Energy Density Functionals Research Articles

The loss of the property of locality of the kernel in high-dimensional Gaussian process regression on the example of the fitting of molecular potential energy surfaces.

Kernel-based methods, including Gaussian process regression (GPR) and generally kernel ridge regression, have been finding increasing use in computational chemistry, including the fitting of potential energy surfaces and density functionals in high-dimensional feature spaces. Kernels of the Matern family, such as Gaussian-like kernels (basis functions), are often used which allow imparting to them the meaning of covariance functions and formulating GPR as an estimator of the mean of a Gaussian distribution. The notion of locality of the kernel is critical for this interpretation. It is also critical to the formulation of multi-zeta type basis functions widely used in computational chemistry. We show, on the example of fitting of molecular potential energy surfaces of increasing dimensionality, the practical disappearance of the property of locality of a Gaussian-like kernel in high dimensionality. We also formulate a multi-zeta approach to the kernel and show that it significantly improves the quality of regression in low dimensionality but loses any advantage in high dimensionality, which is attributed to the loss of the property of locality.

New developments in fission studies within the time-dependent density functional theory framework

We have extended significantly the microscopic description of the fission process by examining a larger set of observables. We extract neutron and proton numbers of fission fragments, their spins and fission fragment relative orbital angular momentum and their correlations, investigate neutrons emitted at or shortly after scission, excitation energy sharing mechanism, total kinetic energy of fission fragments, and the entanglement entropy. I will present a short overview of our simulations obtained with two independent nuclear energy density functionals.

α -particle formation and clustering in nuclei

The nucleonic localization function has been used for a decade to study the formation of $\ensuremath{\alpha}$ particles in nuclei, by providing a measure of having nucleons of a given spin in a single place. However, differences in interpretation remain, compared to the nucleonic density of the nucleus. In order to better understand the respective role of the nucleonic localization function and the densities in the $\ensuremath{\alpha}$-particle formation in cluster states or in the $\ensuremath{\alpha}$-decay mechanism, both an analytic approximation and microscopic calculations, using energy density functionals, are undertaken. The nucleonic localization function is shown to measure the anticentrifugal effect, and is not sensitive to the level of compactness of the $\ensuremath{\alpha}$ particle itself. It probes the purity of the spatial overlap of four nucleons in the four possible (spin, isospin) states. The density provides, in addition, information on the compactness of an $\ensuremath{\alpha}$-particle cluster. The respective roles of the nucleonic localization function and the density are also analyzed in the case of $\ensuremath{\alpha}$ particle emission. More generally, criteria to assess the prediction of $\ensuremath{\alpha}$ cluster in nuclear states are provided.

Isoscalar monopole strength in 100Mo: an indicator for static triaxial deformation in the ground state* *Supported by the National Natural Science Foundation of China (12075068, 11705038), the Fundamental Research Funds for the Central Universities (HIT.BRET.2021003) and the UK STFC (ST/V001108/1)

We perform deformation constraint symmetry-unrestricted three-dimensional time-dependent density functional theory (TDDFT) calculations for the isoscalar monopole (ISM) mode in 100Mo. Monopole moments are obtained as a function of time using time propagating states based on different deformations. A Fourier transform is then performed on the obtained response functions. The resulting ISM strength functions are compared with experimental data. For the static potential-energy-surface (PES) calculations, the results using the SkM* and UNEDF1 energy-density functionals (EDFs) show spherical ground states and considerable softness in the triaxial deformation. The PES obtained with the SLy4 EDF shows static triaxial deformation. The TDDFT results based on different deformations show that a quadrupole deformation (characterized by ) value of 0.25–0.30 gives a two-peak structure of the strength functions. Increasing triaxial deformation (characterized by γ) from 0 to 30 results in the occurrence of an additional peak between the two, making the general shape of the strength functions closer to that of the data. Our microscopic TDDFT analyses suggest that 100Mo is triaxially deformed in the ground state. The calculated isoscalar and strength functions show peaks at lower energies. The coupling of these two modes with the ISM mode is the reason for the three-peak/plateau structure in the strengths of 100Mo.

