Fock Method Research Articles

Hybrid algorithm for the time-dependent Hartree–Fock method using the Yang–Baxter equation on quantum computers**The submitted manuscript has been created by UChicago Argonne, LLC, Operator of Argonne National Laboratory (‘Argonne’). Argonne, a U.S. Department of Energy Office of Science laboratory, operated under Contract No. DE-AC02-06CH11357 and Pacific Northwest National Laboratory (PNNL), a U.S. Department of Energy Office of Science laboratory, operated

Hybrid algorithm for the time-dependent Hartree–Fock method using the Yang–Baxter equation on quantum computers**The submitted manuscript has been created by UChicago Argonne, LLC, Operator of Argonne National Laboratory (‘Argonne’). Argonne, a U.S. Department of Energy Office of Science laboratory, operated under Contract No. DE-AC02-06CH11357 and Pacific Northwest National Laboratory (PNNL), a U.S. Department of Energy Office of Science laboratory, operated

Implementation of time-dependent Hartree–Fock in real space

Abstract Time-dependent Hartree–Fock (TDHF) is one of the fundamental post-Hartree–Fock (HF) methods to describe excited states. In its Tamm-Dancoff form, equivalent to Configuration Interaction Singles, it is still widely used and particularly applicable to big molecules where more accurate methods may be unfeasibly expensive. However, it is rarely implemented in real space, mostly because of the expensive nature of the exact-exchange potential in real space. Compared to widely used Gaussian-type orbitals (GTO) basis sets, real space often offers easier implementation of equations and more systematic convergence of Rydberg states, as well as favorable scaling, effective domain parallelization, flexible boundary conditions, and ability to treat model systems. We implemented TDHF in the Octopus real-space code as a step toward linear-response hybrid time-dependent density-functional theory (TDDFT), other post-HF methods, and ensemble density-functional theory methods involving exact exchange. Calculation of HF’s non-local exact exchange is very expensive in real space. We overcome this limitation with Octopus’ implementation of Adaptively Compressed Exchange, and find the appropriate mixing scheme and starting point to complete the ground-state calculation in a practical amount of time, and thus enable TDHF. We compared our results to those from GTOs on a set of small molecules and confirmed close agreement of results, though with larger deviations than in the case of semi-local TDDFT. We find that convergence of TDHF demands a finer real-space grid than semi-local TDDFT. We also present the subtleties in benchmarking a real-space calculation against GTOs, relating to Rydberg and vacuum states.

Collisional and radiative data for tellurium ions in kilonovae modelling and laboratory benchmarks

ABSTRACT Tellurium is a primary candidate for the identification of the 2.1 $\, \mu$m emission line in kilonovae (KNe) spectra AT2017gfo and GRB230307A. Despite this, there is currently an insufficient amount of atomic data available for this species. We calculate the required atomic structure and collisional data, particularly the data required for accurate non-local-thermodynamic-equilibrium (NLTE) modelling of the low temperatures and densities in KNe. We use a multiconfigurational Dirac–Hartree–Fock method to produce optimized one-electron orbitals for Te i-iii. As a result energy levels and Einstein A-coefficients for Te i-iii have been calculated. These orbitals are then employed within Dirac R-matrix collision calculations to provide electron-impact-excitation collision strengths that were subsequently averaged according to a thermal Maxwellian distribution. Subsequent tardis simulations using this new atomic data reveal no significant changes to the synthetic spectra due to the very minor contribution of Te at early epochs. NLTE simulations with the colradpy package reveal optically thin spectra consistent with the increasing prominence of the Te iii 2.1 $\, \mu$m line as the KNe ejecta cools. This is reinforced by the estimation of luminosities at nebular KNe conditions. New line ratios for both observation and laboratory benchmarks of the atomic data are proposed.

