Reduced Matrix Elements Research Articles

Critically evaluated theoretical atomic properties of Y iii

A systematic study of Y iii atomic properties is carried out using high-precision relativistic all-order method. Recommended values and estimates of their uncertainties are provided for a large number of electric-dipole reduced matrix elements, transition rates, and oscillator strengths for allowed transitions between $ns$, $n{p}_{j}$, $n{d}_{j}$, $n{f}_{j}$, and $n{g}_{j}$ levels with $n\ensuremath{\le}8$. The lifetimes of these levels are also evaluated. Electric-quadrupole and magnetic-dipole matrix elements are calculated to determine lifetimes of the $4{d}_{5/2}$ and $5s$ metastable levels. The ground-state $E1$, $E2$, and $E3$ static polarizabilities are calculated. This work provides recommended values critically evaluated for their accuracy for a number of Y iii atomic properties for use in theoretical modeling as well as planning and analysis of various experiments. We hope that the present study will stimulate further exploration of Y iii for various applications owing to its interesting structure of different low-lying metastable levels.

Reduced Matrix Elements for Symmetry-Constructed Systems

Eigenvalue problems involving symmetry, such as the Schrodinger equation when the Hamilto- nian commutes with a group, can generally be reduced in size using group theoretical techniques such as the Wigner-Eckart theorem. The key step is calculation of the reduced matrix elements followed by ei- genvalue determination by the secular equation. For finite groups it is usual to obtain reduced matrices by transformation to the symmetry adapted basis. Direct determination of reduced matrix elements by some means would be computationally more efficient with better precision.It is shown here that this direct de- termination is possible to some extent for symmetry-constructed systems such as symmetry-generated molecules. A simple illustration is given using the Huckel treatment of the cyclopropenyl radical (doi: 10.5562/cca2307)

Invariants and reduced matrix elements associated with the Lie superalgebra gl(m|n)

We construct explicit formulae for the eigenvalues of certain invariants of the Lie superalgebra gl(m|n) using characteristic identities. We discuss how such eigenvalues are related to reduced Wigner coefficients and the reduced matrix elements of generators, and thus provide a first step to a new algebraic derivation of matrix element formulae for all generators of the algebra.

Analysis of Finite Conformal Frequency Selective Surfaces via the Characteristic Basis Function Method and Spectral Rotation Approaches

An efficient characteristic basis function (CBF)-based method is proposed to analyze conformal frequency selective surfaces (FSS) that are not amenable to analysis by using conventional numerical methods typically used to model infinite, planar, and doubly periodic FSSs. The technique begins by employing the CBFs to describe the currents induced on the elements. The reaction integrals needed to derive the reduced matrix elements are computed either in the spatial domain, or spectral domain, depending on the separation distance between the blocks, so as to make the process numerically efficient. The spectral domain integrals are evaluated by making use of the spectral rotation (SR) on the spectra of the CBFs to alleviate the computational burden to be associated with full three-dimensional (3-D) Fourier transform, which is required in the conventional spectral domain approach applied to nonplanar geometries. Numerical results from the new method have good agreement with the fully spatial characteristic basis function method (CBFM) and the conventional method of moments (MoM). However, the required central processing unit (CPU) time is greatly reduced.

Long-range interactions between excited helium and alkali-metal atoms

The dispersion coefficients for the long-range interaction of the first four excited states of He, i.e., He(2 S-1,S-3) and He(2 P-1,P-3), with the low-lying states of the alkali-metal atoms Li, Na, K, and Rb are calculated by summing over the reduced matrix elements of the multipole transition operators. For the interaction between He and Li the uncertainty of the calculations is 0.1-0.5%. For interactions with other alkali-metal atoms the uncertainty is 1-3% in the coefficient C-5, 1-5% in the coefficient C-6, and 1-10% in the coefficients C-8 and C-10. The dispersion coefficients C-n for the interaction of He(2 S-1,S-3) and He(2 P-1,P-3) with the ground-state alkali-metal atoms and for the interaction of He(2 S-1,S-3) with the alkali-metal atoms in their first P-2 states are presented in this Brief Report. The coefficients for other pairs of atomic states are listed in the Supplemental Material.