Skyrme–Hartree–Fock–Bogoliubov mass models on a 3D mesh: II. Time-reversal symmetry breaking

Models based on nuclear energy density functionals can provide access to a multitude of observables for thousands of nuclei in a single framework with microscopic foundations. Such models can rival the accuracy of more phenomenological approaches, but doing so requires adjusting parameters to thousands of nuclear masses. To keep such large-scale fits feasible, several symmetry restrictions are typically imposed on the nuclear configurations. One such example is time-reversal invariance, which is generally enforced via the Equal Filling Approximation (EFA). Here we lift this assumption, enabling us to access the spin and current densities in the ground states of odd-mass and odd-odd nuclei, which contribute to the total energy of such nuclei through so-called ‘time-odd’ terms. We present here the Skyrme-based BSkG2 model whose parameters were adjusted to essentially all known nuclear masses without relying on the EFA, refining our earlier work [G. Scamps et al., EPJA 57, 333 (2021)]. Moving beyond ground state properties, we also incorporated information on the fission barriers of actinide nuclei in the parameter adjustment. The resulting model achieves a root-mean-square (rms) deviation of (i) 0.678 MeV on 2457 known masses, (ii) 0.027 fm on 884 measured charge radii, (iii) 0.44 MeV and 0.47 MeV, respectively, on 45 reference values for primary and secondary fission barriers of actinide nuclei, and (iv) 0.49 MeV on 28 fission isomer excitation energies. We limit ourselves here to a description of the model and the study the impact of lifting the EFA on ground state properties such as binding energies, deformation and pairing, deferring a detailed discussion of fission to a forthcoming paper.

Implications of parity-violating electron scattering experiments on 48Ca (CREX) and 208Pb (PREX-II) for nuclear energy density functionals

Recent precise parity-violating electron scattering experiments on 48Ca (CREX) and 208Pb (PREX-II) provide a new insight on the formation of neutron skin in nuclei. Within the energy density functional (EDF) framework, we investigate the implications of CREX and PREX-II data on nuclear matter symmetry energy and isovector properties of finite nuclei: neutron skin thickness and dipole polarizability. The weak-charge form factors from the CREX and PREX-II experiments are employed directly in constraining the relativistic density-dependent point coupling EDFs. The EDF established with the CREX data acquires considerably smaller values of the symmetry energy parameters, neutron skin thickness and dipole polarizability both for 48Ca and 208Pb, in comparison to the EDF obtained using the PREX-II data, and previously established EDFs. Presented analysis shows that CREX and PREX-II experiments could not provide consistent constraints for the isovector sector of the EDFs, and further theoretical and experimental studies are required.

Energy density functional for α+12C elastic scattering

In this work, the α-particle elastic scattering from 12C is studied using three different 12C densities based on the cluster model. This work aims to extend our previous analysis of α+12C system to study the energy dependence of the real double folded (DF) potential deduced from the energy density functional (EDF) α-cluster model over a wide energy range. In addition, The EDF α-cluster model is compared with other cluster models to see how reliable is this cluster model for 12C nucleus. The two α-cluster models used for comparison are based on the antisymmetrized molecular dynamics (AMD) and a cubic approximation of the nuclear equation of state (liquid drop type density-LD) models. The 12C densities based on these cluster models are used in the double folding procedure to obtain α+12C real part of the optical model potential (OMP). The imaginary part of the OMP is considered in conventional Wood-Saxon (WS) form. The obtained elastic scattering results show the reliability of the EDF α-cluster model in reproducing the experimental data on equal footing with the other two considered cluster models. Thus the EDF is reliable for studying the ground state α-cluster of the light 4mα-conjugate nuclei.

Macroscopic and microscopic description of phase transition in cerium isotopes

The spherical-to-deformed phase transition in cerium isotopes recently suggested to occur between $^{146}\mathrm{Ce}$ and $^{148}\mathrm{Ce}$ has been examined in the framework of the macroscopic algebraic collective model and two microscopic approaches, namely Skyrme-Hartree-Fock $+$ Bardeen-Cooper-Schrieffer (BCS) calculations and the symmetry conserving configuration mixing method with Gogny energy density functionals applied also to the neighboring nuclei along the cerium isotopic chain. Possible spectral signatures of the phase transition are discussed in more details. The microscopic calculations predict octupole softness manifested by rather flat potential energy curves as a function of the octupole deformation parameter ${\ensuremath{\beta}}_{3}$ for $^{146}\mathrm{Ce}$ and $^{148}\mathrm{Ce}$ and shape coexistence characterized by axially symmetric ${0}^{+}$ states, triaxial ${2}^{+}$ bands, and octupole deformation for the lowest ${1}^{\ensuremath{-}}$ states.

