Bethe-Goldstone Equation Research Articles

Exploration of in-medium hyperon-nucleon interactions

The study focuses on exploring the changes in the hyperon-baryon interaction at various nuclear densities. This approach starts by building a vacuum hyperon-nucleon interaction model based on Boson -Exchange maintaining SU(3) flavor symmetry. Bethe-Goldstone equation is then explored to investigate the medium properties over the bare interaction. A detailed investigation of the density dependence revealed clear changes in the low energy parameters with the variation of the medium density shown for different strangeness channels.

Cluster many-body expansion: A many-body expansion of the electron correlation energy about a cluster mean field reference.

The many-body expansion (MBE) is an efficient tool that has a long history of use for calculating interaction energies, binding energies, lattice energies, and so on. In the past, applications of MBE to correlation energy have been unfeasible for large systems, but recent improvements to computing resources have sparked renewed interest in capturing the correlation energy using the generalized nth order Bethe-Goldstone equation. In this work, we extend this approach, originally proposed for a Slater determinant, to a tensor product state (TPS) based wavefunction. By partitioning the active space into smaller orbital clusters, our approach starts from a cluster mean field reference TPS configuration and includes the correlation contribution of the excited TPSs using the MBE. This method, named cluster MBE (cMBE), improves the convergence of MBE at lower orders compared to directly doing a block-based MBE from a RHF reference. We present numerical results for strongly correlated systems, such as the one- and two-dimensional Hubbard models and the chromium dimer. The performance of the cMBE method is also tested by partitioning the extended π space of several large π-conjugated systems, including a graphene nano-sheet with a very large active space of 114 electrons in 114 orbitals, which would require 1066 determinants for the exact FCI solution.

Microscopic Calculation of the Optical Model Potential from One Boson Exchange Poten­tials

A new method for calculating the optical model potential from One Boson Exchange Potentials (OBEPs) is developed. The G-matrix is calculated by solving the Bethe-Goldstone equation in momentum space. Using vector brackets these G-matrix elements can be transformed from the center of mass representation into the laboratory system. This allows the evaluation of the (r-matrix interaction between nucleons in bound states and those in a plane-wave state. The lowest order contribution to the real part of the potential comes from the Hartree-Fock term, while the lowest order contribution to the imaginary part comes from the two-particle-one-hole (2p1h) diagram. Calculations for 16O and 40Ca have been carried out. Local approximations are obtained by describing the results for the central part in terms of a Woods-Saxon potential and those for the spin-orbit part in terms of the corresponding derivatives. The dependence of these potentials on energy and angular momentum is discussed. The parameters for these local approximations are in good agreement with empirical fits.

Coarse graining the Bethe–Goldstone equation: Nucleon–nucleon high-momentum components

The delta-shell representation of the nuclear force allows a simplified treatment of nuclear correlations. We show how this applies to the Bethe-Goldstone equation as an integral equation in coordinate space with a few mesh points, which is solved by inversion of a 5-dimensional square matrix in the single channel cases and a $10\times10$ matrix for the tensor-coupled channels. This allows us to readily obtain the high momentum distribution, for all partial waves, of a back-to-back correlated nucleon pair in nuclear matter. We find that the probability of finding a high-momentum correlated neutron-proton pair is about 18 times that of a proton-proton one, as a result of the strong tensor force, thus confirming in an independent way previous results and measurements.

Coarse-grained short-range correlations

We develop a scheme to take into account the effects of short-range nucleon-nucleon correlations in the nucleon-pair wave function by solving the Bethe-Goldstone equation for a coarse grained delta shell potential in S-wave configuration. The S-wave delta shell potential has been adjusted to reproduce the $^1$S$_0$ phase shifts of the AV18 potential for this partial wave up to 2 GeV in the laboratory kinetic energy. We show that a coarse grained potential can describe the high momentum tail of the back-to-back correlated pairs and the $G$-matrix in momentum space. We discuss the easiness and robustness of the calculation in coordinate space and the future improvements and utilities of this model. This work suggests the possibility of using perturbation theory for describing the short-range correlations, and related to this, to substitute the $G$-matrix by an appropriate coarse-grained potential.

Towards Efficient Orbital-Dependent Density Functionals for Weak and Strong Correlation.

We present a new paradigm for the design of exchange-correlation functionals in density-functional theory. Electron pairs are correlated explicitly by means of the recently developed second order Bethe-Goldstone equation (BGE2) approach. Here we propose a screened BGE2 (sBGE2) variant that efficiently regulates the coupling of a given electron pair. sBGE2 correctly dissociates H_{2} and H_{2}^{+}, a problem that has been regarded as a great challenge in density-functional theory for a long time. The sBGE2 functional is then taken as a building block for an orbital-dependent functional, termed ZRPS, which is a natural extension of the PBE0 hybrid functional. While worsening the good performance of sBGE2 in H_{2} and H_{2}^{+}, ZRPS yields a remarkable and consistent improvement over other density functionals across various chemical environments from weak to strong correlation.

