A probabilistic evolution approach trilogy, part 1: quantum expectation value evolutions, block triangularity and conicality, truncation approximants and their convergence

This is the first one of three companion papers focusing on the “probabilistic evolution approach (PEA)” which has been developed for the solution of the explicit ODE involving problems under certain consistent impositions. The main purpose here is the determination of the expectation value of a given operator in quantum mechanics by solving only ODEs, not directly using the wave function. To this end we first define a basis operator set over the Kronecker powers of an appropriately defined “system operator vector”. We assume that the target operator’s commutator with the system’s Hamiltonian can be expressed in terms of the above-mentioned basis operators. This assumption leads us to an infinite set of linear homogeneous ODEs over the expectation values of the basis operators. Its coefficient matrix is in block Hessenberg form when the target operator has no singularity, and beyond that, it may become block triangular when certain conditions over the system’s potential function are satisfied. The initial conditions are the basic determining agents giving the probabilistic nature to the solutions of the obtained infinite set of ODEs. They may or may not have fluctuations depending on the nature of the probability density. All these issues are investigated in a phenomenological and constructive theoretical manner in this paper. The remaining two papers are devoted to further details of PEA in quantum mechanics, and, the application of PEA to systems defined by Liouville equation.     

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.     

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.     

反应碰撞的动力学李代数描述

提出一种李代数方法描述分子反应碰撞问题．给出了含有主要动力学参量的S-矩阵元、分子碰撞跃迁几率以及反应体系能量统计平均值随时间演化的解析表达式．讨论了一个简单排斥势场中的原子-双原子分子共线反应体系，以阐明这种新方法的要点。     

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.     

A Hierarchical-Equation-of-Motion Based Semiclassical Approach to Quantum Dissipation?

We present a semiclassical (SC) approach for quantum dissipative dynamics, constructed on basis of the hierarchical-equation-of-motion (HEOM) formalism. The dynamical components considered in the developed SC-HEOM are wavepackets' phase-space moments of not only the primary reduced system density operator but also the auxiliary density operators (ADOs) of HEOM. It is a highly numerically efficient method, meanwhile taking into account the high-order effects of system-bath couplings. The SC-HEOM methodology is exemplified in this work on the hierarchical quantum master equation[J. Chem. Phys. 131 , 214111 (2009)] and numerically demonstrated on linear spectra of anharmonic oscillators.     

Solution reaction space Hamiltonian based on an electrostatic potential representation of solvent dynamics

Quantum chemical solvation models usually rely on the equilibrium solvation condition and is thus not immediately applicable to the study of nonequilibrium solvation dynamics, particularly those associated with chemical reactions. Here we address this problem by considering an effective Hamiltonian for solution-phase reactions based on an electrostatic potential (ESP) representation of solvent dynamics. In this approach a general ESP field of solvent is employed as collective solvent coordinate, and an effective Hamiltonian is constructed by treating both solute geometry and solvent ESP as dynamical variables. A harmonic bath is then attached onto the ESP variables in order to account for the stochastic nature of solvent dynamics. As an illustration we apply the above method to the proton transfer of a substituted phenol-amine complex in a polar solvent. The effective Hamiltonian is constructed by means of the reference interaction site model self-consistent field method (i.e., a type of quantum chemical solvation model), and a mixed quantum/classical simulation is performed in the space of solute geometry and solvent ESP. The results suggest that important dynamical features of proton transfer in solution can be captured by the present approach, including spontaneous fluctuations of solvent ESP that drives the proton from reactant to product potential wells.     

Quantum Mechanical Basis for Mulliken Population Analysis

Using an approach alternative to that of Mayer, this paper shows that a Hermitian operator can be found, such that, in a molecule atomic populations can be obtained as its expectation values. In this way, atomic charges can be computed within a quantum mechanical correct definition. When working within the LCAO MO framework, it is found that Mulliken populations appear as the appropriate expectation values of the charge operator.     

The ground state tunneling splitting of malonaldehyde: accurate full dimensional quantum dynamics calculations

Benchmark calculations of the tunneling splitting in malonaldehyde using the full dimensional potential proposed by Yagi et al. are reported. Two exact quantum dynamics methods are used: the multiconfigurational time-dependent Hartree (MCTDH) approach and the diffusion Monte Carlo based projection operator imaginary time spectral evolution (POITSE) method. A ground state tunneling splitting of 25.7+/-0.3 cm(-1) is calculated using POITSE. The MCTDH computation yields 25 cm(-1) converged to about 10% accuracy. These rigorous results are used to evaluate the accuracy of approximate dynamical approaches, e.g., the instanton theory.     

On a New Formulation of the Real-Time Propagator

The development of theoretical tools for the study of dynamical phenomena of many-particle systems on the quantum level is a fundamental challenge since many decades. A lot of efforts have been invested on Feynman's path integral approach, however, no computationally tractable method for investigating realistic systems could be developed up to now. In this paper we propose an alternative representation of the real-time many-body evolution operator formulated within the framework of the auxiliary field formalism. Our goal is to derive a new auxiliary field functional integral representation, in which the large oscillations of the functional integrand are reduced, in order to render the auxiliary field approach more attractive for real-time computation. This objective is attained using a generalized version of the method of Gaussian equivalent representation of Efimov and Ganbold [Phys. Stat. Sol. 168 (1991) 165], which eliminates the low-order fluctuations of the auxiliary field from the interaction functional.     

