Expressions For Matrix Elements Research Articles

Explicitly correlated Gaussians for high-precision variational calculations of S^{e},P^{e}, and D^{e} states of quantum systems: An efficient algorithm.

In this work we consider an efficient algorithm for variational calculations of quantum few-particle systems in S,P, and D states of the even parity using all-particle explicitly correlated Gaussian (ECG) basis sets. We primarily focus on the description of states where the dominant configuration contains either two particles in p states or a single particle in a d state (all other particles are in s states). The basis functions we consider are products of spherically symmetric ECGs and bipolar harmonics. We introduced a scheme for deriving expressions for matrix elements of the overlap, kinetic and potential energy, as well as their derivatives with respect to the nonlinear parameters of the Gaussians. This allowed us to improve the efficiency of numerical calculations of the matrix elements (which is the most critical part of any code that uses ECGs) by one to two orders of magnitude compared to previous implementations. We provide a complete set of formulas for all basic matrix elements using the formalism of matrix differential calculus and discuss some technical details relevant to their efficient implementation. Lastly, we report a few example calculations of the ^{2}D^{e} states of the Li atom, ^{4}P^{e} and ^{2}D^{e} states of the B atom, and ^{3}P^{e} states of the C atom.

MadFlow: automating Monte Carlo simulation on GPU for particle physics processes

We present MadFlow, a first general multi-purpose framework for Monte Carlo (MC) event simulation of particle physics processes designed to take full advantage of hardware accelerators, in particular, graphics processing units (GPUs). The automation process of generating all the required components for MC simulation of a generic physics process and its deployment on hardware accelerator is still a big challenge nowadays. In order to solve this challenge, we design a workflow and code library which provides to the user the possibility to simulate custom processes through the MadGraph5_aMC@NLO framework and a plugin for the generation and exporting of specialized code in a GPU-like format. The exported code includes analytic expressions for matrix elements and phase space. The simulation is performed using the VegasFlow and PDFFlow libraries which deploy automatically the full simulation on systems with different hardware acceleration capabilities, such as multi-threading CPU, single-GPU and multi-GPU setups. The package also provides an asynchronous unweighted events procedure to store simulation results. Crucially, although only Leading Order is automatized, the library provides all ingredients necessary to build full complex Monte Carlo simulators in a modern, extensible and maintainable way. We show simulation results at leading-order for multiple processes on different hardware configurations.

MadFlow: towards the automation of Monte Carlo simulation on GPU for particle physics processes

In this proceedings we present MadFlow, a new framework for the automation of Monte Carlo (MC) simulation on graphics processing units (GPU) for particle physics processes. In order to automate MC simulation for a generic number of processes, we design a program which provides to the user the possibility to simulate custom processes through the Mad-Graph5_aMC@NLO framework. The pipeline includes a first stage where the analytic expressions for matrix elements and phase space are generated and exported in a GPU-like format. The simulation is then performed using the VegasFlow and PDFFlow libraries which deploy automatically the full simulation on systems with different hardware acceleration capabilities, such as multi-threading CPU, single-GPU and multi-GPU setups. We show some preliminary results for leading-order simulations on different hardware configurations.

Comments on “Decoupling Capacitor Placement on Resonant Parallel-Plates Via Driving Point Impedance” [Jun 19 2162-2171

In [1] , a semianalytical method based on 2-D Newton-Raphson algorithm is proposed to evaluate the effectiveness of decoupling capacitor placement on a rectangular power plane. Lei et al. [2] present analytical derivation of the Jacobian matrix elements and the related partial derivative expressions based on the cavity mode expansion of power-ground plane impedance. Unfortunately, some of the presented Jacobian matrix element expressions presented in [1] does not correspond to the proposed second relation for the application of the Newton-Raphson algorithm. Also, several errors are found in the presented results regarding the power-ground plane impedance, and various partial derivative expressions for the input impedance and the impedance of parallel plates in [1] . These errors are investigated and corrected in this work.