Electron-capture and beta-decay modes with realistic nuclear structure calculations

In this work, we concentrate on the e-capture and β-decay processes in isotopes that are important for searching the explosive nucleosynthesis in massive stars. To this aim, we improved our codes which use compact analytical expressions for the required reduced matrix elements of all basic multipole operators (isospin repre- sentation). The ground state of the nuclear isotopes chosen, is constructed in the context of the BCS method while their excited states are calculated by solving the QRPA equations, using as residual two-body interactions that of the Bonn C-D one-meson exchange potential. We focus on the role of the charged Gamow-Teller and Fermi transitions in Fe group nuclei that are the main constituents of the core in presupernovae formation.

Pseudo-Jahn–Teller origin of icosahedral instability in boron buckyball, B80

A study of Ih to Th pseudo-Jahn–Teller (PJT) distortion in B80 is presented. Reduced matrix elements for vibronic coupling and resulting force constants were calculated in Ih to elucidate the origin of B80 symmetry breaking. Important orbital excitations involved are between inner orbitals 9hg, 5hg, and 6gu and virtual orbitals 8t2u and 8gg. Low-lying occupied orbitals interact more strongly than frontier molecular orbitals due to their density distribution on the capping atoms. The large energy gaps, associated with these excitations, attenuate the PJT effect. The force constant calculations for B80 point to a weak PJT energy stabilization of 304cm−1 at B3LYP/6-31G(d).

Relativistic many-body calculation of energies, oscillator strengths, transition rates, and lifetimes of Sc III ion

A systematic study of Sc iii atomic properties is carried out using a high-precision relativistic all-order method where all single, double, and partial-triple excitations of the Dirac-Fock wave functions are included to all orders of perturbation theory. Reduced matrix elements, oscillator strengths, transition rates, and lifetimes are determined for the $ns$, $n{p}_{j}$, $n{d}_{j}$, $n{f}_{j}$, and $n{g}_{j}$ levels with $n\ensuremath{\leqslant}7$. Recommended values and estimates of their uncertainties are provided for a large number of electric-dipole transitions. Electric-quadrupole and magnetic-dipole matrix elements are evaluated to determine lifetimes of the $3{d}_{5/2}$ and $4s$ metastable levels. These calculations provide recommended values critically evaluated for their accuracy for a number of Sc iii atomic properties for use in theoretical modeling as well as planning and analysis of various experiments. We hope that the present study will stimulate further exploration of Sc iii for various applications owing to its interesting structure of different low-lying metastable levels.

Atomic properties of Pb III

We carry out a systematic study of Pb iii properties using a hybrid method that combines configuration interaction and linearized coupled-cluster approaches. The calculations start from a [Xe]$4{f}^{14}5{d}^{10}$ Dirac-Fock potential. Excitation energies and lifetimes of the $6{p}^{2}$, $6sns$, $6snp$, and $6snd$ ($n\ensuremath{\le}9$) states are evaluated. Reduced matrix elements, oscillator strengths, and transition rates are determined for allowed electric-dipole transitions including these states. Extensive comparison with other theoretical and experimental values is carried out. Electric-dipole polarizabilities of the $6{s}^{2}\phantom{\rule{0.16em}{0ex}}{}^{1}\phantom{\rule{-0.16em}{0ex}}{S}_{0}$, $6s6p{\phantom{\rule{0.16em}{0ex}}}^{3}\phantom{\rule{-0.16em}{0ex}}{P}_{0}$, and $6s6p{\phantom{\rule{0.16em}{0ex}}}^{3}\phantom{\rule{-0.16em}{0ex}}{P}_{1}$ states in Pb iii and ground state polarizability of Pb${}^{4+}$ are reported.

