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Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

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At least 757 records · Page 42
All sourcesNASA NTRSDOE OSTI
DOE OSTI2020

Enabling Large-Scale Condensed-Phase Hybrid Density Functional Theory Based Ab Initio Molecular Dynamics. 1. Theory, Algorithm, and Performance

By including a fraction of exact exchange (EXX), hybrid functionals reduce the self-interaction error in semilocal density functional theory (DFT) and thereby furnish a more accurate and reliable description of the underlying electronic structure in systems throughout biology, chemistry, physics, and materials science. However, the high computational cost associated with the evaluation of all required EXX quantities has limited the applicability of hybrid DFT in the treatment of large molecules and complex condensed-phase materials. To overcome this limitation, we describe a linear-scaling approach that utilizes a local representation of the occupied orbitals (e.g., maximally localized Wannier functions (MLWFs)) to exploit the sparsity in the real-space evaluation of the quantum mechanical exchange interaction in finite-gap systems. In this work, we present a detailed description of the theoretical and algorithmic advances required to perform MLWF-based ab initio molecular dynamics (AIMD) simulations of large-scale condensed-phase systems of interest at the hybrid DFT level. We focus our theoretical discussion on the integration of this approach into the framework of Car–Parrinello AIMD, and highlight the central role played by the MLWF-product potential (i.e., the solution of Poisson’s equation for each corresponding MLWF-product density) in the evaluation of the EXX energy and wave function forces. We then provide a comprehensive description of the exx algorithm implemented in the open-source Quantum ESPRESSO program, which employs a hybrid MPI/OpenMP parallelization scheme to efficiently utilize the high-performance computing (HPC) resources available on current- and next-generation supercomputer architectures. Furthermore, this is followed by a critical assessment of the accuracy and parallel performance (e.g., strong and weak scaling) of this approach when AIMD simulations of liquid water are performed in the canonical (NVT) ensemble. With access to HPC resources, we demonstrate that exx enables hybrid DFT-based AIMD simulations of condensed-phase systems containing 500–1000 atoms (e.g., (H₂O)₂₅₆) with a wall time cost that is comparable to that of semilocal DFT. In doing so, exx takes us one step closer to routinely performing AIMD simulations of complex and large-scale condensed-phase systems for sufficiently long time scales at the hybrid DFT level of theory.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2025

CpFe(CO) 2 Radical Generated from Dinuclear [CpFe(CO) 2 ] 2 and Mononuclear (Cp)(CO) 2 Fe(H): Density Functional Theory Is Accurate for One, But Not Both

Density functional theory (DFT) methods remain the most practical approach to calculating properties and reaction mechanisms of transition metal complexes. While the accuracy of DFT methods has been evaluated for some properties of mononuclear organometallic complexes there has been a general lack of evaluation for dinuclear organometallic complexes, in particular bonding changes related to reaction mechanisms. Here, this work evaluated DFT and coupled cluster methods for the accuracy of calculating the CpFe(CO) 2 radical (Fp•) generated from dinuclear [CpFe(CO) 2 ] 2 (Fp 2 ) and mononuclear [(Cp)(CO) 2 Fe(H)] (Fp-H). This transition metal radical fragment was evaluated because dinuclear complexes built with it have recently shown a variety of unique reactions but has proven challenging to accurately calculate with DFT methods. Here we show that DFT methods provide a surprising wide range of fragmentation energies for Fp 2 and lower and mid rung DFT methods as well as DLPNO–CCSD(T) perform well for this dissociation energy. The highest rung double-hybrid methods have a large range in the Fp 2 dissociation energy, and the energy greatly depends on the amount of MP2 correlation energy included. For generating Fp• from Fp-H the lower and mid rung methods that worked well for Fp 2 showed significant error. Double-hybrid methods unfortunately are only accurate for the Fe–H bond if they are very inaccurate for the Fp 2 dissociation energy. While DLPNO–CCSD(T) is not perfect, and not close to chemically accurate for the Fe–H bond, it does provide reasonable accuracy for both Fp 2 and Fp-H dissociation energies.