An accurate and simple quantum model for liquid water

The path-integral molecular dynamics and centroid molecular dynamics methods have been applied to investigate the behavior of liquid water at ambient conditions starting from a recently developed simple point charge/flexible (SPC/Fw) model. Several quantum structural, thermodynamic, and dynamical properties have been computed and compared to the corresponding classical values, as well as to the available experimental data. The path-integral molecular dynamics simulations show that the inclusion of quantum effects results in a less structured liquid with a reduced amount of hydrogen bonding in comparison to its classical analog. The nuclear quantization also leads to a smaller dielectric constant and a larger diffusion coefficient relative to the corresponding classical values. Collective and single molecule time correlation functions show a faster decay than their classical counterparts. Good agreement with the experimental measurements in the low-frequency region is obtained for the quantum infrared spectrum, which also shows a higher intensity and a redshift relative to its classical analog. A modification of the original parametrization of the SPC/Fw model is suggested and tested in order to construct an accurate quantum model, called q-SPC/Fw, for liquid water. The quantum results for several thermodynamic and dynamical properties computed with the new model are shown to be in a significantly better agreement with the experimental data. Finally, a force-matching approach was applied to the q-SPC/Fw model to derive an effective quantum force field for liquid water in which the effects due to the nuclear quantization are explicitly distinguished from those due to the underlying molecular interactions. Thermodynamic and dynamical properties computed using standard classical simulations with this effective quantum potential are found in excellent agreement with those obtained from significantly more computationally demanding full centroid molecular dynamics simulations. The present results suggest that the inclusion of nuclear quantum effects into an empirical model for water enhances the ability of such model to faithfully represent experimental data, presumably through an increased ability of the model itself to capture realistic physical effects.     

Communication: Partial linearized density matrix dynamics for dissipative, non-adiabatic quantum evolution

An approach for treating dissipative, non-adiabatic quantum dynamics in general model systems at finite temperature based on linearizing the density matrix evolution in the forward-backward path difference for the environment degrees of freedom is presented. We demonstrate that the approach can capture both short time coherent quantum dynamics and long time thermal equilibration in an application to excitation energy transfer in a model photosynthetic light harvesting complex. Results are also presented for some nonadiabatic scattering models which indicate that, even though the method is based on a "mean trajectory" like scheme, it can accurately capture electronic population branching through multiple avoided crossing regions and that the approach offers a robust and reliable way to treat quantum dynamical phenomena in a wide range of condensed phase applications.     

Quantum generalized Langevin equation approach to gas/solid collisions

The classical generalized Langevin equation (GLE) approach to gas/solid collisions is generalized to quantum scattering. Using Feynman's method of partial path integration, the full gas/solid propagator is reduced to a form in which only the dynamics of the incident atom and the surface oscillator(s) directly struck appear explicitly. Solving this effective dynamical problem in the semiclassical limit yields a stationary phase equation of motion identical in form to the classical GLE. The noise, however, is distributed according to quantum rather than classical statistics. From the GLE a quantum phase can be constructed and an S-matrix computed. The resulting theory is capable of describing inelastic-diffractive scattering which has been seen experimentally by Williams.     

Quantum observable homotopy tracking control

This paper presents a new tracking method where the target observable O(s,T) at the final dynamical time T follows a predefined track P(s) with respect to a homotopy tracking variable s>or=0. The procedure calculates the series of control fields E(s,t) required to accomplish observable homotopy tracking by solving a first-order differential equation in s for the evolution of the control field. Controls produced by this technique render the desired track for all s without encountering field singularities. This paper also extends the technique to the case where the field-free Hamiltonian and dipole moment operator change with s in order to explore the control of new physical systems along the track. Several simulations are presented illustrating the various uses for this quantum tracking control technique.     

A probabilistic evolution approach trilogy, part 2: spectral issues for block triangular evolution matrix, singularities, space extension

This is the second part of the trilogy on the probabilistic evolution approach and related to the quantum dynamical systems as the first part is. In this sense this work extends the content of the first part to the perhaps secondary but very important details. The spectral investigation of the evolution matrix reveals important issues first and brings the importance of the zero eigenvalues to the surface. The asymptotic convergence possibility and difficulties arising from there can be softened by redefining the state vector. Beside the redefinition, the dimensional extension by adding new elements to the state vector may facilitate the utilization of evolution matrix by bringing conicality or at least multinomiality. The space extension may also help us to deal with singular Hamiltonian systems. All these issues are focused on rather phenomenologically. Illustrative or not, no comprehensive implementation is given since the main purpose is just conceptuality.     

A probabilistic foundation for dynamical systems: phenomenological reasoning and principal characteristics of probabilistic evolution

This paper is the second in a series of two. The first paper has been devoted to the detailed explanation of the mathematical formulation of the underlying theoretical framework. Specifically, the first paper shows that it is possible to construct an infinite linear ODE set, which describes a probabilistic evolution. The evolution is probabilistic because the unknowns are expectations, with appropriate initial conditions. These equations, which we name, Probabilistic Evolution Equations (PEE) are linear at the level of ODEs and initial conditions. In this paper, we first focus on the phenomenological reasoning that lead us to the derivation of PEE. Second, the aspects of the PEE construction is revisited with a focus on the spectral nature of the probabilistic evolution. Finally, we postulate fruitful avenues of research in the fields of dynamical causal modeling in human neuroimaging and effective connectivity analysis. We believe that this final section is a prime example of how the rigorous methods developed in the context of mathematical chemistry can be influential in other fields and disciplines.