IBAR: Interacting boson model calculations for large system sizes

Scaling the system size of the interacting boson model-1 (IBM-1) into the realm of hundreds of bosons has many interesting applications in the field of nuclear structure, most notably quantum phase transitions in nuclei. We introduce IBAR, a new software package for calculating the eigenvalues and eigenvectors of the IBM-1 Hamiltonian, for large numbers of bosons. Energies and wavefunctions of the nuclear states, as well as transition strengths between them, are calculated using these values. Numerical errors in the recursive calculation of reduced matrix elements of the d-boson creation operator are reduced by using an arbitrary precision mathematical library. This software has been tested for up to 1000 bosons using comparisons to analytic expressions. Comparisons have also been made to the code PHINT for smaller system sizes. Program summaryProgram title: IBARCatalogue identifier: AELI_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AELI_v1_0.htmlProgram obtainable from: CPC Program Library, Queenʼs University, Belfast, N. IrelandLicensing provisions: GNU General Public License version 3No. of lines in distributed program, including test data, etc.: 28 734No. of bytes in distributed program, including test data, etc.: 4 104 467Distribution format: tar.gzProgramming language: C++Computer: Any computer system with a C++ compilerOperating system: Tested under LinuxRAM: 150 MB for 1000 boson calculations with angular momenta of up to L=4Classification: 17.18, 17.20External routines: ARPACK (http://www.caam.rice.edu/software/ARPACK/)Nature of problem: Construction and diagonalization of large Hamiltonian matrices, using reduced matrix elements of the d-boson creation operator.Solution method: Reduced matrix elements of the d-boson creation operator have been stored in data files at machine precision, after being recursively calculated with higher than machine precision. The Hamiltonian matrix is calculated and diagonalized, and the requested transition strengths are calculated using the eigenvectors.Restrictions: The 1000 boson coefficients for L=0 and L=20 have been included in the IBAR distribution and the 7.3 GB of data that make up the remaining coefficients for L=21 to L=2000 are available upon request.Running time: If the provided example is changed to include 100 bosons, the calculation requires about 1 second of run time. For 1000 bosons, the calculation requires about 9 minutes of run time.

Blackbody radiation shift, multipole polarizabilities, oscillator strengths, lifetimes, hyperfine constants, and excitation energies in Hg+

Excitation energies of the [Xe]$4{f}^{14}5{d}^{10}ns$, [Xe]$4{f}^{14}5{d}^{10}n{p}_{j}$, [Xe]$4{f}^{14}5{d}^{10}n{d}_{j}$, [Xe]$4{f}^{14}5{d}^{10}{n}^{\ensuremath{'}}{f}_{j}$, and [Xe]$4{f}^{14}5{d}^{10}{n}^{\ensuremath{'}}{g}_{j}$ states in Hg${}^{+}$ are evaluated ($n\ensuremath{\le}10$, ${n}^{\ensuremath{'}}\ensuremath{\le}9$, and $[\mathrm{Xe}]=1{s}^{2}2{s}^{2}2{p}^{6}3{s}^{2}3{p}^{6}3{d}^{10}4{s}^{2}4{p}^{6}4{d}^{10}5{s}^{2}5{p}^{6}$). First-, second-, third-, and all-order Coulomb energies and first- and second-order Coulomb-Breit energies are calculated. Reduced matrix elements, oscillator strengths, and transition rates are determined for electric-dipole transitions, including the $ns$ ($n=6$--11), $np$ ($n=6$--10), $nd$ ($n=6$--10), $nf$ ($n=5$--9), and $ng$ ($n=5$--9) states. Lifetime values are determined for all above-mentioned states. The ground state $E1$, $E2$, and $E3$ polarizabilities are evaluated. The hyperfine structure in ${}^{199}$Hg${}^{+}$ and ${}^{201}$Hg${}^{+}$ ions is investigated. The hyperfine $A$ and $B$ values are determined for the first low-lying levels up to $n$ $=$ 7. The quadratic Stark effect on hyperfine structure levels of ${}^{199}$Hg${}^{+}$ and ${}^{201}$Hg${}^{+}$ ground states is investigated. The calculated shift for the ${}^{199}$Hg${}^{+}$ ($F$ $=$ 1, $M$ $=$ 0) $\ensuremath{\leftrightarrow}$ ($F$ $=$ 0, $M$ $=$ 0) transition is $\ensuremath{-}$0.0597(2) Hz/(kV/cm)${}^{2}$, in agreement with previous theoretical result $\ensuremath{-}$0.060(3) Hz/(kV/cm)${}^{2}$. These calculations provide a theoretical benchmark for comparison with experiment and theory and provide values of blackbody radiation shifts for microwave frequency standards with ${}^{199}$Hg${}^{+}$ and ${}^{201}$Hg${}^{+}$ ions.

