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

Vibrational energy relaxation of large-amplitude vibrations in liquids

Given the limited intermolecular spaces available in dense liquids, the large amplitudes of highly excited, low frequency vibrational modes pose an interesting dilemma for large molecules in solution. We carry out molecular dynamics calculations of the lowest frequency ("warping") mode of perylene dissolved in liquid argon, and demonstrate that vibrational excitation of this mode should cause identifiable changes in local solvation shell structure. But while the same kinds of solvent structural rearrangements can cause the non-equilibrium relaxation dynamics of highly excited diatomic rotors in liquids to differ substantially from equilibrium dynamics, our simulations also indicate that the non-equilibrium vibrational energy relaxation of large-amplitude vibrational overtones in liquids should show no such deviations from linear response. This observation seems to be a generic feature of large-moment-arm vibrational degrees of freedom and is therefore probably not specific to our choice of model system: The lowest frequency (largest amplitude) cases probably dissipate energy too quickly and the higher frequency (more slowly relaxing) cases most likely have solvent displacements too small to generate significant nonlinearities in simple nonpolar solvents. Vibrational kinetic energy relaxation, in particular, seems to be especially and surprisingly linear.

Dynamics of relaxation and fragmentation in size-selected icosahedral Ar(n)[NO(-)(v = 1)] clusters

We study the vibrational relaxation and solvation dynamics in size-selected icosahedral Ar(n)(NO(-)) at 300 K, where NO(-)(X(3)Σ(-)) is in v = 1 and n = 1-12, using a classical dynamics method and an interaction model consisting of detailed host-guest and host-host interactions. Two relaxation time scales are found: (i) the short-time (<200 ps), in which rate is nearly independent of cluster size, and (ii) the ns scale, in which a slow energy transfer process occurs between NO(-) vibration and argon modes at a rate (~10(8) s(-1)) decreasing slightly from n = 12 to 6 and rapidly from n = 5 to 1 (~10(6) s(-1)). In Ar(12)(NO(-)), less than one-quarter of the host atoms sampled evaporate, nearly 60% of evaporation occurring within 200 ps caused by rapid energy transfer from NO(-) at short time. The fraction of evaporation decreases nearly exponentially with increasing evaporation time, but ~16% of evaporation still occurs on a time scale longer than 1 ns. Evaporation from one hemisphere of Ar(12)(NO(-)) dominates the rest. Final cluster sizes commonly produced from the fragmentation of Ar(12)(NO(-)) are n = 6-11 (evaporation of 6-1 atoms) and n = 12 (no evaporation).

Comparison of two simple models for high frequency friction: exponential versus Gaussian wings

Liquid phase vibrational energy relaxation (VER) times T1 typically depend critically on the relaxing mode's high frequency friction or wing function. The wing function may, in principle, be found from the mode's normalized force autocorrelation function (faf) C(t), since it is proportional to lim formula see text C(t)dt. However, the full form of C(t) is never available. Thus, the wing function is typically estimated from a model faf CM(t) which duplicates the known part of C(t) and which (hopefully) approximates its unknown part with enough realism to yield meaningful formula see text behavior. Unfortunately, apparently realistic CM(t)'s can predict unphysical wing functions, and T1's in error by tens of orders of magnitude. Thus, a condition is needed to discriminate between CM(t)'s which yield meaningful and unphysical forms for the high frequency friction. This condition is shown to be that model faf's CM(t) yield physical wing functions if and only if these functions derive from the short time "heads" of the faf's. This test is applied to the model faf's Cga(t) triple bond exp[-1/2(t/tau)2] and Cse(t) triple bond sech(t/tau). These faf's cannot both be physical, since they yield incompatible Gaussian and exponential wing functions. The test accepts Cga(t) as physical. It, however, rejects Cse(t), since its "tail" lim formula see text Cse(t) = 2 exp(-t/tau) (because of its long range) dominates the wing function.

