Calculation Of Matrix Elements Research Articles

Avoiding matrix exponentials for large transition rate matrices.

Exact methods for the exponentiation of matrices of dimension N can be computationally expensive in terms of execution time (N3) and memory requirements (N2), not to mention numerical precision issues. A matrix often exponentiated in the natural sciences is the rate matrix. Here, we explore five methods to exponentiate rate matrices, some of which apply more broadly to other matrix types. Three of the methods leverage a mathematical analogy between computing matrix elements of a matrix exponential process and computing transition probabilities of a dynamical process (technically a Markov jump process, MJP, typically simulated using Gillespie). In doing so, we identify a novel MJP-based method relying on restricting the number of "trajectory" jumps that incurs improved computational scaling. We then discuss this method's downstream implications on mixing properties of Monte Carlo posterior samplers. We also benchmark two other methods of matrix exponentiation valid for any matrix (beyond rate matrices and, more generally, positive definite matrices) related to solving differential equations: Runge-Kutta integrators and Krylov subspace methods. Under conditions where both the largest matrix element and the number of non-vanishing elements scale linearly with N-reasonable conditions for rate matrices often exponentiated-computational time scaling with the most competitive methods (Krylov and one of the MJP-based methods) reduces to N2 with total memory requirements of N.

Investigation of the Possibility of Using Turn-by-Turn Measurements to Detect Perturbations in the Magnetic Structure of Accelerators

The PCA correction method consists of a global analysis of turn-by-turn data obtained from beam position monitors (BPMs). One of its main advantages is speed of operation and the ability to detect pulsating power supplies. Along with this, a wide range of applications characterizes the PCA method: determination of perturbations of the accelerator magnetic structure, calculation of transport matrix elements and optical functions, and determination of the magnitude of the transverse coupling. A software prototype implementing the proposed correction method was written and tested on the electron-positron collider VEPP-4M for experimental validation. In the experiments, perturbations were introduced one by one into individual elements of the skew-quadrupole and quadrupole corrections. In one case, the excitation of betatron oscillations was carried out by the kick of the inflector. In another case, the oscillations were excited by resonant swing with a depolariser. It was found that, despite the simplicity of practical implementation, the examined method has some nuances that make it difficult to obtain and identify correct results. Nevertheless, it was possible to identify the introduced perturbations of quadrupole and skew-quadrupole corrections. Moreover, it was possible to detect a magnet with low-frequency pulsations by characteristic changes in the estimated invariants. In addition, the possibility of determining the integral coupling parameters in experiments on the observation of resonantly excited eigen modes having the shape of ellipses in the plane transverse to the beam motion axis is considered.

First Study of the PIKACHU Project: Development and Evaluation of High-Purity Gd3Ga3Al2O12:Ce Crystals for 160Gd Double Beta Decay Search

Abstract Uncovering neutrinoless double beta decay (0ν2β) is crucial for confirming neutrinos’ Majorana characteristics. The decay rate of 0νββ is theoretically uncertain, influenced by nuclear matrix elements that vary across nuclides. To reduce this uncertainty, precise measurement of the half-life of neutrino-emitting double beta decay (2ν2β) in different nuclides is essential. We have launched the PIKACHU (Pure Inorganic scintillator experiment in KAmioka for CHallenging Underground sciences) project to fabricate high-purity Ce-doped Gd3Ga3Al2O12 (GAGG) single crystals and use them to study the double beta decay of 160Gd. Predictions from two theoretical models on nuclear matrix element calculations for 2ν2β in 160Gd show a significant discrepancy in estimated half-lives, differing by approximately an order of magnitude. If the lower half-life estimation holds true, detecting 2ν2β in 160Gd could be achievable with a sensitivity enhancement slightly more than an order of magnitude compared to prior investigations using Ce-doped Gd2SiO5 (GSO) crystal. We have successfully developed GAGG crystals with purity levels surpassing previous standards through refined purification and selection of raw materials. Our experiments with these crystals indicate the feasibility of reaching sensitivities exceeding those of earlier studies. This paper discusses the ongoing development and scintillator performance evaluation of High-purity GAGG crystals, along with the anticipated future prospects of the PIKACHU experiment.

