Mixed quantum-classical equilibrium

We present an analysis of the equilibrium limits of the two most widely used approaches for simulating the dynamics of molecular systems that combine both quantum and classical degrees of freedom. For a two-level quantum system connected to an infinite number of classical particles, we derive a simple analytical expression for the equilibrium mean energy attained by the self-consistent-field (Ehrenfest) method and show that it deviates substantially from Boltzmann. By contrast, "fewest switches" surface hopping achieves Boltzmann quantum state populations. We verify these analytical results with simulations.     

Transition probabilities of a string oscillator subject to impulsive collisions with a heavy mass point

Impulsive linear collisions between a string oscillator (a one-dimensional particle in a box) and a mass point are studied quantum mechanically. In the limit of a very heavy mass point (which corresponds classically to many collisions during a single encounter) the transition probabilities are determined exactly. The result permits a discussion of the mixed quantum-classical regime where the collider becomes almost classical while the oscillator remains quantum mechanical. While the average transition probabilities P(m-->n) are well reproduced by the Ehrenfest mean-field approximation, the prediction for the superimposed high-frequency resonance structure is qualitatively wrong for a genuine quantum oscillator. Only if the oscillator is also almost classical and if (m-n)2 square root(mu) < m, where mu is the mass ratio collider/oscillator, this structure is correctly predicted by the Ehrenfest approximation.     

A modified Ehrenfest method that achieves Boltzmann quantum state populations

A modification of the hybrid quantum/classical mean-field or Ehrenfest method is proposed which allows us to reproduce the Boltzmann quantum state populations when simulating the vibrational energy transfer between quantum systems and classical thermal baths. The modified Ehrenfest results are shown to be able to simulate the energy fluxes between the solute vibration and the classical solvent in the proper directions, depending on their relative temperatures.     

Nonadiabatic electron wavepacket dynamics of molecules in an intense optical field: an ab initio electronic state study

A theory of quantum electron wavepacket dynamics that nonadiabatically couples with classical nuclear motions in intense optical fields is studied. The formalism is intended to track the laser-driven electron wavepackets in terms of the linear combination of configuration-state functions generated with ab initio molecular orbitals. Beginning with the total quantum Hamiltonian for electrons and nuclei in the vector potential of classical electromagnetic field, we reduce the Hamiltonian into a mixed quantum-classical representation by replacing the quantum nuclear momentum operators with the classical counterparts. This framework gives equations of motion for electron wavepackets in an intense laser field through the time dependent variational principle. On the other hand, a generalization of the Newtonian equations provides a matrix form of forces acting on the nuclei for nonadiabatic dynamics. A mean-field approximation to the force matrix reduces this higher order formalism to the semiclassical Ehrenfest theory in intense optical fields. To bring these theories into a practical quantum chemical package for general molecules, we have implemented the relevant ab initio algorithms in it. Some numerical results in the level of the semiclassical Ehrenfest-type theory with explicit use of the nuclear kinematic (derivative) coupling and the velocity form for the optical interaction are presented.     

Generalization of classical mechanics for nuclear motions on nonadiabatically coupled potential energy surfaces in chemical reactions

Classical trajectory study of nuclear motion on the Born-Oppenheimer potential energy surfaces is now one of the standard methods of chemical dynamics. In particular, this approach is inevitable in the studies of large molecular systems. However, as soon as more than a single potential energy surface is involved due to nonadiabatic coupling, such a naive application of classical mechanics loses its theoretical foundation. This is a classic and fundamental issue in the foundation of chemistry. To cope with this problem, we propose a generalization of classical mechanics that provides a path even in cases where multiple potential energy surfaces are involved in a single event and the Born-Oppenheimer approximation breaks down. This generalization is made by diagonalization of the matrix representation of nuclear forces in nonadiabatic dynamics, which is derived from a mixed quantum-classical representation of the electron-nucleus entangled Hamiltonian [Takatsuka, K. J. Chem. Phys. 2006, 124, 064111]. A manifestation of quantum fluctuation on a classical subsystem that directly contacts with a quantum subsystem is discussed. We also show that the Hamiltonian thus represented gives a theoretical foundation to examine the validity of the so-called semiclassical Ehrenfest theory (or mean-field theory) for electron quantum wavepacket dynamics, and indeed, it is pointed out that the electronic Hamiltonian to be used in this theory should be slightly modified.     

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.     

