Synthesizing quantum probability by a single chaotic complex‐valued trajectory

The current trajectory interpretation of quantum mechanics is based on an ensemble viewpoint that the evolution of an ensemble of Bohmian trajectories guided by the same wavefunction Ψ converges asymptotically to the quantum probability . Instead of the Bohm's ensemble‐trajectory interpretation, the present paper gives a single‐trajectory interpretation of quantum mechanics by showing that the distribution of a single chaotic complex‐valued trajectory is enough to synthesize the quantum probability. A chaotic complex‐valued trajectory manifests both space‐filling (ergodic) and ensemble features. The space‐filling feature endows a chaotic trajectory with an invariant statistical distribution, while the ensemble feature enables a complex‐valued trajectory to envelop the motion of an ensemble of real trajectories. The comparison between complex‐valued and real‐valued Bohmian trajectories shows that without the participation of its imaginary part, a single real‐valued trajectory loses the ensemble information contained in the wavefunction Ψ, and this explains the reason why we have to employ an ensemble of real‐valued Bohmian trajectories to recover the quantum probability . © 2015 Wiley Periodicals, Inc.     

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.     

Contracted Schrödinger equation in quantum phase‐space

The phase space formulation of quantum mechanics is equivalent to standard quantum mechanics where averages are calculated by way of phase space integration as in the case of classical statistical mechanics. We derive the quantum hierarchy equations, often called the contracted Schrödinger equation, in the phase space representation of quantum mechanics which involves quasi‐distributions of position and momentum. We use the Wigner distribution for the phase space function and the Moyal phase space eigenvalue formulation to derive the hierarchy. We show that the hierarchy equations in the position, momentum, and position‐momentum representations are very similar in structure. © 2017 Wiley Periodicals, Inc.     

New aspects of quantum electrodynamics on electronic structure and dynamics

Application of Alpha-oscillator theory to quantum electrodynamics (QED) solves the mystery (Feynman) of the double-slit phenomenon involved in the foundation of quantum mechanics (QM). Even if with the same initial condition given, different spots on the screen can be predicted deterministically with no introduction of hidden variables. The interference pattern is similar to, but cannot be reproduced quantitatively by, that of the QM wave function, contrary to many-years-anticipation: a new prediction, awaiting experimental test over and above the Bohr–Einstein gedanken experiment. The general proof has already been published in Ref. [3a] and the concrete numerical algorithm of the extended normal mode technique for concrete trajectory of one electron in Ref. [3b]. In this article, (1) the new “interpretation” of the QED wave function is given in section “Interpretation of Wave Function in QED”: the QED wave function used in the extended normal mode technique gives probability density distribution function of the initial values of trajectories. Moreover, (2) for the sake of demonstration of this new interpretation, the time-independent stationary state QM wave function is substituted to the QED wave function in section “Internal Self-Stress of Energetic Particles”: the QED wave function is realized by internal self-stress revealed as energy density at the initial conditions. The renewed energy density is applied to study a unified scheme for generalized chemical reactivity. This is a new kind of chemical force acting in between electrons not in between nuclei. This paves a way for more advanced time-dependent simulation of electronic structure and dynamics in chemical reaction dynamics by tracing trajectories of many electrons. © 2018 Wiley Periodicals, Inc.     

Electronic quantum trajectories in a quantum dot

This article gives a quantum‐trajectory demonstration of the observed electric, magnetic, and thermal effects on a quantum dot with circular or elliptic shape. By applying quantum trajectory method to a quantum dot, we reveal the quantum‐mechanical meanings of the classical concepts of backscattering and commensurability, which were used in the literature to explain the peak locations of the magnetoresistance curve. Under the quantum commensurability condition, electronic quantum trajectories in a circular quantum dot are shown to be stationary like a standing wave, whose presence increases the electrical resistance. A hidden quantum effect called magnetic stagnation is discovered and shown to be the main cause of the observed jumps of the magnetoresistance curve. Quantum trajectories in an elliptic quantum dot are found to be chaotic and an index of chaos called Lyapunov exponent is proposed to measure the irregularity of the various quantum trajectories. It is shown that the response of the Lyapunov exponent to the applied magnetic field captures the main features of the experimental magnetoresistance curve. © 2014 Wiley Periodicals, Inc.     