density functional theory↗
DOE OSTI2017

Accurate Isomerization Enthalpy and Investigation of the Errors in Density Functional Theory for Dihydroazulene/Vinylheptafulvene Photochromism Using Diffusion Monte Carlo

In this paper, we investigate the isomerization enthalpy of the dihydroazulene/vinylheptafulvene (DHA/VHF) molecular photoswitch system derivatives using electronic structure calculation methods including density functional theory (DFT), quantum Monte Carlo (QMC), and coupled cluster (CCSD(T)). Recent efforts have focused on tuning the isomerization enthalpy of the photoswitch for solar thermal energy storage applications using substitutional functional groups on its five- and seven-membered carbon rings, predominantly using DFT for the energy predictions. However, using the higher accuracy QMC and CCSD(T) methods, we show that in many cases DFT incorrectly predicts the isomerization enthalpy, and the errors depend on the functional groups substituted and the choice of the DFT functional. Isomerization of the DHA to VHF molecule is an electrocyclic ring-opening reaction on the five-membered ring of the DHA isomer. We find that the DFT errors are correlated to the electrocyclic ring-opening reactions of cyclobutene and cyclo-1,3-hexadiene, such that the DFT error changes monotonically with the size of the carbon ring, although QMC and CCSD(T) results are in a good agreement irrespective of the ring size. Using the QMC and CCSD(T) isomerization enthalpies, we predict gravimetric energy densities of the DHA derivatives for solar thermal storage applications. Our results show that suitable substitutions on DHA can yield gravimetric storage densities as large as 732 kJ/kg.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2018

Initial Reduction of the PdO(101) Surface: Role of Oxygen Vacancy Formation Kinetics

Here, we investigated the initial thermal reduction of the PdO(101) surface using density functional theory (DFT) calculations. The defect-free PdO(101) surface consists of parallel rows of undercoordinated Pd and O atoms (Pd cus and O cus , respectively). DFT was used to map out the elementary processes of O cus vacancy formation, oxygen surface atom diffusion along the Pd cus rows, and various O 2 formation pathways. Because oxygen vacancies form during the reduction of the surface, the elementary processes were examined with and without the presence of adjacent O cus vacancies. DFT calculations show that the presence of oxygen vacancies strongly affects the barriers of adjacent surface processes. Barriers for O vacancy formation are reduced by nearly 50% with an adjacent O vacancy. The barrier for O 2 formation along the Pd cus row drops from 1.54 eV on the defect-free PdO(101) surface to 0.81 eV in the presence of an O cus vacancy dimer. Yet, the presence of O cus vacancies also leads to more strongly bound O 2 on the Pd cus row. The strong neighbor effect of O vacancies on the PdO(101) surface will favor the growth of O vacancy chains and increase the rate of thermal reduction of the surface, both features that have been observed in recent ultrahigh vacuum isothermal reduction experiments of the PdO(101) surface.

Defects in solids↗
DOE OSTI2018

Insights on the Alumina–Water Interface Structure by Direct Comparison of Density Functional Simulations with X-ray Reflectivity

Density functional theory molecular dynamics (DFT-MD) simulations are frequently used to predict the interfacial structures and dynamical processes at solid-water interfaces in efforts to gain a deeper understanding of these systems. However, the accuracy of these predictions of interfacial structure has not been rigorously quantified. Here, direct comparisons between large-scale DFT-MD simulations and high-resolution X-ray reflectivity (XR) measurements of the well-defined Al2O3(001)-water interface reveal the relative accuracy of these two methods to describe interfacial structure, a comparison that is enabled by XR’s high sensitivity to atomic-scale displacements. The DFT-MD simulated and XR model-fit structures are qualitatively similar, but XR signals calculated directly from the DFT-MD predictions deviate significantly from the experimental data, revealing discrepancies in these two approaches. Differences in the derived interfacial Al2O3 relaxation profiles of ~0.02 A within the top 5 layers are significant to XR but at the limit of the accuracy of DFT. Further differences are found in the surface hydration layer with a simulated average water layer height ~0.2 A higher than that observed experimentally. This is outside the accuracy of both XR and DFT and is not improved by the inclusion of a phenomenological correction for hydrogen bonding (e.g., Grimme).