Wilson loop duality and OPE for super form factors of half-BPS operators

We propose a dual Wilson loop description for the MHV super form factors of half-BPS operators in planar N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 4 super-Yang-Mills theory. In this description, the local operators are represented by on-shell states, made out of zero-momentum particles, that are absorbed by a null periodic super Wilson loop. We present evidence for this duality at weak coupling, by performing an explicit calculation of the Wilson loop matrix elements through one loop. At tree level, the interactions localize at the cusps of the loop, revealing a simple connection between the super form factors and the m = 2 tree amplituhedron. At loop level, we show that the Wilson loop calculation reproduces the known results for the super form factors. Inspired by this duality, we extend the OPE program developed for the form factors of the Lagrangian to the super form factors of the higher-charge operators. We introduce non-perturbative axioms and conjectures for the main building blocks that govern the exchange of the lightest flux-tube excitations. These blocks appear as simple refinements of the form factor transitions introduced in earlier OPE studies. They are expressed at any value of the ’t Hooft coupling in terms of the tilted Beisert-Eden-Staudacher kernel. We carry out checks of our conjectures up to two loops at weak coupling for three- and four-point form factors of half-BPS operators of various lengths, finding perfect agreement with perturbative data.

Spin-Adapted Externally Contracted Multireference Configuration Interaction Method Based on Selected Reference Configurations.

As one kind of approximation of the full configuration interaction solution, the selected configuration interaction (sCI) methods have been shown to be valuable for large active spaces. However, the inclusion of dynamic correlation beyond large active spaces is necessary for more quantitative results. Since the sCI wave function can provide a compact reference for multireference methods, previously, we proposed an externally contracted multireference configuration interaction method using the sCI reference reconstructed from the density matrix renormalization group wave function [J. Chem. Theory Comput. 2018, 14, 4747-4755]. The DMRG2sCI-EC-MRCI method is promising for dealing with more than 30 active orbitals and large basis sets. However, it suffers from two drawbacks: spin contamination and low efficiency when using Slater determinant bases. To solve these problems, in this work, we adopt configuration state function bases and introduce a new algorithm based on the hybrid of tree structure for convenient configuration space management and the graphical unitary group approach for efficient matrix element calculation. The test calculation of naphthalene shows that the spin-adapted version could achieve a speed-up of 6.0 compared with the previous version based on the Slater determinant. Examples of dinuclear copper(II) compound as well as Ln(III) and An(III) complexes show that the sCI-EC-MRCI can give quantitatively accurate results by including dynamic correlation over sCI for systems with large active spaces and basis sets.

A New A Priori Method to Avoid Calculation of Negligible Hamiltonian Matrix Elements in CI Calculation.

Hamiltonian matrices typically contain many elements that are negligibly small compared to the diagonal elements, even with methods to prune the underlying basis. Because for general potentials the calculation of H-matrix elements is a major part of the computational effort to obtain eigenvalues and eigenfunctions of the Hamiltonian, there is strong motivation to investigate locating these negligible elements without calculating them or at least avoid calculating them. We recently demonstrated an effective means to "learn" negligible elements using machine learning classification (J. Chem. Phys. 2023, 159, 071101). Here we present a simple, new method to avoid calculating them by using a cut-off value for the absolute difference in the quantum numbers for the bra and ket. This method is demonstrated for many of the same case studies as were used in the paper above, namely for realistic H-matrices of H2O, the vinyl radical, C2H3, and glycine, C2H5NO2. The new method is compared to the recently reported machine learning approach. In addition, we point out an important synergy between the two methods.