Atomic Data for Astrophysically Important Spectral Lines of Singly Ionized Nitrogen

Nitrogen lines are widely observed in astrophysical spectra and provide important diagnostics for plasma properties. In this work, we present extended calculations for accurate energy levels, electric dipole radiative transition parameters, and lifetimes for the lowest 102 states of the 2s 22p 2, 2s2p 3, 2s2p 23s, 2s 22p{n 1 l, n 2 d, 4f}(n 1 = 3–5, l = s, p, n 2 = 3, 4), and 2p 4 configurations of N ii within the framework of the fully relativistic multiconfiguration Dirac–Hartree–Fock and relativistic configuration interaction methods. These data are useful for modeling astrophysical spectra, for example, for nitrogen abundance determinations in early B-type stars, and for studying the compositions and plasma properties of H ii regions and planetary nebulae. Our computed transition parameters are compared with available experimental and theoretical data. The accuracy of the calculations is also assessed via a statistical analysis of the differences between the transition rates in the Babushkin and Coulomb gauges and by consideration of cancellation factors. In this way, 201 of the 1656 transitions computed in this work are estimated to be from accurate to better than 3%, corresponding to an accuracy class of A.

Fully relativistic energies, transition properties, and lifetimes of lithium-like germanium

Abstract Employing two fully relativistic methods, the multi-reference configuration Dirac–Hartree–Fock (MCDHF) method and the relativistic many-body perturbation theory (RMBPT) method, we report energies and lifetime values for the lowest 35 energy levels of the (1s2)nl configurations (where the principal quantum number n = 2–6 and the angular quantum number l = 0, …, n–1) of lithium-like germanium (Ge XXX), as well as complete data on the transition wavelengths, radiative rates, absorption oscillator strengths, and line strengths between the levels. Both the allowed (E1) and forbidden (magnetic dipole M1, magnetic quadrupole M2, and electric quadrupole E2) ones are reported. The results from the two methods are consistent with each other and align well with previous accurate experimental and theoretical findings. We assess the overall accuracies of present RMBPT results to be likely the most precise ones to date. The present fully relativistic results should be helpful for soft x-ray laser research, spectral line identification, plasma modeling and diagnosing. The datasets presented in this paper are openly available at https://doi.org/10.57760/sciencedb.j00113.00135.

Synthesis, crystal structure, spectroscopic, electronic and vibrational properties of N′-[(E)-3-pyridinylmethylene]nicotinohydrazide trihydrate: Experimental and computational study

N′-[(E)-3-pyridinylmethylene]nicotinohydrazide trihydrate (C12H10N4O·3(H2O)) was synthesized and its structure was determined by single crystal X-ray diffraction method. X-ray analysis showed that the compound crystalized in the triclinic system P-1 with a = 7.1134 (3) Å, b = 10.0208 (4) Å, c = 10.3601 (4) Å, α = 96.386 (3)°, β = 96.995 (3)°, γ = 101.219 (4)°, V = 712.05 (5) Å3 and Z = 2 and exhibits that the title compound (I) consists of N′-[(E)-3-pyridinylmethylene]nicotinohydrazide (II) (C12H10N4O), and three water molecules of crystallization. The two water molecules are attached to the third water molecule through intermolecular two OH…O hydrogen bonds. At the same time, this water molecule is linked to the NH amide group of the hydrazone by an intermolecular NH…O hydrogen bond. Hirshfeld surface (HS) analyses were carried out are to visualise the intermolecular interactions in the crystals of Compound I. The geometric optimization of the titled compound has been carried out by different computational methods such as by Hatree Fock (HF) and Density Functional Theory (DFT) methods with the 6–311++G (d,p) basis set. The vibrational frequency, dipole moment (μ), polarizability (α), hyperpolarizability (β), and electronic properties including the highest occupied molecular orbital-lowest unoccupied molecular orbital (HOMO–LUMO) of the compound have been calculated at the level of both theories. In the results of crystal analysis and geometry calculations of the compound I, it was observed that intermolecular hydrogen bonds, N2H2A…O3, O3H31…O2 and O3H32…O4, were formed. The energy band gap (Eg = ELUMO-EHOMO) values were also obtained by using ELUMO and EHOMO at the level of both theories. The value of equilibrium (ground state) dipole moment of the studied molecule was calculated to be 8.31 Debye in B3LYP and 7.55 Debye in HF. The calculated hyperpolarizability values at the B3LYP/6–311++G(d,p) of the compound are 2.53 × 10−30 esu and this value corresponds to 6.79 times the hyperpolarizability value of urea.