74 ATOMIC AND MOLECULAR PHYSICS↗
DOE OSTI2017

Interplay between Terminal and Bridging Diiron Hydrides in Neutral and Oxidized States

This study describes the structural, spectroscopic, and electrochemical properties of electronically unsymmetrical diiron hydrides. The terminal hydride Cp*Fe(pdt)Fe(dppe)(CO)H ([1(t-H)]0, Cp*– = Me5C5 –, pdt2– = CH2(CH2S–)2, dppe = Ph2PC2H4PPh2) was prepared by hydride reduction of [Cp*Fe(pdt)Fe(dppe)(CO)(NCMe)]+. As established by X-ray crystallography, [1(t- H)]0 features a terminal hydride ligand. Unlike previous examples of terminal diiron hydrides, [1(t-H)]0 does not isomerize to the bridging hydride [1(µ-H)]0. Oxidation of [1(t-H)]0 gives [1(t- H)]+, which was also characterized crystallographically as its BF4 – salt. Density functional theory (DFT) calculations indicate that [1(t-H)]+ is best described as containing an Cp*Fe(III) center. In solution, [1(t-H)]+ isomerizes to [1(µ-H)]+ as anticipated by DFT. Reduction of [1(µ- H)]+ by Cp2Co affords the diferrous bridging hydride [1(µ-H)]0. Electrochemical measurements and DFT calculations indicate that the couples [1(t-H)]+/0 and [1(µ-H)]+/0 differ by 210 mV. Qualitative measurements indicate that [1(t-H)]0 and [1(µ-H)]0 are close in free energy. Protonation of [1(t-H)]0 in MeCN solution affords H2 even with weak acids via hydride transfer.

Yu, Xin↗
DOE OSTI2015

Enhanced O-2 Selectivity versus N-2 by Partial Metal Substitution in Cu-BTC

Here, we describe the homogeneous substitution of Mn, Fe, and Co at various levels into a prototypical metal organic framework (MOP), namely Cu-BTC (HKUST-1), and the effect of that substitution on preferential gas sorption. Using a combination of density functional theory (DFT) calculations, postsynthetic metal substitutions, materials characterization, and gas sorption testing, we demonstrate that the identity of the metal ion has a quantifiable effect on their oxygen and nitrogen sorption properties at cryogenic temperatures. An excellent correlation is found between O-2/N-2 selectivities determined experimentally at 77 K and the difference in O-2 and N-2 binding energies calculated from DFT modeling data: Mn > Fe Co >> Cu. Room temperature gas sorption studies were also performed and correlated with metal substitution. The Fe-exchanged sample shows a significantly higher nitrogen isosteric heat of adsorption at temperatures close to ambient conditions (273-298 K) as compared to all other metals studied, indicative of favorable interactions between N-2 and coordinatively unsaturated Fe metal centers. Interestingly, differences in gas adsorption results at cryogenic and room temperatures are evident; they are explained by comparing experimental results with DFT binding energies (0 K) and room temperature Grand Canonical Monte Carlo simulations.

Sava Gallis, Dorina F.↗
DOE OSTI2018

Facile Dehydrogenation of Ethane on the IrO 2 (110) Surface

Realizing the efficient and selective conversion of ethane to ethylene is important for improving the utilization of hydrocarbon resources, yet remains a major challenge in catalysis. Herein, ethane dehydrogenation on the IrO 2 (110) surface is investigated using temperature-programmed reaction spectroscopy (TPRS) and density functional theory (DFT) calculations. The results show that ethane forms strongly bound σ-complexes on IrO 2 (110) and that a large fraction of the complexes undergo C–H bond cleavage during TPRS at temperatures below 200 K. Further, continued heating causes as much as 40% of the dissociated ethane to dehydrogenate and desorb as ethylene near 350 K, with the remainder oxidizing to CO x species. Both TPRS and DFT show that ethylene desorption is the rate-controlling step in the conversion of ethane to ethylene on IrO 2 (110) during TPRS. Partial hydrogenation of the IrO 2 (110) surface is found to enhance ethylene production from ethane while suppressing oxidation to CO x species. DFT predicts that hydrogenation of reactive oxygen atoms of the IrO 2 (110) surface effectively deactivates these sites as H atom acceptors, and causes ethylene desorption to become favored over further dehydrogenation and oxidation of ethane-derived species. Finally, the study reveals that IrO 2 (110) exhibits an exceptional ability to promote ethane dehydrogenation to ethylene near room temperature, and provides molecular-level insights for understanding how surface properties influence selectivity toward ethylene production.