Quantum bifurcation in a coulombic‐like potential

Because of the lack of nonlinear dynamics, up to now no bifurcation phenomenon in its original sense has been discovered directly in quantum mechanical systems. Based on the formalism of complex‐valued quantum mechanics, this article derives the nonlinear Hamilton equations from the Schrödinger equation to provide the necessary mathematic framework for the analysis of quantum bifurcation. This new approach makes it possible to identify quantum bifurcation by the direct evidence of the sudden change of fixed points and their surrounding trajectories. As a practical application of the proposed approach, we consider the quantum motion in a Coulombic‐like potential modeled by V(r) = A/r² ? B/r, where the first term describes the centrifugal trend and the second deals with the Coulombic attraction. As the bifurcation parameter evolves, we demonstrate how local and global bifurcations in quantum dynamics can be identified by inspecting the changes of fixed points and their surrounding trajectories. © 2011 Wiley Periodicals, Inc. Int J Quantum Chem, 2011     

Quantum tunneling effect in entanglement dynamics

Quantum tunneling effect in entanglement dynamics between two coupled particles with separable Gaussian initial state is investigated using entangled trajectory molecular dynamics method in terms of the reduced‐density linear entropy. It has been presented through showing distinguish contribution of single trajectory to linear entropy between classical trajectory and entangled trajectory with same initial state. We find that quantum tunneling effect makes single trajectory's contribution remarkably decrease under quantum dynamics compared to classical dynamics. The nonlocality of quantum entanglement is presented, and the energy transfer between two coupled particles through quantum correlations and classical ones is also discussed in the end. © 2015 Wiley Periodicals, Inc.     

Quantum dynamics of inelastic scattering with a moving grid

The time‐dependent discrete variable representation (TDDVR) of a wave function with grid points defined by the Hermite part of the Gauss–Hermite (G‐H) basis set introduces quantum corrections to classical mechanics. The grid points in this method follow classical trajectory and the approach converges to the exact quantum formulation with sufficient trajectories (TDDVR points) but just with a single grid point; only classical mechanics performs the dynamics. This newly formulated approach (developed for handling time‐dependent molecular quantum dynamics) has been explored to calculate vibrational transitions in the inelastic scattering processes. Traditional quantum mechanical results exhibit an excellent agreement with TDDVR profiles during the entire propagation when enough grid points are included in the quantum‐classical dynamics. © 2005 Wiley Periodicals, Inc. Int J Quantum Chem, 2005     

Calculating populations of subcellular compartments using density matrix formalism

In our earlier paper (Ref. 1 ) we developed an integrated probabilistic system for predicting the subcellular localization of proteins and estimating the relative populations of the various compartments in yeast. To justify our formulas we show here that there is a one‐to‐one correspondence between our previous calculations and the prediction of a state of a many‐particle quantum system. The equivalence between these two types of predictions can be easily established if one maps the probability of finding a particular protein in a certain subcellular compartment to the probability of measuring the corresponding quantum particle in one of the possible quantum states (the number of proteins being equal to the number of particles in the system, and the number of compartments being equal to the number of achievable quantum states). Once the sought correspondence is established, we can utilize a well‐known formula from quantum statistical mechanics to calculate the overall occupation of a particular quantum state, associating the state with the corresponding subcellular compartment. In the present work we present the details of how we arrived at the formula for the compartment population, borrowing the tools from quantum statistical mechanics. © 2001 John Wiley & Sons, Inc. Int J Quantum Chem, 2001     

