Real-space Electronic Structure Research Articles

Embedded Localized Molecular-Orbital Representations for Periodic Wave Functions.

To more straightforwardly provide local chemical-bonding reasoning in crystalline matter, we introduce a new approach to generate a real-space analogue of periodic electronic structures using "exact" top-down frozen-density embedding calculations. Based on the obtained real-space electronic structure, we then construct localized molecular orbitals and evidence that our technique compares favorably against the commonly used Wannier method, both in terms of numerical efficiency and details of chemical bonding. The new method has been implemented into the LOBSTER software package and designed as a black-box approach, digesting any periodic electronic structure from the currently supported codes, i.e., VASP, Quantum ESPRESSO, and ABINIT.

GPU acceleration of local and semilocal density functional calculations in the SPARC electronic structure code.

We present a Graphics Processing Unit (GPU)-accelerated version of the real-space SPARC electronic structure code for performing Kohn-Sham density functional theory calculations within the local density and generalized gradient approximations. In particular, we develop a modular math-kernel based implementation for NVIDIA architectures wherein the computationally expensive operations are carried out on the GPUs, with the remainder of the workload retained on the central processing units (CPUs). Using representative bulk and slab examples, we show that relative to CPU-only execution, GPUs enable speedups of up to 6× and 60× in node and core hours, respectively, bringing time to solution down to less than 30s for a metallic system with over 14 000 electrons and enabling significant reductions in computational resources required for a given wall time.

Mechanisms behind large Gilbert damping anisotropies

A method with which to calculate the Gilbert damping parameter from a real-space electronic structure method is reported here. The anisotropy of the Gilbert damping with respect to the magnetic moment direction and local chemical environment is calculated for bulk and surfaces of Fe$_{50}$Co$_{50}$ alloys from first principles electronic structure in a real space formulation. The size of the damping anisotropy for Fe$_{50}$Co$_{50}$ alloys is demonstrated to be significant. Depending on details of the simulations, it reaches a maximum-minimum damping ratio as high as 200%. Several microscopic origins of the strongly enhanced Gilbert damping anisotropy have been examined, where in particular interface/surface effects stand out, as do local distortions of the crystal structure. Although theory does not reproduce the experimentally reported high ratio of 400% [Phys. Rev. Lett. 122, 117203 (2019)], it nevertheless identifies microscopic mechanisms that can lead to huge damping anisotropies.

The electronic structure studies of hybrid h-BNC sheets based on a semi-empirical Hamiltonian

The quest for modulating the wide bandgap of a pristine h-BN sheet for device-related applications has prompted the present study of ternary 2-dimensional sheets, h-BNC, that contain carbon domains of different shapes and sizes embedded in the h-BN network. The structural stability and electronic properties of hybrid h-BNC sheets containing rectangular-, circular-, hexagonal-, and triangular-shaped carbon domains are investigated using a real-space electronic structure method, where an environment-dependent semi-empirical Hamiltonian is used within a framework of a linear combination of atomic orbitals and self-consistent charge calculations. This method allows a study of larger carbon domains embedded in the h-BN matrix beyond what is possible via first-principles calculations, and thus serves to complement previous theoretical studies on hybrid h-BNC systems. The electronic density of states reveals mid-gap states for all h-BNC sheets, suggesting a narrowing of the energy gap compared to the pristine h-BN sheet. They arise from the breaking of the hexagonal symmetry due to bond distortions at the interface between the carbon domain and the h-BN network. The features of such mid-gap states strongly depend on the size and shape of carbon domains and the type of bonding (C-N, C-B, or a mixture of both) at the interface. The hybrid h-BNC sheet containing rectangular carbon domains switches from a semiconductor to a gapless semi-metal-like, and to a metal-like system as the size of the carbon domain increases. The energy gap of the h-BNC sheet containing hexagonal carbon domains exhibits a power-law decrease with the size of the carbon domains. On the other hand, the energy gap of the hybrid h-BN sheet containing circular carbon domains oscillates as the size of the carbon domain is increased. For electron-rich hybrid h-BNC sheets containing triangular graphene domains and C-N interfaces, a p-type-like behaviour is obtained. On the other hand, for electron-deficient hybrid h-BNC sheets, containing triangular graphene domains and C-B interfaces, the electronic behaviour switches from a p-type-like semiconductor to a metal-like behaviour as the number of carbon atoms increase. The top of the valence band was found to be half-filled for all triangular domains. The results obtained here may pave the way for novel device concepts based on h-BNC sheets.

