A semiclassical initial-value representation for quantum propagator and boltzmann operator

Starting from the position-momentum integral representation, we apply the correction operator method to the derivation of a uniform semiclassical approximation for the quantum propagator and then extend it to approximate the Boltzmann operator. In this approach, the involved classical dynamics is determined by the method itself instead of given beforehand. For the approximate Boltzmann operator, the corresponding classical dynamics is governed by a complex Hamiltonian, which can be described as a pair of real Hamiltonian systems. It is demonstrated that the semiclassical Boltzmann operator is exact for linear systems. A quantum propagator in the complex time is thus proposed and preliminary numerical results show that it is a reasonable approximation for calculating thermal correlation functions of general systems. © 2018 Wiley Periodicals, Inc.     

Rotational state populations and angular distributions on surface scattered molecules: No on graphite

Rotational state populations and angular distributions of NO molecules were determined after the scattering of a supersonic beam from a graphite surface at different surface temperatures. The angular distributions exhibit an isotropic and a specular part. The rotational population of the scattered molecules can be described by Boltzmann distributions with identical temperatures for both electronic ground states²Π_½ and²Π_3/2 and both scattering components. The rotational temperature agrees with the surface temperature below 170 K and converges to a constant value of 250 K for surface temperatures higher than 350 K.     

Classical theory of collisional entropy production

A classical dynamical theory of elementary collision processes is formulated in analogy to the quantum theory of the dynamical scattering matrix, which can be defined for a pure quantum stationary scattering state. The elements of this matrix are probability amplitudes for transitions between internal states defined for given values of a reaction coordinate. The squared magnitudes of these amplitudes, modeled in the proposed classical theory, define normalized internal state population distributions suitable for information theoretical analysis. Statistical entropy and surprisal are defined as dynamical functions of a reaction coordinate. This formalism differs fundamentally from concepts based on the classical Liouville equation.     

A variational principle in Wigner phase-space with applications to statistical mechanics

We consider the Dirac-Frenkel variational principle in Wigner phase-space and apply it to the Wigner-Liouville equation for both imaginary and real time dynamical problems. The variational principle allows us to deduce the optimal time-evolution of the parameter-dependent Wigner distribution. It is shown that the variational principle can be formulated alternatively as a "principle of least action." Several low-dimensional problems are considered. In imaginary time, high-temperature classical distributions are "cooled" to arrive at low-temperature quantum Wigner distributions whereas in real time, the coherent dynamics of a particle in a double well is considered. Especially appealing is the relative ease at which Feynman's path integral centroid variable can be incorporated as a variational parameter. This is done by splitting the high-temperature Boltzmann distribution into exact local centroid constrained distributions, which are thereafter cooled using the variational principle. The local distributions are sampled by Metropolis Monte Carlo by performing a random walk in the centroid variable. The combination of a Monte Carlo and a variational procedure enables the study of quantum effects in low-temperature many-body systems, via a method that can be systematically improved.     

Angular intensity distribution of a molecular oxygen beam scattered from a graphite surface

The scattering of the oxygen molecule from a graphite surface has been studied using a molecular beam scattering technique. The angular intensity distributions of scattered oxygen molecules were measured at incident energies from 291 to 614 meV with surface temperatures from 150 to 500 K. Every observed distribution has a single peak at a larger final angle than the specular angle of 45° which indicates that the normal component of the translation energy of the oxygen molecule is lost by the collision with the graphite surface. The amount of the energy loss by the collision has been roughly estimated as about 30-41% based on the assumption of the tangential momentum conservation during the collision. The distributions have also been analyzed with two theoretical models, the hard cubes model and the smooth surface model. These results indicate that the scattering is dominated by a single collision event of the particle with a flat surface having a large effective mass. The derived effective mass of the graphite surface for the incoming oxygen is 9-12 times heavier than that of a single carbon atom, suggesting a large cooperative motion of the carbon atoms in the topmost graphene layer.     

