Analytical probability distribution function for a one-dimensional anharmonic oscillator in thermal equilibrium

A simple closed formula is proposed for the probability distribution function of a one-dimensional anharmonic oscillator in thermal equilibrium. It is shown that this formula yields a plausible approximation for the distribution function over wide ranges of the parameter θ = hcωelkT and the degree of anharmonicity. A more general formula with extended limits of applicability is also considered.     

Analytical results for the classical and semiclassical coordinate probability distribution function of anharmonic oscillators

The classical and Wigner—Kirkwood semiclassical distribution functions for single and double-well anharmonic oscillators with ? = p?²/2m + k₂x?² + k_2nx?²ⁿ are given in closed form. Applications in the theory of electron diffraction are discussed briefly.     

Application of the virial theorem to the calculation of anharmonic force constants for the water molecule

On the basis of the virial theorem, relationships have been found between the anharmonic force constants which make it possible to decrease the number of unknowns in the equations obtained from the general quantum-mechanical theory of the vibrations and rotations of molecules. The complete set of anharmonic force constants for the H₂O molecule have been used to calculate all the anharmonic constants X_SS' for the D₂O molecule.     

Classical probability matrix: Prediction of quantum-state distributions by a moment analysis of classical trajectories

A new method is presented for extracting approximate quantum mechanical state-to-state transition probabilities from the results of classical trajectory calculations. The method recognizes quantum discreteness by dealing with the quantum mechanical probability matrix, but all dynamical quantities are evaluated by classical mechanics. It is illustrated by application to the linear atom-diatom collision (vibrational excitation); it is capable of treating both classically allowed and classically forbidden processes.     

Quantum corrections to classical time-correlation functions: hydrogen bonding and anharmonic floppy modes

Several simple quantum correction factors for classical line shapes, connecting dipole autocorrelation functions to infrared spectra, are compared to exact quantum data in both the frequency and time domain. In addition, the performance of the centroid molecular dynamics approach to line shapes and time-correlation functions is compared to that of these a posteriori correction schemes. The focus is on a tunable model that is able to describe typical hydrogen bonding scenarios covering continuously phenomena from tunneling via low-barrier hydrogen bonds to centered hydrogen bonds with an emphasis on floppy modes and anharmonicities. For these classes of problems, the so-called "harmonic approximation" is found to perform best in most cases, being, however, outperformed by explicit centroid molecular dynamics calculations. In addition, a theoretical analysis of quantum correction factors is carried out within the framework of the fluctuation-dissipation theorem. It can be shown that the harmonic approximation not only restores the detailed balance condition like all other correction factors, but that it is the only one that also satisfies the fluctuation-dissipation theorem. Based on this analysis, it is proposed that quantum corrections of response functions in general should be based on the underlying Kubo-transformed correlation functions.     

Infrared line collisional parameters of HCl in argon, beyond the impact approximation: measurements and classical path calculations

Measurement of room temperature absorption by HCl-Ar mixtures in the 1-0 and 2-0 bands have been made for pressures between 10 and 50 atm. Fits of these spectra are made for the determination of the width, spectral shift, asymmetry, and intensity of individual lines. The broadening and shifting parameters are in satisfactory agreement with previous determinations but provide the first complete and self-consistent sets covering P(15)-R(14) and P(7)-R(8) in the 1-0 and 2-0 bands, respectively. The asymmetries of the profiles, which have been studied for the first time, are smaller than typically 10(-3) atm(-1) and cannot be determined experimentally. On the other hand, the intensities of the low j lines show a significant linear decrease with increasing Ar pressure. Calculations of all measured quantities are made with a classical path approach and an accurate vibrational-dependent HCl-Ar potential energy surface (PES). Comparisons with experimental values show that widths and shifts are well predicted, confirming the quality of the PES and of the theoretical model, and the calculations confirm that asymmetries are small. The damping factors of the intensities are analyzed by considering three contributions: The first is due to the formation of van der Waals complexes, the second results from the finite duration of collisions, and the last comes from initial correlations. Calculations indicate that the last process has negligible consequences but that the first two processes lead to effects of the same order and explain most of the observed decrease of the intensities, even if some discrepancies persist for the mid R:mmid R:=1 rotational components.     

Semi-classical eigenvalues using the classical Born-Oppenheimer adiabatic approximation

The classical Born-Oppenheimer method is applied to the two-degrees-of-freedom Hénon-Heiles potential. The resulting one-dimensional potentials are then treated semi-classically to give the eigenvalues for the regular states. No trajectories are needed and the results are in good agreement with the exact results.     

Generalized approximation to the reaction path: the formic acid dimer case

A set of mass-weighted internal coordinates was derived and applied to the double proton transfer reaction in the formic acid dimer (FAD). The coordinate set was obtained starting from the Hirschfelder "mobile" by an optimization procedure consisting of a sequence of kinematic rotations. In FAD, the optimization procedure leads to three coordinates that do change significantly along the reaction path. These coordinates span the reaction space, whereas the remaining modes are treated in a harmonic approximation. The effect that the dimer dissociative motion has on the ground and excited vibrational states dynamics was explored. In the frequency region corresponding to the symmetric OH-stretch vibration four doublets have been identified with splittings of 2.76, 0.07, 0.60, and 4.03 cm(-1).     