Vibrational relaxation of trapped molecules in solid matrices: OH(A 2Sigma+; v = 1)/Ar

The vibrational relaxation of OH(A (2)Sigma(+);v=1) embedded in solid Ar has been studied over 4-80 K. The interaction model is based on OH undergoing local motions in a cage formed by a face-centered cubic stacking where the first shell atoms surround the guest and connect it to the heat bath through 12 ten-atom chains. The motions confined to the cage are the local translation and libration-rotation of OH and internal vibrations in OH...Ar, their energies being close to or a few times the energies of nearby first shell and chain atoms. The cage dynamics are studied by solving the equations of motion for the interaction between OH and first shell atoms, while energy propagation to the bulk phase through lattice chains is treated in the Langevin dynamics. Calculated energy transfer data are used in semiclassical procedure to obtain rate constants. In the early stage of interaction, OH transfers its energy to libration-rotation intramolecularily and then to the vibrations of the first shell and chain atoms on the time scale of several picoseconds. Libration-to-rotational transitions dispense the vibrational energy in small packages comparable to the lattice frequencies for ready flow. Energy propagation from the chains to the heat bath takes place on a long time scale of 10 ns or longer. Over the solid argon temperature range, the rate constant is on the order of 10(6) s(-1) and varies weakly with temperature.

A Systematic Approach for Computing Zero-Point Energy, Quantum Partition Function, and Tunneling Effect Based on Kleinert's Variational Perturbation Theory

In this paper, we describe an automated integration-free path-integral (AIF-PI) method, based on Kleinert's variational perturbation (KP) theory, to treat internuclear quantum-statistical effects in molecular systems. We have developed an analytical method to obtain the centroid potential as a function of the variational parameter in the KP theory, which avoids numerical difficulties in path-integral Monte Carlo or molecular dynamics simulations, especially at the limit of zero-temperature. Consequently, the variational calculations using the KP theory can be efficiently carried out beyond the first order, i.e., the Giachetti-Tognetti-Feynman-Kleinert variational approach, for realistic chemical applications. By making use of the approximation of independent instantaneous normal modes (INM), the AIF-PI method can readily be applied to many-body systems. Previously, we have shown that in the INM approximation, the AIF-PI method is accurate for computing the quantum partition function of a water molecule (3 degrees of freedom) and the quantum correction factor for the collinear H(3) reaction rate (2 degrees of freedom). In this work, the accuracy and properties of the KP theory are further investigated by using the first three order perturbations on an asymmetric double-well potential, the bond vibrations of H(2), HF, and HCl represented by the Morse potential, and a proton-transfer barrier modeled by the Eckart potential. The zero-point energy, quantum partition function, and tunneling factor for these systems have been determined and are found to be in excellent agreement with the exact quantum results. Using our new analytical results at the zero-temperature limit, we show that the minimum value of the computed centroid potential in the KP theory is in excellent agreement with the ground state energy (zero-point energy) and the position of the centroid potential minimum is the expectation value of particle position in wave mechanics. The fast convergent property of the KP theory is further examined in comparison with results from the traditional Rayleigh-Ritz variational approach and Rayleigh-Schrödinger perturbation theory in wave mechanics. The present method can be used for thermodynamic and quantum dynamic calculations, including to systematically determine the exact value of zero-point energy and to study kinetic isotope effects for chemical reactions in solution and in enzymes.

An automated integration-free path-integral method based on Kleinert's variational perturbation theory

Based on Kleinert's variational perturbation (KP) theory [Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets, 3rd ed. (World Scientific, Singapore, 2004)], we present an analytic path-integral approach for computing the effective centroid potential. The approach enables the KP theory to be applied to any realistic systems beyond the first-order perturbation (i.e., the original Feynman-Kleinert [Phys. Rev. A 34, 5080 (1986)] variational method). Accurate values are obtained for several systems in which exact quantum results are known. Furthermore, the computed kinetic isotope effects for a series of proton transfer reactions, in which the potential energy surfaces are evaluated by density-functional theory, are in good accordance with experiments. We hope that our method could be used by non-path-integral experts or experimentalists as a "black box" for any given system.

On the importance of the classically forbidden region in calculations of the relaxation rate for high-frequency vibrations: a model calculation

The quantum mechanical relaxation rate for a high-frequency vibrational mode is evaluated for a one-dimensional model system having two diatomic molecules involved in a collinear collision. The thermally averaged rate is obtained as an integral over energies for the relative translation of the two molecules. These calculations show that energies several times K(B)T make the largest contributions to the rate. Several orders of magnitude of cancellation due to phase interference is found in the evaluation of the coupling matrix elements between the initial and final states, and this is one of the main factors leading to the very small value for the relaxation rate. The region near the classical turning point in the relative translational motion of the colliding molecules dominates the calculation of the contribution to the rate at each energy. Calculations using low-order expansions of the translational potential energy and the interstate coupling about this turning point provide good approximations to the exact quantum mechanical rate. This suggests a possible method for performing calculations of the rate by means of realistic simulations of liquid systems.