Deep-learning density functionals for gradient descent optimization.

Machine-learned regression models represent a promising tool to implement accurate and computationally affordable energy-density functionals to solve quantum many-body problems via density functional theory. However, while they can easily be trained to accurately map ground-state density profiles to the corresponding energies, their functional derivatives often turn out to be too noisy, leading to instabilities in self-consistent iterations and in gradient-based searches of the ground-state density profile. We investigate how these instabilities occur when standard deep neural networks are adopted as regression models, and we show how to avoid them by using an ad hoc convolutional architecture featuring an interchannel averaging layer. The main testbed we consider is a realistic model for noninteracting atoms in optical speckle disorder. With the interchannel average, accurate and systematically improvable ground-state energies and density profiles are obtained via gradient-descent optimization, without instabilities nor violations of the variational principle.

Density dependence of the nuclear symmetry energy and neutron skin thickness in the KIDS framework

The KIDS framework for the nuclear equation of state (EoS) and energy density functional (EDF) offers the possibility to explore systematically the effect of EoS parameters on predictions for a variety of observables. The EoS parameters can be varied independently of each other and independently of assumptions regarding the in-medium nucleon effective mass. Here I present a pilot study of the neutron skin thickness (NST) in nuclei of current interest. The results indicate that variations of the symmetry energy slope parameter L by roughly 10 MeV and variations of the droplet-model counterpart of the curvature parameter Kτ by roughly 20 MeV affect predictions by comparable amounts. However, structural details may also have sizable effects on predictions, notably in the cases of 68Ni and 208Pb. This work is part of a systematic investigation of the NST within the KIDS framework and of a broader effort to constrain the density dependence of the nuclear symmetry energy.

Isotopic Shift in Hg-Isotopes within Brückner versus Relativistic Energy Density Functional

The present study is focused on revealing a characteristic kink of the neutron shell closure N = 126 across the Hg-isotopic chain within the relativistic mean-field (RMF) approach with the IOPB-I, DD-ME2, DD-PC1 and NL3 parameter sets. The RMF densities are converted to their spherical equivalence via the Wood–Saxon approximation and used as input within the parametrization procedure of the coherent density fluctuation model (CDFM). The nuclear matter symmetry energy is calculated using the Brückner energy density functional, and its surface, as well as volume components, are evaluated within Danielwicz’s liquid drop prescription. In addition, a comparison between Brückner and relativistic energy density functionals using the NL3 parameter set is shown as a representative case. The binding energy, charge distribution radius and symmetry energy are used as indicators of the isotopic shift in both ground and isomeric states. We have found the presence of a kink at the shell/sub-shell closure at N = 126 for neutron-rich 206Hg. The formation of the kink is traceable to the early filling of the 1i11/2 orbitals rather than 2g9/2, due to the large spin-orbit splitting. As such, the link between the occupational probability and the magicity of nuclei over the Hg-isotopic chain is established.

Efficient Implementation of Density Functional Theory Based Embedding for Molecular and Periodic Systems Using Gaussian Basis Functions.

A practical and effective implementation of density functional theory based embedding is reported, which allows us to treat both periodic and aperiodic systems on an equal footing. Its essence is the expansion of orbitals and electron density of the periodic system using Gaussian basis functions, rather than plane-waves, which provides a unique all-electron direct-space representation, thus avoiding the need for pseudopotentials. This makes the construction of embedding potential for a molecular active subsystem due to a periodic environment quite convenient, as transformation between representations is far from trivial. The three flavors of embedding, molecule-in-molecule, molecule-in-periodic, and periodic-in-periodic embedding, are implemented using embedding potentials based on non-additive kinetic energy density functionals (approximate) and level-shift projection operator (exact). The embedding scheme is coupled with a variety of correlated wave function theory (WFT) methods, thereby providing an efficient way to study the ground and excited state properties of low-dimensional systems using high-level methods for the region of interest. Finally, an implementation of real time-time-dependent density functional embedding theory (RT-TDDFET) is presented that uses a projection operator-based embedding potential and provides accurate results compared to full RT-TDDFT for systems with uncoupled excitations. The embedding potential is calculated efficiently using a combination of density fitting and continuous fast multipole method for the Coulomb term. The applicability of (i) WFT-in-DFT embedding, in predicting the adsorption and excitation energies, and (ii) RT-TDDFET, in predicting the absorption spectra, is explored for various test systems.