Ab initio electronic structure calculations using a real-space Chebyshev-filtered subspace iteration method

Ab initio electronic structure calculations within Kohn–Sham density functional theory requires a solution for the Kohn–Sham equation. However, the traditional self-consistent field (SCF) approach of solving the equation using iterative diagonalization exhibits an inherent cubic scaling behavior and becomes prohibitive for large systems. The Chebyshev-filtered subspace iteration (CheFSI) method holds considerable promise for large-system calculations by substantially accelerating the SCF procedure. Here, we employed a combination of the real space finite-difference formulation and CheFSI to solve the Kohn–Sham equation, and implemented this approach in ab initio Real-space Electronic Structure (ARES) software in a multi-processor, parallel environment. An improved scheme was proposed to generate the initial subspace of Chebyshev filtering in ARES efficiently, making it suitable for large-scale simulations. The accuracy, stability, and efficiency of the ARES software were illustrated by simulations of large-scale crystalline systems containing thousands of atoms.

Discrete discontinuous basis projection method for large-scale electronic structure calculations.

We present an approach to accelerate real-space electronic structure methods several fold, without loss of accuracy, by reducing the dimension of the discrete eigenproblem that must be solved. To accomplish this, we construct an efficient, systematically improvable, discontinuous basis spanning the occupied subspace and project the real-space Hamiltonian onto the span. In calculations on a range of systems, we find that accurate energies and forces are obtained with 8-25 basis functions per atom, reducing the dimension of the associated real-space eigenproblems by 1-3 orders of magnitude.

Density Functional Theory under the Bubbles and Cube Numerical Framework.

Density functional theory within the Kohn–Sham density functional theory (KS-DFT) ansatz has been implemented into our bubbles and cube real-space molecular electronic structure framework, where functions containing steep cusps in the vicinity of the nuclei are expanded in atom-centered one-dimensional (1D) numerical grids multiplied with spherical harmonics (bubbles). The remainder, i.e., the cube, which is the cusp-free and smooth difference between the atomic one-center contributions and the exact molecular function, is represented on a three-dimensional (3D) equidistant grid by using a tractable number of grid points. The implementation of the methods is demonstrated by performing 3D numerical KS-DFT calculations on light atoms and small molecules. The accuracy is assessed by comparing the obtained energies with the best available reference energies.

Development of the SSPH Method for Real-Space Electronic Structure Calculations

We have applied symmetric smoothed particle hydrodynamics (SSPH) to real-space electronic structure calculations. The electronic structure of a simple atom such as hydrogen, helium and lithium is calculated. The results of SSPH are compared to that of the higher-order finite difference method (FD), and are in good agreement with that of FD. Our results indicate that SSPH can be applicable to the electronic structure calculations that require high accuracy.

Structural and electronic properties of sodium clusters under confinement

Using real-space density functional theory, electronic structure and equilibrium geometries of sodium clusters in the size range of 2--20 atoms have been calculated as a function of confinement. We have examined the evolution of the five lowest isomers as a function of volume for six different compressions. The minimum volume considered is about $1/15$ to $1/10$ of the free-space box volume. We observe a strong tendency for isomeric transitions in many cases, with the higher isomers evolving into the ground state under confinement. In general the clusters tend to become more spherical. The changes in the total energies and the geometries are not significant until the volume gets reduced beyond the 1/3 of the original volume. In this sense, the clusters are not easy to compress. Once the critical volume is reached, the changes in the total energies and geometries are rapid. It turns out that the increase in the total energy is mainly due to the ion-ion and Hartree energies of electrons. We also address how anisotropic confinement affects the geometry of clusters. We further show that geometries obtained with anisotropic confinement are strongly supported by the simulation of clusters inside a carbon nanotube using a hybrid quantum-mechanical and molecular-mechanics approach.

Tensor numerical methods in quantum chemistry: from Hartree-Fock to excitation energies.

We resume the recent successes of the grid-based tensor numerical methods and discuss their prospects in real-space electronic structure calculations. These methods, based on the low-rank representation of the multidimensional functions and integral operators, first appeared as an accurate tensor calculus for the 3D Hartree potential using 1D complexity operations, and have evolved to entirely grid-based tensor-structured 3D Hartree-Fock eigenvalue solver. It benefits from tensor calculation of the core Hamiltonian and two-electron integrals (TEI) in O(n log n) complexity using the rank-structured approximation of basis functions, electron densities and convolution integral operators all represented on 3D n × n × n Cartesian grids. The algorithm for calculating TEI tensor in a form of the Cholesky decomposition is based on multiple factorizations using algebraic 1D "density fitting" scheme, which yield an almost irreducible number of product basis functions involved in the 3D convolution integrals, depending on a threshold ε > 0. The basis functions are not restricted to separable Gaussians, since the analytical integration is substituted by high-precision tensor-structured numerical quadratures. The tensor approaches to post-Hartree-Fock calculations for the MP2 energy correction and for the Bethe-Salpeter excitation energies, based on using low-rank factorizations and the reduced basis method, were recently introduced. Another direction is towards the tensor-based Hartree-Fock numerical scheme for finite lattices, where one of the numerical challenges is the summation of electrostatic potentials of a large number of nuclei. The 3D grid-based tensor method for calculation of a potential sum on a L × L × L lattice manifests the linear in L computational work, O(L), instead of the usual O(L(3) log L) scaling by the Ewald-type approaches.

