Anharmonic Densities Research Articles

Representing molecular ground and excited vibrational eigenstates with nuclear densities obtained from semiclassical initial value representation molecular dynamics.

We present in detail and validate an effective Monte Carlo approach for the calculation of the nuclear vibrational densities via integration of molecular eigenfunctions that we have preliminary employed to calculate the densities of the ground and the excited OH stretch vibrational states in the protonated glycine molecule [Aieta et al., Nat Commun 11, 4348 (2020)]. Here, we first validate and discuss in detail the features of the method on a benchmark water molecule. Then, we apply it to calculate on-the-fly the ab initio anharmonic nuclear densities in the correspondence of the fundamental transitions of NH and CH stretches in protonated glycine. We show how we can gain both qualitative and quantitative physical insight by inspection of different one-nucleus densities and assign a character to spectroscopic absorption peaks using the expansion of vibrational states in terms of harmonic basis functions. The visualization of the nuclear vibrations in a purely quantum picture allows us to observe and quantify the effects of anharmonicity on the molecular structure, also to exploit the effect of IR excitations on specific bonds or functional groups, beyond the harmonic approximation. We also calculate the quantum probability distribution of bond lengths, angles, and dihedrals of the molecule. Notably, we observe how in the case of one type of fundamental NH stretching, the typical harmonic nodal pattern is absent in the anharmonic distribution.

Anharmonic densities of states: A general dynamics-based solution.

Density of states is a fundamental physical characteristic that lies at the foundation of statistical mechanics and theoretical constructs that derive from them (e.g., kinetic rate theories, phase diagrams, and others). Even though most real physical systems are anharmonic, the vibrational density of states is customarily treated within the harmonic approximation, or with some partial, often limited, account for anharmonicity. The reason for this is that the problem of anharmonic densities of states stubbornly resisted a general and exact, yet convenient and straightforward in applications, solution. Here we formulate such a solution within both classical and quantum mechanics. It is based on actual dynamical behavior of systems as a function of energy and as observed, or monitored, on a chosen time scale, short or long. As a consequence, the resulting anharmonic densities of states are fully dynamically informed and, in general, time-dependent. As such, they lay the ground for formulation of new statistical mechanical frameworks that incorporate time and are ergodic, by construction, with respect to actual dynamical behavior of systems.

Anharmonic Vibrational Properties from Intrinsic n-Mode State Densities

A method for calculating fully anharmonic vibrational state counts, state densities, and partition functions for molecules is presented. The method makes use of a new quantity, the intrinsic density of states, which is associated with the states that uniquely arise from a given mode, mode pairing, or higher-order mode coupling. By using only low-order intrinsic densities, the fully coupled anharmonic vibrational result can be constructed, as shown by our application of the method to methane, CH4, and cyclopropene, C3H4. Truncation of the intrinsic expansion at the coupling of pairs of modes yields greatly improved scaling over direct evaluation of the full-dimensional result and recovers a large fraction of the total anharmonicity. We also discuss the relation of the new quantities to the structure of the potential energy surface.

Anharmonic state counts and partition functions for molecules via classical phase space integrals in curvilinear coordinates

An algorithm is presented for calculating fully anharmonic vibrational state counts, state densities, and partition functions for molecules using Monte Carlo integration of classical phase space. The algorithm includes numerical evaluations of the elements of the Jacobian and is general enough to allow for sampling in arbitrary curvilinear or rectilinear coordinate systems. Invariance to the choice of coordinate system is demonstrated for vibrational state densities of methane, where we find comparable sampling efficiency when using curvilinear z-matrix and rectilinear Cartesian normal mode coordinates. In agreement with past work, we find that anharmonicity increases the vibrational state density of methane by a factor of ∼2 at its dissociation threshold. For the vinyl radical, we find a significant (∼10×) improvement in sampling efficiency when using curvilinear z-matrix coordinates relative to Cartesian normal mode coordinates. We attribute this improved efficiency, in part, to a more natural curvilinear coordinate description of the double well associated with the H2C-C-H wagging motion. The anharmonicity correction for the vinyl radical state density is ∼1.4 at its dissociation threshold. Finally, we demonstrate that with trivial parallelizations of the Monte Carlo step, tractable calculations can be made for the vinyl radical using direct ab initio potential energy surface evaluations and a composite QCISD(T)/MP2 method.