Dipole polarizability for confined two-electron atomic systems: simple approximations

The electric dipole polarizability is analyzed for a two-electron system embedded in an impenetrable spherically symmetric cavity. Methods to find simple estimates of polarizability and corrections to them are considered. The expressions for matrix elements in Hylleraas coordinates required for energy and polarizability variational calculations are presented for the two-electron atomic system. The results of simple polarizability estimates are presented for the ground state of the helium atom and are compared with polarizabilities calculated using Hylleraas variational perturbation theory and with recently calculated polarizabilities reported in the literature for confined systems.

Some dipole shower studies

Parton showers have become a standard component in the description of high-energy collisions. Nowadays most final-state ones are of the dipole character, wherein a pair of partons branches into three, with energy and momentum preserved inside this subsystem. For initial-state showers a dipole picture is also possible and commonly used, but the older global-recoil strategy remains a valid alternative, wherein larger groups of partons share the energy–momentum preservation task. In this article we introduce and implement a dipole picture also for initial-state radiation in Pythia, and compare with the existing global-recoil one, and with data. For the case of Deeply Inelastic Scattering we can directly compare with matrix element expressions and show that the dipole picture gives a very good description over the whole phase space, at least for the first branching.

Algebraic evaluation of matrix elements in the Laguerre function basis

The Laguerre functions constitute one of the fundamental basis sets for calculations in atomic and molecular electron-structure theory, with applications in hadronic and nuclear theory as well. While similar in form to the Coulomb bound-state eigenfunctions (from the Schroedinger eigenproblem) or the Coulomb-Sturmian functions (from a related Sturm-Liouville problem), the Laguerre functions, unlike these former functions, constitute a complete, discrete, orthonormal set for square-integrable functions in three dimensions. We construct the SU(1,1)xSO(3) dynamical algebra for the Laguerre functions and apply the ideas of factorization (or supersymmetric quantum mechanics) to derive shift operators for these functions. We use the resulting algebraic framework to derive analytic expressions for matrix elements of several basic radial operators (involving powers of the radial coordinate and radial derivative) in the Laguerre function basis. We illustrate how matrix elements for more general spherical tensor operators in three dimensional space, such as the gradient, may then be constructed from these radial matrix elements.

A computer code for calculations in the algebraic collective model of the atomic nucleus

