Nuclear Cusps Research Articles

Exact nuclear cusps in all-electron orbital-free density functional theory

Exact nuclear cusps in all-electron orbital-free density functional theory

Nuclear cusps and singularities in the nonadditive kinetic potential bifunctional from analytical inversion

The non-additive kinetic potential $v^{\text{NAD}}$ is a key quantity in density-functional theory (DFT) embedding methods, such as frozen density embedding theory and partition DFT. $v^{\text{NAD}}$ is a bi-functional of electron densities $\rho_{\rm B}$ and $\rho_{\rm tot} = \rho_{\rm A} + \rho_{\rm B}$. It can be evaluated using approximate kinetic-energy functionals, but accurate approximations are challenging. The behavior of $v^{\text{NAD}}$ in the vicinity of the nuclei has long been questioned, and singularities were seen in some approximate calculations. In this article, the existence of singularities in $v^{\text{NAD}}$ is analyzed analytically for various choices of $\rho_{\rm B}$ and $\rho_{\rm tot}$, using the nuclear cusp conditions for the density and Kohn-Sham potential. It is shown that no singularities arise from smoothly partitioned ground-state Kohn-Sham densities. We confirm this result by numerical calculations on diatomic test systems HeHe, HeLi$^+$, and H$_2$, using analytical inversion to obtain a numerically exact $v^{\rm NAD}$ for the local density approximation. We examine features of $v^{\rm NAD}$ which can be used for development and testing of approximations to $v^{\rm NAD}[\rho_{\rm B},\rho_{\rm tot}]$ and kinetic-energy functionals.

Using Pauli energy to appraise the quality of approximate semilocal non-interacting kinetic energy density functionals.

It is well-known that the kinetic energy density (KED) functional is the most difficult to approximate in density functional theory (DFT), yet to take full advantage of DFT with its density-based descriptive capability of molecular properties, an accurate account of KED is a must. To have a better idea of how an approximate KED formula behaves and where we should focus in the future development of better approximate KEDs, in this work we propose to employ the Pauli energy to assess their quality. We tested the performance of a total of 22 approximate semilocal noninteracting KED functionals from the literature for 18 neutral atoms and 20 small molecules. We found that generalized gradient approximation formulas of the KED functional can often reasonably accurately predict the total kinetic energy value for atoms and molecules but failed miserably to forecast the integrated values for Pauli energy related properties. The reason behind this is that presently available approximate KED functionals are unable to accurately account for the kinetic energy distribution in the medium range away from nuclei, where the Pauli energy plays a crucial role. Our results strongly suggest that the key information missing in approximate KED functionals comes from the medium regions, not nuclear cusps nor asymptotic areas, and the Pauli energy is a reliable measure of the quality of approximate KED functionals. Future efforts in developing better KED approximations should be invested in the regions of molecules where chemical bonds are formed in order to accurately account for the Pauli energy.

The any particle molecular orbital grid-based Hartree-Fock (APMO-GBHF) approach

The any particle molecular orbital grid-based Hartree-Fock approach (APMO-GBHF) is proposed as an initial step to perform multi-component post-Hartree-Fock, explicitly correlated, and density functional theory methods without basis set errors. The method has been applied to a number of electronic and multi-species molecular systems. Results of these calculations show that the APMO-GBHF total energies are comparable with those obtained at the APMO-HF complete basis set limit. In addition, results reveal a considerable improvement in the description of the nuclear cusps of electronic and non-electronic densities.

Coupled-Cluster in Real Space. 1. CC2 Ground State Energies Using Multiresolution Analysis.

A framework to calculate CC2 approximated coupled-cluster ground state correlation energies in a multiresolution basis is derived and implemented into the MADNESS library. The CC2 working equations are formulated in first quantization which makes them suitable for real-space methods. The first quantized equations can be interpreted diagrammatically using the usual diagrams from second quantization with adjusted interpretation rules. Singularities arising from the nuclear and electronic potentials are regularized by explicitly taking the nuclear and electronic cusps into account. The regularized three- and six-dimensional cluster functions are represented directly on an adaptive grid. The resulting equations are free of singularities and virtual orbitals, which results in a low intrinsic scaling. Correlation energies close to the basis set limit are computed for small molecules. This work is the first step toward CC2 excitation energies in a multiresolution basis.

Coupled-Cluster in Real Space. 2. CC2 Excited States Using Multiresolution Analysis.