NISQ algorithm for the matrix elements of a generic observable

The calculation of off-diagonal matrix elements has various applications in fields such as nuclear physics and quantum chemistry. In this paper, we present a noisy intermediate scale quantum algorithm for estimating the diagonal and off-diagonal matrix elements of a generic observable in the energy eigenbasis of a given Hamiltonian without explicitly preparing its eigenstates. By means of numerical simulations we show that this approach finds many of the matrix elements for the one and two qubits cases. Specifically, while in the first case, one can initialize the ansatz parameters over a broad interval, in the latter the optimization landscape can significantly slow down the speed of convergence and one should therefore be careful to restrict the initialization to a smaller range of parameters.

QERaman: An open-source program for calculating resonance Raman spectra based on Quantum ESPRESSO

We present an open-source program QERaman that computes first-order resonance Raman spectroscopy of materials using the output data from Quantum ESPRESSO. Complex values of Raman tensors are calculated based on the quantum description of the Raman scattering from calculations of electron-photon and electron-phonon matrix elements, which are obtained by using the modified Quantum ESPRESSO. Our program also calculates the resonant Raman spectra as a function of incident laser energy for linearly- or circularly-polarized light. Hands-on tutorials for graphene and MoS2 are given to show how to run QERaman. All codes, examples, and scripts are available on the GitHub repository. Program summaryProgram Title:QERamanCPC Library link to program files:https://doi.org/10.17632/ddcrrrmxw9.1Developer's repository link:https://github.com/nguyen-group/QERamanLicensing provisions: GNU General Public Licence 3.0Programming language: FortranExternal routines:Quantum ESPRESSO v7.2Nature of problem: Resonance Raman spectra with first-principles calculations.Solution method: The Raman intensity formula is given by the quantum theory, in which the electron-photon and the electron-phonon matrix elements are obtained from the modified Quantum ESPRESSO package.

Primitive to conventional geometry projection for efficient phonon transport calculations

The primitive Wigner-Seitz cell and corresponding first Brillouin zone (FBZ) are typically used in calculations of lattice vibrational and transport properties as they contain the smallest number of degrees of freedom and thus have the cheapest computational cost. However, in complex materials, the FBZ can take on irregular shapes where lattice symmetries are not apparent. Thus, conventional cells (with more atoms and regular shapes) are often used to describe materials, though dynamical and transport calculations are more expensive. Here we discuss an efficient anharmonic lattice dynamic method that maps conventional cell dynamics to primitive cell dynamics based on translational symmetries. Such symmetries have not been utilized in typical lattice dynamical calculations. This leads to phase interference conditions that act like conserved quantum numbers and a conservation rule for phonon scattering that is hidden in conventional dynamics which significantly reduces the computational cost. We demonstrate this method for phonon transport in a variety of materials with inputs from first-principles calculations and attribute its efficiency to reduced scattering phase space and fewer summations in scattering matrix element calculations.

Algorithms on low energy spectra of the Hubbard model pertinent to molecular nanomagnets

SummaryComputer supported design of materials with tailored properties is an important part of computational science so that developments of new effective algorithms is a key issue. Molecular magnetism deals with complex objects described by the quantum physics and their simulations come up against the computational constraints. In this work, we consider a numerical version of a recently developed hybrid approach that combines the ab initio DFT method with the exact diagonalization of the microscopic Hubbard model (HM). The latter avoids the prior perturbative framework but increases computing resources needed for solving the eigenvalue problem of the matrix representation of HM. We demonstrate that in the case of the ring‐shape molecular nanomagnets the computational demands can be curbed due to efficient numerical matrix construction algorithms provided. These algorithms pertain both to the calculation of the single nonzero matrix elements and to their localization in the final sparse matrix that is subsequently diagonalized. Their implementation executed on a simple six‐core‐computer system and the Mathematica environment leads to a significant gain in the computing time for the exemplary chromium‐based molecule denoted Cr, running the code sequentially. We claim, referring to the numerical diagonalization of the spin Hamiltonian matrices, that the parallelization flaws related to the high‐level programming approach can be all removed, porting the code to the appropriate HPC environment. A prospective implementation of the code, based on some dedicated and optimized mathematical libraries, will ultimately open a window for further applications of the Hubbard model in molecular magnetism.

