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.     

Variational reactive scattering calculations: computational optimization strategies

Summary We have developed efficient and accurate techniques for the calculation of quantum mechanical reaction probabilities of atom-diatom exchange reactions in the gas phase, and we have optimized a computer code employing these techniques and applied it sucessfully to several systems. In this paper we consider further strategies for improving the algorithm to allow even more demanding applications. In this context, improvement means that equivalent results can be obtained using fewer computational resources (computer time or storage) or that an equivalent expenditure of resources can yield higher accuracy. The new strategies discussed here lead to improvement in both of these areas. Two areas of special focus in the present paper are (i) the finite difference boundary value method used for calculating distorted wave Green's functions and regular solutions for scattering by the distortion potential and (ii) the choice of the distortion potential itself. Among other results included here is the first application of the outgoing wave or scattered wave variational principle to reactive scattering.     

Multidimensional quantum trajectories: applications of the derivative propagation method

In a previous publication [J. Chem. Phys. 118, 9911 (2003)], the derivative propagation method (DPM) was introduced as a novel numerical scheme for solving the quantum hydrodynamic equations of motion (QHEM) and computing the time evolution of quantum mechanical wave packets. These equations are a set of coupled, nonlinear partial differential equations governing the time evolution of the real-valued functions C and S in the complex action, S=C(r,t) + iS(r,t)/Planck's over 2pi, where Psi(r,t)=exp(S). Past numerical solutions to the QHEM were obtained via ensemble trajectory propagation, where the required first- and second-order spatial derivatives were evaluated using fitting techniques such as moving least squares. In the DPM, however, equations of motion are developed for the derivatives themselves, and a truncated set of these are integrated along quantum trajectories concurrently with the original QHEM equations for C and S. Using the DPM quantum effects can be included at various orders of approximation; no spatial fitting is involved; there is no basis set expansion; and single, uncoupled quantum trajectories can be propagated (in parallel) rather than in correlated ensembles. In this study, the DPM is extended from previous one-dimensional (1D) results to calculate transmission probabilities for 2D and 3D wave packet evolution on coupled Eckart barrier/harmonic oscillator surfaces. In the 2D problem, the DPM results are compared to standard numerical integration of the time-dependent Schrodinger equation. Also in this study, the practicality of implementing the DPM for systems with many more degrees of freedom is discussed.     

Quantum crystallography: A perspective

Extraction of the complete quantum mechanics from X‐ray scattering data is the ultimate goal of quantum crystallography. This article delivers a perspective for that possibility. It is desirable to have a method for the conversion of X‐ray diffraction data into an electron density that reflects the antisymmetry of an N‐electron wave function. A formalism for this was developed early on for the determination of a constrained idempotent one‐body density matrix. The formalism ensures pure‐state N‐representability in the single determinant sense. Applications to crystals show that quantum mechanical density matrices of large molecules can be extracted from X‐ray scattering data by implementing a fragmentation method termed the kernel energy method (KEM). It is shown how KEM can be used within the context of quantum crystallography to derive quantum mechanical properties of biological molecules (with low data‐to‐parameters ratio). © 2017 Wiley Periodicals, Inc.     

Higher-order split operator schemes for solving the Schrödinger equation in the time-dependent wave packet method: applications to triatomic reactive scattering calculations

The efficiency of the numerical propagators for solving the time-dependent Schr?dinger equation in the wave packet approach to reactive scattering is of vital importance. In this Perspective, we first briefly review the propagators used in quantum reactive scattering calculations and their applications to triatomic reactions. Then we present a detailed comparison of about thirty higher-order split operator propagators for solving the Schr?dinger equation with their applications to the wave packet evolution within a one-dimensional Morse potential, and the total reaction probability calculations for the H + HD, H + NH, H + O(2), and F + HD reactions. These four triatomic reactions have quite different dynamic characteristics and thus provide a comprehensive picture of the relative advantages of these higher-order propagation methods for describing reactive scattering dynamics. Our calculations reveal that the most often used second-order split operator method is typically more efficient for a direct reaction, particularly for those involving flat potential energy surfaces. However, the optimal higher-order split operator methods are more suitable for a reaction with resonances and intermediate complexes or a reaction experiencing potential energy surface with fluctuations of considerable amplitude. Three 4th-order and one 6th-order split operator methods, which are most efficient for solving reactive scattering in various conditions among the tested ones, are recommended for general applications. In addition, a brief discussion on the relative performance between the Chebyshev real wave packet method and the split operator method is given. The results in this Perspective are expected to stimulate more applications of (high-order) split operators to the quantum reactive scattering calculation and other related problems.     