Investigating empirical and theoretical calculations for intensity ratios of L-Shell X-ray transitions in atoms with 39 ≤ Z ≤ 94

The significance of theoretical, experimental, and analytical methods in calculating the intensity ratios of L-shell transitions for diverse elements lies in their widespread applications across various domains, including physical chemistry and medical research. In the present paper, empirical values for intensity ratios were computed through polynomial interpolations using experimental databases within the scope of atomic number 39 ≤ Z ≤ 92 for ILβILα and ILγILα, and extending to the range of 39 ≤ Z ≤ 94 for ILlILα. Additionally, new theoretical calculations were conducted using the Multiconfiguration Dirac–Fock Method for specific elements. The obtained results were compared with standard theoretical, experimental, and empirical values, showing a reasonable agreement with them.

Comparative study of kilonova opacities for three elements of the sixth period (hafnium, osmium, and gold) from new atomic structure calculations in Hf I–IV, Os I–IV, and Au I–IV

Aims. It is now well established that a large amount of heavy (trans-iron) elements are produced during neutron star (NS) mergers. These elements can be detected in the spectra of the kilonova emitted from the post-merger ejected materials. Due to the high level densities that characterize the complex configurations belonging to heavy elements, thus giving rise to millions of absorption lines, the kilonova ejecta opacity is of significant importance. The elements that contribute the most to the latter are those with an unfilled nd subshell belonging to the fifth and the sixth rows of the periodic table, and those with an unfilled nf subshell belonging to the lanthanide and actinide groups. The aim of the present work is to make a new contribution to this field by performing large-scale atomic structure calculations in three specific sixth-row 5d elements, namely hafnium, osmium, and gold, in the first four charge stages (I–IV), and by computing the corresponding opacities, while focusing on the importance of the atomic models used. Methods. The pseudo-relativistic Hartree–Fock (HFR) method, including extended sets of interacting configurations, was used for the atomic structure and radiative parameter calculations, while the expansion formalism was used to estimate the opacities. Results. Theoretical energy levels, wavelengths, and oscillator strengths were computed for millions of spectral lines in Hf I–IV, Os I–IV, and Au I–IV ions, the reliability of these parameters being assessed through detailed comparisons with previously published experimental and theoretical results. The newly obtained atomic data were then used to calculate expansion opacities for typical kilonova conditions expected one day after the NS merger; these are a density of ρ = 10−13 g cm−3 and temperatures ranging from T = 5000 K to T = 15 000 K. Some agreements and differences were found when comparing our results with available data, highlighting the importance of using sufficiently complete atomic models for the determination of opacities.

Prediction of the 1st excitation energy of odd–odd nuclei with the Bayesian neural network approach

A Bayesian neural network (BNN) is developed to predict the 1st excitation energy of odd–odd nuclei. Aside from the proton number and neutron number, we introduce two empirical physical quantities into the input layer. δ=[−1N+−1Z]/2 is introduced to distinguish even–even, odd–odd and odd-A nuclei; and the so-called Casten factor P≡vpvnvp+vn is introduced to stand for collectivity. The BNN is trained with an experimental dataset of the 1st excitation energy for 434 odd–odd, 649 even–even and 1050 odd-A nuclei. After training, the BNN predicts the 1st excitation energy of odd–odd nuclei with a rms of 0.21 MeV. Examples of Dy, Gd, Eu and Cs isotopes are also shown. The BNN results show moderate predictive ability, in comparison with results from the projected Hartree–Fock method.