Stacking effect on the orbital characters and transition probability of bilayer MoS2

In this paper, employing the van der Waals (vdW) corrected density functional method, the electronic and optical properties of bilayer MoS2 are calculated for five different stacking patterns ( and ). In this regard, the structural stability and the van der Waals interlayer distance are investigated. The minimum formation energy has indicated the overall trend of relative stability to be as This means and have the most and and have the least stable structures. The band structure and probability distribution for finding electrons in atomic orbitals have also been calculated in the Brillouin zone. The results obtained from band structures and orbital characters show the same trend in the three most stable and a different but the same trend for the two least stable orders. In addition, the oscillator strength has been derived from the calculation of the momentum operator matrix elements using first-principles methods. The calculation of transition probabilities from matrix elements indicates that and have higher transition probabilities than that of and The study of transition probability also shows the y-polarized light is stronger than the x-polarized light for all stacking patterns.

Study of the quenching of the GT-decay operator in a microscopic shell-model approach

The need of a reliable calculation of the nuclear matrix elements for the 0νββ decay has ignited a new interest about the quenching of the axial coupling constant gA , a procedure introduced to reproduce experimental results connected with GT decays. The goal of this work is to present a preliminary study to tackle this problem within the framework of the realistic shell model.

Measurements of jet multiplicity and jet transverse momentum in multijet events in proton–proton collisions at {sqrt{s}=13, text {TeV}}

Multijet events at large transverse momentum (p_{textrm{T}}) are measured at sqrt{s}=13,text {TeV} using data recorded with the CMS detector at the LHC, corresponding to an integrated luminosity of 36.3{,text {fb}^{-1}} . The multiplicity of jets with p_{textrm{T}} >50,text {GeV} that are produced in association with a high-p_{textrm{T}} dijet system is measured in various ranges of the p_{textrm{T}} of the jet with the highest transverse momentum and as a function of the azimuthal angle difference varDelta phi _{1,2} between the two highest p_{textrm{T}} jets in the dijet system. The differential production cross sections are measured as a function of the transverse momenta of the four highest p_{textrm{T}} jets. The measurements are compared with leading and next-to-leading order matrix element calculations supplemented with simulations of parton shower, hadronization, and multiparton interactions. In addition, the measurements are compared with next-to-leading order matrix element calculations combined with transverse-momentum dependent parton densities and transverse-momentum dependent parton shower.

Calculation of dipole matrix elements for the modeling of H-[formula omitted] Stark broadening in the presence of large magnetic field

B-spline basis sets are applied to the calculation of dipole matrix elements for the hydrogen atom placed in an intense magnetic field, in view of applications to the spectroscopic analysis of white dwarf atmospheres. A focus is put on the H-α line and magnetic field values in the kilotesla range are considered. The dipole matrix elements are shown to be significantly sensitive to the magnetic field. In the framework of plasma diagnostic, this means that the Stark broadening of the line, which is involved in density measurements, can be misinterpreted if the atomic structure is not well accounted for in numerical codes. As an illustration, we consider the H-ασ− component and apply a computer simulation method to the modeling of the Stark broadening of this line. The calculated dipole matrix elements are used therein as input.