Computation of correlation functions and wave function projections in the context of quantum trajectory dynamics

The de Broglie-Bohm formulation of the Schrodinger equation implies conservation of the wave function probability density associated with each quantum trajectory in closed systems. This conservation property greatly simplifies numerical implementations of the quantum trajectory dynamics and increases its accuracy. The reconstruction of a wave function, however, becomes expensive or inaccurate as it requires fitting or interpolation procedures. In this paper we present a method of computing wave packet correlation functions and wave function projections, which typically contain all the desired information about dynamics, without the full knowledge of the wave function by making quadratic expansions of the wave function phase and amplitude near each trajectory similar to expansions used in semiclassical methods. Computation of the quantities of interest in this procedure is linear with respect to the number of trajectories. The introduced approximations are consistent with approximate quantum potential dynamics method. The projection technique is applied to model chemical systems and to the H+H(2) exchange reaction in three dimensions.     

A chemical bond theory of quantum size effects of semiconductor clusters

Size dependence effects in semiconductor clusters have been a subject of extensive studies for the last two decades. However, it is still difficult to employ the existing theoretical models to give reliable results of energies for clusters in the whole nanometer region. Here we offer a new theoretical method for the quantum size effects based on the idea that the energy gap shift of the cluster arises from the sum of the surface effect shift and quantum effect shift parts. We express the effects through algebraic relations rather than through variational solutions of the wave equation, without the use of any special adjustable parameter. Results reveal for the first time that the shape of the energy gap shift curve is dominated by the surface energy shift. Our method can also predict quantitatively the size dependence of dielectric constant. The new theoretical findings in the ultrasmall (<1 nm) anatase TiO(2) and the silicon clusters cannot be explained using previous theories.     

Quantum mechanical generalized phase-shift approach to atom-surface scattering: a Feshbach projection approach to dealing with closed channel effects

We have developed a new method for solving quantum dynamical scattering problems, using the time-independent Schr?dinger equation (TISE), based on a novel method to generalize a "one-way" quantum mechanical wave equation, impose correct boundary conditions, and eliminate exponentially growing closed channel solutions. The approach is readily parallelized to achieve approximate N(2) scaling, where N is the number of coupled equations. The full two-way nature of the TISE is included while propagating the wave function in the scattering variable and the full S-matrix is obtained. The new algorithm is based on a "Modified Cayley" operator splitting approach, generalizing earlier work where the method was applied to the time-dependent Schr?dinger equation. All scattering variable propagation approaches to solving the TISE involve solving a Helmholtz-type equation, and for more than one degree of freedom, these are notoriously ill-behaved, due to the unavoidable presence of exponentially growing contributions to the numerical solution. Traditionally, the method used to eliminate exponential growth has posed a major obstacle to the full parallelization of such propagation algorithms. We stabilize by using the Feshbach projection operator technique to remove all the nonphysical exponentially growing closed channels, while retaining all of the propagating open channel components, as well as exponentially decaying closed channel components.     

Dynamics of nonadiabatic chemical reactions

New methods are proposed to treat nonadiabatic chemical dynamics in realistic large molecular systems by using the Zhu-Nakamura (ZN) theory of curve-crossing problems. They include the incorporation of the ZN formulas into the Herman-Kluk type semiclassical wave packet propagation method and the trajectory surface hopping (TSH) method, formulation of the nonadiabatic transition state theory, and its application to the electron transfer problem. Because the nonadiabatic coupling is a vector in multidimensional space, the one-dimensional ZN theory works all right. Even the classically forbidden transitions can be correctly treated by the ZN formulas. In the case of electron transfer, a new formula that can improve the celebrated Marcus theory in the case of normal regime is obtained so that it can work nicely in the intermediate and strong electronic coupling regimes. All these formulations mentioned above are demonstrated to work well in comparison with the exact quantum mechanical numerical solutions and are expected to be applicable to large systems that cannot be treated quantum mechanically numerically exactly. To take into account another quantum mechanical effect, namely, the tunneling effect, an efficient method to detect caustics from which tunneling trajectories emanate is proposed. All the works reported here are the results of recent activities carried out in the author's research group. Finally, the whole set of ZN formulas is presented in Appendix.     

