Discretising the Herman–Kluk propagator

The Herman–Kluk (HK) semiclassical propagator is applied to two-dimensional (2D) and four-dimensional (4D) activated reactive scattering of H2 from a Cu(100) surface. Flux analysis is used to calculate collision-energy-dependent reaction probabilities from the time dependence of the semiclassical wave function; several different variations of this analysis method are implemented. Direct comparison is made between the reaction probabilities calculated using the HK method, and numerically exact quantum results, as well as quasiclassical (QC) trajectory calculations. Although aspects of results obtained using the HK propagator are an improvement over the QC results—for example, reproduction of the quantum reaction threshold and quantum resonances—overall the HK reaction probabilities are no more accurate than the QC probabilities. Furthermore, the HK method is relatively expensive to apply, and the computational expense does not scale with dimension as favorably as might be expected. The reasons for this are discussed.

Within a differential expression of the Heisenberg operator, the forward and backward evolution can be joined together along a closed time contour. This manipulation leads to a dramatic cancellation of oscillations due to the two individual propagators in the Heisenberg operator and the resulting forward-backward propagator is more tractable to semiclassical approximations. This article gives a detailed description of the forward-backward semiclassical dynamics (FBSD) formalism. The semiclassical propagators, especially those of the initial value representations (IVRs), are briefly discussed. The derivation of the FBSD based on the Herman–Kluk propagator is reviewed. Different FBSD formulations with other semiclassical IVRs are worked out and numerical calculations show that they are also capable of describing quantum dynamics semiquantitatively and all display accuracy similar to the classical Wigner method.

The semiclassical determination of the specific density of quantum states, ρ(E;J), at energy E with fixed total angular momentum J is discussed for small molecules. Monte Carlo integration allows the accurate numerical determination of the phase space volume of systems with J>0 and arbitrary anharmonicity. The corresponding semiclassical number of states can be corrected for the effects of zero point motion in analogy to the well-known Whitten–Rabinovitch procedure. In this paper, the procedures are tested by comparison with rigid rotor harmonic oscillator models, while a comparison with recent exact quantum calculations on H+3 and HD+2 is described in the following paper. We conclude that, if the intramolecular potential is known or assumed, this numerical semiclassical procedure is a viable and simple way to get state densities of a much improved accuracy.

Synthesis of multi-qudit hybrid and d-valued quantum logic circuits by decomposition

Relativistic quantum molecular dynamics and eta production in nucleus-nucleus collisions

The solvation properties of the Zn(2+) ion in methanol solution have been investigated using a combined approach based on molecular dynamics (MD) simulations and extended X-ray absorption fine structure (EXAFS) experimental results. The quantum mechanical potential energy surface for the interaction of the Zn(2+) ion with a methanol molecule has been calculated taking into account the effect of bulk solvent by the polarizable continuum model (PCM). The effective Zn-methanol interactions have been fitted by suitable analytical potentials, and have been utilized in the MD simulation to obtain the structural properties of the solution. The reliability of the whole procedure has been assessed by comparing the theoretical structural results with the EXAFS experimental data. The structural parameters of the first solvation shells issuing from the MD simulations provide an effective complement to the EXAFS experiments.

The structural and dynamic properties of the Ba2+ cation in water have been studied by combining quantum mechanical (QM) calculations, molecular dynamics (MD) simulations, and extended X-ray absorption fine structure (EXAFS) spectroscopy. An effective Ba2+-water interaction potential, to be used in the MD simulation of a Ba2+ aqueous solution, has been developed by means of QM methods, and the validity of the whole procedure has been assessed by comparing the theoretical structural results with the EXAFS experimental data. By combining distance and angular distribution functions it was possible to unambiguously identify the geometry adopted by the water molecules surrounding the ion in the solution. The Ba2+ ion was found to preferentially form an 8-fold first shell complex with a bicapped trigonal prism (BTP) geometry. The 8-fold complex is in equilibrium with a 9-fold structure having a tricapped trigonal prism (TTP) geometry, and the hydration shell is very diffuse and flexible, being characterized by a very fast solvent exchange process on the picosecond time scale.

The frozen Gaussian approximation, proposed in [J. Lu and X. Yang, Commun. Math. Sci., 9 (2011), pp. 663–683], is an efficient computational tool for high frequency wave propagation. We continue in this paper the development of frozen Gaussian approximation. The frozen Gaussian approximation is extended to general linear strictly hyperbolic systems. Eulerian methods based on frozen Gaussian approximation are developed to overcome the divergence problem of Lagrangian methods. The proposed Eulerian methods can also be used for the Herman–Kluk propagator in quantum mechanics. Numerical examples verify the performance of the proposed methods.