Accurate evaporation rates of pure and doped water clusters in vacuum: A statistico-dynamical approach

Unimolecular evaporation of selected pure (H(2)O)(n) and heterogeneous (H(2)O)(n-1)X(+) water clusters containing a single hydronium or ammonium impurity is investigated in the framework of phase space theory (PST) in its orbiting transition state version. Using the many-body polarizable Kozack-Jordan potential and its extensions for X(+)=H(3)O(+) and NH(4) (+), the thermal evaporation of clusters containing 21 and 50 molecules is simulated at several total energies. Numerous molecular dynamics (MD) trajectories at high internal energies provide estimates of the decay rate constant, as well as the kinetic energy and angular momentum released upon dissociation. Additional Monte Carlo simulations are carried out to determine the anharmonic densities of vibrational states, which combined with suitable forms for the rotational densities of states provide expressions for the energy-resolved differential rates. Successful comparison between the MD results and the independent predictions of PST for the distributions of kinetic energy and angular momentum released shows that the latter statistical approach is quantitative. Using MD data as a reference, the absolute evaporation rates are calculated from PST over broad energy and temperature ranges. Based on these results, the presence of an ionic impurity is generally found to decrease the rate, however the effect is much more significant in the 21-molecule clusters. Our calculations also suggest that due to backbendings in the microcanonical densities of states the variations of the evaporation rates may not be strictly increasing with energy or temperature.

Stepwise hydration and evaporation of adenosine monophosphate nucleotide anions: a multiscale theoretical study

The structure and finite-temperature properties of hydrated nucleotide anion adenosine 5'-monophosphate (AMP) have been theoretically investigated with a variety of methods. Using a polarizable version of the Amber force field and replica-exchange molecular dynamics simulations, putative lowest-energy structures have been located for the AMP(-)(H(2)O)(n) cluster anions with n = 0-20. The hydration energies obtained with the molecular mechanics potential slightly overestimate experimental measurements. However, closer values are found after reoptimizing the structures locally at more sophisticated levels, namely semi-empirical (PM6) and density-functional theory (B3LYP/6-31+G*). Upon heating the complexes, various indicators such as the heat capacity, number of hydrogen bonds or surface area provide evidence that the water cluster melts below 200 K but remains bonded to the AMP anion. The sequential loss of water molecules after sudden heating has been studied using a statistical approach in which unimolecular evaporation is described using the orbiting transition state version of phase space theory, together with anharmonic densities of vibrational states. The evaporation rates are calibrated based on the results of molecular dynamics trajectories at high internal energy. Our results indicate that between 4 and 10 water molecules are lost from AMP(-)(H(2)O)(20) after one second depending on the initial heating in the 250-350 K range, with a concomitant cooling of the remaining cluster by 75-150 K.

Anharmonic Rovibrational Numbers and Densities of States for HO2, H2CO, and H2O2

Anharmonic vibrational numbers and densities of states are calculated by quantum-corrected Monte Carlo integration of phase space volumes on ab initio potential energy surfaces for HO2, H2CO, and H2O2. Rotational dependences of the anharmonicity are also determined for H2CO. The total numbers of states W(E,J = 0) are approximated in analytical form. Using Whitten-Rabinovitch expressions as a reference, anharmonicity factors Fanh,rho(E,J = 0) for the densities of states are also given analytically. The results are used as a benchmark for comparisons with an empirical anharmonicity model (Troe, J. Chem. Phys. 1995, 190, 381). It was found that some modifications of this model are necessary for applications at high energies. The improved empirical model is intended to serve for estimates of anharmonic densities of states of larger molecular systems at high energies.