Oscillator brackets at non equal particle masses

Abstract. Efficient programs to calculate two-particle oscillator brackets (or Talmi-Moshinsky-Smirnov coefficients) have been constructed recently, Comput. Phys. Comm. 265 (2021) 108005. The corresponding Talmi-Smirnov transformation is mostly performed in case of particles of equal masses and the constructed programs have been tested in detail as to this case. However, it occurred that those programs included a bug revealing itself in case when particle masses are different. The bug is eliminated in the present version of the programs. All the various tests employed previously at equal masses of particles now reproduce the required results for the case of different masses as well. Program summaryProgram Title: OSBRACKETSCPC Library link to program files:https://doi.org/10.17632/4m594wzv94.2Licensing provisions: GPLv3Programming language: Fortran-90Journal Reference of previous version: Comput. Phys. Comm. 265 (2021) 108005Does the new version supersede the previous version?: YesReasons for the new version: Necessity to fix a bug in the programs which affected the brackets in case of the Talmi-Smirnov transformation at non equal particle masses. The author is thankful to M.A. Sharaf who pointed out the incorrect results pertaining to the previous version of the programs.Summary of revisions: The subroutine to calculate the quantity (23) from Ref. [1] is corrected. This quantity includes powers of tan⁡φ where φ is the angle specifying the kinematic rotation. In case of equal mass particles one has |tan⁡φ|=1, see [1]. Most applications of the brackets in the literature refer to this case and the previous version of the programs has been tested at |tan⁡φ|=1. However, the mentioned subroutine included a bug that reveals itself in the |tan⁡φ|≠1 case. In the present version of the programs, the bug is eliminated. All the various tests described in [1] now reproduce the required results also at |tan⁡φ|≠1, i.e., at arbitrary mass ratio of particles.Nature of problem: Single-particle basis oscillator states are widely used for studying the structure of various many-body systems. To compute matrix elements of two-body operators, the Talmi-Smirnov transformation of oscillator states is performed. Coefficients of this transformation are called oscillator brackets. Often it is necessary to retain large sets of basis oscillator states in calculations. Therefore, a fast program to compute the brackets is needed. The program should provide accurate results up to high oscillator excitations.Solution method: At zero radial quantum numbers, oscillator brackets are calculated using an explicit expression that includes few summations. Starting from such brackets, recurrence relations are employed to calculate the brackets of the general type. These relations prove to work perfectly up to very high oscillator excitations.

NuHamil : A numerical code to generate nuclear two- and three-body matrix elements from chiral effective field theory

The applicability of nuclear ab initio calculations has rapidly extended over the past decades. However, starting research projects is still challenging due to the required numerical expertise in the generation of underlying nuclear interaction matrix elements and many-body calculations. To ease the first issue, in this paper we introduce the numerical code NuHamil to generate the nucleon-nucleon (NN) and three-nucleon (3N) matrix elements expressed in a spherical harmonic-oscillator basis, inputs of many-body calculations. The ground-state energies for the selected doubly closed shell nuclei are calculated with the no-core shell-model (NCSM) and in-medium similarity renormalization group (IMSRG). The code is written in modern Fortran, and OpenMP+MPI hybrid parallelization is available for the 3N matrix-element calculations.

Measurement of single top-quark production in the s-channel in proton–proton collisions at sqrt{s} = 13 TeV with the ATLAS detector

A measurement of single top-quark production in the s-channel is performed in proton–proton collisions at a centre-of-mass energy of 13 TeV with the ATLAS detector at the CERN Large Hadron Collider. The dataset corresponds to an integrated luminosity of 139 fb−1. The analysis is performed on events with an electron or muon, missing transverse momentum and exactly two b-tagged jets in the final state. A discriminant based on matrix element calculations is used to separate single-top-quark s-channel events from the main background contributions, which are top-quark pair production and W-boson production in association with jets. The observed (expected) signal significance over the background-only hypothesis is 3.3 (3.9) standard deviations, and the measured cross-section is sigma ={8.2}_{-2.9}^{+3.5} pb, consistent with the Standard Model prediction of {sigma}^{textrm{SM}}={10.32}_{-0.36}^{+0.40} pb.