A Maple code is presented for algebraic collective model (ACM) calculations. The ACM is an algebraic version of the Bohr model of the atomic nucleus, in which all required matrix elements are derived by exploiting the model’s SU(1,1)×SO(5) dynamical group. This paper reviews the mathematical formulation of the ACM, and serves as a manual for the code.The code enables a wide range of model Hamiltonians to be analysed. This range includes essentially all Hamiltonians that are rational functions of the model’s quadrupole moments qˆM and are at most quadratic in the corresponding conjugate momenta πˆN (−2≤M,N≤2). The code makes use of expressions for matrix elements derived elsewhere and newly derived matrix elements of the operators [πˆ⊗qˆ⊗πˆ]0 and [πˆ⊗πˆ]LM. The code is made efficient by use of an analytical expression for the needed SO(5)-reduced matrix elements, and use of SO(5)⊃SO(3) Clebsch–Gordan coefficients obtained from precomputed data files provided with the code. Program summaryProgram title: ACMCatalogue identifier: AEYO_v1_0Program summary URL: http://cpc.cs.qub.ac.uk/summaries/AEYO_v1_0.htmlProgram obtainable from: CPC Program Library, Queen’s University, Belfast, N. IrelandLicensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.htmlNo. of lines in distributed program, including test data, etc.: 3873526No. of bytes in distributed program, including test data, etc.: 46345414Distribution format: tar.gzProgramming language: Maple 18 (or versions 17, 16, 15).Computer: Any.Operating system: Any which supports Maple; tested under Linux, Max OSX, Windows 7.RAM: 500MbClassification: 17.20.Nature of problem: The calculation of energy eigenvalues, transition rates and amplitudes of user specified Hamiltonians in the Bohr model of the atomic nucleus.Solution method: Exploit the model’s SU(1,1)×SO(5) dynamical group to calculate analytic (as far as possible) expressions for matrix elements, making use of extensive files (supplied) of SO(5)⊃SO(3) Clebsch–Gordan coefficients. Diagonalisation of the resulting matrices (once the entries are converted to floating point) is carried out using the Maple library procedure Eigenvectors. (Maple [1] makes use of the NAG [2] and CLAPACK [3] linear algebra libraries.)Additional comments: 1.The dimension of the Hilbert space that can be handled is limited only by the available computer memory and the available SO(5)⊃SO(3) Clebsch–Gordan coefficients (v1α1L1v2α2L2∥v3α3L3).2.The supplied data files provide coefficients (v1α1L1v2α2L2∥v3α3L3) for 1≤v2≤6, and contain all non-zero coefficients for v1<v3≤50 when v2∈1,3, for v1≤v3≤30 when v2∈2,4, and for v1≤v3≤25 when v2∈5,6. (Once calculated, further coefficients can be readily made available to the code without changing the code.) Thus, depending on the model Hamiltonian being analysed, the states in the Hilbert space used are limited in their seniority. For analysis of the more typical types of model Hamiltonian, only the coefficients with v2∈{1,3} are required, and therefore, with the supplied files, the seniority limit is 50. More exotic Hamiltonians having terms with seniority v2∈{2,4,5,6} would have the seniority limited to 30 or 25 accordingly.3.The code provides lower level procedures that give ready access to the Clebsch–Gordan coefficients and the SU(1, 1) and SO(5) matrix elements. These procedures are described in the manuscript and enable extensions to the code and model to be made easily.4.The accuracy to which Maple performs numerical calculations is determined by the Maple parameter Digits, which specifies the number of significant decimal digits used. The default value of 10 is more than adequate for most ACM calculations. Note, however, that if Digits is increased beyond a certain value (obtained from the Maple command evalhf(Digits), and usually 15 on modern computers) then the code can no longer take advantage of hardware mathematical operations, and is significantly slower.Documents included1.The code makes use of SO(5)⊃SO(3) Clebsch–Gordan coefficients which are supplied in zip files, and must be installed by the user.2.A Maple worksheet that gives various example calculations and tests carried out using procedures from the code is provided.3.A 162 page PDF file containing everything displayed in the worksheet (input, output and comments, and making use of colour) is also provided. !!!!! The distribution file for this program is over 46 Mbytes and therefore is not delivered directly when download or Email is requested. Instead a html file giving details of how the program can be obtained is sent. !!!!!Running time: For a fixed value of the parameter Digits, the running time depends on the dimension of the Hilbert space on which the diagonalisation is performed, and this in turn is governed by the number of eigenvalues required and the accuracy required. Note that diagonalisation is performed separately in each L-space. For typical ACM calculations (such as those carried out in [4]), the matrices being diagonalised are usually of dimension at most a few hundred, and often much smaller. On a modest personal computer, the computation for the smallest cases takes at most a few seconds. The worksheet contains a range of examples for which the calculation time varies between a few seconds and 750s. In the latter case, diagonalisation is performed on L-spaces for 0≤L≤8, the dimensions of these spaces being between 154 and 616.

Two-electron exchange interaction between polar molecules and atomic ions — Asymptotic approach

We have described the asymptotic approach for calculation of the two-electron exchange interaction between atomic ion and polar molecule responsible for direct double electron transfer processes. The closed analytic expression for matrix element of exchange interaction has been obtained in the framework of the semiclassical version of the asymptotic theory and point-dipole approximation for description of the polar molecule.