SummaryWe propose a model for the analysis of non-stationary point processes with almost periodic rate of occurrence. The model deals with the arrivals of events which are unequally spaced and show a pattern of periodicity or almost periodicity, such as stock transactions and earthquakes. We model the rate of occurrence of a non-homogeneous Poisson process as the sum of sinusoidal functions plus a baseline. Consistent estimates of frequencies, phases and amplitudes which form the sinusoidal functions are constructed mainly by the Bartlett periodogram. The estimates are shown to be asymptotically normally distributed. Computational issues are discussed and it is shown that the frequency estimates must be resolved with order o(T−1) to guarantee the asymptotic unbiasedness and consistency of the estimates of phases and amplitudes, where T is the length of the observation period. The prediction of the next occurrence is carried out and the mean-squared prediction error is calculated by Monte Carlo integration. Simulation and real data examples are used to illustrate the theoretical results and the utility of the model.

To investigate the molecular details of the phosphoryl-transfer mechanism catalyzed by cAMP-dependent protein kinase, we performed quantum mechanical (QM) calculations on a cluster model of the active site and molecular dynamics (MD) simulations of a ternary complex of the protein with Mg(2)ATP and a 20-residue peptide substrate. Overall, our theoretical results confirm the participation of the conserved aspartic acid, Asp(166), as an acid/base catalyst in the reaction mechanism catalyzed by protein kinases. The MD simulation shows that the contact between the nucleophilic serine side chain and the carboxylate group of Asp(166) is short and dynamically stable, whereas the QM study indicates that an Asp(166)-assisted pathway is structurally and energetically feasible and is in agreement with previous experimental results.

It is by now well known, that at thermal equilibrium, the dynamics of multimoded optical configurations eventually settle into a Rayleigh-Jeans (RJ) distribution in the presence of weak nonlinearities. Yet, in spite of intense research, a general methodology to quantify and predict the complete statistical response of optical microcanonical settings remains elusive. Toward this end, we here develop a universal theory of fluctuation statistics for nonlinear multimode optical systems. Our results reveal a transition from narrow quasi-Lorentzian statistics in the low temperature regime, to an exponential distribution at higher temperatures. We show that this peculiar morphing of the underlying photostatistics is unique to microcanonical systems and has no analog in grand canonical configurations. A comparison between direct phase space integration and numerical simulations shows excellent agreement with our theoretical results, providing a testament to ergodicity even in nonlinear systems with very few modes. Through our formal methodology, we are able to accurately quantify the nonlinear equilibria of small sized configurations, demonstrating a strong deviation from RJ statistics and an equilibrium response that defies equipartition at infinite temperatures.

Time Evolution in an External Field: The Unitarity Paradox

The Wei–Norman technique allows one to express the solution of a system of linear non-autonomous differential equations in terms of product of exponentials with time-dependent exponents being solutions of a system of nonlinear differential equations. We show that in the unitary case, i.e. when the solution of the linear system is given by a unitary evolution operator, the nonlinear system, by an appropriate choice of ordering, can be reduced to a hierarchy of matrix Riccati equations. To this end, we consider a general linear non-autonomous dynamical system on the special linear group . The unitary case, of particular significance for quantum optimal control problems, is then obtained by restriction to anti-Hermitian generators. We also point to the connections of the obtained results with the theory of the so-called Lie systems.

Given a quantum Hamiltonian and its evolution time, the corresponding unitary evolution operator can be constructed in many different ways, corresponding to different trajectories between the desired end-points. A choice among these trajectories can then be made to obtain the best computational complexity and control over errors. As an explicit example, Grover's quantum search algorithm is described as a Hamiltonian evolution problem. It is shown that the computational complexity has a power-law dependence on error when a straightforward Lie-Trotter discretisation formula is used, and it becomes logarithmic in error when reflection operators are used. The exponential change in error control is striking, and can be used to improve many importance sampling methods. The key concept is to make the evolution steps as large as possible while obeying the constraints of the problem. In particular, we can understand why overrelaxation algorithms are superior to small step size algorithms.

This work, concentrated on the effect of nuclear equation of state on the elliptic flow (v2) in heavy ion collision by using quantum molecular dynamics model (QMD). The elliptic flow of proton in 197Au+197Au collision at incident energy 0.25A GeV and impact parameter from 0 to 0.25 and from 0.25 to 0.45 within the quantum molecular dynamics model was studied. The theoretical calculations were performed with the nuclear equation of state (soft and hard EoS). The elliptic flow of proton as a function of the mid-rapidity (y0) was computed and compared with those from experiments. The calculated results show thet the behaviour of matter at high density and high temperature are described by a soft EoS. With the theoretical result, the elliptic flow of proton as a function of (y0) is consistent with the experimental FOPI data.