B-spline Basis Set Research Articles

Calculation of dipole matrix elements for the modeling of H-[formula omitted] Stark broadening in the presence of large magnetic field

B-spline basis sets are applied to the calculation of dipole matrix elements for the hydrogen atom placed in an intense magnetic field, in view of applications to the spectroscopic analysis of white dwarf atmospheres. A focus is put on the H-α line and magnetic field values in the kilotesla range are considered. The dipole matrix elements are shown to be significantly sensitive to the magnetic field. In the framework of plasma diagnostic, this means that the Stark broadening of the line, which is involved in density measurements, can be misinterpreted if the atomic structure is not well accounted for in numerical codes. As an illustration, we consider the H-ασ− component and apply a computer simulation method to the modeling of the Stark broadening of this line. The calculated dipole matrix elements are used therein as input.

Photoionization and core resonances from range-separated time-dependent density-functional theory for open-shell states: Example of the lithium atom.

We consider the calculations of photoionization spectra and core resonances of open-shell systems using range-separated time-dependent density-functional theory. Specifically, we use the time-dependent range-separated hybrid (TDRSH) scheme, combining a long-range Hartree-Fock exchange potential and kernel with a short-range potential and kernel from a local density-functional approximation, and the time-dependent locally range-separated hybrid (TDLRSH) scheme, which uses a local range-separation parameter. To efficiently perform the calculations, we formulate a spin-unrestricted linear-response Sternheimer approach in a non-orthogonal B-spline basis set using appropriate frequency-dependent boundary conditions. We illustrate this approach on the Li atom, which suggests that TDRSH and TDLRSH are adequate simple methods for estimating the single-electron photoionization spectra of open-shell systems.

Photoionization and core resonances from range-separated density-functional theory: General formalism and example of the beryllium atom.

We explore the merits of linear-response range-separated time-dependent density-functional theory (TDDFT) for the calculation of photoionization spectra. We consider two variants of range-separated TDDFT, namely, the time-dependent range-separated hybrid (TDRSH) scheme, which uses a global range-separation parameter, and the time-dependent locally range-separated hybrid (TDLRSH), which uses a local range-separation parameter, and compare with standard time-dependent local-density approximation (TDLDA) and time-dependent Hartree-Fock (TDHF). We show how to calculate photoionization spectra with these methods using the Sternheimer approach formulated in a non-orthogonal B-spline basis set with appropriate frequency-dependent boundary conditions. We illustrate these methods on the photoionization spectrum of the Be atom, focusing, in particular, on the core resonances. Both the TDRSH and TDLRSH photoionization spectra are found to constitute a large improvement over the TDLDA photoionization spectrum and a more modest improvement over the TDHF photoionization spectrum.

HeTDSE: A GPU based program to solve the full-dimensional time-dependent Schrödinger equation for two-electron helium subjected to strong laser fields* *Project supported the National Natural Science Foundation of China (Grant Nos. 11904192, 11604119, 11627807, and 11604131), the Natural Science Basic Research Plan of Shaanxi Province of China (Grant No. 2016JM1012), the Natural Science Foundation of the Education Committee of Shaanxi

Electron–electron correlation plays an important role in the underlying dynamics in physics and chemistry. Helium is the simplest and most fundamental two-electron system. The dynamic process of helium in a strong laser field is still a challenging issue because of the large calculation cost. In this study, a graphic processing unit (GPU) openACC based ab initio numerical simulations package HeTDSE is developed to solve the full-dimensional time-dependent Schrödinger equation of helium subjected to a strong laser pulse. HeTDSE uses B-spline basis sets expansion method to construct the radial part of the wavefunction, and the spherical harmonic functions is used to express for the angular part. Adams algorithm is employed for the time propagation. Our example shows that HeTDSE running on an NVIDIA Kepler K20 GPU can outperform the one on an Intel E5-2640 single CPU core by a factor of 147. HeTDSE code package can be obtained from the author or from the author’s personal website (doi: 10.13140/RG.2.2.15334.45128) directly under the GPL license, so HeTDSE can be downloaded, used and modified freely.

Determination of structure parameters in strong-field ionization models of atoms

The Ammosov–Delone–Krainov (ADK) and Perelomov–Popov–Terent’ev (PPT) ionization models were widely used in strong-field physics and attosecond science due to their many attractive advantages such as simpler analytical formula, less computational demands, and satisfied accuracy of ionization rate. Based on the density-functional theory, we systematically determine accurate structure parameters of 25 atoms, 24 positive ions and 13 negative ions and tabulate for future applications. The wave function with correct asymptotic behavior is obtained by solving the time-independent Schrödinger equation with B-spline basis sets and the accurate structure parameters are extracted from this wave function in the asymptotic region. The accuracies of structure parameters are carefully examined by comparing the ionization probabilities (or yields) calculated by PPT and ADK models with those of solving the three-dimensional time-dependent Schrödinger equation and the experimental data.