Explicit expressions for totally symmetric spherical functions and symmetry-dependent properties of multipoles

Closed expressions for matrix elements 〈 lm' | A (G)| lm 〉, where | lm 〉 are spherical functions and A (G) is the average of all symmetry operators of point group G, are derived for all point groups (PGs) and then used to obtain linear combinations of spherical functions that are totally symmetric under all symmetry operations of G. In the derivation, we exploit the product structure of the groups. The obtained expressions are used to explore properties of multipoles of symmetric charge distributions. We produce complete lists of selection rules for multipoles Q l and their moments Q lm , as well as of numbers of independent moments in a multipole, for any l and m and for all PGs. Periodicities and other trends in these properties are revealed.

Unitary group approach to the many-electron correlation problem: spin-dependent operators

Following a brief overview of the unitary group approach (UGA) to the many-electron correlation problem, focusing in particular on Shavitt’s contribution via his graphical unitary group approach, we present a short review of our earlier results for the evaluation of matrix elements (MEs) of unitary group generators or products of generators in the electronic Gel’fand–Tsetlin basis with the help of spin-adapted second-quantization-like creation and annihilation vector operators at the unitary group level. This formalism is then extended to a spin-dependent case that is required when accounting for relativistic effects by developing explicit expressions for MEs of spin-orbital creation and annihilation operators in terms of the standard spin-adapted UGA basis. This leads naturally to a segmentation of these MEs and enables the evaluation of spin-dependent one-body operators while relying largely on the segment values of the standard spin-independent UGA.

Structure and properties of the algebra of partially transposed permutation operators

We consider the structure of algebra of operators, acting in n-fold tensor product space, which are partially transposed on the last term. Using purely algebraical methods we show that this algebra is semi-simple and then, considering its regular representation, we derive basic properties of the algebra. In particular, we describe all irreducible representations of the algebra of partially transposed operators and derive expressions for matrix elements of the representations. It appears that there are two kinds of irreducible representations of the algebra. The first one is strictly connected with the representations of the group S(n − 1) induced by irreducible representations of the group S(n − 2). The second kind is structurally connected with irreducible representations of the group S(n − 1).

The Monte Carlo event generator AcerMC versions 2.0 to 3.8 with interfaces to PYTHIA 6.4, HERWIG 6.5 and ARIADNE 4.1

