Mapping Coriolis-coupled quantum dynamics onto parallel computer architectures

The present article is devoted to Chemistry or Physics undergraduate students, given their difficulty to understand fundamental concepts and technical language used in atomic spectroscopy and quantum mechanics. An easy approach is shown in the treatment of the emission spectrum of the sodium atom without any involved calculations. In a previous article, the hydrogen spectrum was considered and the energy degeneracy of the angular momentum quantum number was observed. For the sodium spectrum, due to the valence electron penetration into internal shells, a breakdown of this degeneracy occurs and a dependence of this penetration on the angular momentum quantum number is observed. The eigenvalues are determined introducing the quantum defect correction (Rydberg correction) in the denominator of the Balmer equation, and the energy diagram is obtained. The intensity ratio for the observed doublets is explained by introducing new wave functions, containing the magnetic quantum number of the total angular momentum.

As we have already indicated, experimental knowledge on the proton-neutron system is acquired by studying the scattering of neutrons by protons and by investigations of the properties of the deuteron. An analysis of these data, the goal of which is the determination of phase shifts, requires, first of all, a resolution of the scattered waves in terms of spherical functions. Because of the spins of the proton and neutron this procedure turns out to be somewhat more complicated than in the case of spinless particles which have been described. First of all we must introduce a system for classifying the possible states of the system. In the case of spinless particles the state was determined by the azimuthal quantum number L. In the presence of spin the state of the system is also determined by the spin quantum number S. If the spin and coordinates are independent, the orbital angular momentum and the spin angular momentum are conserved individually. Any coupling between the spin and orbital motion means that the vectors L and S are not conserved individually; rather the total angular momentum J is conserved;* this is the sum of the orbital and spin angular momenta.

Theoretical models of computation (automata, Turing machines, random access machines, etc.) define not only what is computable but determine how much time and/or space is required for a computation. The parallel extensions of the sequential models usually offer significant speedup in algorithm execution (not all algorithms can be parallelized, however). Real parallel computer architectures based on these parallel computation models should exhibit the same algorithmic speedup. A complexity hierarchy can be defined for parallel computation models based on the space and/or time speedup of certain classes of algorithms. In particular the parallel shared memory models1 are some of the more powerful of the parallel computation models. Their close resemblance to actual computer architectures provide guidance for real architectural design and their massive interconnection requirements are well suited for implementation in optical computing architectures. In addition, the shared memory models have recently been shown to simulate deterministic Boltzmann machines and threshold-logic Turing machines. Thus optical machines based on these parallel models can potentially exhibit the computational power of the shared memory models. These parallel computation models and their implications for optical computing are discussed.

Much of the physics determined for hydrogen and hydrogen-like ions is relevant to atoms and ions with many electrons. In a multi-electron atom or ion, the quantum state energies and wavefunctions are dominated by the central potential arising from the nuclear charge as is the case for hydrogen and hydrogen-like ions. Consequently, quantum wavefunctions similar in form to those found for hydrogen or hydrogen-like ions are produced. There are some corrections needed to the exact electron energies associated with the nuclear electric field experienced by each electron due to shielding effects by other electrons – and some perturbing energy effects due to electron–electron Coulomb repulsion. The electron wavefunctions have similar orbital or angular shapes as for hydrogen and hydrogen-like ions (see Figures 7.2 and 7.3). The same electron-configuration system (i.e. 1s, 2s, 2p, …) can be used and the same degeneracies are associated with different values of principal quantum number n (degeneracy 2 n 2 ) and orbital angular momentum quantum number l (degeneracy 2×2 l(l +1 ) ) as found for hydrogen and hydrogen-like ions. As multi-electron atoms and ions have quantum states following the ‘rules’ established for hydrogen and hydrogen-like ions, the quantum state energy values are primarily determined by a principal quantum number n with values n =1, 2, 3, … . Angular quantum numbers l with l =0, 1, 2, … n − 1 are found to have a small effect on energy values due to shielding of the nuclear charge. As for hydrogen, the number of individual quantum states at a particular energy involves the different magnetic quantum numbers m with values m = − l ,− l + 1, … , 0, 1 … + l and the two possible spin orientations (relative to the orbital angular momentum). The states which the electrons occupy are called the electronic configuration for the atom or ion. An important feature of multi-electron atoms is established by the Pauli exclusion principle which requires that only one electron can occupy an individual quantum state. In the ground state of an atom or ion, the required number of electrons effectively fills the lowest energy quantum states designated by the n , l , m and s values according to the hydrogen degeneracies.