We report a first quantized approach to calculate approximate coupled-cluster singles and doubles CC2 excitation energies in real space. The cluster functions are directly represented on an adaptive grid using multiresolution analysis. Virtual orbitals are neither calculated nor needed in this approach. The nuclear and electronic cusps are taken into account explicitly regularizing the corresponding equations exactly. First calculations on small molecules are in excellent agreement with the best available LCAO results.

Mesh-free canonical tensor products for six-dimensional density matrix: computation of kinetic energy

The computation of a six-dimensional density matrix is the crucial step for the evaluation of kinetic energy in electronic structure calculations. For molecules with heavy nuclei, one has to consider a very refined mesh in order to deal with the nuclear cusps. This leads to high computational time and needs huge memory for the computation of the density matrix. To reduce the computational complexity and avoid discretization errors in the approximation, we use mesh-free canonical tensor products in electronic structure calculations. In this paper, we approximate the six-dimensional density matrix in an efficient way and then compute the kinetic energy. Accuracy is examined by comparing our computed kinetic energy with the exact computation of the kinetic energy.

SRG/eROSITA prospects for the detection of stellar tidal disruption flares

We discuss the potential of the eROSITA telescope on board the Spectrum-Roentgen-Gamma (SRG) observatory to detect stellar tidal disruption events (TDE) during its 4-year all-sky survey. These events are expected to reveal themselves as luminous flares of UV/soft X-ray emission associated with the centres of previously non-active galaxies and fading by few orders of magnitude on time-scales of several months to years. Given that eROSITA will complete an all-sky survey every 6 months and a total of 8 such scans will be performed over the course of the mission, we propose to distinguish TDEs from other X-ray transients using two criteria: I) large (more than a factor of 10) X-ray variation between two subsequent 6-month scans and ii) soft X-ray spectrum. The expected number of TDE candidates is $\sim 10^3$ per scan (with most of the events being new discoveries in a given scan), so that a total of several thousand TDE candidates could be found during the 4-year survey. The actual number may significantly differ from this estimate, since it is based on just a few TDEs observed so far. The eROSITA all-sky survey is expected to be nearly equally sensitive to TDEs occurring near supermassive black holes (SMBH) of mass between $\sim 10^6$ and $\sim 10^7 M_\odot$ and will thus provide a unique census of quiescent SMBHs and associated nuclear stellar cusps in the local Universe ($z\lesssim 0.15$). Information on TDE candidates will be available within a day after their detection and localization by eROSITA, making possible follow-up observations that may reveal peculiar types of TDEs.

Nonlinear approximations for electronic structure calculations

We present a new method for electronic structure calculations based on novel algorithms for nonlinear approximations. We maintain a functional form for the spatial orbitals as a linear combination of products of decaying exponentials and spherical harmonics centred at the nuclear cusps. Although such representations bare some resemblance to the classical Slater-type orbitals, the complex-valued exponents in the representations are dynamically optimized via recently developed algorithms, yielding highly accurate solutions with guaranteed error bounds. These new algorithms make dynamic optimization an effective way to combine the efficiency of Slater-type orbitals with the adaptivity of modern multi-resolution methods. We develop numerical calculus suitable for electronic structure calculations. For any spatial orbital in this functional form, we represent its product with the Coulomb potential, its convolution with the Poisson kernel, etc., in the same functional form with optimized parameters. Algorithms for this purpose scale linearly in the number of nuclei. We compute electronic structure by casting the relevant equations in an integral form and solving for the spatial orbitals via iteration. As an example, for several diatomic molecules we solve the Hartree–Fock equations with speeds competitive to those of multi-resolution methods and achieve high accuracy using a small number of parameters.

AGN and Starbursts in Dusty Galaxy Mergers: Insights from the Great Observatories All-sky LIRG Survey