Electron trajectories in molecular orbitals

The time-dependent Schrödinger equation can be rewritten so that its interpretation is no longer probabilistic. Two well-known and related reformulations are Bohmian mechanics and quantum hydrodynamics. In these formulations, quantum particles follow real, deterministic trajectories influenced by a quantum force. Generally, trajectory methods are not applied to electronic structure calculations as they predict that the electrons in a ground-state, real, molecular wavefunction are motionless. However, a spin-dependent momentum can be recovered from the nonrelativistic limit of the Dirac equation. Therefore, we developed new, spin-dependent equations of motion for the quantum hydrodynamics of electrons in molecular orbitals. The equations are based on a Lagrange multiplier, which constrains each electron to an isosurface of its molecular orbital, as required by the spin-dependent momentum. Both the momentum and the Lagrange multiplier provide a unique perspective on the properties of electrons in molecules.     

Force‐field parameters of the Ψ and Φ around glycosidic bonds to oxygen and sulfur atoms

The Ψ and Φ torsion angles around glycosidic bonds in a glycoside chain are the most important determinants of the conformation of a glycoside chain. We determined force‐field parameters for Ψ and Φ torsion angles around a glycosidic bond bridged by a sulfur atom, as well as a bond bridged by an oxygen atom as a preparation for the next study, i.e., molecular dynamics free energy calculations for protein‐sugar and protein‐inhibitor complexes. First, we extracted the Ψ or Φ torsion energy component from a quantum mechanics (QM) total energy by subtracting all the molecular mechanics (MM) force‐field components except for the Ψ or Φ torsion angle. The Ψ and Φ energy components extracted (hereafter called “the remaining energy components”) were calculated for simple sugar models and plotted as functions of the Ψ and Φ angles. The remaining energy component curves of Ψ and Φ were well represented by the torsion force‐field functions consisting of four and three cosine functions, respectively. To confirm the reliability of the force‐field parameters and to confirm its compatibility with other force‐fields, we calculated adiabatic potential curves as functions of Ψ and Φ for the model glycosides by adopting the Ψ and Φ force‐field parameters obtained and by energetically optimizing other degrees of freedom. The MM potential energy curves obtained for Ψ and Φ well represented the QM adiabatic curves and also these curves' differences with regard to the glycosidic oxygen and sulfur atoms. Our Ψ and Φ force‐fields of glycosidic oxygen gave MM potential energy curves that more closely represented the respective QM curves than did those of the recently developed GLYCAM force‐field. © 2009 Wiley Periodicals, Inc., J Comput Chem, 2009     

Quantum interference effects in the Br2 electronic predissociation: Dependence upon the molecular properties

The occurrence of quantum interference in the Br₂ electronic predissociation has been observed for the first time. It is also shown that the quantum interference effects are strongly dependent upon the representation of the molecular properties, such as the kinetic coupling and the transition dipole moment. The fast Fourier transform method implemented within the wave packet formalism was used to simulate the time evolution of the molecular system under the influence of single monochromatic ultrashort laser pulses. The quantum interference has been inferred from the oscillatory behavior of the autocorrelation functions, which have also been used to calculate the cross-sections, the branching ratios, and the selectivity. © 2003 Wiley Periodicals, Inc. Int J Quantum Chem, 2004     

Electronic quantum motions in hydrogen molecule ion

The hydrogen molecule ion is a two‐center force system expressed under the prolate spheroidal coordinates, whose quantum motions and quantum trajectories have never been addressed in the literature before. The momentum operators in this coordinate system are derived for the first time from the Hamilton equations of motion and used to construct the Hamiltonian operator. The resulting Hamiltonian comprises a kinetic energy T and a total potential V_Total consisting of the Coulomb potential and a quantum potential. It is shown that the participation of the quantum potential and the accompanied quantum forces in the force interaction within H₂⁺ is essential to develop an electronic motion consistent with the prediction of the probability density function |Ψ|². The motion of the electron in H₂⁺ can be either described by the Hamilton equations derived from the Hamiltonian H = T_K + V_Total or by the Lagrange equations derived from the Lagrangian H = T_K ? V_Total. Solving the equations of motion with different initial positions, we show that the solutions yield an assembly of electronic quantum trajectories whose distribution and concentration reconstruct the σ and π molecular orbitals in H₂⁺. © 2010 Wiley Periodicals, Inc. Int J Quantum Chem, 2011     