Projection Operator on Rational Noncommutative Orbifold T2/Z4

In this paper, we introduce the reduced matrix in kq representation and provide the reduced matrix elements of a projection operator P̂ on the rational noncommutative orbifold T2/Z4. we give the closed form for the projector by Jacobi elliptical functions. Since projectors correspond to soliton solutions of the field theory on the noncommutative orbifold, we thus present a corresponding soliton solution.

Critically evaluated theoretical energies, lifetimes, hyperfine constants, and multipole polarizabilities inRb87

Systematic study of Rb atomic properties is carried out using a high-precision relativistic all-order method. Excitation energies of the $\mathit{ns}$, $\mathit{np}$, $\mathit{nd}$, and $\mathit{nf}$ ($n\ensuremath{\leqslant}10$) states in neutral rubidium are evaluated. Reduced matrix elements, oscillator strengths, transition rates, and lifetimes are determined for the levels up to $n=8$. Recommended values and estimates of their uncertainties are provided for a large number of electric-dipole transitions. Electric-dipole ($5s\ensuremath{-}\mathit{np}$, $n=5\ensuremath{-}26$), electric-quadrupole ($5s\ensuremath{-}{\mathit{nd}}_{j}$, $n=4\ensuremath{-}26$), and electric-octupole ($5s\ensuremath{-}{\mathit{nf}}_{j}$, $n=4\ensuremath{-}26$) matrix elements are calculated to obtain the ground state $E1$, $E2$, and $E3$ static polarizabilities. Scalar polarizabilities of the $\mathit{ns}$, $\mathit{np}$, and $\mathit{nd}$ states, and tensor polarizabilities of the ${\mathit{np}}_{3/2}$ and $\mathit{nd}$ excited states of Rb are evaluated. The hyperfine $A$ and $B$ values in $^{87}\mathrm{Rb}$ are determined for the first low-lying levels up to $n=9$. These calculations provide recommended values critically evaluated for their accuracy for a number of Rb atomic properties useful for a variety of applications.

Excitation energies, E1, M1, and E2 transition rates, and lifetimes in Ca+, Sr+, Cd+, Ba+, and Hg+ 1This article is part of a Special Issue on the 10th International Colloquium on Atomic Spectra and Oscillator Strengths for Astrophysical and Laboratory Plasmas.

Excitation energies of ns1/2, npj, and ndj states in Cd+ (n = 5), Hg+ (n = 5) and ns1/2, npj, and (n – 1)dj states in Ca+ (n = 4), Sr+ (n = 5), and Ba+ (n = 6) are evaluated using the linearized coupled-cluster (all-order) method. Reduced matrix elements, oscillator strengths, and transition rates are determined for the ns–npj–ndj (or ns–npj–(n – 1)dj) possible electric dipole transitions in Ca+, Sr+, Ba+, Cd+, and Hg+. Electric quadrupole matrix elements are evaluated to obtain ns1/2–(n – 1)dj transition rates in Ca+ (n = 5), Sr+ (n = 5), and Ba+ (n = 6). The matrix elements are calculated using both relativistic many-body perturbation theory, complete through third order, and the relativistic all-order method restricted to single and double (SD) excitations. The SD lifetime results for the np and nd states in Ca+, Sr+, Ba+, Cd+, and Hg+, are compared with the latest available experimental measurements. The contribution of the magnetic dipole nd3/2–nd5/2 transition to the lifetimes of the lowest nd5/2 level ln Ca+, Sr+, and Ba+ ions is discussed. These calculations provide a theoretical benchmark for comparison with experiment and theory as well as data needed for various applications.