The role of transition dipole phase in atomic attosecond transient absorption from the multi-level model

Based on a multilevel model considering enough bound electronic states of atoms, we theoretically study the role of the transition dipole phase (TDP) in the attosecond transient absorption (ATA) spectrum of helium in intense laser fields. By solving the stationary Schrödinger equation with B-spline basis sets, we first calculate the transition dipole moments with well-defined phases between the bound states. Using the modified multilevel model, we reveal that the TDP plays an important role in determining the spectral structures if two or more paths populate the excited states from the ground state. Our multilevel model with the accurate TDP is convenient to address the origin of atomic ATA spectral structures by freely removing or adding specific electronic states and has been justified by comparing with the ATA spectra via directly solving the time-dependent Schrödinger equation. Hopefully, further incorporating macroscopic propagation into the model will provide indepth physical insights into experimental ATA spectra.

Linear-response range-separated density-functional theory for atomic photoexcitation and photoionization spectra.

We investigate the performance of the range-separated hybrid (RSH) scheme, which combines long-range Hartree-Fock (HF) and a short-range density-functional approximation (DFA), for calculating the photoexcitation/photoionization spectra of the H and He atoms, using a B-spline basis set in order to correctly describe the continuum part of the spectra. The study of these simple systems allows us to quantify the influence on the spectra of the errors coming from the short-range exchange-correlation DFA and from the missing long-range correlation in the RSH scheme. We study the differences using the long-range HF exchange (nonlocal) potential and the long-range exact exchange (local) potential. Contrary to the former, the latter supports a series of Rydberg states and gives reasonable photoexcitation/photoionization spectra, even without applying linear-response theory. The most accurate spectra are obtained with the linear-response time-dependent RSH (TDRSH) scheme. In particular, for the He atom at the optimal value of the range-separation parameter, TDRSH gives slightly more accurate photoexcitation and photoionization spectra than the standard linear-response time-dependent HF. More generally, this work shows the potential of range-separated density-functional theory for calculating linear and nonlinear optical properties involving continuum states.

Restricted Correlation Space B-Spline ADC Approach to Molecular Ionization: Theory and Applications to Total Photoionization Cross-Sections.

Herein is presented a new approach to the ab initio algebraic diagrammatic construction (ADC) schemes for the polarization propagator, which is explicitly designed to accurately and efficiently describe molecular ionization. The restricted correlation space (RCS) version of the ADC methods up to second order of perturbation theory is derived via the intermediate state representation (ISR) and implemented in the multicenter B-spline basis set for the electronic continuum. Remarkably a general close-coupling structure of the RCS-ADC many-electron wave function, connecting the N-particle to the ( N - 1)-particle ADC intermediate states, emerges naturally as a nontrivial result of the RCS ansatz. Moreover, the introduced RCS-ADC schemes prove to be particularly manageable from a computational point of view, overcoming the practical limitations of the conventional ADC approaches. The quality of the new RCS-ADC( n) approaches is verified by performing a series of total photoionization cross-section calculations on a test set of molecules. The excellent agreement of the results with existing accurate benchmarks demonstrates that the RCS versions of the ADC schemes are optimal and quantitatively accurate methods for studying multichannel molecular photoionization.

Kinetic energy classification and smoothing for compact B-spline basis sets in quantum Monte Carlo.

Quantum Monte Carlo calculations of defect properties of transition metal oxides have become feasible in recent years due to increases in computing power. As the system size has grown, availability of on-node memory has become a limiting factor. Saving memory while minimizing computational cost is now a priority. The main growth in memory demand stems from the B-spline representation of the single particle orbitals, especially for heavier elements such as transition metals where semi-core states are present. Despite the associated memory costs, splines are computationally efficient. In this work, we explore alternatives to reduce the memory usage of splined orbitals without significantly affecting numerical fidelity or computational efficiency. We make use of the kinetic energy operator to both classify and smooth the occupied set of orbitals prior to splining. By using a partitioning scheme based on the per-orbital kinetic energy distributions, we show that memory savings of about 50% is possible for select transition metal oxide systems. For production supercells of practical interest, our scheme incurs a performance penalty of less than 5%.

Ab initio calculations of spectroscopic constants and vibrational state lifetimes of diatomic alkali-alkaline-earth cations.