Accurate ab initio calculation and neural network prediction of the atomic properties of Os LXXIII, Ir LXXIV, Pt LXXV, and Au LXXVI

In this paper, we present an ab initio theoretical calculation of the atomic parameters, including energy levels, wavelengths, lifetimes, and transition parameters, associated with the 1s22snl and 1s22pnl (n= 2–4, l≤n−1) configurations of Be-like sequence: Os LXXIII, Ir LXXIV, Pt LXXV, and Au LXXVI. Our analysis is based on the relativistic wavefunctions derived from the multi-configuration Dirac–Fock (MCDF) method, which are implemented in the relativistic atomic structure package GRASP2018. The correlations within the (principal quantum number) n= 10 complex are accounted for. The Breit interaction and quantum electrodynamical effects are added in the relativistic configuration interaction (RCI) calculation. Our results are compared with the existing data. The uncertainties of the dipole transition line strengths are assessed. In addition, we propose a feed forward neural network to predict the energy levels of Au LXXVI. This is a powerful machine learning tool capable of learning from existing data and predicting unknown data. The network is trained using theoretical energy levels of Os LXXIII, Ir LXXIV, and Pt LXXV obtained from the MCDF method. Our neural network exhibits average relative deviations of approximately 0.01% and 0.02% for trained and predicted energy levels, respectively, when compared to the MCDF results. This level of deviation suggests that our results are reasonably accurate. The data set presented in this study is useful for modeling fusion plasmas.

Limitations of mean-field approximations in describing shift-current and injection-current in materials

We theoretically and computationally investigate bulk photovoltaic effects, with a specific focus on shift-current and injection-current. Initially, we perform a numerical analysis of the direct current (dc) induced by a laser pulse with a one-dimensional model, utilizing mean-field theories such as time-dependent Hartree–Fock and time-dependent Hartree methods. Our numerical results, obtained with mean-field theories, reveal that the dc component of the current, as a second-order nonlinear effect, exists even after irradiation with linearly polarized light as a second-order nonlinear effect, indicating the generation of injection-current. Conversely, when we employ the independent-particle approximation, no injection-current is generated by linearly polarized light. To develop the microscopic understanding of injection-current within the mean-field approximation, we further analyze the dc component of the current with the perturbation theory, employing the mean-field approximations, the independent-particle approximation, and the exact solution of the many-body Schrödinger equation. The perturbation analysis clarifies that the injection-current induced by linearly polarized light under the mean-field approximations is an artifact caused by population imbalance, created through quantum interference from unphysical self-excitation pathways. Therefore, investigation of many-body effects on the bulk photovoltaic effects have to be carefully conducted in mean-field schemes due to potential contamination by unphysical dc current. Additionally, we perform the first-principles electron dynamics calculation for BaTiO3 based on the time-dependent density functional theory, and we confirm that the above findings from the one-dimensional model calculation and the perturbation analysis apply to realistic systems. Published by the American Physical Society 2024

The excitation energies and hyperfine structures for 2l, 3l states in lithiumlike ions

Combining the multi-configuration Dirac–Hartree–Fock method and the model-quantum electrodynamics (QED) approach, the wave functions, transition energies and hyperfine structure constants for 2l, 3l states in Li-like Ne7+, Mg9+, Al10+, P12+, Ar15+ and Ca17+ ions are calculated. The effects of electron correlation, Breit interactions, nuclear recoil and QED effects are analyzed in detail. We find that the contribution of the non-diagonal elements of the self-energy is significant for the 2p1/2→2s1/2 and 2p3/2→2s1/2 transitions. However, for the 3l→2s1/2 transitions, the contribution of non-diagonal elements is small. The accuracy of the currently calculated hyperfine structure constants is expected to be within 1%.