The AcerMC Monte Carlo generator is dedicated to the generation of Standard Model background processes which were recognised as critical for the searches at LHC, and generation of which was either unavailable or not straightforward so far. The program itself provides a library of the massive matrix elements (coded by MADGRAPH) and native phase space modules for generation of a set of selected processes. The hard process event can be completed by the initial and the final state radiation, hadronisation and decays through the existing interface with either PYTHIA, HERWIG or ARIADNE event generators and (optionally) TAUOLA and PHOTOS. Interfaces to all these packages are provided in the distribution version. The phase-space generation is based on the multi-channel self-optimising approach using the modified Kajantie–Byckling formalism for phase space construction and further smoothing of the phase space was obtained by using a modified ac-VEGAS algorithm. An additional improvement in the recent versions is the inclusion of the consistent prescription for matching the matrix element calculations with parton showering for a select list of processes. Program summaryProgram title: AcerMC version 3.8Catalogue identifier: ADQQ_v2_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/ADQQ_v2_0.htmlProgram obtainable from: CPC Program Library, Queen’s University, Belfast, N. IrelandLicensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.htmlNo. of lines in distributed program, including test data, etc.: 3853309No. of bytes in distributed program, including test data, etc.: 68045728Distribution format: tar.gzProgramming language: FORTRAN 77 with popular extensions (g77, gfortran).Computer: All running Linux.Operating system: Linux.Classification: 11.2, 11.6.External routines: CERNLIB (http://cernlib.web.cern.ch/cernlib/), LHAPDF (http://lhapdf.hepforge.org/)Catalogue identifier of previous version: ADQQ_v1_0Journal reference of previous version: Comput. Phys. Comm. 149(2003)142Does the new version supersede the previous version?: YesNature of problem: Despite a large repertoire of processes implemented for generation in event generators like PYTHIA [1] or HERWIG [2] a number of background processes, crucial for studying the expected physics of the LHC experiments, is missing. For some of these processes the matrix element expressions are rather lengthy and/or to achieve a reasonable generation efficiency it is necessary to tailor the phase space selection procedure to the dynamics of the process. That is why it is not practical to imagine that any of the above general purpose generators will contain every, or even only observable, processes which will occur at LHC collisions. A more practical solution can be found in a library of dedicated matrix-element-based generators, with the standardised interfaces like that proposed in [3], to the more universal one which is used to complete the event generation.Solution method: The AcerMC EventGenerator provides a library of the matrix-element-based generators for several processes. The initial- and final-state showers, beam remnants and underlying events, fragmentation and remaining decays are supposed to be performed by the other universal generator to which this one is interfaced. We will call it a supervising generator. The interfaces to PYTHIA 6.4, ARIADNE 4.1 and HERWIG 6.5, as such generators, are provided. Provided is also an interface to TAUOLA [4] and PHOTOS [5] packages for τ-lepton decays (including spin correlations treatment) and QED radiations in decays of particles. At present, the following matrix-element-based processes have been implemented: gg,qq̄→tt̄bb̄, qq̄→W(→ℓν)bb̄; qq̄→W(→ℓν)tt̄; gg,qq̄→Z/γ∗(→ℓℓ)bb̄; gg,qq̄→Z/γ∗(→ℓℓ,νν,bb̄)tt̄; complete EW gg,qq̄→(Z/W/γ∗→)tt̄bb̄; gg,qq̄→tt̄tt̄; gg,qq̄→(tt̄→)ff̄bff̄b̄; gg,qq̄→(WWbb→)ff̄ff̄bb̄. Both interfaces allow the use of the LHAPDF/LHAGLUE library of parton density functions. Provided is also a set of control processes: qq̄→W→ℓν; qq̄→Z/γ∗→ℓℓ; gg,qq̄→tt̄ and gg→(tt̄→)WbWb̄;Reasons for new version: Implementation of several new processes and methods.Summary of revisions: Each version added new processes or functionalities, a detailed list is given in the section “Changes since AcerMC 1.0”.Restrictions: The package is optimised for the 14 TeV pp collision simulated in the LHC environment and also works at the achieved LHC energies of 7 TeV and 8 TeV. The consistency between results of the complete generation using PYTHIA 6.4 or HERWIG 6.5 interfaces is technically limited by the different approaches taken in both these generators for evaluating αQCD and αQED couplings and by the different models for fragmentation/hadronisation. For the consistency check, in the AcerMC library contains native coded definitions of the QCD and αQED. Using these native definitions leads to the same total cross-sections both with PYTHIA 6.4 or HERWIG 6.5 interfaces.Additional comments: !!!!! The distribution file for this program is over 67 Mbytes and therefore is not delivered directly when download or Email is requested. Instead an html file giving details of how the program can be obtained is sent. !!!!!Running time: On an PIII 800 MHz PC it amounts to ∼0.05→1.1 events/sec, depending on the choice of process.

Analysis of phonon-induced spin relaxation processes in silicon

We study all of the leading-order contributions to spin relaxation of conduction electrons in silicon due to the electron-phonon interaction. Using group theory, the $k\ifmmode\cdot\else\textperiodcentered\fi{}p$ perturbation method, and the rigid-ion model, we derive an extensive set of matrix element expressions for all of the important spin-flip transitions in the multivalley conduction band. The scattering angle has an explicit dependence on the electron wave vectors, phonon polarization, valley position, and spin orientation of the electron. Comparison of the derived analytical expressions with results of empirical pseudopotential and adiabatic band charge models shows excellent agreement.