Full-dimensional quantum dynamics calculations on ArI2(B,vi) vibrational predissociation with total angular momenta J=0, 1, and 2 are presented. Models based on a few interacting states are shown to reproduce important aspects of the dynamics, confirming that vibrational predissociation is mediated by a few-state intramolecular vibrational energy redistribution effect. As a consequence, vibrational predissociation rate constants exhibit large oscillations with vi, the initial I2 vibrational quantum number in the complex. The qualitative effect persists when alternative choices for the interaction potential parameters are considered. Similarly, despite the importance of Coriolis coupling, the effect persists as J is varied from 0 to 2. We also discuss how the effect may be difficult to observe in typical experiments that involve higher J values.

The CN(B 2Σ+–X 2Σ+) tail band system was observed in the reaction of argon metastable atoms with BrCN. The observed transitions in the 17-14 and 17-16 bands were analyzed in terms of the B 2Σ+,v=17∼A 2Πinv,v=34 and 4Π,v=p+11 perturbations, and the molecular constants for these levels and the interaction constants were determined. Intensity anomalies were also observed in the 9-8, 9-9, and 12-12 bands. The quantum numbers of the total angular momentum, J, of the perturbed rotational levels of B 2Σ+ were assigned as (vB,N;J)=(9,56;55.5) and (12,10;10.5). The vibrational quantum number was determined as p=0, and the crossing spin sublevels at the anomalies of vB=9 and 12 were determined as 4Π5/2 and 4Π1/2 by examination of the electronic part of the B 2Σ+∼4Π perturbation constant, HSOel, for the (vB,vΠ)=(14,6) and (17,11) perturbations estimated by use of the Rydberg–Klein–Rees (RKR) potentials for B 2Σ+ and 4Π.

We present fast and highly scalable parallel computations for a number of important and fundamental matrix problems on distributed memory systems (DMS). These problems include matrix multiplication, matrix chain product, and computing the powers, the inverse, the characteristic polynomial, the determinant, the rank, the Krylov matrix, and an LU- and a QR-factorization of a matrix, and solving linear systems of equations. Our highly scalable parallel computations for these problems are based on a highly scalable implementation of the fastest sequential matrix multiplication algorithm on DMS. We show that compared with the best known parallel time complexities on parallel random access machines (PRAM), the most powerful but unrealistic shared memory model of parallel computing, our parallel matrix computations achieve the same speeds on distributed memory parallel computers (DMPC), and have an extra polylog factor in the time complexities on DMS with hypercubic networks. Furthermore, our parallel matrix computations are fully scalable on DMPC and highly scalable over a wide range of system size on DMS with hypercubic networks. Such fast (in terms of parallel time complexity) and highly scalable (in terms of our definition of scalability) parallel matrix computations were rarely seen before on any distributed memory systems.

We present fast and scalable parallel computations for a number of important and fundamental matrix problems on distributed memory systems (DMS). These problems include computing the powers, the inverse, the characteristic polynomial, the determinant, the rank, the Krylov matrix, and an LU- and a QR-factorization of a matrix, and solving linear systems of equations. These parallel computations are based on efficient implementations of the fastest sequential matrix multiplication algorithm on DMS. We show that compared with the best known time complexities on PRAM, our parallel matrix computations achieve the same speeds on distributed memory parallel computers (DMPC), and have an extra polylog factor in the time complexities on DMS with hypercubic networks. Furthermore, our parallel matrix computations are fully scalable on DMPC and highly scalable over a wide range of system size on DMS with hypercubic networks. Such fast and scalable parallel matrix computations were not seen before on any distributed memory systems.

Cross sections for single electron capture (SEC), or charge exchange (CX), in collisions of Ne(8–10)+ and Mg(8–12)+ with H and He, are computed using an approximate multichannel Landau–Zener (MCLZ) formalism. Final-state-resolved cross sections for the principal (n), orbital angular momentum (ℓ), and where appropriate, total spin angular momentum (S) quantum numbers are explicitly computed, except for the incident bare ions Ne10+ and Mg12+. In the latter two cases, -resolution is obtained from analytical ℓ-distribution functions applied to n-resolved MCLZ cross sections. In all cases, the cross sections are computed over the collision energy range 1 meV/u to 50 keV/u with LZ parameters estimated from atomic energies obtained from experiment, theory, or, in the case of high-lying Rydberg levels, estimated with a quantum defect approach. Errors in the energy differences in the adiabatic potentials at the avoided crossing distances give the largest contribution to the uncertainties in the cross sections, which are expected to increase with decreasing cross section magnitude. The energy differences are deduced here with the Olson–Salop–Tauljberg radial coupling model. Proper selection of an ℓ-distribution function for bare ion collisions introduces another level of uncertainty into the results. Comparison is made to existing experimental or theoretical results when available, but such data are absent for most considered collision systems. The -resolved SEC cross sections are used in an optically thin cascade simulation to predict X-ray spectra and line ratios that will aid in modeling the X-ray emission in environments where CX is an important mechanism. Details on a MCLZ computational package, Stueckelberg, are also provided.