bond cleavage oxidation↗
DOE OSTI2013

The Influence of Functionals on Density Functional Theory Calculations of the Properties of Reducible Transition Metal Oxide Catalysts

Reducible transition metal oxides (RTMOs) comprise an important class of catalytic materials that are used for the selective oxidation and electro- and photochemical splitting of water, and as supports for metal nanoparticles. It is highly desirable to model the properties of these materials accurately using density functional theory (DFT) in order to understand how oxide structure and performance are related and to guide the search for materials exhibiting superior performance. Unfortunately, accurate description of the structural and electronic properties of RTMOs using DFT has proven particularly challenging. The M06-L density functional, which has been shown to be broadly accurate for calculations of gas phase clusters, has recently become available to researchers carrying out calculations in the solid state, but its performance in determining the properties RTMOs has been little investigated. The aim of this work was to assess the performance of the M06-L functional for describing the structural and electronic properties of a family of RTMOs: MoO 2 , MoO 3 , and Bi 2 Mo 3 O 12 . Lattice constants, band gaps, and densities of states calculated using the M06-L functional are compared to those obtained from DFT+U. We have also used the M06-L functional to determine the reaction barrier for propene activation over Bi 2 Mo 3 O 12 , the rate-limiting step in the oxidation of propene to acrolein. We find that while DFT calculations carried out with the M06-L functional are roughly five times more expensive computationally than those performed with DFT+U, the results obtained using the M06-L functional provide sensible results for all properties investigated, while avoiding the necessary trade-off between accurate electronic structure and accurate thermochemistry that occurs in DFT+U.

Getsoian, Andrew “Bean”↗
DOE OSTI2018

The diamine cation is not a chemical example where density functional theory fails

In a recent communication, Weber et al.1 presented a surprising study on charge-localization effects in the N,N’- dimethylpiperazine (DMP+) diamine cation to provide a stringent test of density functional theory (DFT) methods. Within their study, the authors examined various DFT methods and concluded that “all DFT functionals commonly used today, including hybrid functionals with exact exchange, fail to predict a stable chargelocalized state.”1 This surprising conclusion is based on the authors’ use of a self-interaction correction (namely, complexvalued Perdew–Zunger Self-Interaction Correction (PZ-SIC))2,3 to DFT, which appears to give excellent agreement with experiment and other wavefunction-based benchmarks. Since the publication of this recent communication, the same DMP+ molecule has been cited in numerous subsequent studies4–13 as a prototypical example of the importance of self-interaction corrections for accurately calculating other chemical systems. In this correspondence, we have carried out new high-level CCSD(T) analyses on the DMP+ cation to show that DFT actually performs quite well for this system (in contrast to their conclusion that all DFT functionals fail), whereas the PZ-SIC approach used by Weber et al. is the outlier that is inconsistent with the high-level CCSD(T) (coupled-cluster with single and double excitations and perturbative triples) calculations. Our new findings and analysis for this system are briefly discussed in this correspondence.

Ali, Zulfikhar A.↗
DOE OSTI2015

On the accuracy of density functional theory and wave function methods for calculating vertical ionization energies

The best practice in computational methods for determining vertical ionization energies (VIEs) is assessed, via reference to experimentally determined VIEs that are corroborated by highly accurate coupled-cluster calculations. These reference values are used to benchmark the performance of density functional theory (DFT) and wave function methods: Hartree-Fock theory, second-order Møller-Plesset perturbation theory, and Electron Propagator Theory (EPT). The core test set consists of 147 small molecules. An extended set of six larger molecules, from benzene to hexacene, is also considered to investigate the dependence of the results on molecule size. The closest agreement with experiment is found for ionization energies obtained from total energy difference calculations. In particular, DFT calculations using exchange-correlation functionals with either a large amount of exact exchange or long-range correction perform best. The results from these functionals are also the least sensitive to an increase in molecule size. In general, ionization energies calculated directly from the orbital energies of the neutral species are less accurate and more sensitive to an increase in molecule size. For the single-calculation approach, the EPT calculations are in closest agreement for both sets of molecules. For the orbital energies from DFT functionals, only those with long-range correction give quantitative agreement with dramatic failing for all other functionals considered. The results offer a practical hierarchy of approximations for the calculation of vertical ionization energies. In addition, the experimental and computational reference values can be used as a standardized set of benchmarks, against which other approximate methods can be compared.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2015