AbstractThe Great Observatories All-sky LIRG Survey (GOALS) is combining imaging and spectroscopic data from the Herschel, Spitzer, Hubble, GALEX, Chandra, and XMM-Newton space telescopes augmented with extensive ground-based observations in a multiwavelength study of approximately 180 Luminous Infrared Galaxies (LIRGs) and 20 Ultraluminous Infrared Galaxies (ULIRGs) that comprise a statistically complete subset of the 60μm-selected IRAS Revised Bright Galaxy Sample. The objects span the full range of galaxy environments (giant isolated spirals, wide and close pairs, minor and major mergers, merger remnants) and nuclear activity types (Seyfert 1, Seyfert 2, LINER, starburst/HII), with proportions that depend strongly on the total infrared luminosity. I will review the science motivations and present highlights of recent results selected from over 25 peer-reviewed journal articles published recently by the GOALS Team. Statistical investigations include detection of high-ionization Fe K emission indicative of deeply embedded AGN, comparison of UV and far-IR properties, investigations of the fraction of extended emission as a function of wavelength derived from mid-IR spectroscopy, mid-IR spectral diagnostics and spectral energy distributions revealing the relative contributions of AGN and starbursts to powering the bolometric luminosity, and quantitative structure analyses that delineate the evolution of stellar bars and nuclear stellar cusps during the merger process. Multiwavelength dissections of individual systems have unveiled large populations of young star clusters and heavily obscured AGN in early-stage (II Zw 96), intermediate-stage (Mrk 266, Mrk 273), and late-stage (NGC 2623, IC 883) mergers. A recently published study that matches numerical simulations to the observed morphology and gas kinematics in mergers has placed four systems on a timeline spanning 175-260 million years after their first passages, and modeling of additional (U)LIRGs is underway. A very brief description of ongoing work with Herschel and ALMA will be given. The talk will conclude with a discussion of the demographics of dual AGN (kpc-scale orbits) in the GOALS sample, including the difficulty of their detection and confirmation, a proposed sequence representing a progression from dual AGN to binary AGNs (sub-pc scale orbits), and implications for the scarcity of confirmed binary QSOs in recent large surveys. Grant support from NASA is gratefully acknowledged.

The build-up of nuclear stellar cusps in extreme starburst galaxies and major mergers

Nuclear stellar cusps are defined as central excess light component in the stellar light profiles of galaxies and are suggested to be stellar relics of intense compact starbursts in the central ∼100–500 pc region of gas-rich major mergers. Here, we probe the build-up of nuclear cusps during the actual starburst phase for a complete sample of luminous infrared galaxy (LIRG) systems (85 LIRGs, with 11.4 < log [LIR/L⊙] < 12.5) in the Great Observatories All-sky LIRG Survey sample. Cusp properties are derived via 2D fitting of the nuclear stellar light imaged in the near-infrared (NIR) by the Hubble Space Telescope and have been combined with mid-infrared (IR) diagnostics for active galactic nucleus (AGN)/starburst characterization. We find that nuclear stellar cusps are resolved in 76 per cent of LIRGs (merger and non-interacting galaxies). The cusp strength and luminosity increase with far-IR luminosity (excluding AGN) and merger stage, confirming theoretical models that starburst activity is associated with the build-up of nuclear stellar cusps. Evidence for ultracompact nuclear starbursts is found in ∼13 per cent of LIRGs, which have a strong unresolved central NIR light component but no significant contribution of an AGN. The nuclear NIR surface density (measured within 1 kpc radius) increases by a factor of ∼5 towards late merger stages. A careful comparison to local early-type galaxies with comparable masses reveals (a) that local (U)LIRGs have a significantly larger cusp fraction and (b) that the majority of the cusp LIRGs have host galaxy luminosities (H band) similar to core ellipticals which are roughly one order in magnitude larger than those for cusp ellipticals.

A divide and conquer real-space approach for all-electron molecular electrostatic potentials and interaction energies

A computational scheme to perform accurate numerical calculations of electrostatic potentials and interaction energies for molecular systems has been developed and implemented. Molecular electron and energy densities are divided into overlapping atom-centered atomic contributions and a three-dimensional molecular remainder. The steep nuclear cusps are included in the atom-centered functions making the three-dimensional remainder smooth enough to be accurately represented with a tractable amount of grid points. The one-dimensional radial functions of the atom-centered contributions as well as the three-dimensional remainder are expanded using finite element functions. The electrostatic potential is calculated by integrating the Coulomb potential for each separate density contribution, using our tensorial finite element method for the three-dimensional remainder. We also provide algorithms to compute accurate electron-electron and electron-nuclear interactions numerically using the proposed partitioning. The methods have been tested on all-electron densities of 18 reasonable large molecules containing elements up to Zn. The accuracy of the calculated Coulomb interaction energies is in the range of 10(-3) to 10(-6) E(h) when using an equidistant grid with a step length of 0.05 a(0).