Theoretical investigation of energy levels and transitions for Ce iii with applications to kilonova spectra

ABSTRACT Doubly ionized cerium (Ce2+) is one of the most important ions to understand the kilonova spectra. In particular, near-infrared (NIR) transitions of Ce iii between the ground (5p6 4f2) and first excited (5p6 4f 5d) configurations are responsible for the absorption features around 14 500 Å. However, there is no dedicated theoretical studies to provide accurate transition probabilities for these transitions. We present energy levels of the ground and first excited configurations and transition data between them for Ce iii. Calculations are performed using the grasp2018 package, which is based on the multiconfiguration Dirac–Hartree–Fock and relativistic configuration interaction methods. Compared with the energy levels in the NIST data base (Kramida et al. 2024), our calculations reach the accuracy with the root-mean-square (rms) of 2732 or 1404 cm−1 (excluding one highest level) for ground configuration, and rms of 618 cm−1 for the first excited configuration. We extensively study the line strengths and find that the Babushkin gauge provides the more accurate values. By using the calculated gf values, we show that the NIR spectral features of kilonova can be explained by the Ce iii lines.

Relativistic configuration-interaction calculations on K α X-ray satellites of medium-charge Be-like ions

This study puts forward a calculation of the transitions of 1s2s22p 1P1−1s22s2 1S0 and 1s2s22p 3P1−1s22s2 1S0 for Be-like ions (Z = 6–30) using the multiconfiguration Dirac-Hartree–Fock (MCDHF) and relativistic configuration interaction methods. The computational results include transition wavelengths, transition probabilities, line strengths, and weighted oscillator strengths. The calculations also consider the contributions of the Breit interaction and quantum electrodynamics corrections (QED) to these levels. The results show that the contribution of the Breit interaction, self-energy, and vacuum polarization increase rapidly with increasing nuclear charge for certain configurations. The present calculated values are in good agreement with previous theoretical and experimental results, with errors better than 0.3%. A V/A L comparisons demonstrate the reliability of the data.

Overview of the contributions from all lanthanide elements to kilonova opacity in the temperature range from 25 000 to 40 000 K

Context. It is now well established that the neutron star (NS) merger is at the origin of the production of trans-iron heavy elements in the universe. These elements are therefore present in large quantities in the ejected matter, whose electromagnetic radiation, called kilonova, is characterized by a significant opacity due to the high density of spectral lines belonging to many heavy ions. Among these, the lanthanide ions play an essential role since, with their open 4f subshell, they have a considerable number of transitions that can absorb emitted light. The knowledge of the atomic structure and the radiative parameters of these ions as well as the determination of the corresponding opacities is therefore of paramount importance for the spectral analysis of kilonovae. Aims. The main goal of the present work is to determine the relative contributions of the different lanthanide elements to the opacity of the emission spectrum of a kilonova in its early phase, that is, a few hours after the NS merger, where the conditions are such that the temperature is between 25 000 and 40 000 K. At these temperatures, the lanthanide ions whose charge states are between V and VII are predominant. Methods. We used the pseudo-relativistic Hartree–Fock (HFR) method extensively to calculate the relevant atomic data (energy levels, wavelengths, and oscillator strengths) in La-Lu V-VII ions. The corresponding monochromatic opacities were estimated from the expansion formalism. Results. We calculated the spectroscopic parameters for a total of more than 800 million radiative transitions in all the ions considered. These data were used to estimate the expansion opacities and Planck mean opacities for all the lanthanide elements at early-phase kilonova conditions between 25 000 and 40 000 K, making it possible to deduce the respective contributions of each element as a function of temperature. Atomic calculations were also carried out with the fully relativistic Multiconfiguration Dirac-Hartree-Fock (MCDHF) method in the specific case of the Yb V ion, as the available experimental data had not yet been compared with the theoretical calculations in our previous studies on lanthanide ions.

Biexcitons and quadrons in self-assembled quantum dots

We theoretically study biexcitons and quadrons in quantum dots with parabolic confinement and give a complete comparison between the two excitations. The calculation of quadron and biexciton binding energies as functions of electron-to-hole confinement potentials and mass ratios, using the unrestricted Hartree–Fock method, shows the essential differences between biexcitons and quadrons. The crossover between the negative and positive binding energies is indicated. The effect of an external magnetic field on the quadron and biexciton binding energies has also been investigated. In addition, the crossover between anti-binding and binding of both excited quadron and biexciton states in a certain range of the electron-to-hole oscillator length ratios has been found.

