Quantum analogue of the Kolmogorov–Arnold–Moser transition in different quantum anharmonic oscillators

The results of numerical computations are presented for the Bohmian trajectories of the family of different one‐ and two‐dimensional anharmonic oscillators, which exhibit regular or chaotic motion in both classical and quantum domains, depending on the values of the parameters appearing in the respective Hamiltonians. Quantum signatures of the Kolmogorov–Arnold–Moser (KAM) transition from the regular to chaotic classical dynamics of these oscillators are studied using a quantum theory of motion (QTM) as developed by de Broglie and Bohm. A phase space distance function between two initially close Bohmian trajectories, the associated Kolmogorov–Sinai–Lyapunov (KSL) entropy, the phase space volume, the autocorrelation function, the associated power spectrum, and the nearest‐neighbor spacing distribution, clearly differentiate the quantum analogues of the corresponding regular and chaotic motions in the classical domain. These quantum anharmonic oscillators are known to be useful in several diverse branches of science. © 2004 Wiley Periodicals, Inc. Int J Quantum Chem, 2004     

Collision dynamics of non-integrable systems: Validity of classical scaling

The classical collision dynamics of a model atom—molecule non-integrable collision system is studied, and the energy transfer (ET) moment is examined as a function of the initial semiclassical level of the molecule. A recently derived classical scaling theory is shown to be valid in the case when the molecular motion remains regular throughout the collision, and the ET variation is then characterized by a polynomial dependence on the initial (semiclassical) quantum numbers. When chaotic motions participate, the ET no longer follows the scaling law. The utility of the scaling theory in providing the proper interpolation form for extending classical trajectory data in non-integrable collision systems is discussed.     

The search for quantum chaos: From celestial mechanics to the helium atom

The classical problem of three bodies interacting under their mutual gravitational force has long been known to exhibit a mixture of regular and chaotic dynamics. Three bodies interacting under the influence of their mutual electric force should exhibit the same dynamical behavior, because the gravitational force and the electric force both obey the same inverse-square power law. However, an atomic-scale three-body electrical system—the helium atom—is also governed by quantum mechanics. The question is how the underlying chaotic classical behavior of the three-body electrical problem manifests in a quantum system. Or, how large does an atom have to be to show classical behavior? This question is addressed by experiments performed using an ultrabright beam of photons from the Advanced Light Source to study doubly excited autoionizing states of the helium atom.     

Stripped ion–helium atom collision dynamics within a time-dependent quantum fluid density functional theory

A nonperturbative, time-dependent (TD) quantum mechanical approach is described for studying the collision dynamics between the He atom and a fully stripped ion. The method combines quantum fluid dynamics and density functional theory to solve two coupled equations: one for the trajectory of the projectile nucleus and the other for the electronic charge distribution of the target atom. The computed TD and frequency-dependent properties provide detailed features of the collision process. Inelastic and ionization cross sections are also reported. © 1998 John Wiley & Sons, Inc. Int J Quant Chem 67: 251–271, 1998     

Quantum manifestations of classical trajectories in molecular systems

Quantum chaos, understood as the effect of the underlying classical dynamics on the stationary quantum properties in classically chaotic systems, is examined in two molecular floppy systems. Realistic models of two degrees of freedom for HO₂ and HCN/HNC are considered. The structure of the classical phase space is studied using Poincaré surfaces of section and the dynamical characteristics of the corresponding wave functions analyzed also in phase space with the aid of Husimi functions. Some wave functions show strong localization along periodic orbits. © 2002 John Wiley & Sons, Inc. Int J Quantum Chem, 2001     

Quantum molecular dynamics

Electron or ion dynamics are treated using spin‐dependent quantum trajectories. These trajectories are inferred from the Dirac current, which contributes Schroedinger's current and additional spin‐dependent terms, all of which are of order c⁰ in the nonrelativistic regime of particle velocity, where c is the speed of light. The many‐body problem is treated precisely as in classical dynamics. Each electron or ion has its own equation of motion, which is the time‐dependent Dirac or the time‐dependent Schroedinger equation in the relativistic or nonrelativistic regime of particle velocity, respectively. As an example the theory is applied to the electronic structure of the helium atom, in which two electrons with opposite spin states are shown to correlate such that their quantum trajectories keep them on average on opposite sides of the nucleus. As the theory is time dependent, excited states are also generated. © 2009 Wiley Periodicals, Inc. Int J Quantum Chem, 2011     

Quantum tunneling in a periodically driven double well system: Entangled trajectory molecular dynamics method

We study a wavepacket tunneling in one‐dimensional periodically driven double‐well system using entangled trajectory molecular dynamics method. The tunneling dynamics dependents on the amplitude and frequency of the driven force are present. Both resonant and nonresonant tunneling process are enhanced by the driven force when the system is chaotic under classical dynamics. We give entangled trajectory in phase space compared to corresponding classical trajectory with same initial state to visually show quantum tunneling process. The average values of quantum tunneling probability after long time evolution have been shown in the parameter spaces, the effect of resonance and chaos on the tunneling dynamics are present. The relation between chaos and the uncertainly product is discussed in the end. © 2016 Wiley Periodicals, Inc.     