Quantum anharmonic densities of states using the Wang–Landau method

The Wang-Landau sampling method is adapted to the calculation of quantum densities of states for fully coupled anharmonic systems. The accuracy of the method is illustrated against exact counting for two molecules with separable oscillators, namely, the Zundel complex H(5)O(2) (+) and the Na(11) cluster. Application to the fully coupled naphthalene molecule (C(10)H(8)) reveals significant deviations in the finite temperature thermodynamical properties that are not captured by simple perturbation theory. There are no limitations in the size of the molecules that can be treated with this method.

Simplified models for anharmonic numbers and densities of vibrational states. Part III: Resonance states of HO 2

Anharmonic rovibrational numbers and densities of resonance states of HO 2 are determined for a number of potential energy surfaces and employing a variety of methods. It is shown that the results present smooth analytical continuations of the expressions derived for the bound states [Part II of this series, J. Troe, V.G. Ushakov, Chem. Phys. 346 (2008) 186] and that no major changes of these expressions arise at the energies where the dissociations to H + O 2 and HO + O open up. The results are also used to test the empirical local mode model of part I of this series [J. Troe, Chem. Phys. 190 (1995) 381] which allows one to approximately predict anharmonic densities of states without using explicit potential energy surfaces.

Simplified models for anharmonic numbers and densities of vibrational states. II. All the bound states of HO 2

Anharmonic rovibrational densities of states play an important role in statistical unimolecular rate theory and suitable methods for their determination are in large demand. This article compares different approaches to their calculation using the bound states of HO 2 as an example. Nearly accurate results for this system are becoming available on the basis of ab initio calculations. After empirical correction of these results, the present work predicts a total number of bound vibrational states of 315 ± 5 (for the non-rotating molecule). Approximate methods at different levels of complexity can be validated against the results obtained. In particular, the empirical method from part I [J. Troe, Chem. Phys. 190 (1995) 381] is tested. Gaining confidence in the performance of the various methods is a necessary condition for further applications to larger molecules for which ab initio calculations are out of reach.

Approximate calculation of anharmonic densities of vibrational states for very large molecules

A fast linear-scaling method for calculating the density of vibrational states for a system of coupled anharmonic oscillators, applicable to very large molecules, is presented. Instead of discrete counting of molecular eigenstates, convolution of one-dimensional densities of states is performed in order to obtain the numbers and densities of states as a function of total internal energy. An efficient numerical integration scheme over an n -simplex is used. The density of states is given as the harmonic density multiplied by a correction factor that exclusively depends on dissociation energies and the bonding topology of the molecule. While the harmonic contribution to the density of states relies on the usual normal-mode picture, local modes are employed to account for anharmonicity and intermode coupling. Mode assignment problems encountered in previous models are absent due to permutational invariance of the integrand in the numerical integration. One-dimensional densities of states for coupled systems are defined as partial derivatives. Stretch-bend coupling is accounted for through the empirical model proposed by Troe [J. Troe, Chem. Phys. 90 (1995) 381]. The proposed method reproduces very well the results obtained from discrete counting for triatomic molecules. Extension to energies beyond the first dissociation limit is possible. Calculations on very large alkanes indicate anharmonic corrections to the number and density of states in the order of 2 at the lowest dissociation limit, while explicit consideration of hindered rotations increases this value to about 4. These large corrections have far-reaching consequences for all processes that directly depend on the density of vibrational states.

Evaporation of a mixed cluster: KrXe 13

The competition between two cluster fragments (KrXe 12 and Xe 13) in the evaporative process of the non-rotating mixed Lennard–Jones cluster KrXe 13 has been investigated both from molecular dynamics simulations and Phase Space Theory (PST). The branching ratio between the two fragments is seen to be strongly altered by the thermodynamical changes, especially the preliminary core/surface isomerization of the Kr atom. Such a multistep melting behavior is well reproduced by PST when numerically exact radial potentials along the dissociative coordinate and anharmonic vibrational densities of states are considered.