Group-theoretical approach to interaction renormalization for ab initio solutions of the nuclear problem

For many problems in nuclear, high-energy and solid-state physics a crucial step toward a solution involves a diagonalization or simplification of the Hamiltonian of a many-body quantum-mechanical system. Problems of interest include renormalization of the Hamiltonian to accommodate excluded basis spaces, decoupling of a model space region of special interest from irrelevant configurations, or near-diagonalization to invoke perturbative treatments. In such applications, the Similarity Renormalization Group (SRG) approach (or flow equations) can provide solutions where more conventional methods fail. We show that when SRG employs a near symmetry of a system, utilizing the Casimir invariant operators, it evolves the Hamiltonian of the system toward a (block-)diagonal form that can be realized in terms of a finite number of integrity basis operators. In particular, for atomic nuclei, an SRG flow equation that utilizes the SU(3) symmetry, which often dominates the nuclear dynamics, is found to naturally evolve a nucleon-nucleon (NN) interaction toward a block-diagonal form. A remarkable consequence is that a renormalized SU(3)-preserving many-body Hamiltonian expressed in terms of only the SU(3) integrity basis realizes energy spectra identical to the ones of the original realistic NN interaction, which reflects the low-energy symmetries as well as symmetry-breaking patterns of quantum chromodynamics. Such a scheme is especially suitable for ab initio SU(3) symmetry-adapted shell model calculations, which can further allow for perturbative treatments of the nuclear interaction. Moreover, the use of symmetries dramatically reduces the size of the problem, as the flow equation can be rewritten in terms of SU(3) reduced matrix elements of the NN interaction. This is crucial for maintaining the unitarity of the SRG transformations that generate many-body interactions.

Polarization-dependent atomic dipole traps behind a circular aperture for neutral-atom quantum computing

The neutral-atom quantum computing community has successfully implemented almost all necessary steps for constructing a neutral-atom quantum computer. We present computational results of a study aimed at solving the remaining problem of creating a quantum memory with individually addressable sites for quantum computing. The basis of this quantum memory is the diffraction pattern formed by laser light incident on a circular aperture. Very close to the aperture, the diffraction pattern has localized bright and dark spots that can serve as red-detuned or blue-detuned atomic dipole traps. These traps are suitable for quantum computing even for moderate laser powers. In particular, for moderate laser intensities ($~100$ W/${\mathrm{cm}}^{2}$) and comparatively small detunings ($~1000$--10 000 linewidths), trap depths of $~1$ mK and trap frequencies of several to tens of kilohertz are achieved. Our results indicate that these dipole traps can be moved by tilting the incident laser beams without significantly changing the trap properties. We also explored the polarization dependence of these dipole traps. We developed a code that calculates the trapping potential energy for any magnetic substate of any hyperfine ground state of any alkali-metal atom for any laser detuning much smaller than the fine-structure splitting for any given electric field distribution. We describe details of our calculations and include a summary of different notations and conventions for the reduced matrix element and how to convert it to SI units. We applied this code to these traps and found a method for bringing two traps together and apart controllably without expelling the atoms from the trap and without significant tunneling probability between the traps. This approach can be scaled up to a two-dimensional array of many pinholes, forming a quantum memory with single-site addressability, in which pairs of atoms can be brought together and apart for two-qubit gates for quantum computing.

SHAPE EVOLUTION IN HEAVIEST STABLE EVEN-EVEN MOLYBDENUM ISOTOPES STUDIED VIA COULOMB EXCITATION

The quadrupole shape evolution in heaviest stable Mo isotopes is studied in terms of the shape coexistence phenomenon occurrence in this region of nuclear chart. Quadrupole deformation parameters of the 96,98,100 Mo isotopes in the low-lying (ground and excited) 0+ states were deduced using Coulomb excitation method. In all cases rich sets of electromagnetic reduced matrix elements were determined using the GOSIA code and then analysed using the Quadrupole Sum Rules formalism. Discussion of the experimental results in comparison with the predictions of microscopic calculations within the general quadrupole collective Bohr Hamiltonian model is presented for 100 Mo .