There has been a recent emergence of many interesting and highly efficient hierarchical (multilevel) algorithms (e.g. multigrid, domain decomposition, wavelets, multilevel preconditioning, the fast multipole algorithms, etc.) for solving numerical problems in scientific computing. These algorithms derive their efficiency from using a hierarchical approach to capture the sharing of global information which is inherent in the physical processes being modelled. In addition to being computationally efficient, these algorithms also possess relatively high degrees of parallelism. I therefore argue that the architectures of parallel computers (especially massively parallel ones) should be designed to support hierarchical communication and synchronization needs of these algorithms. Hierarchical architectures are also more universal because they go beyond supporting a particular class of algorithms to supporting the underlying physical processes being modelled. Finally, I propose that algorithm designers take a critical look at traditional kernel algorithms and re-examine their cost-effectiveness in a massively parallel computing environment.

We examine how the theoretical analysis of configuration dependences arising from crossedbeam molecular scattering in an intense laser field can be tractably handled. Difficulties in analysing such dependences derive from dynamical equations which do not conserve the total angular momentum number J and moreover are parametrized by the total angular momentum projection quantum number M onto the field axis. We first examine simplifications which arise in the weak-field limit; for a given initial value of J we establish under broad conditions that a single weak-field close-coupling calculation suffices to determine all M dependent elements. We next extend this weak-field analysis to deal with more intense fields; essentially we intensity scale the elements from the weak-field analysis and then mix these in a new procedure for preserving unitarity in scattering matrices. Our resulting methodology is examined for a model laser-modified collision problem; results are very encouraging for utilization in realistic studies of configuration dependences.

Using the technique of stimulated emission pumping (SEP) spectroscopy, highly excited vibration–rotation states of the CH3O (X̃ 2E) molecule were probed up to energies of E≤10 000 cm−1. The highest excitation energies exceed the asymptotic H–H2CO dissociation limit of the molecule [ΔrH00(H–H2CO)≊6900 cm−1]. Work was carried out at different experimental resolutions. First, low resolution survey SEP spectra were found to exhibit persistent vibrational structure up to energies far above the dissociation limit. The observed main features were found to be assignable, in a zero-order picture that leaves aside possible mode-to-mode couplings, to the progression of the excited C–O stretch vibration states (ν3). The widths of the respective features correspond to localized short-time vibrational motion for times of ≥0.3 ps (≥10 C–O vibrational periods). Second, in high resolution scans over the coarse vibrational features, characteristic clumps of individual vibration–rotation eigenstates were revealed. These clumps are ascribed to distinctive Franck–Condon active bright zero-order levels which are mixed with the large number of Franck–Condon inactive dark bath states. Under carefully selected conditions, the clumps could be attributed to states with defined and well known values of the total angular momentum quantum number J, which remains as a good quantum number in different coupling cases. These clump spectra will be analyzed quantitatively in the following paper with respect to their bearing for the intramolecular vibrational dynamics of highly excited CH3O (X̃) as a function of vibrational and rotational excitation. From the observed spectra, quantitative data can be obtained on the rate and extent of collision-free intramolecular vibrational and rovibrational energy redistribution (IVR and IRVR) processes, which would result after coherent ultrashort pulse excitation of the molecules.

A program for generating complete active spaces for relativistic atomic structure calculations

ADVERTISEMENT RETURN TO ISSUEPREVArticleNEXTComparison of classical simulations of the H + H2 reaction to accurate quantum mechanical state-to-state partial cross sections with total angular momenta J = 0-4 and to experiment for all JMeishan Zhao, Mirjana Mladenovic, Donald G. Truhlar, David W. Schwenke, Yan Sun, Donald J. Kouri, and Normand C. BlaisCite this: J. Am. Chem. Soc. 1989, 111, 3, 852–859Publication Date (Print):February 1, 1989Publication History Published online1 May 2002Published inissue 1 February 1989https://pubs.acs.org/doi/10.1021/ja00185a011https://doi.org/10.1021/ja00185a011research-articleACS PublicationsRequest reuse permissionsArticle Views92Altmetric-Citations35LEARN ABOUT THESE METRICSArticle Views are the COUNTER-compliant sum of full text article downloads since November 2008 (both PDF and HTML) across all institutions and individuals. These metrics are regularly updated to reflect usage leading up to the last few days.Citations are the number of other articles citing this article, calculated by Crossref and updated daily. Find more information about Crossref citation counts.The Altmetric Attention Score is a quantitative measure of the attention that a research article has received online. Clicking on the donut icon will load a page at altmetric.com with additional details about the score and the social media presence for the given article. Find more information on the Altmetric Attention Score and how the score is calculated. Share Add toView InAdd Full Text with ReferenceAdd Description ExportRISCitationCitation and abstractCitation and referencesMore Options Share onFacebookTwitterWechatLinked InRedditEmail Other access optionsGet e-Alertsclose Get e-Alerts

Electronic Eigenenergies of the Hydrogen Molecular Ion