Nuclear incompressibility and speed of sound in uniform matter and finite nuclei

We have extended the compressible liquid-drop model (CLDM) with a density-dependent surface term (eCLDM), which allows for a unified description of both the nuclear ground state energies and the incompressibility modulus in finite nuclei $K_A$. We analyse the role of the nuclear empirical parameters, e.g., $K_{sat}$, $Q_{sat}$, $L_{sym}$ and $K_{sym}$, which contribute to the bulk properties, as well as the role of the finite size contributions. For the bulk properties, the density and isospin dependencies of the nuclear incompressibility in infinite matter are characterized by introducing new empirical parameters, and two new constraints for the value of $K_{sym}$ are suggested. For finite nuclei, we employ a Bayesian approach coupled to a Markov-Chain Monte-Carlo (MCMC) exploration of the parameter space to confront the model predictions of $K_A$ in Zr, Sn and Pb isotopes to the experimental data. We show that $Q_{sat}\approx -950\pm 200$~MeV describes the experimental measurements of $K_A$ in these isotopes. This value is different from the ones deduced from phenomenological nuclear energy density functionals, suggesting a possible explanation of their difficulty to accurately describe Zr, Sn and Pb data all together. In addition we explore the impact of a fictitious measurement of the Giant Monopole Resonance energy in $^{132}$Sn. We show that this measurement, provided it is accurate enough, will allow to better determine $K_{sym}$ and $K_{\tau}$. Finally we explore the properties of the sound speed around saturation density and show the important role of finite size terms in finite nuclei since they reduce the sound speed to approximately half compared to nuclear matter.

Applications of reduced-basis methods to the nuclear single-particle spectrum

Reduced basis methods provide a powerful framework for building efficient and accurate emulators. Although widely applied in many fields to simplify complex models, reduced basis methods have only been recently introduced into nuclear physics. In this letter we build an emulator to study the single-particle structure of atomic nuclei. By scaling a suitable mean-field Hamiltonian, a "universal" reduced basis is constructed capable of accurately and efficiently reproduce the entire single-particle spectrum of a variety of nuclei. Indeed, the reduced basis model reproduces both ground- and excited-state energies as well as the associated wave-functions with remarkable accuracy. Our results bode well for more demanding applications that use Bayesian optimization to calibrate nuclear energy density functionals.

Neutrinoless double- β decay from an effective field theory for heavy nuclei

We study neutrinoless double-beta decay in an effective field theory (EFT) for heavy nuclei, which are treated as a spherical core coupled to additional neutrons and/or protons. Since the low-energy constants of the EFT cannot be fitted to data for this unobserved decay, we follow an alternative strategy to constrain these through a correlation with double Gamow-Teller transitions. This correlation was recently found to hold for shell-model calculations, energy-density functionals, and other nuclear structure models. We therefore first calculate the nuclear matrix elements for double Gamow-Teller transitions in the EFT for heavy nuclei. The combination of the EFT uncertainty with the correlation uncertainty enables predictions of nuclear matrix elements for neutrinoless double-beta decay for a broad range of isotopes with quantified uncertainties. Generally the EFT predicts smaller nuclear matrix elements compared to other approaches, but our EFT results are consistent with recent ab initio calculations.

Enhanced moments of inertia for rotation in neutron-rich nuclei

Ground-state moments of inertia (MoI) are investigated for approximately 1700 even–even nuclei from the proton drip line to the neutron drip up to Z=120 and N=184. The cranked Hartree–Fock–Bogoliubov equation is solved in the coordinate space using the Skyrme and pairing energy density functionals. This model well describes the available experimental data of the first excited Iπ=2+ state energy of more than 300 nuclides possessing appreciable deformation. The MoI is found to greatly increase near the drip line, whereas the deformation is found to be not as strong as estimated by the empirical relation when the pairing functional including the isovector-density dependence is adopted. The currently used pairing functionals produce large uncertainties in the prediction of E(21+) values near the neutron drip line.

