian centered at ω=ωD which mediates VV processes. Vector integral expressions for MD(t), ωD, and ΩD are presented which are similar to the expressions for ωl and 〈ℱ̃(t)ℱ̃〉0 given in Paper I. These expressions permit the evaluation of the VV contribution to T1 from the atomic masses, bondlengths, vibrational frequencies, and site–site interaction potentials of the solute and solvent molecules and from specified rigid solvent equilibrium site–site pair correlation functions of the liquid solution.

Time correlation function approach to vibrational energy relaxation in liquids: Revised results for monatomic solvents and a comparison with the isolated binary collision model

A refined version of a molecular theory of liquid phase vibrational energy relaxation (VER) [S. A. Adelman and R. H. Stote, J. Chem. Phys. 88, 4397, 4415 (1988)] is presented and compared to the isolated binary collision (IBC) model. The theory is based on the Gaussian model for the fluctuating force autocorrelation function of the solute vibrational coordinate. Within the Gaussian model, the VER rate constant may be constructed in terms of solute–solvent site–site potential energy and equilibrium pair correlation functions. In the present refined treatment, crossfrictional contributions to the fluctuating force autocorrelation function are retained and its initial value 〈F̃2〉0 is evaluated from an exact rather than an approximate formula. Applications of the theory are made to model Lennard-Jones systems designed to simulate molecular iodine dissolved in liquid xenon at T=298 K and molecular bromine dissolved in liquid argon at T=295 K and T=1500 K. The refinements, along with an improved polynomial fitting procedure for the solute–solvent pair correlation functions, are found to yield significant changes in both the absolute VER rates and in their isothermal density dependencies. Moreover, it is found for all three solutions that the Gaussian decay time is nearly independent of density from ideal gas to the dense fluid regimes. This condition is sufficient for the emergence of an IBC-like factorization of the VER rate constant kliq(T) into a liquid phase structural contribution proportional to 〈F̃2〉0 and a dynamical contribution which is nearly density independent. The liquid structural contribution is, in general, not well-approximated by a contact collisional assumption but rather depends on a range of solute–solvent interatomic separations. For the Br2/Ar solutions, the rate constant isotherms show a superlinear deviation from the low density extrapolation kliq(T)≂ρ0kgas(T) which is qualitatively similar to that observed for a number of cryogenic and pressurized fluids. For the I2/Xe solution a qualitatively different sublinear rate isotherm is found. This ‘‘nonclassical’’ isotherm is correlated with the nonmonotonic density dependence of the magnitudes of the solute–solvent pair correlation function in the region important for the determination of 〈F̃2〉0 found for the I2/Xe system.

Theory of vibrational energy relaxation in liquids: Diatomic solutes in monatomic solvents

A numerical study of the solute, solvent, and energy flow pathway dependence of the vibrational energy relaxation (VER) time T1 of a harmonic solute vibrational mode is presented for the prototype case of diatomic solutes in monatomic solvents. This study is based on formulas for T1 developed in the preceding paper, especially the relationship T1=β−1(ωl) where β(ω) is the frequency-dependent friction kernel of the solute normal mode and where ωl is its liquid state frequency. These formulas permit evaluation of T1 and its energy flow pathway dependence from equilibrium solute–solvent pair correlation functions. Applications are made to VER of ground electronic state molecular iodine and bromine in the fluids xenon and argon and in model Lennard-Jones solvents designed to simulate ethane and carbon tetrachloride. Satisfactory agreement between the present treatment and experimental and computer simulation results for 15 thermodynamic states is found. The VER rates (∼T−11) were found to increase with increase in the degree of resonance overlap between ωl and the solvent frequency spectrum ρℱ(ω)∼β(ω). Moreover indirect energy flow pathways, i.e., those which involve solute vibration ↔ solute translation–rotation ↔ solvent energy transfer, are found to play a qualitatively essential role for many of the systems studied here. Finally a study of the temperature and density dependence of T1 for iodine in xenon in the experimentally accessible range is presented.