Electronic spectra from TDDFT and machine learning in chemical space

Due to its favorable computational efficiency, time-dependent (TD) density functional theory (DFT) enables the prediction of electronic spectra in a high-throughput manner across chemical space. Its predictions, however, can be quite inaccurate. We resolve this issue with machine learning models trained on deviations of reference second-order approximate coupled-cluster (CC2) singles and doubles spectra from TDDFT counterparts, or even from DFT gap. We applied this approach to low-lying singlet-singlet vertical electronic spectra of over 20 000 synthetically feasible small organic molecules with up to eight CONF atoms. The prediction errors decay monotonously as a function of training set size. For a training set of 10 000 molecules, CC2 excitation energies can be reproduced to within ±0.1 eV for the remaining molecules. Analysis of our spectral database via chromophore counting suggests that even higher accuracies can be achieved. Based on the evidence collected, we discuss open challenges associated with data-driven modeling of high-lying spectra and transition intensities.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2015

Quantitative first-principles theory of interface absorption in multilayer heterostructures

The unique chemical bonds and electronic states of interfaces result in optical properties that are different from those of the constituting bulk materials. In the nanoscale regime, the interface effects can be dominant and impact the optical response of devices. Using density functional theory (DFT), the interface effects can be calculated, but DFT is computationally limited to small systems. In this paper, we describe a method to combine DFT with macroscopic methodologies to extract the interface effect on absorption in a consistent and quantifiable manner. The extracted interface effects are an independent parameter and can be applied to more complicated systems. Finally, we demonstrate, using NiSi 2 /Si heterostructures, that by varying the relative volume fractions of interface and bulk, we can tune the spectral range of the heterostructure absorption.

36 MATERIALS SCIENCE↗
DOE OSTI2016

A polarizable QM/MM approach to the molecular dynamics of amide groups solvated in water

The infrared (IR) spectra of polypeptides are dominated by the so-called amide bands. Because they originate from the strongly polar and polarizable amide groups (AGs) making up the backbone, their spectral positions sensitively depend on the local electric fields. Aiming at accurate computations of these IR spectra by molecular dynamics (MD) simulations, which derive atomic forces from a hybrid quantum and molecular mechanics (QM/MM) Hamiltonian, here we consider the effects of solvation in bulk liquid water on the amide bands of the AG model compound N-methyl-acetamide (NMA). As QM approach to NMA we choose grid-based density functional theory (DFT). For the surrounding MM water, we develop, largely based on computations, a polarizable molecular mechanics (PMM) model potential called GP6P, which features six Gaussian electrostatic sources (one induced dipole, five static partial charge distributions) and, therefore, avoids spurious distortions of the DFT electron density in hybrid DFT/PMM simulations. Bulk liquid GP6P is shown to have favorable properties at the thermodynamic conditions of the parameterization and beyond. Lennard-Jones (LJ) parameters of the DFT fragment NMA are optimized by comparing radial distribution functions in the surrounding GP6P liquid with reference data obtained from a “first-principles” DFT-MD simulation. Finally, IR spectra of NMA in GP6P water are calculated from extended DFT/PMM-MD trajectories, in which the NMA is treated by three different DFT functionals (BP, BLYP, B3LYP). Method-specific frequency scaling factors are derived from DFT-MD simulations of isolated NMA. The DFT/PMM-MD simulations with GP6P and with the optimized LJ parameters then excellently predict the effects of aqueous solvation and deuteration observed in the IR spectra of NMA. As a result, the methods required to accurately compute such spectra by DFT/PMM-MD also for larger peptides in aqueous solution are now at hand.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2016

The energy level alignment at metal–molecule interfaces using Wannier–Koopmans method

We apply a recently developed Wannier-Koopmans method (WKM), based on density functional theory (DFT), to calculate the electronic energy level alignment at an interface between a molecule and metal substrate. We consider two systems: benzenediamine on Au (111), and a bipyridine-Au molecular junction. The WKM calculated level alignment agrees well with the experimental measurements where available, as well as previous GW and DFT + Σ results. These results suggest that the WKM is a general approach that can be used to correct DFT eigenvalue errors, not only in bulk semiconductors and isolated molecules, but also in hybrid interfaces.