Comparison between the Landau–Teller and flux-flux methods for computing vibrational energy relaxation rate constants in the condensed phase

The calculation of vibrational energy relaxation (VER) rate constants in the condensed phase is usually based on the Landau-Teller formula, which puts them in terms of the Fourier transform, at the vibrational frequency, of the autocorrelation function of the force exerted on the relaxing mode by the bath modes. An alternative expression for the VER rate constant puts it in terms of the autocorrelation function of the vibrational energy flux. In this paper, we compare the predictions obtained via those two methods in the case of iodine in liquid xenon. We find that the computational cost underlying both methods is comparable and that they predict similar VER rates. However, while the calculation of the VER rate via the Landau-Teller formula is somewhat more direct, the predictions obtained via the flux-flux formula are in somewhat better agreement with the VER rates obtained from nonequilibrium molecular dynamics simulations.

Vibrational energy relaxation of azide in water

Vibrational lifetimes of the asymmetric stretch fundamental of azide anion in normal and heavy water have been measured experimentally, with results in the range of a few picoseconds. This is an interesting problem for theoretical study because of the competition between intramolecular (relaxation to the other excited vibrational states of azide) and purely intermolecular (relaxation to azide's ground vibrational state) pathways. In addition it is important to understand the origin of the solvent isotope effect. Building on the seminal work of Morita and Kato [J. Chem. Phys. 109, 5511 (1998)], the authors develop a simple model based on a two-dimensional description of the azide stretching vibrations. A novel aspect of their theory is the use of an "on-the-fly" optimized quantum mechanical/molecular mechanical approach to calculate the system-bath coupling. Their theoretical lifetimes are in good agreement with experiment for azide in both normal and heavy water. They find that the predominant relaxation pathway is intramolecular. The solvent isotope effect arises from the different librational frequencies in normal and heavy water.

Vibrational Energy Relaxation Rates of H2 and D2 in Liquid Argon via the Linearized Semiclassical Method

The vibrational energy relaxation (VER) rates for H2 and D2 in liquid argon (T=152 K, rho=1.45x1022 cm-3) are calculated using the linearized semiclassical (LSC) method (J. Phys. Chem. 2003, 107, 9059, 9070). The calculation is based on Fermi's golden rule. The VER rate constant is expressed in terms of the quantum-mechanical force-force correlation function, which is then estimated using the LSC method. A local harmonic approximation (LHA) is employed in order to compute the multidimensional Wigner integrals underlying the LSC approximation. The H2-Ar and D2-Ar interactions are described by the three-body potential of Bissonette et al. (J. Phys. Chem. A 1996, 105, 2639). The LHA-LSC-based VER rate constants for both D2 and H2 are found to be about 2-3 orders of magnitude slower than those obtained experimentally. However, their ratio agrees quantitatively with the corresponding experimental result. In contrast, the classical VER rate constants are found to be 8-9 orders of magnitude slower than those obtained experimentally, and their ratio is found to be qualitatively different from the corresponding experimental result.

Classical vs Quantum Vibrational Energy Relaxation Pathways in Solvated Polyatomic Molecules

Vibrational energy relaxation (VER) of solvated polyatomic molecules can occur via different pathways. In this paper, we address the question of whether treating VER classically or quantum-mechanically can lead to different predictions with regard to the preferred pathway. To this end, we consider the relaxation of the singly excited asymmetric stretch of a rigid, symmetrical, and linear triatomic molecule (A-B-A) in a monatomic liquid. In this case, VER can occur either directly to the ground state or indirectly via intramolecular vibrational relaxation (IVR) to the symmetric stretch. We have calculated the rates of these two different VER pathways via classical mechanics and the linearized semiclassical (LSC) method. When the mass of the terminal A atoms is significantly larger than that of the central B atom, we find that LSC points to intermolecular VER as the preferred pathway, whereas the classical treatment points to IVR. The origin of this trend reversal appears to be purely quantum-mechanical and can be traced back to the significantly weaker quantum enhancement of solvent-assisted IVR in comparison to that of intermolecular VER.