The surface temperature dependence of the inelastic scattering and dissociation of hydrogen molecules from metal surfaces

High-dimensional, wave packet calculations have been carried out to model the surface temperature dependence of rovibrationally inelastic scattering and dissociation of hydrogen molecules from the Cu(111) surface. Both the molecule and the vibrating surface are treated fully quantum-mechanically. It is found, in agreement with experimental data, that the surface temperature dependence of a variety of dynamical processes has an Arrhenius form with an activation energy dependent on molecular translational energy and on the initial and final molecular states. The activation energy increases linearly with decreasing translational energy below the threshold energy. Above threshold the behavior is more complex. A quasianalytical model is proposed that faithfully reproduces the Arrhenius law and the translational energy dependence of the activation energy. In this model, it is essential to include quantized energy transfer between the surface and the molecule. It further predicts that for any process characterized by a large energy barrier and multiphonon excitation, the linear change in activation energy up to threshold has slope-1. This explains successfully the universal nature of the unit slope found experimentally for H2 and D2 dissociation on Cu.     

Vibrational spectrum of a 1D oscillator: The quantum,the Wigner,and the classical ways

Infrared spectra have been used in many chemical applications, and theoretical calculations have been useful for analyzing these experimental results. While quantum mechanics is used for calculating the spectra for small molecules, classical mechanics is used for larger systems. However, a systematic understanding of the similarities and differences between the two approaches is not clear. Previous studies focused on peak position and relative intensities of the spectra obtained by various quantum and classical methods, but here, we included “absolute” intensities in the evaluation. The infrared spectrum of a one-dimensional (1D) harmonic oscillator (HO) and Morse oscillator were examined using four treatments: quantum, Wigner, truncated Wigner, and classical microcanonical treatments. For a 1D HO with a linear dipole moment function (DMF), the quantum and Wigner treatments give nearly the same spectra. On the other hand, the truncated Wigner underestimates the fundamental transition's intensity by half. In the case of cubic DMF, the truncated Wigner and classical methods fail to reproduce the relative intensity between the fundamental and second overtone transitions. Unfortunately, all the Wigner and classical methods fail to agree with the quantum results for a Morse oscillator with just 1% anharmonicity.     

Quantized Hamilton Dynamics

The paper describes the quantized Hamilton dynamics (QHD) approach that extends classical Hamiltonian dynamics and captures quantum effects, such as zero point energy, tunneling, decoherence, branching, and state-specific dynamics. The approximations are made by closures of the hierarchy of Heisenberg equations for quantum observables with the higher order observables decomposed into products of the lower order ones. The technique is applied to the vibrational energy exchange in a water molecule, the tunneling escape from a metastable state, the double-slit interference, the population transfer, dephasing and vibrational coherence transfer in a two-level system coupled to a phonon, and the scattering of a light particle off a surface phonon, where QHD is coupled to quantum mechanics in the Schrödinger representation. Generation of thermal ensembles in the extended space of QHD variables is discussed. QHD reduces to classical mechanics at the first order, closely resembles classical mechanics at the higher orders, and requires little computational effort, providing an efficient tool for treatment of the quantum effects in large systems.     

Rotationally inelastic molecule–surface scattering: dynamical lie algebraic method

The rotationally inelastic molecule–surface scattering is analyzed using dynamical Lie algebraic method. We treat, by example, the simple model of the scattering of NO from a rigid, flat Ag(111) surface. The explicit expressions of transition probability and the probability current density are obtained. It is proved that dynamical Lie algebraic method can be useful for describing the scattering problems. © 2000 John Wiley & Sons, Inc. Int J Quant Chem 76: 500–510, 2000     

Dissociative chemisorption of H2 on the Cu(110) surface: a quantum and quasiclassical dynamical study

Six-dimensional quantum dynamical and quasiclassical trajectory (QCT) calculations are reported for the reaction and vibrationally inelastic scattering of (v = 0,1,j = 0) H(2) scattering from Cu(110), and for the reaction and rovibrationally elastic and inelastic scattering of (v = 1,j = 1) H(2) scattering from Cu(110). The dynamics results were obtained using a potential energy surface obtained with density functional theory using the PW91 functional. The reaction probabilities computed with quantum dynamics for (v = 0,1,j = 0) were in excellent agreement with the QCT results obtained earlier for these states, thereby validating the QCT approach to sticking of hydrogen on Cu(110). The vibrational de-excitation probability P(v=1,j = 0 --> v = 0) computed with the QCT method is in remarkably good agreement with the quantum dynamical results for normal incidence energies E(n) between 0.2 and 0.6 eV. The QCT result for the vibrational excitation probability P(v = 0,j = 0 --> v = 1) is likewise accurate for E(n) between 0.8 and 1 eV, but the QCT method overestimates vibrational excitation for lower E(n). The QCT method gives probabilities for rovibrationally (in)elastic scattering, P(v = 1,j = 1 --> v('),j(')), which are in remarkably good agreement with quantum dynamical results. The rotationally averaged, initial vibrational state-selective reaction probability obtained with QCT agrees well with the initial vibrational state-selective reaction probability extracted from molecular beam experiments for v = 1, for the range of collision energies for which the v=1 contribution to the measured total sticking probability dominates. The quantum dynamical probabilities for rovibrationally elastic scattering of (v = 1,j = 1) H(2) from Cu(110) are in good agreement with experiment for E(n) between 0.08 and 0.25 eV.     