36 MATERIALS SCIENCE↗
DOE OSTI2016

Global and local curvature in density functional theory

Piecewise linearity of the energy with respect to fractional electron removal or addition is a requirement of an electronic structure method that necessitates the presence of a derivative discontinuity at integer electron occupation. Semi-local exchange-correlation (xc) approximations within density functional theory (DFT) fail to reproduce this behavior, giving rise to deviations from linearity with a convex global curvature that is evidence of many-electron, self-interaction error and electron delocalization. Popular functional tuning strategies focus on reproducing piecewise linearity, especially to improve predictions of optical properties. In a divergent approach, Hubbard U-augmented DFT (i.e., DFT+U) treats self-interaction errors by reducing the local curvature of the energy with respect to electron removal or addition from one localized subshell to the surrounding system. Although it has been suggested that DFT+U should simultaneously alleviate global and local curvature in the atomic limit, no detailed study on real systems has been carried out to probe the validity of this statement. In this work, we show when DFT+U should minimize deviations from linearity and demonstrate that a “+U” correction will never worsen the deviation from linearity of the underlying xc approximation. However, we explain varying degrees of efficiency of the approach over 27 octahedral transition metal complexes with respect to transition metal (Sc–Cu) and ligand strength (CO, NH{sub 3}, and H{sub 2}O) and investigate select pathological cases where the delocalization error is invisible to DFT+U within an atomic projection framework. Finally, we demonstrate that the global and local curvatures represent different quantities that show opposing behavior with increasing ligand field strength, and we identify where these two may still coincide.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2016

Angle-dependent strong-field molecular ionization rates with tuned range-separated time-dependent density functional theory

Strong-field ionization and the resulting electronic dynamics are important for a range of processes such as high harmonic generation, photodamage, charge resonance enhanced ionization, and ionization-triggered charge migration. Modeling ionization dynamics in molecular systems from first-principles can be challenging due to the large spatial extent of the wavefunction which stresses the accuracy of basis sets, and the intense fields which require non-perturbative time-dependent electronic structure methods. In this paper, we develop a time-dependent density functional theory approach which uses a Gaussian-type orbital (GTO) basis set to capture strong-field ionization rates and dynamics in atoms and small molecules. This involves propagating the electronic density matrix in time with a time-dependent laser potential and a spatial non-Hermitian complex absorbing potential which is projected onto an atom-centered basis set to remove ionized charge from the simulation. For the density functional theory (DFT) functional we use a tuned range-separated functional LC-PBE*, which has the correct asymptotic 1/r form of the potential and a reduced delocalization error compared to traditional DFT functionals. Ionization rates are computed for hydrogen, molecular nitrogen, and iodoacetylene under various field frequencies, intensities, and polarizations (angle-dependent ionization), and the results are shown to quantitatively agree with time-dependent Schrödinger equation and strong-field approximation calculations. This tuned DFT with GTO method opens the door to predictive all-electron time-dependent density functional theory simulations of ionization and ionization-triggered dynamics in molecular systems using tuned range-separated hybrid functionals.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2012

Reflection-asymmetric nuclear deformations within the Density Functional Theory

Within the nuclear density functional theory (DFT) we study the effect of reflection-asymmetric shapes on ground-state binding energies and binding energy differences. To this end, we developed the new DFT solver AXIALHFB that uses an approximate second-order gradient to solve the Hartree-Fock-Bogoliubov equations of superconducting DFT with the quasi-local Skyrme energy density functionals. Illustrative calculations are carried out for even-even isotopes of radium and thorium.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