VSCF in internal coordinates and the calculation of anharmonic torsional mode transitions

The vibrational self-consistent field (VSCF) method assumes separability in normal modes in its usual version. However, the method fails in cases such as soft torsional modes which are better treated by angular variables. We develop VSCF equations based on the assumption of wave function separability in internal coordinates. To test the method, simple illustrative applications to small systems are provided: trans-HONO, cis-HONO, H₂S₂, and H₂O₂. The code directly uses points from ab initio calculations, and the method proves to be accurate for all types of transitions. For typical torsional transitions, the error in the computed frequency is smaller than that of VSCF in normal coordinates. The wave functions for the torsional mode are compared with the corresponding normal mode wave functions. The differences are substantial. The results are encouraging for extension of the model for large polyatomic systems. Work along these lines is in progress.     

On the application of the geometric approximation to the calculation of ring-current properties

Burrows' formula giving an upper bound for the error in the geometric approximation is applied to the calculation of the ring-current contribution to the diamagnetic anisotropies of conjugated molecules. It is also emphasized that this approximation is easily applied in practice.     

Phase-controlled one-dimensional shape evolution of InSe nanocrystals

Cubic-phase InSe nanowires were synthesized. Obtaining an appropriate high reaction temperature for formation of nanocrystals using poor solubility of Se powder in oleylamine induced the generation of unusual cubic phase of InSe to form nanowires. In comparison, hexagonal-phase nanoplates were formed by dissolving Se powder in oleylamine before increase of reaction temperature. The diameter of highly monodisperse wires was controlled by varying the amount of Se.     

Use of the probability density function in the anharmonic theory of electron scattering by molecules

A scheme for the obtaining the electron density function W(Q ^s ) in a curvilinear vibrational coordinate system and scattering equation sM(s) is described in the framework of a vibrational model.Engels Anti-Aircraft Rocket Command Institute. Translated from Zhurnal Strukturnoi Khimii, Vol. 30, No. 6, pp. 33–37, November–December, 1989.     

On the wigner-kirkwood coordinate probability distribution of an asymmetric anharmonic oscillator; an addendum

In a recent publication we calculated the Wigner—Kirkwood coordinate distribution function for a symmetric single- or double-well anharmonic oscillator. By a simple transformation, the results are generalized for an asymmetric anharmonic oscillator.     

On the classical mechanical reactive infinite order sudden approximation

The classical mechanical formulation of the reactive infinite order sudden approximation (RIOSA) is reconsidered. It is shown that the classical RIOSA conserves the total energy of the system not only in each arrangement channel, but also on the borderline separating the different arrangement channel configuration spaces. A unique interchannel transformation rule for the RIOSA angular momenta parameters is derived, which is valid for any masses of the interacting atoms and which explicitly incorporates the specificity of the RIOSA approach.     

An single program multiple data strategy for calculation of anharmonic vibrations

This paper presents a general perturbational and variational scheme to calculate solutions of the spectral problem for the vibrational molecular Hamiltonian. A parallel strategy in the ongoing development of our software P_Anhar is presented, in order to calculate the vibrational spectrum of medium sized molecules. The efficiency of this approach is checked on ethylene oxide.     

Application of the constrained fluid lambda-integration path to the calculation of high temperature Au(110) surface free energies

Recently a method termed constrained fluid lambda-integration was proposed for calculating the free energy difference between bulk solid and liquid reference states via the construction of a reversible thermodynamic integration path; coupling the two states in question. The present work shows how the application of the constrained fluid lambda-integration concept to solid/liquid slab simulation cells makes possible a generally applicable computer simulation methodology for calculating the free energy of any surface and/or surface defect structure, including surfaces requiring variations in surface atom or density number, such as the (1 x 5) Au(100) or (1 x 2) missing row Au(110) reconstructed surfaces or excess adatom/vacancy/step populated surfaces. We evaluate the methodology by calculating the free energy of various disordered high temperature Au(110) embedded atom method surfaces constrained to differing excess surface atom numbers [including those corresponding to the (1 x 2) missing row reconstructed surface] and obtained the interesting result that at 1000 K (as distinct from lower temperatures) the free energy difference between these surfaces is reduced to zero; a result which is consistent with an expected order-disorder phase transition for the Au(110) surface at such high temperatures.     

Orientational invariance of the rotational transition probability in the sudden approximation

Semiclassical collisions of an atom with a rigid-rotor molecule are examined in the sudden approximation. The rotational transition probability is shown to be variant with respect to the choice of orientation for the molecular coordinate system; this fact contradicts recently reported results of a computer analysis. The present analysis may lead to an improved interpretation of recent molecular beam measurements.     

Transition path sampling study of classical rate-promoting vibrations

It is now widely accepted that there is a class of enzymatic proton transfer reactions, which proceed through quantum tunneling. In a series of papers we have argued that some experimental features of these reactions can be explained by assuming the presence of a "rate-promoting" vibration which brings donor and acceptor closer together, thus leading to rate enhancement. There has never been a study of this effect for classical systems. We used transition path sampling to study the equivalent classical problem and found a complicated dynamical behavior that cannot be captured by transition state theory. Slow promoting vibrations lead to reactive trajectories that overshoot the saddle point, but on the other hand the short period of fast oscillations allows the reactants to stay only briefly in a low-barrier regime. There is a competition between these effects, which results to an intermediate value for the frequency of the rate-promoting vibration that is optimal for enhancing the rate.