Pure-state noninteractingv-representability of electron densities from Kohn-Sham calculations with finite basis sets

Within the linear combination of atomic orbitals (LCAO) approximation, one can distinguish two different Kohn-Sham potentials. One is the potential available numerically in calculations, and the other is the exact potential corresponding to the LCAO density. The latter is usually not available, but can be obtained from the total density by a numerical inversion procedure or, as is done here, analytically using only one LCAO Kohn-Sham orbital. In the complete basis-set limit, the lowest-lying Kohn-Sham orbital suffices to perform the analytical inversion, and the two potentials differ by no more than a constant. The relation between these two potentials is investigated here for diatomic molecules and several atomic basis sets of increasing size and quality. The differences between the two potentials are usually qualitative (wrong behavior at nuclear cusps and far from the molecule even if Slater-type orbitals are used) and $\ensuremath{\delta}$-like features at nodal planes of the lowest-lying LCAO Kohn-Sham orbital. Such nodes occur frequently in LCAO calculations and are not physical. Whereas the behavior of the potential can be systematically improved locally by the increase of the basis sets, the occurrence of nodes is not correlated with the size of the basis set. The presence of nodes in the lowest-lying LCAO orbital can be used to monitor whether the effective potential in LCAO Kohn-Sham equations can be interpreted as the potential needed for pure-state noninteracting $v$-representability of the LCAO density. Squares of such node-containing lowest-lying LCAO Kohn-Sham orbitals are nontrivial examples of two-electron densities which are not pure-state noninteracting $v$-representable.

Multigrid Accelerated Tensor Approximation of Function Related Multidimensional Arrays

In this paper, we describe and analyze a novel tensor approximation method for discretized multidimensional functions and operators in $\mathbb{R}^d$, based on the idea of multigrid acceleration. The approach stands on successive reiterations of the orthogonal Tucker tensor approximation on a sequence of nested refined grids. On the one hand, it provides a good initial guess for the nonlinear iterations to find the approximating subspaces on finer grids; on the other hand, it allows us to transfer from the coarse-to-fine grids the important data structure information on the location of the so-called most important fibers in directional unfolding matrices. The method indicates linear complexity with respect to the size of data representing the input tensor. In particular, if the target tensor is given by using the rank-$R$ canonical model, then our approximation method is proved to have linear scaling in the univariate grid size $n$ and in the input rank $R$. The method is tested by three-dimensional (3D) electronic structure calculations. For the multigrid accelerated low Tucker-rank approximation of the all electron densities having strong nuclear cusps, we obtain high resolution of their 3D convolution product with the Newton potential. The accuracy of order $10^{-6}$ in max-norm is achieved on large $n\times n\times n$ grids up to $n=1.6\cdot10^4$, with the time scale in several minutes.

Orbital-free effective embedding potential at nuclear cusps

A strategy to construct approximants to the kinetic-energy-functional dependent component (v(t)[rho(A),rho(B)](r)) of the effective potential in one-electron equations for orbitals embedded in a frozen-density environment [Eqs. (20) and (21) in Wesolowski and Warshel, J. Phys. Chem. 97, (1993) 8050] is proposed. In order to improve the local behavior of the orbital-free effective embedding potential near nuclei in the environment, the exact behavior of v(t)[rho(A),rho(B)](r) at rho(A)-->0 and integralrho(B)dr=2 is taken into account. As a result, the properties depending on the quality of this potential are invariably improved compared to the ones obtained using conventional approximants which violated the considered exact condition. The approximants obtained following the proposed strategy and especially the simplest one constructed in this work are nondecomposable, i.e., cannot be used to obtain the analytic expression for the functional of the total kinetic energy.

Bulge Formation from Super Star Clusters in a Responding Cuspy Dark Matter Halo

We simulate the bulge formation in very late-type dwarf galaxies from circumnuclear super star clusters (SSCs) moving in a responding cuspy dark matter halo (DMH). The simulations show that (1) the response of DMH to sinking of SSCs is detectable only in the region interior to about 200 pc. The mean logarithmic slope of the responding DM density profile over that area displays two different phases: the very early descent followed by ascent till approaching to 1.2 at the age of 2 Gyrs. (2) the detectable feedbacks of the DMH response on the bulge formation turned out to be very small, in the sense that the formed bulges and their paired nuclear cusps in the fixed and the responding DMH are basically the same, both are consistent with $HST$ observations. (3) the yielded mass correlation of bulges to their nuclear (stellar) cusps and the time evolution of cusps' mass are accordance with recent findings on relevant relations. In combination with the consistent effective radii of nuclear cusps with observed quantities of nuclear clusters, we believe that the bulge formation scenario that we proposed could be a very promising mechanism to form nuclear clusters.