We investigate the lifetimes of vibrational states of diatomic alkali-alkaline-earth cations to determine their suitability for ultracold experiments where long decoherence time and controllability by an external electric field are desirable. The potential energy and permanent dipole moment curves for the ground electronic states of LiBe+, LiMg+, NaBe+, and NaMg+ are obtained using the coupled cluster with singles doubles and triples and multireference configuration interaction methods in combination with large all-electron cc-pCVQZ and aug-cc-pCV5Z basis sets. The energies and wave functions of all vibrational states are obtained by solving the Schrödinger equation for nuclei with the B-spline basis set method. To predict the lifetimes of vibrational states, the transition dipole moments, as well as the Einstein coefficients describing spontaneous emission, and the stimulated absorption and emission induced by black body radiation are calculated. Surprisingly, in all studied ions, the lifetimes of the highest excited vibrational states are similar to the lifetimes of the ground vibrational states indicating that highly vibrationally excited ions could be useful for the ultracold experiments requiring long decoherence time.

Electronic-nuclear entanglement in H2+ : Schmidt decomposition of non-Born-Oppenheimer wave functions expanded in nonorthogonal basis sets

We compute the entanglement between the electronic and vibrational motions in the simplest molecular system, the hydrogen molecular ion, considering the molecule as a bipartite system, electron and vibrational motion. For that purpose we compute an accurate total non-Born-Oppenheimer wave function in terms of a huge expansion using nonorthogonal B-spline basis sets that expand separately the electronic and nuclear wave functions. According to the Schmidt decomposition theorem for bipartite systems, widely used in quantum-information theory, it is possible to find a much shorter but equivalent expansion in terms of the natural orbitals or Schmidt bases for the electronic and nuclear half spaces. Here we extend the Schmidt decomposition theorem to the case in which nonorthogonal bases are used to span the partitioned Hilbert spaces. This extension is first illustrated with two simple coupled systems, the former without an exact solution and the latter exactly solvable. In these model systems of distinguishable coupled particles it is shown that the entanglement content does not increase monotonically with the excitation energy, but only within the manifold of states that belong to an existing excitation mode, if any. In the hydrogen molecular ion the entanglement content for each non-Born-Oppenheimer vibronic state is quantified through the von Neumann and linear entropies and we show that entanglement serves as a witness to distinguish vibronic states related to different Born-Oppenheimer molecular energy curves or electronic excitation modes.

Comparison of fMRI analysis methods for heterogeneous BOLD responses in block design studies

A large number of fMRI studies have shown that the temporal dynamics of evoked BOLD responses can be highly heterogeneous. Failing to model heterogeneous responses in statistical analysis can lead to significant errors in signal detection and characterization and alter the neurobiological interpretation. However, to date it is not clear that, out of a large number of options, which methods are robust against variability in the temporal dynamics of BOLD responses in block-design studies. Here, we used rodent optogenetic fMRI data with heterogeneous BOLD responses and simulations guided by experimental data as a means to investigate different analysis methods’ performance against heterogeneous BOLD responses. Evaluations are carried out within the general linear model (GLM) framework and consist of standard basis sets as well as independent component analysis (ICA). Analyses show that, in the presence of heterogeneous BOLD responses, conventionally used GLM with a canonical basis set leads to considerable errors in the detection and characterization of BOLD responses. Our results suggest that the 3rd and 4th order gamma basis sets, the 7th to 9th order finite impulse response (FIR) basis sets, the 5th to 9th order B-spline basis sets, and the 2nd to 5th order Fourier basis sets are optimal for good balance between detection and characterization, while the 1st order Fourier basis set (coherence analysis) used in our earlier studies show good detection capability. ICA has mostly good detection and characterization capabilities, but detects a large volume of spurious activation with the control fMRI data.

Hyperpolarizabilities of alkaline-earth metal ions Be+, Mg+, and Ca+

The knowledge of the hyperpolarizabilities of atoms and ions is helpful for the analysis of the high order effects of the frequency shifts in precision spectroscopy experiments. Liu et al. [Phys. Rev. Lett. 114, 223001 (2015)] proposed to establish all-optical trapped ion clocks using laser at the magic wavelength for clock transition. To evaluate the high-order frequency shifts in this new scheme of optical clocks, hyperpolarizabilities are needed, but absent. Using the finite field method based on the B-spline basis set and model potentials, we calculated the electric-field-dependent energy shifts of the ground and low-lying excited states in Be+, Mg+, and Ca+ in the field strength range of 0.0-6×10−5 a.u.. The scalar and tensor polarizabilities (α 0, α 2) and hyperpolarizabilities (γ 0, γ 2, γ 4) were deduced. The results of the hyperpolarizabilities for Be+ showed good agreement with the values in literature, implying that the present method can be applied for the effective estimation of the atomic hyperpolarizabilities, which are rarely reported but needed in experiments. The feasibility of optical trapping of Ca+ is discussed, and the contributions of hyperpolarizabilities to the transition frequency shift for Ca+ in the optical dipole trap are estimated using quasi-electrostatic approximation.