Vibration of Diatomic System in One-Dimensional Nanomaterials

By means of the hypergeometric series method, the explicit expressions of energy eigenvalues and eigenfunctions of bound states for a diatomic system with a hyperbolic potential function are obtained in the one-dimensional nanomaterials. The eigenfunctions of a one-dimensional diatomic system, expressed in terms of the Jacobi polynomial, are employed as an orthonormal basis set, and the analytic expressions of matrix elements for position and momentum operators are given in a closed form.

Quantum electrodynamic problem of interaction of two quasimolecular electrons within the limits of effects of the 3-rd order of quantum electrodynamics

Effects of the third order of quantum electrodynamics, within the limits of which, the problem of interaction of two quasimolecular electrons has been solved, which are situated on any distance one from another at different atoms (nuclei) are being considered. The general expression for matrix elements of the operator of effective energy of two quasimolecular electrons interaction with the external field of radiation has been received. It is shown, that consecutive carrying out of procedure of symmetrisations of the factor of delay concerning to both electrons leads to occurrence the relativistic operator of interaction of two quasimolecular electrons.

The relativistic operator of interaction of two quasimolecular electrons as a third-order effect of quantum electrodynamics

We solve the problem of interaction two quasimolecular electrons located at an arbitrary separation near different atoms (nuclei). We consider third-order effects in quantum electrodynamics, which include the virtual photon exchange between electrons with emission (absorption) of a real photon. We obtain the general expression for matrix elements of the operator of the effective interaction energy of two quasimolecular electrons with the external radiation field, which allows calculating probabilities of inelastic processes with rearrangement at slow collisions of multicharge ions with relativistic atoms. We demonstrate that consistently taking the natural condition of the interaction symmetry with respect to the two electrons into account results in the appearance of additional terms in the operators of spin-orbit, spin-spin, and retarded interactions compared with the previously obtained expressions for these operators. We construct the operator of the dipole-dipole interaction of two neutral atoms located at an arbitrary separation.

Closed expressions for matrix elements of the trigonometric Pöschl–Teller potential

Analytical matrix elements of the x n ( n > 0 ) and [ tan ( x ) ] m [ cos ( x ) ] m ′ d n / d x n operators are derived using the eigenfunctions of the symmetric trigonometric Pöschl–Teller potential. The closed formulas are written in terms of Gauss hypergeometric functions and could be used in variational calculations to describe vibrational energy levels associated with bending modes. Multiprecision computational packages are considered in order to obtain an arbitrary level of precision.

Electronic structure and optical properties of quantum-confined lead salt nanowires

In the framework of four-band envelope-function formalism, developed earlier for spherical semiconductor nanocrystals, we study the electronic structure and optical properties of quantum-confined lead-salt (PbSe and PbS) nanowires (NWs) with a strong coupling between the conduction and the valence bands. We derive spatial quantization equations, and calculate numerically energy levels of spatially quantized states of a transverse electron motion in the plane perpendicular to the NW axis, and electronic subbands developed due to a free longitudinal motion along the NW axis. Using explicit expressions for eigenfunctions of the electronic states, we also derive analytical expressions for matrix elements of optical transitions and study selection rules for interband absorption. Next we study a two-particle problem with a conventional long-range Coulomb interaction and an interparticle coupling via medium polarization. The obtained results show that due to a large magnitude of the high-frequency dielectric permittivity of PbSe material, and hence, a high dielectric NW/vacuum contrast, the effective coupling via medium polarization significantly exceeds the effective direct Coulomb coupling at all interparticle separations along the NW axis. Furthermore, the strong coupling via medium polarization results in a bound state of the longitudinal motion of the lowest-energy electron-hole pair (a longitudinal exciton), while fast transverse motions of charge carriers remain independent of each other.

Displacement matrix elements of Deng-Fan oscillators

Analytic expressions for matrix elements of integral powers of the displacement of the coordinate from equilibrium for the Deng-Fan oscillator have been derived. A comparison is made with the corresponding matrix elements for the Morse oscillator. These matrix elements are useful in high overtone spectroscopy and in models for DNA melting.