量子控制论在化学中的应用

控制量子现象是化学研究中的一个重要目标,量子控制论对实现该目标具有积极的指导意义.本文综述了量子控制论在化学中的应用及其进展,重点分析了量子相干控制、量子优化控制、闭环学习控制和能控性观念在化学研究中的应用,介绍了它们的研究现状,并对其未来研究进行了展望.     

Computational method for the quantum Hamilton-Jacobi equation: bound states in one dimension

An accurate computational method for the one-dimensional quantum Hamilton-Jacobi equation is presented. The Mobius propagation scheme, which can accurately pass through singularities, is used to numerically integrate the quantum Hamilton-Jacobi equation for the quantum momentum function. Bound state wave functions are then synthesized from the phase integral using the antithetic cancellation technique. Through this procedure, not only the quantum momentum functions but also the wave functions are accurately obtained. This computational approach is demonstrated through two solvable examples: the harmonic oscillator and the Morse potential. The excellent agreement between the computational and the exact analytical results shows that the method proposed here may be useful for solving similar quantum mechanical problems.     

聚酯胺改善聚甲基丙烯酸甲酯电纺纤维均匀性研究

研究比较线型聚酯胺和超支化聚酯胺作为添加剂用于静电纺丝时,对低浓度聚甲基丙烯酸甲酯溶液可纺性的改善效果及其机理.结果表明,只需加入1wt%的添加剂,无论是线型的还是超支化聚酯胺均能够提高低浓度聚甲基丙烯酸甲酯溶液的可纺性,得到无"串珠"结构的均匀纤维,其直径比不加添加剂而在高浓度纺丝时得到的均匀纤维细很多.通过溶液性能的测试,发现提高可纺性的原因均是由于溶液电导率的提高.超支化聚酯胺因其多枝的结构而含有较多的极性端基,致使本身电导率较高,因而对可纺性的改善效果好于线型聚酯胺.     

The eigenvalue problem in phase space

We formulate the standard quantum mechanical eigenvalue problem in quantum phase space. The equation obtained involves the c‐function that corresponds to the quantum operator. We use the Wigner distribution for the phase space function. We argue that the phase space eigenvalue equation obtained has, in addition to the proper solutions, improper solutions. That is, solutions for which no wave function exists which could generate the distribution. We discuss the conditions for ascertaining whether a position momentum function is a proper phase space distribution. We call these conditions psi‐representability conditions, and show that if these conditions are imposed, one extracts the correct phase space eigenfunctions. We also derive the phase space eigenvalue equation for arbitrary phase space distributions functions. © 2017 Wiley Periodicals, Inc.     

A numerical method for the determination of atom—atom scattering amplitudes from the measured differential cross sections

A numerical method is given for the determination of the scattering amplitude, hence all the phase shifts, from the differential cross section at fixed energy. To obtain the phase of the scattering amplitude, the unitarity equation of scattering theory is solved, using the (experimental) cross section as input information. A straightforward iterative approach diverges for atom—atom input data, thus an appropriately modified method of solution is introduced to overcome this difficulty. The method was applied to two test cases, in both of which calculated atom—atom cross sections were used as simulated input data. Convergence to the correct phase was obtained in both examples, starting in each case with a guess phase function that differed considerably from the true solution. Convergence cannot be obtained, however, from an extremely poor starting phase. This study shows that the scattering amplitude for atomic scattering at thermal energies can be determined systematically, without use of parametrized functions, if a sufficiently accurate experimental cross section is available. A subsequent article describes a quantum mechanical procedure whereby the interaction potential can be constructed from the determined scattering amplitude.     