Impact of nuclear matrix element calculations for current and future neutrinoless double beta decay searches

Nuclear matrix elements (NME) are a crucial input for the interpretation of neutrinoless double beta decay data. We consider a representative set of recent NME calculations from different methods and investigate the impact on the present bound on the effective Majorana mass mββ by performing a combined analysis of the available data as well as on the sensitivity reach of future projects. A crucial role is played by the recently discovered short-range contribution to the NME, induced by light Majorana neutrino masses. Depending on the NME model and the relative sign of the long- and short-range contributions, the current 3σ bound can change between mββ< 40 meV and 600 meV. The sign-uncertainty may either boost the sensitivity of next-generation experiments beyond the region for mββ predicted for inverted mass ordering or prevent even advanced setups to reach this region. Furthermore, we study the possibility to distinguish between different NME calculations by assuming a positive signal and by combining measurements from different isotopes. Such a discrimination will be impossible if the relative sign of the long- and short-range contribution remains unknown, but can become feasible if mββ ≳ 40 meV and if the relative sign is known to be positive. Sensitivities will be dominated by the advanced 76Ge and 136Xe setups assumed here, but NME model-discrimination improves if data from a third isotope is added, e.g., from 130Te or 100Mo.

Robust formulation of Wick's theorem for computing matrix elements between Hartree-Fock-Bogoliubov wavefunctions.

Numerical difficulties associated with computing matrix elements of operators between Hartree-Fock-Bogoliubov (HFB) wavefunctions have plagued the development of HFB-based many-body theories for decades. The problem arises from divisions by zero in the standard formulation of the nonorthogonal Wick's theorem in the limit of vanishing HFB overlap. In this Communication, we present a robust formulation of Wick's theorem that stays well-behaved regardless of whether the HFB states are orthogonal or not. This new formulation ensures cancellation between the zeros of the overlap and the poles of the Pfaffian, which appears naturally in fermionic systems. Our formula explicitly eliminates self-interaction, which otherwise causes additional numerical challenges. A computationally efficient version of our formalism enables robust symmetry-projected HFB calculations with the same computational cost as mean-field theories. Moreover, we avoid potentially diverging normalization factors by introducing a robust normalization procedure. The resulting formalism treats even and odd number of particles on equal footing and reduces to Hartree-Fock as a natural limit. As proof of concept, we present a numerically stable and accurate solution to a Jordan-Wigner-transformed Hamiltonian, whose singularities motivated the present work. Our robust formulation of Wick's theorem is a most promising development for methods using quasiparticle vacuum states.

Riemannian Trust Region Method for Minimization of the Fourth Central Moment for Localized Molecular Orbitals.

The importance of localized molecular orbitals (MOs) in correlation treatments beyond mean-field calculation and in the illustration of chemical bonding (and antibonding) can hardly be overstated. However, the generation of orthonormal localized occupied MOs is significantly more straightforward than obtaining orthonormal localized virtual MOs. Orthonormal MOs allow facile use of highly efficient group theoretical methods (e.g., graphical unitary group approach) for calculation of Hamiltonian matrix elements in multireference configuration interaction calculations (such as MRCISD) and in quasi-degenerate perturbation treatments, such as the Generalized Van Vleck Perturbation Theory. Moreover, localized MOs can elucidate qualitative understanding of bonding in molecules, in addition to high-accuracy quantitative descriptions. We adopt the powers of the fourth moment cost function introduced by Jørgensen and coworkers. Because the fourth moment cost functions are prone to having multiple negative Hessian eigenvalues when starting from easily available canonical (or near-canonical) MOs, standard optimization algorithms can fail to obtain the orbitals of the virtual or partially occupied spaces. To overcome this drawback, we applied a trust region algorithm on an orthonormal Riemannian manifold with an approximate retraction from the tangent space built into the first and second derivatives of the cost function. Moreover, the Riemannian trust region outer iterations were coupled to truncated Conjugate Gradient inner loops, which avoided any costly solutions of simultaneous linear equations or eigenvector/eigenvalue solutions. Numerical examples are provided on model systems, including the high-connectivity H10 set in 1-, 2-, and 3-dimensional arrangements, and on a chemically realistic description of cyclobutadiene (c-C4H4) and the propargyl radical (C3H3). In addition to demonstrating the algorithm on occupied and virtual blocks of orbitals, the method is also shown to work on the active space at the MCSCF level of theory.