The determination of anharmonic probability densities from static and dynamic disorder by neutron powder diffraction

Abstract Atomic probability densities as derived from a careful analysis of the Debye-Waller factors contain important information about the average static and dynamic disorder in crystals. Although the inclusion of anharmonic terms often requires single crystal data, in favourable cases, in particular at high temperatures, it is possible or even necessary to rely on neutron powder experiments too. This is illustrated by several examples including the anharmonic motion of the atoms in ZnS, the anionic diffusion and the static disorder in zirconia based super-ionic conductors, and the critical behaviour at the structural phase transition in PbTiO3. Anharmonic and split models are compared on the example of Zr—Mg—O—N.

Intramolecular vibrational energy redistribution in the acetylenic C–H and hydroxyl stretches of propynol

The high-resolution infrared spectra of the acetylenic C–H and O–H stretches of propynol have been measured using an electric-resonance optothermal molecular beam spectrometer (EROS). Both spectra display extensive fragmentation of the hydride-stretch oscillator strength characteristic of the intramolecular vibrational energy redistribution (IVR) process. The IVR lifetime is strongly mode-specific. The IVR lifetime of the acetylenic C–H stretch is approximately 400 ps, with a slight increase in the lifetime with increasing values of the Ka quantum number. The lifetime of the O–H stretch is 60 ps and is independent of the rotational quantum numbers. The experimental upper limit for the anharmonic state densities are 30 and 40 states/cm−1 for the acetylenic C–H and O–H stretches, respectively. These values are in good agreement with the values obtained by a direct state count (19 and 32 states/cm−1, respectively). The measured density of states increases with an approximate (2J+1)-dependence. These results indicate that all energetically accessible states are involved in the IVR dynamics. However, neither the acetylenic C–H nor the O–H stretch shows a decrease in lifetime as the total angular momentum (J) increases. This result shows that Coriolis coupling of these two hydride stretches to the near-resonant bath states is much weaker than the anharmonic coupling. For the O–H stretch, we are able to obtain the root-mean-squared (rms) matrix element for the Coriolis coupling prefactor, 0.0015(5) cm−1. The rms anharmonic coupling matrix element is 0.03 cm−1. For the low J values measured in the O–H spectrum, the Coriolis-induced IVR rate is much slower than the initial redistribution rate resulting from the stronger anharmonic interactions leading to an IVR process with two distinct time scales.

Statistical anharmonic unimolecular rate constants for the dissociation of fluxional molecules: Application to aluminum clusters

Anharmonic densities of state are determined for the Aln (n=5,6,12,13) clusters using a model analytic potential energy function. Relative anharmonic densities of state are calculated by the multiple histogram/Nosé dynamics method. Absolute densities for Al5 and Al6 are determined by Monte Carlo evaluation of the phase integral, while for Al12 and Al13 they are determined by adiabatic switching. The anharmonic densities of state are orders of magnitude larger than harmonic values based on the deepest potential energy minimum. At an energy equal to the cluster dissociation threshold, the anharmonic density is 56 and 4600 times larger than the harmonic density for Al6 and Al13, respectively. The anharmonic densities of state are used to determine anharmonic phase space theory rate constants for Al6→Al5+Al and Al13→Al12+Al dissociation. These rate constants are within a factor of 2 of the anharmonic microcanonical rate constants determined by using classical trajectories to calculate the initial decay rates for microcanonical ensembles of Al6 and Al13 clusters. The trajectories also show that the Al6 and Al13 dissociations have ergodic unimolecular dynamics. At the Aln→Aln−1+Al dissociation threshold, where only one Aln−1 conformation is energetically accessible and the harmonic model is accurate for the Aln−1 density of states, the anharmonic correction to the unimolecular rate constant is that for the Aln density of states. However, at higher energies anharmonicity for Aln−1 also becomes important and the anharmonic correction to the unimolecular rate constant becomes smaller. A modified Rice–Ramsperger–Kassel rate constant expression, with all degrees of freedom active and A and/or E0 made energy dependent, fits anharmonic microcanonical unimolecular rate constants for Al3, Al6, and Al13 dissociation. A simple Rice–Ramsperger–Kassel–Marcus model, used to analyze the experimental studies of aluminum cluster dissociation, gives accurate rate constants as a result of a fortuitous cancellation of errors.