The molecular origins of nonlinear response in solute energy relaxation: The example of high-energy rotational relaxation

A key step in solution-phase chemical reactions is often the removal of excess internal energy from the product. Yet, the way one typically studies this process is to follow the relaxation of a solute that has been excited into some distribution of excited states quite different from that produced by any reaction of interest. That the effects of these different excitations can frequently be ignored is a consequence of the near universality of linear-response behavior, the idea that relaxation dynamics is determined by the solvent fluctuations (which may not be all that different for different kinds of solute excitation). Nonetheless, there are some clear examples of linear-response breakdowns seen in solute relaxation, including a recent theoretical and experimental study of rapidly rotating diatomics in liquids. In this paper we use this rotational relaxation example to carry out a theoretical exploration of the conditions that lead to linear-response failure. Some features common to all of the linear-response breakdowns studied to date, including our example, are that the initial solute preparation is far from equilibrium, that the subsequent relaxation promotes a significant rearrangement of the liquid structure, and that the nonequilibrium response is nonstationary. However, we show that none of these phenomena is enough to guarantee a nonlinear response. One also needs a sufficient separation between the solute time scale and that of the solvent geometry evolution. We illustrate these points by demonstrating precisely how our relaxation rate is tied to our liquid-structural evolution, how we can quantitatively account for the initial nonstationarity of our effective rotational friction, and how one can tune our rotational relaxation into and out of linear response.

Vibrational Energy Relaxation of Polyatomic Molecules in Liquid Solution via the Linearized Semiclassical Method

Vibrational energy relaxation (VER) of polyatomic, as opposed to diatomic, molecules can occur via different, often solvent assisted, intramolecular and/or intermolecular pathways. In this paper, we apply the linearized semiclassical (LSC) method for calculating VER rates in the prototypical case of a rigid, symmetrical and linear triatomic molecule (A-B-A) in a monatomic liquid. Starting at the first excited state of either the symmetric or asymmetric stretches, VER can occur either directly to the ground state or indirectly via intramolecular vibrational relaxation (IVR). The VER rate constants for the various pathways are calculated within the framework of the Landau-Teller formalism, where they are expressed in terms of two-time quantum-mechanical correlation functions. The latter are calculated by the LHA-LSC method, which puts them in a "Wignerized" form, and employs a local harmonic approximation (LHA) in order to compute the necessary multidimensional Wigner integrals. Results are reported for the LHL/Ar model of Deng and Stratt [J. Chem. Phys. 2002, 117, 1735], as well as for CO(2) in liquid argon and in liquid neon. The LHA-LSC method is shown to give rise to significantly faster VER and IVR rates in comparison to the classical treatment, particularly at lower temperatures. We also find that the type and extent of the quantum rate enhancement is strongly dependent on the particular VER pathway. Finally, we find that the classical and semiclassical treatments can give rise to opposite trends when it comes to the dependence of the VER rates on the solvent.

Host-assisted intramolecular vibrational relaxation at low temperatures: OH in an argon cage

The vibrational relaxation of hydroxyl radicals in the A (2)Sigma(+) (v=1) state has been studied using the semiclassical perturbation treatment at cryogenic temperatures. The radical is considered to be trapped in a closest packed cage composed of the 12 nearest argon atoms and undergoes local translation and hindered rotation around the cage center. The primary relaxation pathway is towards local translation, followed by energy transfer to rotation through hindered-to-free rotational transitions. Free-to-free rotational transitions are found to be unimportant. All pathways are accompanied by the propagation of energy to argon phonon modes. The deexcitation probability of OH(v=1) is 1.3 x 10(-7) and the rate constant is 4.7 x 10(5) s(-1) between 4 and 10 K. The negligible temperature dependence is attributed to the presence of intermolecular attraction (>>kT) in the guest-host encounter, which counteracts the T(2) dependence resulting from local translation. Calculated relaxation time scales are much shorter than those of homonuclear molecules, suggesting the importance of the hindered and free motions of OH and strong guest-host interactions.

