On the ground state of the half-filled Hubbard model

We calculate the ground state of the half-filled Hubbard model and its energy by starting from a spindensity wave approximation and improving it by incorporating transverse spin fluctuations. The calculations are done by employing a projection method. The quality of the proposed approximation is particularly high for intermediate and large Coulomb repulsionU, where it exceeds considerably e.g. that of the Gutzwiller projected spin-density wave state. To ordert ²/U (wheret is the hopping matrix element), our approximation is shown to be equivalent to a recent Coupled Cluster calculation for the Heisenberg antiferromagnet. Finally we show how to ordert ²/U the linear spin-wave approximation for the Heisenberg antiferromagnet may be obtained.     

Large-basis no-core shell-model calculations and their application to10C →10B fermi beta decay

We use a 4ħΩ shell-model calculation with a two-body effective interaction derived microscopically from the Reid93 potential to calculate the isospin-mixing correction for the¹⁰C→¹⁰B superallowed Fermi transition. The effective interaction takes into account the Coulomb potential as well as the charge dependence ofT=1 partial waves. Our results suggest the isospin-mixing correctionδ _c ≈ 0.1%, which is compatible with previous calculations obtained by other methods. The obtainedδ _C correction is about a factor of four too small to obtain unitarity of the Cabibbo-Kobayashi-Maskawa matrix with the present experimental data. Presented by B.R. Barrett at the International Conference on “Atomic Nuclei and Metallic Clusters”, Prague, September 1–5, 1997. This work is supported in part by NSF grant No. PHY96-05192.     

Two- and Three-Body Coulomb Solution Method in a Momentum Space

We present a method to calculate two- and three-body amplitudes including the Coulomb potential in a momentum space. Our aim is to obtain the exact two-body Coulomb amplitudes used in three-body calculations, which reproduce the analytic phase shifts. For the purpose, our theory is based on the modified Coulomb potential (MCP) whose Fourier transformation is equivalent to the pure Coulomb potential in a configuration space, and the two-potential theory with an auxiliary potential. Moreover, one can analytically determine a decisive range R _dec in the MCP. By using the MCP, we obtain the two-body Coulomb modified nuclear amplitude as well as the pure Coulomb amplitude. The calculated phase shift are very good fitting with the experimental data.     

Fine-structure effects in relativistic calculations of the static polarizability of the helium atom

We use the relativistic configuration-interaction method and the model potential method to calculate the scalar and tensor components of the dipole polarizabilities for the excited states 1s3p ³ P ₀ and 1s3p ³ P ₂ of the helium atom. The calculations of the reduced matrix elements for the resonant terms in the spectral expansion of the polarizabilities are derived using two-electron basis functions of the relativistic Hamiltonian of the atom, a Hamiltonian that incorporates the Coulomb and Breit electron-electron interactions. We formulate a new approach to determining the parameters of the Fuss model potential. Finally, we show that the polarizability values are sensitive to the choice of the wave functions used in the calculations. Zh. éksp. Teor. Fiz. 115, 494–504 (February 1999)     

Isospin-forbidden β-decay of 28Mg

We have measured the branching ratio for the isospin-forbidden Fermi β-decay of ²⁸Mg to the 0.972 MeV 0⁺, T = 1 level in ²⁸Al. This decay occurs through an admixture of the 0⁺, t = 2 analog state at 5.992 MeV into the anti-analog level at 0.972 MeV. A charge-dependent matrix element of 17.0_?5.8^+4.3 keV is deduced from the observed (0.21 ± 0.12)% branch. Comparisons are made with matrix elements deduced in other nuclei. An analysis based on a simple shell model with fourfold degenerate Orbitals indicates the importance of the two-body Coulomb interaction in isospin mixing in nuclei with more than one valence proton.     

Spin-orbit interactions from self consistent field wavefunctions

Effects of the spin-orbit Hamiltonian H_LS , including both the spin-same orbit and spin-other orbit terms, are studied at the self consistent field (SCF) level of theory. Separate calculations are carried out for each state considered, and biorthogonal orbitals are constructed for the evaluation of matrix elements. Doublet-doublet and singlet-triplet interactions are discussed. The evaluation of the Gaussian basis function integrals is described in a Cartesian component representation, these integrals being directly related to one and two electron second derivative integrals. This new SCF spin-orbit code is used to (i) determine the spatial dependence of the spin-orbit parameters for the Renner-Teller ²B₁, ²A₁ states of H₂O⁺, (ii) determine the spin-orbit splitting of the ²Π states of OH as a function of bond-length and (iii) calculate the radiative lifetime of the a ³Σ⁺ state of NO⁺. In each case these calculations are compared with more sophisticated configuration interaction studies, but it is found that the evaluation of these effects near equilibrium geometries at the SCF level is sufficiently accurate, provided that a large basis set is used.     