On the estimation of anharmonic densities of states of molecules

A simulation method for the estimation of anharmonic densities of states of classical molecular models is described. The method is based on the equilibrium energy distribution established in an uncoupled dimer of the anharmonic molecule and a reference molecule whose density of states is known analytically as a function of energy. Applications to one-dimensional chain molecules and small clusters of atoms joined by Morse bonds indicate that the method is simple and reliable.

Simplified models for anharmonic numbers and densities of vibrational states. I. Application to NO2 and H3+

Numbers and densities of vibrational states of non-rotating polyatomic molecules are calculated by a simplified model which accounts for Morse anharmonicities of stretching and for generalized empirical stretch-bend couplings of bending modes. At energies above the dissociation limit, adiabatic channel maxima restrict the molecular phase space. The model is applied to the NO2(X̃ 2A1), NO2(Ã 2B2), and H3+ systems for which satisfactory agreement with the available experimental or calculational results is obtained.

Absolute classical densities of states for very anharmonic systems and applications to the evaporation of rare gas clusters

We have simulated the cluster dissociation reaction Arn→Arn−1+Ar (12≤n≤14) using molecular dynamics (MD) with well defined internal energy and total angular momentum. Reaction rates and kinetic energy release distributions are compared to the predictions of several statistical theories: Rice, Ramsperger, and Kassel (RRK), Engelking, and phase space theory (PST). We employ the Nosé prescription for constant temperature dynamics coupled with the multiple histogram method of Labastie and Whetten to obtain highly accurate vibrational densities of states for the clusters. The absolute densities are determined by the adiabatic switching method of Reinhardt. Incorporation of these accurate anharmonic vibrational densities of states into classical PST allows us to make a direct comparison with the simulation results and eliminates any parameters from the theory. Then PST predictions for the kinetics of evaporation are given for the low energy (long time scale) regime where MD simulations are prohibitively expensive. A critical evaluation of the approximate statistical theories is presented.

Is cluster evaporation statistical? A comparison of simulation results for Ar13 with exact classical phase space theory

We have simulated the dissociation reactionA n →An−1+A for small clusters (L-J argon) with well defined internal energy and total angular momentum. Reaction rates and kinetic energy release distributions are compared to the predictions of several statistical theories, including RRK, the “Engelking” model, and phase space theory (PST). We have applied classical phase space theory in an essentially exact formulation using accurate anharmonic vibrational densities of states (and no adjustable parameters). We present a critical evaluation of the different theories and sensitivity of the results to the underlying assumptions.

Statistical approaches for the treatment of anharmonic motion in crystals. II. Anharmonic thermal vibrations and effective atomic potentials in the fast ionic conductor lithium nitride (Li3N)

Results of X-ray diffraction experiments on lithium nitride (Li3N) in the temperature range between 294 and 888 K show strong anharmonic effects. The deviations from harmonicity cannot be interpreted by interstitial sites or split positions. The application of an anharmonic temperature factor which is based on the Gram-Charlier expansion leads to an excellent fit of the data, whereas an anharmonic temperature factor, based on the Edgeworth series expansion, cannot fit the measurements in a satisfactory way. The corresponding anharmonic probability densities and the effective one-particle potentials are presented. The activation energy of the ionic conduction in Li3N perpendicular to the c axis and the thermal expansion of the lattice constant are derived from the potentials. These results agree well with results obtained by other experimental techniques. Therefore it is concluded that the potentials derived from elastic scattering experiments are physically meaningful.