Free iterative-complement-interaction calculations of the hydrogen molecule

The free iterative-complement-interaction (ICI) method based on the scaled Schr\"odinger equation proposed previously has been applied to the calculations of very accurate wave functions of the hydrogen molecule in an analytical expansion form. All the variables were determined with the variational principle by calculating the necessary integrals analytically. The initial wave function and the scaling function were changes to see the effects on the convergence speed of the ICI calculations. The free ICI wave functions that were generated automatically were different from the existing wave functions, and this difference was shown to be physically important. The best wave function reported in this paper seems to be the best worldwide in the literature from the variational point of view. The quality of the wave function was examined by calculating the nuclear and electron cusps.

Adaptive local refinement of the electron density, one-particle density matrices, and electron orbitals by hierarchical wavelet decomposition

The common experience that the distribution and interaction of electrons widely vary by scanning over various parts of a molecule is incorporated in the atomic-orbital expansion of wave functions. The application of Gaussian-type atomic orbitals suffers from the poor representation of nuclear cusps, as well as asymptotic regions, whereas Slater-type orbitals lead to unmanageable computational difficulties. In this contribution we show that using the toolkit of wavelet analysis it is possible to find an expansion of the electron density and density operators which is sufficiently precise, but at the same time avoids unnecessary complications at smooth and slightly detailed parts of the system. The basic idea of wavelet analysis is a coarse description of the system on a rough grid and a consecutive application of refinement steps by introducing new basis functions on a finer grid. This step could highly increase the number of required basis functions, however, in this work we apply an adaptive refinement only in those regions of the molecule, where the details of the electron structure require it. A molecule is split into three regions with different detail characteristics. The neighborhood of a nuclear cusp is extremely well represented by a moderately fine wavelet expansion; the domains of the chemical bonds are reproduced at an even coarser resolution level, whereas the asymptotic tails of the electron structure are surprisingly precise already at a grid distance of 0.5 a.u. The strict localization property of wavelet functions leads to an especially simple calculation of the electron integrals.

Formation of bulges in very late-type galaxies from super star clusters

The dynamical evolution of super star clusters (SSCs) moving in the background of a dark matter halo has been investigated as a possible event responsible for the formation of bulges in late-type spirals. The underlying physical processes include sinking of SSCs due to the dynamical friction and stripping of SSCs on their way to the center. Our model calculations show that only sinking of circumnuclear SSCs contributes to the formation of galactic bulges at the early stage. Based on the assumption of a universal density profile for the dark matter halo, and an isothermal model for the SSCs, our simulations have yielded bulges that are similar in many aspects to the observational ones. In particular, the derived surface density profiles can be well fitted by anexponential structure with nuclear cusps, which is consistent with HST observations.

Spiral Galaxies with [ITAL]HST[/ITAL]/NICMOS. II. Isophotal Fits and Nuclear Cusp Slopes

We present surface brightness profiles for 56 of the 78 spiral galaxies observed in the HST/NICMOS2 F160W snapshot survey introduced in Paper I of this series, as well as surface brightness profiles for 23 objects out of the 41 that were also observed in the F110W filter. We fit these surface brightness profiles with the Nuker law of Lauer et al. and use the smooth analytical descriptions of the data to compute the average nuclear stellar cusp slopes γ in the 01–05 radial range. Our main result is the startling similarity between the nuclear stellar cusp slopes γ in the near-infrared compared with those derived in the visual passband. This similarity has several implications: (1) Despite the significant local color variations that are found in the nuclear regions of spirals and that are documented in Paper I, there are typically little or no optical-NIR global color gradients, and thus no global stellar population variations, inside ~50–100 pc from the nucleus in nearby spirals. (2) The large observed range of the strength of the nuclear stellar cusps seen in the HST optical study of spiral galaxies reflects a physical difference between galaxies and is not an artifact caused by nuclear dust and/or recent star formation. (3) The dichotomy between R1/4 bulges, with steep nuclear stellar cusps γ ~ 1, and exponential bulges, with shallow nuclear stellar cusps γ < 0.3, is also not an artifact of the effects of dust or recent star formation. (4) The presence of a surrounding massive disk appears to have no effect on the rise of the stellar density distribution within the innermost hundred parsecs of the R1/4 spheroids. These results imply a breakdown within the family of exponential bulges of the nuclear versus global relationships that have been found for the R1/4 spheroids. Such a breakdown is likely to have significant implications concerning the formation of exponential bulges and their connection with the R1/4 spheroids.