Numerical solution of the Sinanoǧlu equation using a multicentre radial-angular grid

ABSTRACT

The pair functions that minimise the correlation energy of second-order many-body perturbation (MP2) theory (the Hylleraas functional) are obtained as solutions to the corresponding Sinano?lu equation by expanding them on a six-dimensional, multicentre, radial-angular grid of two electrons. Cusps in the pair functions at the nuclei are described numerically accurately by the multicentre grid. A cusp in each singlet pair function at the coalescence of the two electrons is taken into account analytically by a correlation factor. With a grid of approximately 10,000 points per atom, the MP2 correlation energies for atoms and polyatomic molecules are obtained usually within 0.1 mE _h of the complete-basis-set results. The correlation factor, auxiliary basis functions, and a judicious choice of integration algorithms are all necessary to stabilise the grid-based MP2 and underlying Hartree–Fock (HF) calculations. The auxiliary basis set, in particular, largely restores the hermiticity and diagonal dominance of the Fock matrix as well as furnishes virtual orbitals used in a resolution-of-the-identity approximation to lower the dimension of some integrals. The results of the grid-based HF and MP2 calculations without a correlation factor are found to suffer from large, nonsystematic errors frequently.     

Hilbert Modules and Stochastic Dilation of a Quantum Dynamical Semigroup on a von Neumann Algebra

A general theory for constructing a weak Markov dilation of a uniformly continuous quantum dynamical semigroup T _t on a von Neumann algebra ? with respect to the Fock filtration is developed with the aid of a coordinate-free quantum stochastic calculus. Starting with the structure of the generator of T _t , existence of canonical structure maps (in the sense of Evans and Hudson) is deduced and a quantum stochastic dilation of T _t is obtained through solving a canonical flow equation for maps on the right Fock module ?⊗Γ(L ²(ℝ₊,k ₀)), where k ₀ is some Hilbert space arising from a representation of ?^′. This gives rise to a *-homomorphism j _t of ?. Moreover, it is shown that every such flow is implemented by a partial isometry-valued process. This leads to a natural construction of a weak Markov process (in the sense of [B-P]) with respect to Fock filtration. Received: 15 June 1998/ Accepted: 4 March 1999     

A Symmetric-Impulse Approximation Applied to the Faddeev Theory on the Charge Transfer Amplitude

Off-the-shell anomalous factors of the two-body Coulomb transition matrices appear in the integral form of the Faddeev second-order nuclear-electronic amplitude, for proton-hydrogen charge transfer scattering in a typical nlm → n′l′m′ transition. A symmetric-impulse approximation (SIA) is applied to eliminate these factors and an induction method is proposed to analytically calculate the remaining integrals. The nuclear-electronic amplitude is derived for the general case, and for totally symmetric collisions, in terms of generalized hypergeometric functions of two variables, F ₄, and of one variable, ₃ F ₂, respectively. The angular distribution of the second-order nuclear-electronic charge transfer amplitude shows the Thomas mechanism as a peak or a hump for symmetric and asymmetric collisions. There also exists a peak in the forward angular distribution of the second-order nuclear-electronic amplitude, which partly cancels the kinematic peak in the angular distribution of the charge transfer differential cross sections.     

Eikonal analysis of Coulomb distortion in quasi-elastic electron scattering

An eikonal expansion is used to provide systematic corrections to the eikonal approximation through order 1/k ², where k is the wave number. Electron wave functions are obtained for the Dirac equation with a Coulomb potential. They are used to investigate distorted-wave matrix elements for quasi-elastic electron scattering from a nucleus. A form of effective-momentum approximation is obtained using trajectory-dependent eikonal phases and focusing factors. Fixing the Coulomb distortion effects at the center of the nucleus, the often-used ema approximation is recovered. Comparisons of these approximations are made with full calculations using the electron eikonal wave functions. The ema results are found to agree well with the full calculations.     

A model for short range correlation in the three-body system

A method by means of which the two-body short-range correlations in the nucleon-nucleon interaction are taken accurately into account in three-body bound-state calculations is described. The method is independent of the nucleon-nucleon interaction. We employ our method to calculate meson-exchange corrections to the magnetic moments of ³H and ³He and the Coulomb energy of ³He.     

A comparison of finite basis set and finite difference Hartree—Fock calculations for the open-shell (X 2Σ+) molecules BaF and YbF

A comparison is made of the accuracy with which the total electronic energy can be calculated by using the finite basis set approach (the algebraic approximation) and the finite difference method in calculations employing the Hartree—Fock model for the open shell ground (X ²Σ⁺) states of the fluorides BaF and YbF. The convergence of the calculations carried out within the algebraic approximation is monitored by employing systematically constructed basis sets of increasing size. The difference between the finite basis set and finite difference Hartree—Fock energies is 2.6μE _h for BaF and 2.8μE _h for YbF. Dipole moments determined within the algebraic approximation are also compared with the corresponding finite difference expectation values.     