Semiclassical initial value calculations of the collinear helium atom

Semiclassical calculations using the Herman-Kluk initial value treatment are performed to determine energy eigenvalues of bound and resonance states of the collinear helium atom. Both the eZe configuration (where the classical motion is fully chaotic) and the Zee configuration (where the classical dynamics is nearly integrable) are treated. The classical motion is regularized to remove singularities that occur when the electrons collide with the nucleus. Very good agreement is obtained with quantum energies for bound and resonance states calculated by the complex rotation method.     

Computer simulation of quantum dynamics in a classical spin environment

In this paper, a formalism for studying the dynamics of quantum systems coupled to classical spin environments is reviewed. The theory is based on generalized antisymmetric brackets and naturally predicts open-path off-diagonal geometric phases in the evolution of the density matrix. It is shown that such geometric phases must also be considered in the quantum–classical Liouville equation for a classical bath with canonical phase space coordinates; this occurs whenever the adiabatics basis is complex (as in the case of a magnetic field coupled to the quantum subsystem). When the quantum subsystem is weakly coupled to the spin environment, non-adiabatic transitions can be neglected and one can construct an effective non-Markovian computer simulation scheme for open quantum system dynamics in classical spin environments. In order to tackle this case, integration algorithms based on the symmetric Trotter factorization of the classical-like spin propagator are derived. Such algorithms are applied to a model comprising a quantum two-level system coupled to a single classical spin in an external magnetic field. Starting from an excited state, the population difference and the coherences of this two-state model are simulated in time while the dynamics of the classical spin is monitored in detail. It is the author’s opinion that the numerical evidence provided in this paper is a first step toward developing the simulation of quantum dynamics in classical spin environments into an effective tool. In turn, the ability to simulate such a dynamics can have a positive impact on various fields, among which, for example, nanoscience.     

Hybrid quantum/classical molecular dynamics for a proton transfer reaction coupled to a dissipative bath

A hybrid quantum/classical molecular dynamics approach is applied to a proton transfer reaction represented by a symmetric double well system coupled to a dissipative bath. In this approach, the proton is treated quantum mechanically and all bath modes are treated classically. The transition state theory rate constant is obtained from the potential of mean force, which is generated along a collective reaction coordinate with umbrella sampling techniques. The transmission coefficient, which accounts for dynamical recrossings of the dividing surface, is calculated with a reactive flux approach combined with the molecular dynamics with quantum transitions surface hopping method. The hybrid quantum/classical results agree well with numerically exact results in the spatial-diffusion-controlled regime, which is most relevant for proton transfer in proteins. This hybrid quantum/classical approach has already been shown to be computationally practical for studying proton transfer in large biological systems. These results have important implications for future applications to hydrogen transfer reactions in solution and proteins.     

Current‐density functional study of the HeH+ molecular ion under a strong ultrashort magnetic field

The HeH⁺ molecular ion under an ultrashort magnetic field on the order of 10⁹ G is investigated through quantum fluid dynamics and a current‐density functional theory (CDFT) based approach, employing a vector exchange–correlation (XC) potential which depends on the electronic charge‐density as well as on the current‐density. The behavior of the exchange and correlation energies of the HeH⁺ ion is analyzed and compared with those obtained using an approach based on the time‐dependent density functional theory (TD‐DFT) under similar computational constraints but employing a scalar XC potential dependent only on the electronic charge‐density. The CDFT‐based approach yields exchange and correlation energies as well as TD electronic charge‐densities drastically different from those obtained using the TD‐DFT‐based approach particularly, at typical TD magnetic field strengths. This is attributed to the nonadiabatic effects induced by the vector XC potential of the CDFT in the oscillating charge‐density of the HeH⁺ ion, which are further explained in the terminology of quantum fluid dynamics. The vector XC potential of the CDFT‐based approach is observed to augment the magnetic interactions in the H₂ molecule and in the He ion, whereas it opposes the magnetic interactions in the HeH⁺ ion particularly, at the intermediate magnetic field strengths. © 2012 Wiley Periodicals, Inc.     

On the hydrogen atom beyond the Born–Oppenheimer approximation

Recently a new formulation of quantum mechanics has been suggested which is based on the concept of signed particles, that is, classical objects provided with a position, a momentum and a sign simultaneously. In this article, we comment on the plausibility of simulating atomic systems beyond the Born–Oppenheimer approximation by means of the signed particle formulation of quantum mechanics. First, to show the new perspective offered by this new formalism, we provide an example studying quantum tunnelling through a simple Gaussian barrier in terms of the signed particle formulation. Then, we perform a direct simulation of the hydrogen atom as a full quantum two‐body system, showing that the formalism can be a very promising tool for first‐principle‐only quantum chemistry.     