Surface-hopping dynamics and decoherence with quantum equilibrium structure

In open quantum systems, decoherence occurs through interaction of a quantum subsystem with its environment. The computation of expectation values requires a knowledge of the quantum dynamics of operators and sampling from initial states of the density matrix describing the subsystem and bath. We consider situations where the quantum evolution can be approximated by quantum-classical Liouville dynamics and examine the circumstances under which the evolution can be reduced to surface-hopping dynamics, where the evolution consists of trajectory segments exclusively evolving on single adiabatic surfaces, with probabilistic hops between these surfaces. The justification for the reduction depends on the validity of a Markovian approximation on a bath averaged memory kernel that accounts for quantum coherence in the system. We show that such a reduction is often possible when initial sampling is from either the quantum or classical bath initial distributions. If the average is taken only over the quantum dispersion that broadens the classical distribution, then such a reduction is not always possible.     

Probabilistic evolution approach to the expectation value dynamics of quantum mechanical operators,part I: integral representation of Kronecker power series and multivariate Hausdorff moment problems

This is the first one of two companion papers focusing on the establishment of a new path for the expectation value dynamics of the quantum mechanical operators. The main goal of these studies is to do quantum mechanics without explicitly solving Schrödinger wave equation, in other words, without using wave functions except their initially given forms. This goal is achieved by using Ehrenfest theorem and utilizing probabilistic evolution approach (PEA). PEA, first introduced by Metin Demiralp, is a method providing solutions to the nonlinear ordinary differential equations by transforming them to a set of linear ODEs at the cost of denumerably infinite dimensionality. It is recently shown that this method produces analytic solutions, if the initial conditions are given appropriately at some special cases. However, generalization of these conditions to the quantum mechanical applications is not straightforward due to the dispersion of the quantum mechanical systems. For this purpose, multivariate moment problems for the integral representation of the Kronecker power series are introduced and then solved yielding to more specific and precise convergence analysis for the quantum mechanical applications.     

Non-Born-Oppenheimer trajectories with self-consistent decay of mixing

A semiclassical trajectory method, called the self-consistent decay of mixing (SCDM) method, is presented for the treatment of electronically nonadiabatic dynamics. The SCDM method is a modification of the semiclassical Ehrenfest (SE) method (also called the semiclassical time-dependent self-consistent-field method) that solves the problem of unphysical mixed final states by including decay-of-mixing terms in the equations for the evolution of the electronic state populations. These terms generate a force, called the decoherent force (or dephasing force), that drives the electronic component of each trajectory toward a pure state. Results for several mixed quantum-classical methods, in particular the SCDM, SE, and natural-decay-of-mixing methods and several trajectory surface hopping methods, are compared to the results of accurate quantum mechanical calculations for 12 cases involving five different fully dimensional triatomic model systems. The SCDM method is found to be the most accurate of the methods tested. The method should be useful for the simulation of photochemical reactions.     

NMR implementation of adiabatic SAT algorithm using strongly modulated pulses

NMR implementation of adiabatic algorithms face severe problems in homonuclear spin systems since the qubit selective pulses are long and during this period, evolution under the Hamiltonian and decoherence cause errors. The decoherence destroys the answer as it causes the final state to evolve to mixed state and in homonuclear systems, evolution under the internal Hamiltonian causes phase errors preventing the initial state to converge to the solution state. The resolution of these issues is necessary before one can proceed to implement an adiabatic algorithm in a large system where homonuclear coupled spins will become a necessity. In the present work, we demonstrate that by using "strongly modulated pulses" (SMPs) for the creation of interpolating Hamiltonian, one can circumvent both the problems and successfully implement the adiabatic SAT algorithm in a homonuclear three qubit system. This work also demonstrates that the SMPs tremendously reduce the time taken for the implementation of the algorithm, can overcome problems associated with decoherence, and will be the modality in future implementation of quantum information processing by NMR.     

A new implementation of ab initio ehrenfest dynamics using electronic configuration basis: Exact formulation with molecular orbital connection and effective propagation scheme with locally quasi‐diabatic representation

We propose a new implementation of Ehrenfest molecular dynamics based on the configuration interaction theory using configuration state functions (CSF) as basis set originally proposed by Amano and Takatsuka (J. Chem. Phys. 2005 , 122, 084113). Our development consists of two independet new features. The first one deals with the problem on how to “identify” the molecular orbitals at a simulation time step in terms of those at the previous time step. By giving an exact expression of CSF Ehrenfest method, this problem naturaly vanishes. To actually perform this method, the concept of molecular orbital connection which allows the MOs to be noncanonical is necessary. The second feature of our method is aimed to reduce the computational cost. We propose an approximaion to effectively perform the time propagation of the electron wavefunction. Due to the analogy to the locally diabatic representation method, we name our method locally quasi‐diabatic representation method. In the present work, these two new features were combined and employed to perform test computations. © 2016 Wiley Periodicals, Inc.     

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