Discrete wave-packet representation in nuclear matter calculations

The Lippmann-Schwinger equation for the nucleon-nucleon $t$-matrix as well as the corresponding Bethe-Goldstone equation to determine the Brueckner reaction matrix in nuclear matter are reformulated in terms of the resolvents for the total two-nucleon Hamiltonians defined in free space and in medium correspondingly. This allows to find solutions at many energies simultaneously by using the respective Hamiltonian matrix diagonalization in the stationary wave packet basis. Among other important advantages, this approach simplifies greatly the whole computation procedures both for coupled-channel $t$-matrix and the Brueckner reaction matrix. Therefore this principally novel scheme is expected to be especially useful for self-consistent nuclear matter calculations because it allows to accelerate in a high degree single-particle potential iterations. Furthermore the method provides direct access to the properties of possible two-nucleon bound states in the nuclear medium. The comparison between reaction matrices found via the numerical solution of the Bethe-Goldstone integral equation and the straightforward Hamiltonian diagonalization shows a high accuracy of the method suggested. The proposed fully discrete approach opens a new way to an accurate treatment of two- and three-particle correlations in nuclear matter on the basis of three-particle Bethe-Faddeev equation by an effective Hamiltonian diagonalization procedure.

Wave-function inspired density functional applied to the H2/ challenge

We start from the Bethe–Goldstone equation (BGE) to derive a simple orbital-dependent correlation functional—BGE2—which terminates the BGE expansion at the second-order, but retains the self-consistent coupling of electron-pair correlations. We demonstrate that BGE2 is size consistent and one-electron ‘self-correlation’ free. The electron-pair correlation coupling ensures the correct H2 dissociation limit and gives a finite correlation energy for any system even if it has a no energy gap. BGE2 provides a good description of both H2 and dissociation, which is regarded as a great challenge in density functional theory (DFT). We illustrate the behavior of BGE2 analytically by considering H2 in a minimal basis. Our analysis shows that BGE2 captures essential features of the adiabatic connection path that current state-of-the-art DFT approximations do not.

Relativistic effects in three-nucleon forces for nuclear matter and finite nuclei

In order to simulate the relativistic effects of the Dirac-Brueckner-Hartree-Fock approach for finite nuclei, the part of the Urbana three-nucleon (3N) force is considered, which represents the enhancement of the small components of the Dirac spinors for the nucleons in the nuclear medium. This 3N force is included in a Brueckner-Hartree-Fock calculation with rearrangement terms using a realistic model for the NN interaction. The strength of the 3N force is adjusted to reproduce the empirical saturation point of nuclear matter and then used in corresponding studies of the closed shell nuclei $^{16}\mathrm{O}$ and $^{40}\mathrm{Ca}$. Special attention is paid to a consistent treatment of the spectrum of particle states in the NN propagator of the Bethe-Goldstone equation.

Short-range correlations in isospin symmetric and asymmetric nuclear matter: A microscopic perspective

Short-range correlations in nuclear and neutron matter are examined through the properties of the correlated wave function obtained by solving the Bethe-Goldstone equation. Tensor correlations are explored through the dominant tensor-driven transition and central correlations through the singlet and triplet S waves. Predictions from a popular meson-theoretic nucleon-nucleon potential employed in the Dirac-Brueckner-Hartree-Fock approach are compared with those from two- and three-body high-quality chiral interactions in Brueckner G-matrix calculations. Short-range correlations in symmetric matter are remarkably stronger than in neutron matter. It is found that short-range correlations are very model dependent and have a large impact on the symmetry energy above normal density.

Solution of the Bethe-Goldstone equation without partial-wave decomposition

We present a method for solving the nucleon-nucleon scattering equation without the use of a partial wave expansion of the scattering amplitude. After verifying the accuracy of the numerical solutions, we proceed to apply the method to the in-medium scattering equation (the Bethe-Goldstone equation) in three dimensions. A focal point is a study of Pauli blocking effects calculated in the (angle-dependent) three-dimensional formalism as compared to the usual spherical approximation. We discuss our results and their implications.

Brueckner–AMD study of light nuclei

In many states of light nuclei, the cluster structure is observed in addition to the shell structure. To understand the mechanism of clustering, we apply the Brueckner theory to the Antisymmetrized Molecular Dynamics (AMD) based on realistic nuclear interactions. The Bethe-Goldstone equation in the Brueckner theory is solved for every nucleon pair described by wave packets of AMD, and the G-matrix is calculated with single-particle orbits in AMD self-consistently. We show applicability of this method not only to self-conjugate nuclei but also to N ≠ Z nuclei with A ≤ 12. It is confirmed that these results present reasonable description of cluster structures and energy-level schemes in comparison with the experimental ones in light nuclei. For 8Be having a typical α + α cluster structure, the structure dependence of the G-matrix is investigated and the role of tensor force is shown to be important in understanding the clustering mechanism.