Finite-temperature calculation of the isovector giant dipole resonance in Sn120 using self-consistent inputs

Background: Although microscopic models are frequently used to investigate giant resonances built on the ground state of a nucleus, similar calculations at nonzero temperature are limited. Approaches based on the classical thermal shape-fluctuation model (TSFM) are often utilized to analyze the properties of giant dipole resonance at finite temperature. However, TSFM fails to predict the correct trend of resonance width over a wide range of temperature starting from zero.Purpose: To study the isovector giant dipole resonance (IVGDR) at finite temperature, we present an improved version of TSFM, where driving potential and IVGDR centroid energies are obtained self-consistently using the nuclear energy density functional formalism. The temperature dependence of IVGDR width in $^{120}\mathrm{Sn}$ is calculated to benchmark the proposed method.Method: Nuclear free energy surfaces and entropy surfaces are simulated at different temperatures for two different parametrizations of Skyrme energy density functionals. Moreover, IVGDR centroid energies are calculated by incorporating nuclear deformations extracted from the corresponding self-consistent densities. Subsequently, IVGDR widths are obtained within TSFM. Also, we demonstrate the role of pairing fluctuation in improving the low temperature behavior of IVGDR width.Result: We found good overall agreement of our results with the measured IVGDR widths. Specifically, except at a very low temperature, unprecedented accuracy has been achieved in TSFM. Calculated IVGDR widths are shown to be robust against the choices of driving potential and Skyrme parametrization.Conclusion: The present paper provides guidance on using TSFM based calculations in predicting the IVGDR width over a broad range of temperature. For a more consistent calculation, pairing fluctuations are required to be included as additional degrees of freedom.

Accreting neutron stars from the nuclear energy-density functional theory

Context. The accretion of matter onto the surface of a neutron star in a low-mass X-ray binary triggers X-ray bursts, whose ashes are buried and further processed thus altering the composition and the properties of the stellar crust. Aims. In this second paper of a series, the impact of accretion on the equation of state and on the global properties of neutron stars is studied in the framework of the nuclear energy-density functional theory. Methods. Considering ashes made of 56Fe, we calculated the equations of state using the same Brussels-Montreal nuclear energy-density functionals BSk19, BSk20, and BSk21, as those already employed for determining the crustal heating in our previous study for the same ashes. All regions of accreting neutron stars were treated in a unified and thermodynamically consistent way. With these equations of state, we determined the mass, radius, moment of inertia, and tidal deformability of accreted neutron stars and compared with catalyzed neutron stars for which unified equations of state based on the same functionals are available. Results. The equation of state of accreted neutron stars is found to be significantly stiffer than that of catalyzed matter, with an adiabatic index Γ ≈ 4/3 throughout the crust. For this reason, accreting neutron stars have larger radii. However, their crustal moment of inertia and their tidal deformability are hardly changed provided density discontinuities at the interface between adjacent crustal layers are properly taken into account. Conclusions. The enhancement of the stiffness of the equation of state of accreting neutron stars is mainly a consequence of nuclear shell effects, thus confirming the importance of a quantum treatment as stressed in our first study. With our previous calculations of crustal heating using the same functionals, we have thus obtained consistent microscopic inputs for simulations of accreting neutron stars.

Constraint‐based analysis of a physics‐guided kinetic energy density expansion

AbstractAn approach guided by physical consistency in determining the general forms of D‐dimensional kinetic energy density functionals (KEDF) has been demonstrated previously, producing an expansion which contains the majority of the known one‐point KEDF forms. It has also been shown that any noninteracting KEDF must necessarily have a homogeneity degree of 2 in coordinate scaling, and that the ratio of the collective KED to electron density must approach the ionization energy as . This article demonstrates that the scaling condition is already satisfied in the general expansion despite not being conceived with the scaling as a constraint, and that the second condition places a restriction on the expansion terms of the KED. The discussion is extended as well for some known KEDs for comparison.

Calculations for nuclear matter and finite nuclei within and beyond energy-density-functional theories through interactions guided by effective field theory

We propose a novel idea to construct an effective interaction under energy-density-functional (EDF) theories which is adaptive to the enlargement of the model space. Guided by effective field theory principles, iterations of interactions as well as enlargements of the model space through particle-hole excitations are carried out for infinite nuclear matter and selected closed-shell nuclei ($^{4}\mathrm{He}, ^{16}\mathrm{O}, ^{40}\mathrm{Ca}, ^{56}\mathrm{Ni}$, and $^{100}\mathrm{Sn}$) up to next-to-leading order. Our approach provides a new way for handling the nuclear matter and finite nuclei within the same scheme, with advantages from both EDF and ab initio approaches.