Theory of vibrational energy relaxation in liquids: Construction of the generalized Langevin equation for solute vibrational dynamics in monatomic solvents

Algorithms which permit the explicit, albeit approximate, construction of a physically realistic generalized Langevin equation of motion for the energy relaxation dynamics of a specified solute normal mode coordinate y in a monatomic solvent are developed. These algorithms permit the construction, from equilibrium solute–solvent pair correlation functions, of the liquid state frequency ωl of the normal mode and of the Gaussian model approximation to the autocorrelation function 〈ℱ̃(t)ℱ̃〉0 of the fluctuating generalized force exerted by the solvent on the normal mode. From these quantities one may compute, from equilibrium solute–solvent pair correlation functions, the vibrational energy relaxation time T1 of the solute normal mode and also related quantities which permit one to assess the relative importance of direct [y coordinate→solvent] and indirect [y coordinate→solute translation–rotational coordinates→solvent] energy flow pathways in solute vibrational energy relaxation. The basis of the construction of T1 is the formula T1=β−1(ωl) where β(ω)=∫∞0 β(t)cos ω dt and where β(t)=[kBT]−1 〈ℱ̃(t)ℱ̃〉0 is the friction kernel of the solute normal mode. This formula is valid if T1≫T2=vibrational phase relaxation time. The approximate formulas for T1 are worked out in detail for diatomic solutes. The approximations are tested for this diatomic solute case by comparing with molecular dynamics results.

The effect of soft cores and anisotropic forces on vibrational energy relaxation in liquids

Abstract Starting from the isolated-binary-collision hypothesis, simple explicit expressions for the vibrational energy relaxation time in liquids are obtained. We stress the role of soft repulsive forces and of anisotropic interactions. In both cases, deviations from a model developed by Delalande and Gale, assuming an attractive hard-sphere potential, are obtained and discussed. The specific case of liquid HCl and DCl is treated as an application.

Vibrational energy relaxation in liquids

The de-excitation of the vibrational population of small molecules in the liquid state is considered. Experimental techniques applicable to the measurement of relaxation times in dense phases are first described. Theoretical approaches are subsequently developed with special emphasis on the relationship between ab-initio quantum methods and binary interaction models. Finally, a selection of experimental results is analysed in the light of these theories. Special attention is given to the dependence of the relaxation time on experimental parameters such as density, temperature or the concentration of a mixture. The behaviour of the relaxation time across the liquid/solid phase transition is also treated.

Non-Markovian effects on vibrational energy relaxation in liquids

We consider the non-Markovian effects on vibrational relaxation which can occur when the bath relaxation rate is comparable to or slower than the relaxation of the vibrational degrees of freedom. We carry out quantum mechanical simulations of a two level system strongly coupled to a bath of known statistical properties (either a Poisson or a Gaussian process) and show that the relaxation is surprisingly insensitive to the nature of the bath. Small oscillations in the relaxation are found for the Poisson simulation which maybe observable in high resolution picosecond experiments. We investigate both cumulant expansion and memory function descriptions of relaxation in the non-Markovian limit and show that neither approach provides quantitatively accurate results.

Vibrational energy relaxation in liquids

A theory of vibrational relaxation of polyatomic molecules in liquids is presented which expresses the rate of decay of a vibrational state in terms of atomic force autocorrelation functions. These force correlation functions can be estimated from hydrodynamics or from an independent binary collision model. The results of calculations on small molecules dissolved in inert, cryogenic solvents are discussed, including the dependence of the rates on the details of the intramolecular potential surface.

Vibrational Relaxation in Liquids. I: 1D‐Molecular‐Dynamics Studies of Mixtures of Bromine and Argon

AbstractVibrational energy relaxation in liquids is investigated using the molecular‐dynamics method. A one‐dimensional ensemble of liquid bromine and of mixtures of bromine and argon is simulated with different compositions and temperatures. The vibrational motion of the bromine molecules is described by a harmonic oscillator, and the intermolecular interactions are given by Lennard‐Jones (12, 6) potentials with the parameters being derived from different experiments. The observed energy relaxation times of initially excited bromine molecules decrease at higher temperature, and are longer in mixtures of bromine and argon than in the pure liquid. The effective collision strength is also calculated, the results disagree with the strong collision assumption used in reaction rate theories. It is demonstrated that the main aspects of vibrational energy relaxation in liquids can be adequately studied with the aid of one‐dimensional models.