Linear-scaling computation of excited states in time-domain

The applicability of quantum mechanical methods is severely limited by their poor scaling.To circumvent the problem,linearscaling methods for quantum mechanical calculations had been developed.The physical basis of linear-scaling methods is the locality in quantum mechanics where the properties or observables of a system are weakly influenced by factors spatially far apart.Besides the substantial efforts spent on devising linear-scaling methods for ground state,there is also a growing interest in the development of linear-scaling methods for excited states.This review gives an overview of linear-scaling approaches for excited states solved in real time-domain.     

Quantum mechanics of solvated complexes: A test for positronium

A hybrid quantum mechanics/molecular mechanics approach is developed for calculating properties of solvated complexes. The method is based on self‐consistent calculations of the Kohn–Sham equations for electronic wave functions and integral equations for solvent structure. The method is applied to solvated positronium (Ps). Using a simple parameterization for the Ps wave function, we calculate the local solvent structure around a hydrated Ps. The results are consistent with the experimental data. © 2002 Wiley Periodicals, Inc. Int J Quantum Chem, 2002     

Two more approaches for generating trajectory-based dynamics which conserves the canonical distribution in the phase space formulation of quantum mechanics

We show two more approaches for generating trajectory-based dynamics in the phase space formulation of quantum mechanics: "equilibrium continuity dynamics" (ECD) in the spirit of the phase space continuity equation in classical mechanics, and "equilibrium Hamiltonian dynamics" (EHD) in the spirit of the Hamilton equations of motion in classical mechanics. Both ECD and EHD can recover exact thermal correlation functions (of even nonlinear operators, i.e., nonlinear functions of position or momentum operators) in the classical, high temperature, and harmonic limits. Both ECD and EHD conserve the quasi-probability within the infinitesimal volume dx(t)dp(t) around the phase point (x(t), p(t)) along the trajectory. Numerical tests of both approaches in the Wigner phase space have been made for two strongly anharmonic model problems and a double well system, for each potential auto-correlation functions of both linear and nonlinear operators have been calculated. The results suggest EHD and ECD are two additional potential useful approaches for describing quantum effects for complex systems in condense phase.     

Calculation of wave‐functions with frozen orbitals in mixed quantum mechanics/molecular mechanics methods. II. Application of the local basis equation

The application of the local basis equation (Ferenczy and Adams, J. Chem. Phys. 2009 , 130, 134108) in mixed quantum mechanics/molecular mechanics (QM/MM) and quantum mechanics/quantum mechanics (QM/QM) methods is investigated. This equation is suitable to derive local basis nonorthogonal orbitals that minimize the energy of the system and it exhibits good convergence properties in a self‐consistent field solution. These features make the equation appropriate to be used in mixed QM/MM and QM/QM methods to optimize orbitals in the field of frozen localized orbitals connecting the subsystems. Calculations performed for several properties in divers systems show that the method is robust with various choices of the frozen orbitals and frontier atom properties. With appropriate basis set assignment, it gives results equivalent with those of a related approach [G. G. Ferenczy previous paper in this issue] using the Huzinaga equation. Thus, the local basis equation can be used in mixed QM/MM methods with small size quantum subsystems to calculate properties in good agreement with reference Hartree–Fock–Roothaan results. It is shown that bond charges are not necessary when the local basis equation is applied, although they are required for the self‐consistent field solution of the Huzinaga equation based method. Conversely, the deformation of the wave‐function near to the boundary is observed without bond charges and this has a significant effect on deprotonation energies but a less pronounced effect when the total charge of the system is conserved. The local basis equation can also be used to define a two layer quantum system with nonorthogonal localized orbitals surrounding the central delocalized quantum subsystem. © 2013 Wiley Periodicals, Inc.