Magic Wavelengths for the 1S—2S and 1S—3S Transitions in Hydrogen Atoms**Supported by the National Basic Research Program of China under Grant No 2012CB821305, and the National Natural Science Foundation of China under Grant No 91536102.

The dynamic dipole polarizabilities for 1S, 2S and 3S states of the hydrogen atom are calculated using the finite B-spline basis set method, and the magic wavelengths for 1S—2S and 1S—3S transitions are identified. In comparison of the solutions from the Schrödinger and Dirac equations, the relativistic corrections on the magic wavelengths are of the order of 10−2 nm. The laser intensities for a 300-Er-deep optical trap and the heating rates at 514 and 1371 nm are estimated. The reliable prediction of the magic wavelengths would be helpful for the experimental design on the optical trapping of the hydrogen atoms, and in turn, it would be helpful to improve the accuracy of the measurements of the hydrogen 1S—2S and 1S—3S transitions.

High-order-harmonic generation from periodic potentials driven by few-cycle laser pulses

We investigate the high harmonic generation (HHG) from solids by simulating the dynamics of a single active electron in periodic potentials. The corresponding time-dependent Schr\"odinger equations (TDSE) are solved numerically by using B-spline basis sets in coordinate space. The energy band structure and wave vectors can be directly retrived from the eigenfunctions. The harmonic spectra obtained agree well with the results simulated by TDSE in $k$ space using Bloch states and show a two-plateau structure. Both of the cutoff energies of the two plateaus in the harmonic spectrum scale linearly with the field strength. We also study HHG driven by intense few-cycle laser pulses and find that the cutoff energy of the harmonic spectrum is as sensitive to the changes of the carrier envelope phase, as to HHG from gas samples, which suggests recollision pictures in HHG as found by recent experiments (Nature {\bf 522}, 462 (2015)).

Dissociating H₂⁺(²Σg⁺,JM=00) ion as an exploding quantum bubble.

The nuclear and electronic probability and flux densities for a vibrating and dissociating H2(+) molecular ion in the electronic and rotational ground state (corresponding to the quantum numbers ²Σ(g)⁺,JM=00) are calculated. As a consequence of the isotropy of the scenario, the vibrating H2(+) appears as a pulsating quantum bubble, while the dissociating H2(+) appears as an exploding quantum bubble. The dissociating part is represented by a discretization of the continuum through use of £2 integrable B-spline basis set. It is shown that the vibrating part (the pulsating quantum bubble) interferes with the dissociating part (the exploding quantum bubble) and that the interference is much more noticeable in the probability density than in the flux density.

Multi-electron systems in strong magnetic fields I: The 2D Landau–Hartree–Fock–Roothaan method

We present a 2-dimensional Hartree–Fock–Roothaan code to calculate wave functions and energies of light to heavy atoms in strong external magnetic fields, as they occur in the vicinity of neutron stars. The code enhances the previously presented HFFERII method, resulting in a very high precision for the energies with typical deviations less than 1% compared to extremely precise fixed-phase diffusion quantum Monte Carlo calculations. Despite this high precision the code is highly optimized regarding speed and reliability, which allows calculating large amounts of states in short time, even with small-scale computing clusters. Program summaryProgram title: 2DLHFRCatalogue identifier: AETE_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AETE_v1_0.htmlProgram obtainable from: CPC Program Library, Queen’s University, Belfast, N. IrelandLicensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.htmlNo. of lines in distributed program, including test data, etc.: 121429No. of bytes in distributed program, including test data, etc.: 561163Distribution format: tar.gzProgramming language: Fortran 95.Computer: Cluster of 1-15 Fujitsu ESPRIMO P920.Operating system: Linux.Has the code been vectorized or parallelized?: Yes, parallelized using MPI. Tested on 2–60 processors.RAM: At least 1 GByte per coreClassification: 2.1.External routines: GFortran, LAPACK, BLAS, MPINature of problem:The modeling of highly magnetized atmospheres of neutron stars and magnetic white dwarfs is a difficult task and is further complicated by the lack of atomic data. The absorption features in the thermal emission spectra of neutron stars are still not fully understood, which leads to different interpretations and thus large uncertainties for the atmospheric parameters, such as the magnetic field strength, the gravitational redshift, or the predominant atomic composition. Therefore, a fast and reliable program to scan through the large parameter space is necessary.Solution method:The strong magnetic fields present on neutron stars favor a wave function expansion in terms of Landau channels. Contrary to previous attempts we use a full 2-dimensional basis and assign individual z-wave functions to each Landau channel. This allows for an accurate description of the single-particle orbitals. These are combined in a Slater determinant, resulting in Hartree–Fock–Roothaan equations, which are solved iteratively. As initial wave functions we rely on the solutions calculated by the HFFERII program and reuse the optimized B-spline basis sets and Landau coefficients to maximize the speed of the program presented here.Restrictions:Intense magnetic field strengths B/Z2≳5×104T are required to yield accurate results.Unusual features:2DLHFR is based upon the wave functions calculated with the HFFERII program package, presented in [C. Schimeczek, D. Engel, G. Wunner, Comp. Phys. Comm. 183 (2012) 1502]. In turn, the results of this program may be enhanced beyond the Hartree–Fock limit with quantum Monte Carlo methods, as is shown in the accompanying paper.Additional comments:The gfortran compiler is recommended for this program (http://gcc.gnu.org/onlinedocs/gfortran/).Running time:Seconds to minutes