Relativistic calculations of energy levels, lifetimes, transition parameters, and electron impact excitation cross sections for Kr XIX

In this study, we present a detailed analysis of atomic properties for Kr XIX, specifically focusing on the lowest 128 fine-structure levels originating from the 3s23p6, 3s23p53d, 3s3p63d and 3s23p43d2 configurations. We report energy levels, lifetimes, wavelengths, weighted oscillator strengths, and transition probabilities for multipole transition types (E1, E2, M1, and M2). To achieve this, we employed the GRASP2018 code, which implements the fully relativistic multiconfigurational Dirac–Hartree–Fock method and accounts for Breit interaction and quantum electrodynamic effects. We performed another set of calculations using the many-body perturbation theory implemented in the flexible atomic code (FAC). By comparing the results obtained from GRASP2018 and FAC, we established the accuracy and consistency of our calculations. Furthermore, using the relativistic distorted wave theory, we studied electron impact excitation of all transitions to upper levels from the ground and metastable levels and reported excitation cross sections for incident electron energies up to 5 keV. To enhance the practical utility of our findings, we also provided analytical fittings of these cross sections and excitation rate coefficients for their applications in plasma modeling. This work contributes to the atomic properties and excitation cross sections of Kr XIX, addressing a marked gap in the available atomic data.

Calculation of forbidden transitions in doubly ionized neodymium (Nd III) of interest for kilonova nebular phase analysis

In this paper, we present new radiative rate calculations for forbidden transitions, namely magnetic dipole (M1) and electric quadrupole (E2) transitions, involving all the experimentally known energy levels within the 4f4 ground configuration of doubly ionized neodymium (Nd III). To do this, and in order to estimate the accuracy of the results obtained, two independent computational approaches based on the pseudo-relativistic Hartree–Fock and the fully relativistic Dirac-Hartree–Fock methods were used. The transition probabilities calculated with these two approaches showed good overall agreement, in particular for the most intense forbidden lines for which the relative differences did not exceed 25%. From these new atomic data, some astrophysical implications were deduced such as the possibility (or not) of observing some [Nd III] lines on the infrared spectra recorded by the James Webb Space Telescope, more precisely for the analysis of nebular phase kilonova spectra following compact object mergers.

The Landé g factors of highly charged Sn47+ and Bi80+ ions

The wavefunctions and eigenvalues of the ground states of Sn47+, and Bi80+ ions are calculated using the fully relativistic multiconfiguration Dirac-Hartree–Fock (MCDHF) method. Detailed investigation are carried out to study the effects of electron correlation, Breit interaction, quantum electrodynamics (QED), and nuclear recoil. Based on these calculations, the theoretical predictions for the g factors of the ground states of Sn47+ and Bi80+ ions are 1.980429 and 1.934759, respectively. The accuracy of these calculations is expected to be on the order of 10−5.

The 3d 2 D 5/2−2 D 3/2 magnetic dipole transition in potassium-like ions

Spectral lines of the magnetic-dipole transition between the 3d2D3/2,5/2 levels of K-like Kr17+, Zr21+, Nb22+, Mo23+, and Ru25+ ions were measured in a compact electron beam ion trap. By means of dedicated calibration methods, the transition wavelengths of Kr17+, Zr21+, and Mo23+ ions reduce uncertainties by at least one order of magnitude and the wavelengths of Nb22+ and Ru25+ ions are measured for the first time at a few ppm level. In addition, the fine-structure energy splittings were systematically calculated using the multiconfiguration Dirac–Hartree–Fock method combined with the relativistic configuration interaction approach. The contributions of the core-valence and core-core electron correlations, the Breit interaction and QED effect were discussed in detail. The results show that it is essential to include deep core-valence correlation of L-shell and core-core valence correlation of 3p subshell to obtain an agreement better than 0.15% with existing experimental values. The calculation model can be performed to predict the wavelength within a high accuracy for a wide range of potassium-like ions.