Conformational Properties of and a Reorientation Triggered by Sugar−Water Vibrational Resonance in the Hydroxymethyl Group in Hydrated β-Glucopyranose

In this paper, we discuss the conformational properties of the hydroxymethyl group of beta-glucopyranose in aqueous solution and its reorientation mechanism. First, using the values for the hydroxymethyl torsion (O5-C5-C6-O6) angle obtained by our ab initio simulations, we reestimate the experimental ratio of the hydroxymethyl rotamer populations. The reestimated ratio is found to be in agreement with those previously reported in several computational studies, which probably partly explains the discrepancies between theoretical and experimental studies that have been discussed in the literature. Second, our time-frequency analysis on a reorientation in the hydroxymethyl group in an ab initio molecular dynamics trajectory suggests that, before the reorientation, the O6-H6 stretching mode is vibrationally coupled with a proton-accepting first-hydration-shell water molecule, whereas the C6-O6 stretching mode is vibrationally coupled with a proton-donating one. The amount of the total vibrational energy induced by these vibrational couplings is estimated to be comparable to typical values for the potential barriers between hydroxymethyl rotamers. To elucidate the vibrational couplings, we investigate the hydrogen-bonding properties around the hydroxymethyl group during the pretransition period. The implications, validity, and limitation of a possible reorientation mechanism based on these findings are also discussed.

A study of molecular vibrational relaxation mechanism in condensed phase based upon mixed quantum-classical molecular dynamics. I. A test of IBC model for the relaxation of a nonpolar solute in nonpolar solvent at high density

In order to investigate vibrational relaxation mechanism in condensed phase, a series of mixed quantum-classical molecular dynamics calculations have been executed for nonpolar solute in nonpolar solvent and polar solute in polar solvent. In the first paper (Paper I), relaxation mechanism of I2 in Ar, where Lennard-Jones force is predominant in the interaction, is investigated as a function of density and temperature, focusing our attention on the isolated binary collision (IBC) model. The model was originally established for the relaxation in gas phase. A key question, here, is "can we apply the IBC model to the relaxation in the high-density fluid?" Analyzing the trajectory of solvent molecule as well as its interaction with the solute, we found that collisions between them may be defined clearly even in the high-density fluid. Change of the survival probability of the vibrationally first excited state on collision was traced. The change caused by collisions with a particular solvent molecule was also traced together with the interaction between them. Each collision makes a contribution to the relaxation by a stepwise change in the probability. The analysis clearly shows that the relaxation is caused by collisions even in the high-density fluid. The difference between stepwise relaxation and the continuous one found for the total relaxation in the low-density fluid and in the high-density one, respectively, was clarified to come from just the difference in frequency of the collision. The stronger the intensity of the collision is, the greater the relaxation caused by the collision is. Further, the shorter the collision time is, the greater the resultant relaxation is. The discussion is followed by the succeeding paper (Paper II), where we report that molecular mechanism of the relaxation of a polar molecule in supercritical water is significantly different from that assumed in the IBC model despite that the density dependence of the relaxation rate showed a linear correlation with the local density of water around the solute, the linear correlation being apparently in good accordance with the IBC model. The puzzle will be solved in Paper II.

A study of molecular vibrational relaxation mechanism in condensed phase based upon mixed quantum-classical molecular dynamics. II. Noncollisional mechanism for the relaxation of a polar solute in supercritical water

Mixed quantum-classical molecular dynamics method has been applied to vibrational relaxation of a hydrophilic model NO in supercritical water at various densities along an isotherm above the critical temperature. The relaxation rate was determined based on Fermi's golden rule at each state point and showed an inverse S-shaped curve as a function of bulk density. The hydration number was also calculated as a function of bulk density based on the calculated radial distribution function, which showed a good correlation with the relaxation rate. Change of the survival probability of the solute vibrational state was analyzed as a function of time together with the trajectory of the solvent water and the interaction with it. We will show that the solvent molecule resides near the solute molecule for a while and the solvent contributes to the relaxation by the random-noiselike Coulombic interaction only when it stays near the solute. After the solvent leaves the solute, it shows no contribution to the relaxation. The relaxation mechanism for this system is significantly different from the collisional one found for a nonpolar solute in nonpolar solvent in Paper I. Then, the relaxation rate is determined, on average, by the hydration number or local density of the solvent. Thus, the density dependence of the relaxation rate for the polar solute in supercritical water is apparently similar to that found for the nonpolar solute in nonpolar solvent, although the molecular process is quite different from each other.