Classical and quantum studies of the photodissociation of a HX (X=Cl,F) molecule adsorbed on ice

The photodissociation dynamics of a HX (X = Cl,F) molecule adsorbed on a hexagonal ice surface at T = 0 K is studied using time-dependent quantum wave packets and quasiclassical trajectories. The relevant potential energy surfaces are calculated using high-level ab initio methods. We present here two dimensional calculations for the dynamics of the hydrogen photofragment for both HCl and HF molecules. The purpose of this paper is to compare the photodissociation dynamics of the two molecules which are adsorbed on the ice surface with different equilibrium geometries. The total photodissociation cross section and the angular distribution are calculated. The comparison with classical trajectory calculations provides evidence for typical quantum effects and reveals rainbow structures.     

Ab initio potential energy surface of CH and reaction dynamics of H + CH+

This work presents a new ground state potential energy surface (PES) for CH. The potential is tested using quasi classical trajectory (QCT) and quantum reactive scattering methods for the H + CH(+) reaction. Cross sections and rate coefficients for all reaction channels up to 300 K are calculated. The abstraction rate coefficients follow the expected slightly decreasing behaviour above 90 K, but have a positive gradient with lower temperatures. The inelastic collision and exchange reaction rate constants are increasing monotonically with temperature. The rate coefficients of the exchange reaction differ significantly between QCT and quantum reactive scattering, due to intrinsic shortcomings of the QCT final state distributions.     

Molecular dynamics simulation study of water adsorption on hydroxylated graphite surfaces

In this paper, we present results from molecular dynamic simulations devoted to the characterization of the interaction between water molecules and hydroxylated graphite surfaces considered as models for surfaces of soot emitted by aircraft. The hydroxylated graphite surfaces are modeled by anchoring several OH groups on an infinite graphite plane. The molecular dynamics simulations are based on a classical potential issued from quantum chemical calculations. They are performed at three temperatures (100, 200, and 250 K) to provide a view of the structure and dynamics of water clusters on the model soot surface. These simulations show that the water-OH sites interaction is quite weak compared to the water-water interaction. This leads to the clustering of the water molecules above the surface, and the corresponding water aggregate can only be trapped by the OH sites when the temperature is sufficiently low, or when the density of OH sites is sufficiently high.     

The He-LiH potential energy surface revisited. II. Rovibrational energy transfer on a three-dimensional surface

We use our rigid rotor He-LiH potential energy surface [B. K. Taylor and R. J. Hinde, J. Phys. Chem. 111, 973 (1999)] as a starting point to develop a three-dimensional potential surface that describes the interaction between He and a rotating and vibrating LiH molecule. We use a fully quantum treatment of the collision dynamics on the current potential surface to compute rovibrational state-to-state cross sections. We compute excitation and relaxation vibrational rate constants as a function of temperature by integrating these cross sections over a Maxwell-Boltzmann translational energy distribution and summing over Boltzmann-weighted initial rotational levels. The rate constants for vibrational excitation of LiH are very small for temperatures below 300 K. Rate constants for vibrational relaxation of excited LiH molecules, however, are several orders of magnitude larger and show very little temperature dependence, suggesting that the collisions that result in vibrational relaxation are governed by long-range attractive interactions.     