Ultrasonic non-destructive evaluation of selectively laser-sintered polymeric nanocomposites

Ultrasonic testing as a non-destructive evaluation (NDE) technique is newly introduced to characterize additively manufactured composite materials to identify their anisotropic mechanical properties, being especially facile, useful and accurate approach for dimensional dependent measurement. In this study, the immersion ultrasonic technique is employed to measure the energy loss of ultrasonic elastic waves, and wave propagation speed in the laser-sintered nanocomposite of carbon nanotube reinforced polyamine 12. The relationship of process-structure-property is revealed to establish the correlations between process parameters and energy loss of ultrasound, as well as mechanical moduli. The orientation-dependent wave attenuation and mechanical moduli of nanocomposites along three orthogonal directions are strongly associated with the layer-by-layer fusion induced microstructures and internal imperfections. This technique is capable of quantifying orientation-dependent mechanical properties such as moduli and attenuation without compromising additively manufactured parts, showing a high potential of quality control and safety inspection in end-use applications.     

基于量子波包方法的态-态分辨反应散射动力学计算

本文回顾了最近十几年利用量子波包方法研究气相分子反应散射动力学的工作进展,特别是在态-态分辨水平上的工作进展。比较详细地讨论了目前存在的利用量子波包方法计算态-态微分截面的几种方法。目前态-态分辨的波包动力学计算可以精确地预测三原子和四原子分子反应散射的各种信息,文章最后对几个典型的利用波包方法在态-态分辨水平上研究过的三原子和四原子反应散射体系做了讨论。     

Determination of urinary beta-phenylethylamine as its N-benzenesulphonamide derivative by gas chromatography with flame photometric detection.

A selective and sensitive method for the determination of urinary beta-phenylethylamine (PEA) by gas chromatography (GC) has been developed. After extraction of the urine sample with n-pentane, PEA was converted into its N-benzenesulphonamide derivative and then determined by GC with flame photometric detection using a DB-1301 capillary column. By using this method, nanogram amounts of PEA in urine could be accurately and precisely determined without any influence from coexisting substances. Analytical results for the determination of PEA in urine samples from normal subjects are presented.     

Influence of the nature of the porous confining network on the sorption, diffusion and mechanical properties of hydrogel IPNs

Two series of amphiphilic hydrogels of various compositions were prepared by sequentially interpenetrating two polymer networks, a poly(2-hydroxyethyl acrylate) (PHEA) network inside either a macroporous matrix of poly(methyl methacrylate) (PMMA) or a macroporous poly(ethyl acrylate) (PEA) network. In both cases poly(2-hydroxyethyl acrylate) (PHEA) served as network II, and the firstly formed porous network was a hydrophobic homonetwork, PMMA or PEA, that conferred mechanical strength to the hydrogel. In order to obtain hydrogels with high hydrophilic content, the first network was prepared in the presence of a solvent, thus yielding a macroporous network. The two families of IPNs thus obtained were: (net-PMMA)-ipn-(net-PHEA) and (net-PEA)-ipn-(net-PHEA), with a PHEA content ranging from 36% to 87% and from 64% to 94%, respectively. The novelty of the work consisted in comparing the effect of using as the first macroporous network a polymer which is glassy at room temperature (PMMA) and another of the same family (PEA) but which is in the rubber state at room temperature. Swelling studies showed that the specific equilibrium water content of PHEA falls from 1.6 for pure (unconfined) PHEA to values that range from 0.4 to 1, for the (net-PMMA)-ipn-(net-PHEA), whereas in the second IPNs family, the equilibrium water uptake of PHEA phase is, at least, the same as that of the pure PHEA (in some cases it is greater). This means that the expansion of the PHEA phase is not restricted by the confining hydrophobic component when this last is in the rubber state at room temperature. Whereas for the first IPNs the mechanical properties significantly increased (storage modulus at 37 °C from 0.25 to 2.5 GPa) compared with those of pure PHEA (25.12 MPa), little if any reinforcing effect was observed in the second type of IPNs. This is due to the fact that the glass transition of the PEA network takes place at a lower temperature than that of PHEA, so both components are in the rubbery state at room temperature. Both series behave differently also in dynamic water sorption experiments: the rigid PMMA network hinders the diffusion of water, yielding lower values of the apparent diffusion coefficients. By contrast, with the PEA polymer as network I this diffusion is similar to that of the pure PHEA homonetwork.