The Three-Boson System at Next-to-Next-to-Leading Order

We discuss effective field-theory treatments of the problem of three particles interacting via short-range forces (range R ≪ a ₂, with a ₂ the two-body scattering length). We show that forming a once-subtracted scattering equation yields a scattering amplitude whose low-momentum part is renormalization-group invariant up to corrections of O(R ³/a ₂ ³). Since corrections of O(R/a ₂) and O(R ²/a ₂ ²) can be straightforwardly included in the integral equation’s kernel, a unique solution for 1 + 2 scattering phase shifts and three-body bound-state energies can be obtained up to this accuracy. We use our equation to calculate the correlation between the binding energies of Helium-4 trimers and the atom-dimer scattering length. Our results are in excellent agreement with recent three-dimensional Faddeev calculations that used phenomenological inter-atomic potentials.     

Visualization of deficiencies in approximate molecular wave functions: the orbital amplitude difference function for the matrix Hartree-Fock description of the ground state of the boron fluoride molecule

The orbital amplitude difference function is used to assess the quality of Hartree–Fock orbitals obtained by invoking the algebraic approximation for the BF ground-state. Systematic sequences of even-tempered, spherical-harmonic Gaussian-type basis functions are used to generate orbitals for which the corresponding total Hartree–Fock energy approaches the 1 μE _h level of accuracy. Exact orbitals are obtained from finite difference calculations using a grid based on spheroidal coordinates. The finite basis set approximations for the orbital are discretized. The accuracy of the discretization is assessed. For each occupied orbital a discretized representation of the orbital amplitude difference function is generated and analysed.     

Representation of the Coulomb Matrix Elements by Means of Appell Hypergeometric Function <Emphasis Type="Italic">F</Emphasis><Subscript>2</Subscript>

Exact analytical representation for the Coulomb matrix elements by means of Appell’s double series F₂ is derived. The finite sum obtained for the Appell function F₂ allows us to evaluate explicitly the matrix elements of the two-body Coulomb interaction in the lowest Landau level. An application requiring the matrix elements of Coulomb potential in quantum Hall effect regime is presented.     

Alpha Cluster in Hartree-Fock Calculations

Hartree-Fock (HF) calculations have been performed fcr ¹²C nucleus assuming a trigonal D_3h equilibrium configuration. Different aspects including symmetry, kinds of two-body interactions are investigated. The HF calculations have been carried out using a velocity dependent effective potential of s-wave interaction. The results are compared with previous calculations. In all HF calculations no deviation from axial symmetry is obtained.     

Theoretical analysis of the astrophysical S-factor for the capture reaction α + d → 6Li + γ in the two-body model

Theoretical estimates for the astrophysical S-factor and the rate of the reaction d(α, γ)⁶Li were obtained on the basis of the two-body model involving an α−d potential that has a simple Gaussian form and which describes correctly S-, P-, and D-wave phase shifts, the binding energy, and the asymptotic normalization coefficient for the S-wave bound state. The wave functions for the bound and continuum channels were calculated with the aid of the highly precise Numerov algorithm. The results for the contributions of the E1 and E2 transition components reveal a good convergence as the upper limit in the effective integrals increases up to 40 fm. The results obtained for the astrophysical S-factor and the rate of the reaction d(α, γ)⁶Li in the temperature range of 10⁶ K ≤ T ≤ 10¹⁰ K agree well with the results of the calculations performed by A.M. Mukhamedzhanov and his coauthors [Phys. Rev. C 83, 055805 (2011)] by using the known asymptotic form of the wave function at low energies and a complicated two-body potential at higher energies.

     

Effective interactions and charges in58Ni

The structure of the low-lying states of⁵⁸Ni has been calculated in shell model by assuming an inert⁵⁶Ni core plus two valence nucleons in the p_3/2, f_5/2 and p_1/2 orbitals. The two-body matrix elements are first expressed in terms of seven radial matrix elements and these are then parametrized to give best fit between the computed and the observed energies of the levels below 4 MeV. The wave-functions obtained using these two-body matrix elements are used to study the concept of effective charges. It is found that a single effective charge is not sufficient to predict theB(E2) rates equally well for the thirteen known transitions for which experimental values are available. Assumption of state-dependent effective charges gives a far better agreement. An analysis using wavefunctions obtained with Kuo’s two-body matrix elements also gives a similar result.     

A novel treatment of the proton-proton Coulomb force in proton-deuteron Faddeev calculations: Breakup

We extend our approach to incorporate the proton-proton (ppCoulomb force into the three-nucleon (3NFaddeev calculations from elastic proton-deuteron (pdscattering to the breakup process. The main new ingredient is a 3-dimensional screened pp Coulomb t -matrix obtained by a numerical solution of the 3-dimensional Lippmann-Schwinger (LS) equation. We demonstrate numerically that the proton-deuteron (pdbreakup observables can be determined from the resulting on-shell 3N amplitudes increasing the screening radius. However, contrary to the pd elastic scattering, the screening limit exists only after renormalisation of the pp t -matrices.     

Stretched exponential relaxation in the Coulomb glass

The relaxation of the specific heat and the entropy to their equilibrium values is investigated numerically for the three-dimensional Coulomb glass at very low temperatures. The long time relaxation follows a stretched exponential function, f (t) = f ₀exp - (t/τ)^β , with the exponent β increasing with the temperature. The relaxation time diverges as an Arrhenius law when T→ 0. Received 24 May 2001 and Received in final form 12 September 2001