Blackbody radiation shift, multipole polarizabilities, oscillator strengths, lifetimes, hyperfine constants, and excitation energies in Ca+

A systematic study of Ca${}^{+}$ atomic properties is carried out using a high-precision relativistic all-order method where all single, double, and partial triple excitations of the Dirac-Fock wave functions are included to all orders of perturbation theory. Reduced matrix elements, oscillator strengths, transition rates, and lifetimes are determined for the levels up to $n=7$. Recommended values and estimates of their uncertainties are provided for a large number of electric-dipole transitions. Electric-dipole scalar polarizabilities for the $5s$, $6s$, $7s$, $8s$, $4{p}_{j}$, $5{p}_{j}$, $3{d}_{j}$, and $4{d}_{j}$ states and tensor polarizabilities for the $4{p}_{3/2}$, $5{p}_{3/2}$, $3{d}_{j}$, and $4{d}_{j}$ states in Ca${}^{+}$ are calculated. Methods are developed to accurately treat the contributions from highly excited states, resulting in significant (factor of 3) improvement in the accuracy of the $3{d}_{5/2}$ static polarizability value, $31.8(3){a}_{0}^{3}$, in comparison with the previous calculation [Arora et al., Phys. Rev. A 76, 064501 (2007).]. The blackbody radiation shift of the $4s--3{d}_{5/2}$ clock transition in Ca${}^{+}$ is calculated to be $0.381(4)$ Hz at room temperature, $T=300$ K. Electric-quadrupole $4s--\mathit{nd}$ and electric-octupole $4s--\mathit{nf}$ matrix elements are calculated to obtain the ground-state multipole $E2$ and $E3$ static polarizabilities. Excitation energies of the $\mathit{ns}$, $\mathit{np}$, $\mathit{nd}$, $\mathit{nf}$, and $\mathit{ng}$ states with $n\ensuremath{\leqslant}$ 7 in are evaluated and compared with experiment. Recommended values are provided for the $7{p}_{1/2}$, $7{p}_{3/2}$, $8{p}_{1/2}$, and $8{p}_{3/2}$ removal energies for which experimental measurements are not available. The hyperfine constants $A$ are determined for the low-lying levels up to $n=7$. The quadratic Stark effect on hyperfine structure levels of $^{43}\mathrm{Ca}$${}^{+}$ ground state is investigated. These calculations provide recommended values critically evaluated for their accuracy for a number of Ca${}^{+}$ atomic properties for use in planning and analysis of various experiments as well as theoretical modeling.

Charge-transfer transitions and optic spectra of chromites: A model computation

In the cluster approach, we consider the peculiarities of charge-transfer (CT) states and CT O 2p Cr 3d transitions in the octahedral (CrO6)9− complex. We have computed the reduced matrix elements of electric-dipole transition operator on many-electron wave functions — the initial and final states of CT transitions. We have parameterized the obtained results and computed the relative intensities of various allowed CT transitions in the absence of the mixing of CT configurations having the same symmetry. Using the Tanabe-Sugano technique, we have taken into account this mixing and obtained the energies of many-electron CT transitions and their actual intensities as well. We have also allowed for the Coulomb interaction between the 2p-electrons of the O2− ligands and the 3d-electrons of the central Cr3+ ion in the (CrO6)9− cluster. This interaction proved insignificant for the optic spectra. Modeling the optic spectrum of chromium-based oxides has yielded a complicated CT band consisting of 33 lines with the main maximum at about 7 eV and satellites in the range of 4–5 and 8–9 eV. The total extent of the CT band is about 8 eV. The model spectrum is in satisfactory agreement with experimental data, which shows the limited validity of the generally accepted notion of a simple structure of CT spectra. © 2010 Wiley Periodicals, Inc. Int J Quantum Chem, 2011

Coupled tensorial forms of the second-order effective Hamiltonian for open-subshell atoms in jj-coupling

General expressions for the second-order effective atomic Hamiltonian are derived for open-subshell atoms in jj-coupling. The expansion terms are presented as N-body ( N = 0 , 1 , 2 , 3 ) effective operators given in the second quantization representation in coupled tensorial form. Two alternative coupled tensorial forms for each expansion term have been developed. To reduce the number of expressions of the effective Hamiltonian, the reduced matrix elements of antisymmetric two-particle wavefunctions are involved in the consideration. The general expressions presented allow the determination of the spin-angular part of expansion terms when studying correlation effects dealing with a number of problems in atomic structure calculations.