Accurate potential energy, dipole moment curves, and lifetimes of vibrational states of heteronuclear alkali dimers.

We calculate the potential energy curves, the permanent dipole moment curves, and the lifetimes of the ground and excited vibrational states of the heteronuclear alkali dimers XY (X, Y = Li, Na, K, Rb, Cs) in the X(1)Σ(+) electronic state using the coupled cluster with singles doubles and triples method. All-electron quadruple-ζ basis sets with additional core functions are used for Li and Na, and small-core relativistic effective core potentials with quadruple-ζ quality basis sets are used for K, Rb, and Cs. The inclusion of the coupled cluster non-perturbative triple excitations is shown to be crucial for obtaining the accurate potential energy curves. A large one-electron basis set with additional core functions is needed for the accurate prediction of permanent dipole moments. The dissociation energies are overestimated by only 14 cm(-1) for LiNa and by no more than 114 cm(-1) for the other molecules. The discrepancies between the experimental and calculated harmonic vibrational frequencies are less than 1.7 cm(-1), and the discrepancies for the anharmonic correction are less than 0.1 cm(-1). We show that correlation between atomic electronegativity differences and permanent dipole moment of heteronuclear alkali dimers is not perfect. To obtain the vibrational energies and wave functions the vibrational Schrödinger equation is solved with the B-spline basis set method. The transition dipole moments between all vibrational states, the Einstein coefficients, and the lifetimes of the vibrational states are calculated. We analyze the decay rates of the vibrational states in terms of spontaneous emission, and stimulated emission and absorption induced by black body radiation. In all studied heteronuclear alkali dimers the ground vibrational states have much longer lifetimes than any excited states.

Calculations on polarization properties of alkali metal atoms using Dirac—Fock plus core polarization method

A semi-empirical atomic structure model method is developed in the framework of a relativistic case. This method starts from Dirac—Fock calculations using B-spline basis set. The core-valence electron correction is then treated in a semi-empirical core polarization potential. As an application, the polarization properties of alkali metal atoms, including the static polarizabilities and long-range two-body dispersion coefficients, have been calculated. Our results are in good agreement with the results obtained from ab initio relativistic many-body perturbation method and the available experimental measurements.

Towards an accurate representation of the continuum in calculations of electron, positron and laser field interactions with molecules

Gaussian-type orbitals (GTOs) are the most common choice of basis functions in calculations of electronic structure of molecules, i.e. for the description of bound electrons. The main advantage of this approach is the analytic form of the multicentre molecular integrals. For the same reason GTOs have been adopted as basis functions for the description of the unbound particle in many ab-initio calculations of electron, positron and laser fields interacting with molecules. However, the accurate description of the unbound particle using GTOs may become very difficult and in some cases numerically unstable. We describe an approach for the representation of the continuum in which the unbound particle is described using a mixed GTO and B-spline basis set in a manner which exploits the best features of these functions. Analytical expressions for the GTO-only molecular integrals allow us to accurately represent the part of the wavefunction close to the target, while the B-splines enable us to represent accurately the wavefunction's tail, corresponding to the unbound particle, over a wide range of kinetic energies. The main challenge posed by this approach is the accurate and rapid numerical evaluation of a large number of mixed BTO/GTO molecular integrals. We propose a scheme for this integral calculation in which the overlap integrals between GTOs and B-spline functions play a central role and demonstrate that they can be calculated rapidly and accurately.