Ultrafast Dynamics of the Azobenzene−Coumarin Complex: Investigation of Cooling Dynamics Measured by an Integrated Molecular Thermometer

The energy dissipation mechanism from photoexcited azobenzene (Az) was studied by femtosecond time-resolved UV absorption spectroscopy using 7-amino-4-trifluoromethylcoumarin (ATC) as a probe. The distance between the probe molecule and Az was fixed by covalently linking them together through a rigid proline spacer. Picosecond dynamics in THF solutions were studied upon excitation into the S1 state by a 100 fs laser pulse at 480 nm. Transient absorption spectra obtained for Az-Pro-ATC combined the S1 state absorption and vibrationally excited ground-state absorption of ATC. Correction of the transient spectrum of Az-Pro-ATC for the S1 absorption provided the time-resolved absorption spectrum of the ATC hot band. Three major components were observed in the transient kinetics of Az-Pro-ATC vibrational cooling. It is proposed that in ca. 0.25 ps after the excitation, the S1 state of azobenzene decays to form an initial vibrationally excited nonthermalized ground state of Az-Pro-ATC that involves vibrational modes of both azobenzene and coumarin. This hot ground state decays in ca. 0.32 ps to the next, vibrationally equilibrated, transient state by redistributing the energy within the molecule. Subsequently, the latter state cools by transferring its energy to the closest solvent molecules in ca. 5 ps; then, the energy diffuses to the bulk solvent in 13 ps.

Vibrational Energy Relaxation Rates via the Linearized Semiclassical Approximation: Applications to Neat Diatomic Liquids and Atomic−Diatomic Liquid Mixtures

We report the results obtained from the application of our previously proposed linearized semiclassical method for computing vibrational energy relaxation (VER) rates (J. Phys. Chem. A 2003, 107, 9059, 9070) to neat liquid oxygen, neat liquid nitrogen, and liquid mixtures of oxygen and argon. Our calculations are based on a semiclassical approximation for the quantum-mechanical force-force correlation function, which puts it in terms of the Wigner transforms of the force and the product of the Boltzmann operator and the force. The calculation of the multidimensional Wigner integrals is made feasible by the introduction of a local harmonic approximation. A systematic analysis has been performed of the temperature and mole-fraction dependences of the VER rate constant, as well as the relative contributions of centrifugal and potential forces, and of different types of quantum effects. The results were found to be in very good quantitative agreement with experiment, and they suggest that this semiclassical approximation can capture the quantum enhancement, by many orders of magnitude, of the experimentally observed VER rate constants over the corresponding classical predictions.

The workings of a molecular thermometer: The vibrational excitation of carbon tetrachloride by a solvent

An intriguing energy-transfer experiment was recently carried out in methanol/carbon tetrachloride solutions. It turned out to be possible to watch vibrational energy accumulating in three of carbon tetrachloride's modes following initial excitation of O-H and C-H stretches in methanol, in effect making those CCl(4) modes "molecular thermometers" reporting on methanol's relaxation. In this paper, we use the example of a CCl(4) molecule dissolved in liquid argon to examine, on a microscopic level, just how this kind of thermal activation occurs in liquid solutions. The fact that even the lowest CCl(4) mode has a relatively high frequency compared to the intermolecular vibrational band of the solvent means that the only solute-solvent dynamics relevant to the vibrational energy transfer will be extraordinarily local, so much so that it is only the force between the instantaneously most prominent Cl and solvent atoms that will significantly contribute to the vibrational friction. We use this observation, within the context of a classical instantaneous-pair Landau-Teller calculation, to show that energy flows into CCl(4) primarily via one component of the nominally degenerate, lowest frequency, E mode and does so fast enough to make CCl(4) an excellent choice for monitoring methanol relaxation. Remarkably, within this theory, the different symmetries and appearances of the different CCl(4) modes have little bearing on how well they take up energy from their surroundings--it is only how high their vibrational frequencies are relative to the solvent intermolecular vibrational band edge that substantially favors one mode over another.