Effect of surface modes on the six-dimensional molecule-surface scattering dynamics of H2-Cu(100) and D2-Cu(111) systems

We include the phonon modes originating from the three layers of Cu(100)/Cu(111) surface atoms on the dynamics of molecular [H(2)(v,j)/D(2)(v,j)] degrees of freedom (DOFs) through a mean field approach, where the surface temperature is incorporated into the effective Hamiltonian (potential) either by considering Boltzmann probability (BP) or by including the Bose-Einstein probability (BEP) factor for the initial state distribution of the surface modes. The formulation of effective potential has been carried out by invoking the expression of transition probabilities for phonon modes known from the "stochastic" treatment of linearly forced harmonic oscillator (LFHO). We perform four-dimensional (4D?2D) as well as six-dimensional (6D) quantum dynamics on a parametrically time and temperature-dependent effective Hamiltonian to calculate elastic/inelastic scattering cross-section of the scattered molecule for the H(2)(v,j)-Cu(100) system, and dissociative chemisorption-physisorption for both H(2)(v,j)-Cu(100) and D(2)(v,j)-Cu(111) systems. Calculated sticking probabilities by either 4D?2D or 6D quantum dynamics on an effective potential constructed by using BP factor for the initial state distribution of the phonon modes could not show any surface temperature dependence. In the BEP case, (a) both 4D?2D and 6D quantum dynamics demonstrate that the phonon modes of the Cu(100) surface affect the state-to-state transition probabilities of the scattered H(2) molecule substantially, and (b) the sticking probabilities due to the collision of H(2) on Cu(100) and D(2) on Cu(111) surfaces show noticeable and substantial change, respectively, as function of surface temperature only when the quantum dynamics of all six molecular DOFs are treated in a fully correlated manner (6D).     

Quantum sum formulae for the collision-induced spectroscopies: Molecular systems as H2-H2

Quantum expressions for the second moment of collision-induced spectra are developed in the low-density limit. Previous work by Moraldi is extended to account for the rotational structure of colliding linear molecules; isotropic interaction is assumed. Computations of the lowest three moments are presented for the case of infrared absorption and Raman scattering of molecular hydrogen pairs at temperatures from 77 to 300 K. The radial distribution functions of pairs, mean energy and angular momentum, which are required for that purpose, are obtained for the case of H₂ molecules interacting with H₂, and compared with their classical counterparts. The simple classical approximations with lowest-order Wigner-Kirkwood quantum corrections serve as an accurate representation of the quantum expressions at large separations and may be sufficient for massive systems at high temperatures.     

Steric effects and quantum interference in the inelastic scattering of NO(X) + Ar

Rotationally inelastic collisions of NO(X) with Ar are investigated in unprecedented detail using state-to-state, crossed molecular beam experiments. The NO(X) molecules are selected in the Ω = 0.5, j = 0.5, f state and then oriented such that either the ‘N’ or ‘O’ end of the molecule is directed towards the incoming Ar atom. Velocity map ion imaging is then used to probe the scattered NO molecules in well-defined quantum states. We show that the fully quantum state-resolved differential steric asymmetry, which quantifies how the relative efficiency for scattering off the ‘O’ and the ‘N’ ends of the molecule varies with scattering angle, is strongly affected by quantum interference. Significant changes in both integral and differential cross sections are found depending on whether collisions occur with the N or O ends of the molecule. The results are well accounted for by rigorous quantum mechanical calculations, in contrast to both classical trajectory calculations and more simplistic models that provide, at best, an incomplete picture of the dynamics.     

Dynamical changes of hemoglobin and its surrounding water during thermal denaturation as studied by quasielastic neutron scattering and temperature modulated differential scanning calorimetry

We have investigated the dynamical behavior of both the protein hemoglobin and its surrounding water during the denaturation process using modulated temperature differential scanning calorimetry, quasielastic neutron scattering, and frequency dependent conductivity measurements. To distinguish between the scattering from the protein and its surrounding water, neutron scattering measurements were performed on both a fully hydrogenated sample as well as a sample where the water and the exchangeable hydrogen atoms on the protein surface were deuterated. The experimental data show that the unfolding and aggregation processes are substantially overlapping in temperature. The unfolding process occurs in the approximate temperature range of 315-345 K, whereas the aggregation process starts around 330-335 K and is completed at 360 K. Furthermore, the results suggest that the secondary structure of the protein unfolds at about 325 K, and that this leads to an increased number of water molecule hydrogen bonded to the protein. Thus, the unfolding of the secondary structure reduces the number of mobile (on the experimental time scale of about 50-100 ps) water molecules. In contrast, the aggregation of protein molecules seems to have a minor effect on the dynamics of its surrounding water. In the case of the protein dynamics there are competing effects from unfolding and aggregation. The unfolding process increases the flexibility of the protein, whereas the initial aggregation reduces its dynamics. The conductivity seems to be negatively affected by both reduced water mobility and an aggregation of protein molecules.