Dissipation of classical energy in nonlinear quantum systems

We show using two simple nonlinear quantum systems that the infinite set of quantum dynamical variables, as introduced in quantized Hamilton dynamics [O. V. Prezhdo and Y. V. Pereverzev, J. Chem. Phys. 113, 6557 (2000)], behave as a thermostat with respect to the finite number of classical variables. The coherent classical component of the evolution decays by coupling to the chaotic quantum reservoir. The classical energy, understood as the part of system energy expressible through the average values of coordinates and momenta, is transferred to the quantum energy expressible through the higher moments of coordinates and momenta and other quantum variables. At long times, the classical variables reach equilibrium, and the classical energy fluctuates around the equilibrium value. These phenomena are illustrated with the exactly solvable Jaynes-Cummings model and a nonlinear oscillator.     

Time-dependent probability of quantum tunneling in terms of the quasisemiclassical method

In view of the rapid progress in experiments of the tunneling dynamics in the time domain, we develop a quasisemiclassical method that is aimed at a study of the proton-transfer dynamics in a large system such as tropolone and its interesting derivatives, to which not only full quantum mechanics, but even a standard semiclassical theory is never easy to apply. In our very tractable method for multidimensional systems, the tunneling paths are generated in terms of the generalized classical mechanics, but the quantum phases arising from the action integral, the Maslov index, and the semicalssical amplitude factor as well in the semiclassical kernels are entirely neglected. This approach is called the quasisemiclassical method. One of the technical issues involved in the general semiclassical scheme is how to locate points from which a tunneling path emanates. Hence the studies of such tunneling points and the quasisemiclassical method should be examined collectively. We test several ways of determining the tunneling point, including those already proposed in the literature and a newly proposed one. It is shown numerically that the quasisemiclassical method with an appropriate choice of tunneling points reproduces the full quantum mechanical tunneling probability reasonably well. This case study indicates that the present conventional approach is promising to the study of large systems. The role of tunneling points in the initial process of tunneling is also discussed.     

A structurally driven spectral transform Lanczos method for mixed quantum—classical canonical simulations (MQCC): the thermodynamics of Kr10–H and the energies of Krn–H, (n:1→9)

We develop and test an approximate approach for canonical simulations of weakly bound atomic or molecular systems for which some degrees of freedom can be treated separately by quantum mechanics. The system chosen for testing is Kr₁₀–H, for which the adiabatic approximation applied to separate the hydrogen degrees of freedom works reasonably well. The hydrogen atom is bound to the Kr clusters at cold temperatures and we calculate several bound states for clusters in the n=1–9 range, in the global minimum configuration. The structural character of the mixed quantum classical simulation is substantially different than the classical simulation for Kr₁₀–H as a result of zero point energy effects. When quantum effects are included, the low temperature dynamics of Kr₁₀–H are dominated by a significant well to well hopping about an incomplete icosahedral krypton core.     

Local physical quantities for spin based on the relativistic quantum field theory in molecular systems

Local physical quantities for spin are investigated on the basis of the four‐ and two‐component relativistic quantum theory. In the quantum field theory, local physical quantities for spin such as the spin angular momentum density, spin torque density, zeta force density, and zeta potential play important roles in spin dynamics. We discuss how to calculate these local physical quantities based on the two‐component relativistic quantum theory. Some different types of relativistic numerical calculations of local physical quantities in Li atom and C₆H₆ are demonstrated and compared. Local physical quantities for each orbital are also discussed, and it is seen that a total local zeta potential is given as a result of some cancellation of large contributions from each orbital. © 2016 Wiley Periodicals, Inc.     

Semiclassical quantum unimolecular reaction rate theory revisited

We describe a semiclassical quantum unimolecular reaction rate theory derived from the corresponding classical theory developed by Davis, Gray, Rice and Zhao (DGRZ). The analysis retains the intuitively useful mechanistic distinctions between intramolecular energy transfer and reaction, with the consequence that the semiclassical quantum theory version neglects some interference effects in the reaction dynamics. In the limiting case that intramolecular energy transfer is very fast compared to the rate of reaction we show that the DGRZ representation of the rate constant can be transformed, using the Weyl correspondence between quantum operators and classical variables, to the quantum flux–flux correlation function representation of the rate constant. In the more general case that the rate of intramolecular energy transfer influences the reaction dynamics, the semiclassical representation of the Wigner function for a classical system with both quasiperiodic and chaotic motion is used to obtain the reaction rate constant. Our analysis identifies the quantum analogue of the classical bottleneck to intramolecular energy transfer with the scars of unstable periodic orbits; it leads to a flux–flux correlation function representation of the rate constant for intramolecular energy transfer.     

Quantum dynamics in simple fluids

We use quantum-correction factors to calculate approximately the quantum velocity time-correlation function (TCF) of supercritical Lennard-Jones argon from the classical TCF. We find that for this quite classical system, several different quantum-correction schemes yield essentially identical results for the real and imaginary parts of the quantum TCF, and also agree well with the recent forward-backward semiclassical dynamics (FBSD) results of Wright and Makri [J. Chem. Phys. 119, 1634 (2003)]. We also consider a more quantum-mechanical fluid of lighter atoms (neon) at a lower temperature. In this case different quantum-correction schemes give different results. FBSD calculations show that the harmonic quantum correction factor works the best for this system