Influence of Electron-Electron Correlations and Lattice Effects on Positron-Electron Enhancement Factors

We present an approach taking into account the effect of electron-electron (e-e) correlations on electron-positron (e-p) momentum density distributions. The approach bases on the modification of the Bethe-Goldstone (B-G) equation for the positron in the electron gas due to self-energy effects. The example calculations have been performed for selected parameters corresponding to simple metals. The calculated dependencies exhibit the increase of the e-p enhancement factors below Fermi momentum, like Kahana enhancements, and a decrease above the Fermi sphere, leading to a many-body “tail” in the e-p momentum density distributions. Moreover, the influence of lattice effects on enhancement factors (EF) is taken into account. This decreases by a few percent the absolute values of the e-p momentum distributions and the corresponding annihilation rates and for real metals such as Mg or Cu evidently improve the agreement with experiment.

Particle-hole optical model and strength functions for high-energy giant resonances

A formulation of the particle-hole optical model is proposed for describing the contribution of the fragmentation effect to the formation of strength functions for high-energy giant resonances. The model is based on the Bethe-Goldstone equation for the energy-averaged particle-hole Green’s function. In this equation, the particle-hole interaction that is induced by a virtual excitation of multiquasiparticle configurations and in which, upon averaging over energy, an imaginary part is contained is taken into account. An analogy with the single-quasiparticle optical model is discussed.

Description of nuclear structures in light nuclei with Brueckner-AMD

We develop the new antisymmetrized molecular dynamics (AMD) method, Brueckner-AMD, which makes us perform the AMD calculations starting from the realistic nuclear force. In the Brueckner-AMD, the single-particle orbits of AMD can be applied straightforward to the Bethe-Goldstone equation in the Brueck- ner theory by using the AMD+Hartree-Fock method, and the G-matrices are determined with the single-particle energies of AMD self-consistently. In that sense, in this framework, the G-matrix in AMD can be solved theo- retically without any corrections. We present the applicability of the Brueckner-AMD to describe not only the ground states but also the excited states for some light nuclei, especially the excited 0 + state in 12 C which is not solved suffi ciently by the present shell model approaches, starting from the realistic nuclear force.

Recent CI methods for the calculation of pair correlations and more-electron clusters in ground-state atoms

The equations and approximations of (1) ground-state “Many-Electron Theory of Atoms and Molecules” (MET) of Sinanoǧlu (2) a 1955 atomic theory of Yilmaz, and (3) the equations and approximations of Brueckner theory and the Bethe-Goldstone equations for nuclei in which all correlations are coupled into a superdouble self-consistency and its atomic form by Nesbet are compared. It is shown that as noted recently by Condon and others, recent atomic calculations by the partial CI procedures of MET are in agreement with MET approximations, while calculations with Brueckner theory and the Bethe-Goldstone equations, though essential for nuclei, have so far not been performed for atoms.

CLUSTER STRUCTURE STUDY BASED ON REALISTIC NUCLEAR FORCE: BRUECKNER-AMD

We propose a new approach of the Brueckner-AMD that makes the antisymmetrized molecular dynamics (AMD) calculations possible with realistic nuclear forces. In this method, we solve the Bethe-Goldstone equation and calculate the G-matrix for every nucleon pair described by wave packets of AMD correctly. We report the results of applications to light nuclei with the Brueckner-AMD.

Description of Nuclear Structures with Brueckner-AMD plus J Projection

We propose a new approach that makes the antisymmetrized molecular dynamics (AMD) available to us with realistic nuclear interactions. In this method, the Bethe-Goldstone equation in the Brueckner theory is solved for every nucleon pair described by wave packets of AMD, and the G-matrix is calculated with single-particle orbits in AMD self-consistently. Furthermore, the spin and parity projection is performed correctly using the two-body correlation functions derived from the solutions of the Bethe-Goldstone equation. In this work, we apply this framework to not only α-nuclei but also N �= Z nuclei with A ∼ 10. It is confirmed that these results present the description of reasonable cluster structures and energy-level schemes comparable with the experimental ones in light nuclei. Subject Index: 206, 210

THE DESCRIPTION OF NUCLEAR STRUCTURES WITH THE BRUECKNER-AMD

We propose a new approach of the Brueckner-AMD that makes the antisymmetrized molecular dynamics (AMD) calculations possible with realistic nuclear forces. In this method, we solve the Bethe-Goldstone equation and calculate the G-matrix for every nucleon pair described by wave packets of AMD. In addition, we calculate the G-matrix in the spin and parity projection correctly using the two-body correlation functions derived from the solutions of the Bethe-Goldstone equation. We report the results of applications to light nuclei with the Brueckner-AMD plus the spin and parity projection.

The study of light nuclei with the Brueckner-AMD

We propose a new approach of the Brueckner-AMD that makes the antisymmetrized molecular dynamics (AMD) calculations possible with realistic nuclear forces. In this method, we solve the Bethe-Goldstone equation and calculate the G-matrix for every nucleon pair described by wave packets of AMD. In addition, we perform the spin and parity projection correctly using the two-body correlators derived from the solutions of the Bethe-Goldstone equation. We report the results of applications to light nuclei with the Brueckner-AMD plus the spin and parity projection.