Multipole-preserving quadratures for the discretization of functions in real-space electronic structure calculations.

Discretizing an analytic function on a uniform real-space grid is often done via a straightforward collocation method. This is ubiquitous in all areas of computational physics and quantum chemistry. An example in density functional theory (DFT) is given by the external potential or the pseudo-potential describing the interaction between ions and electrons. The accuracy of the collocation method used is therefore very important for the reliability of subsequent treatments like self-consistent field solutions of the electronic structure problems. By construction, the collocation method introduces numerical artifacts typical of real-space treatments, like the so-called egg-box error, which may spoil the numerical stability of the description when the real-space grid is too coarse. As the external potential is an input of the problem, even a highly precise computational treatment cannot cope this inconvenience. We present in this paper a new quadrature scheme that is able to exactly preserve the moments of a given analytic function even for large grid spacings, while reconciling with the traditional collocation method when the grid spacing is small enough. In the context of real-space electronic structure calculations, we show that this method improves considerably the stability of the results for large grid spacings, opening up the path towards reliable low-accuracy DFT calculations with a reduced number of degrees of freedom.

Smoothed Particle Method for Real-Space Electronic Structure Calculations

We present a meshfree particle method for real-space electronic structure calculations based on symmetric smoothed particle hydrodynamics (SSPH). As a simple example, we applied SSPH to the Schrödinger equation for a one-dimensional harmonic oscillator. The results calculated using SSPH are in good agreement with analytical solutions. We compared results for different cases of kernel function, smoothing length, number of particles, and the order of the Taylor series expansion. The results using SSPH and the higher-order finite-difference method are also compared. Our results indicate that SSPH can be applied to real-space electronic structure calculations that require high accuracy.

Higher-order adaptive finite-element methods for Kohn-Sham density functional theory

We present an efficient computational approach to perform real-space electronic structure calculations using an adaptive higher-order finite-element discretization of Kohn-Sham density-functional t...

Higher-order adaptive finite-element methods for Kohn–Sham density functional theory

We present an efficient computational approach to perform real-space electronic structure calculations using an adaptive higher-order finite-element discretization of Kohn–Sham density-functional theory (DFT). To this end, we develop an a priori mesh-adaption technique to construct a close to optimal finite-element discretization of the problem. We further propose an efficient solution strategy for solving the discrete eigenvalue problem by using spectral finite-elements in conjunction with Gauss–Lobatto quadrature, and a Chebyshev acceleration technique for computing the occupied eigenspace. The proposed approach has been observed to provide a staggering 100–200-fold computational advantage over the solution of a generalized eigenvalue problem. Using the proposed solution procedure, we investigate the computational efficiency afforded by higher-order finite-element discretizations of the Kohn–Sham DFT problem. Our studies suggest that staggering computational savings—of the order of 1000-fold—relative to linear finite-elements can be realized, for both all-electron and local pseudopotential calculations, by using higher-order finite-element discretizations. On all the benchmark systems studied, we observe diminishing returns in computational savings beyond the sixth-order for accuracies commensurate with chemical accuracy, suggesting that the hexic spectral-element may be an optimal choice for the finite-element discretization of the Kohn–Sham DFT problem. A comparative study of the computational efficiency of the proposed higher-order finite-element discretizations suggests that the performance of finite-element basis is competing with the plane-wave discretization for non-periodic local pseudopotential calculations, and compares to the Gaussian basis for all-electron calculations to within an order of magnitude. Further, we demonstrate the capability of the proposed approach to compute the electronic structure of a metallic system containing 1688 atoms using modest computational resources, and good scalability of the present implementation up to 192 processors.

Theoretical analysis of electronic band structure of 2- to 3-nm Si nanocrystals

We introduce a general method which allows reconstruction of electronic band structure of nanocrystals from ordinary real-space electronic structure calculations. A comprehensive study of band structure of a realistic nanocrystal is given including full geometric and electronic relaxation with the surface passivating groups. In particular, we combine this method with large scale density functional theory calculations to obtain insight into the luminescence properties of silicon nanocrystals of up to 3 nm in size depending on the surface passivation and geometric distortion. We conclude that the band structure concept is applicable to silicon nanocrystals with diameter larger than $\approx$ 2 nm with certain limitations. We also show how perturbations due to polarized surface groups or geometric distortion can lead to considerable moderation of momentum space selection rules.

Inhomogenous Electronic Structure, Transport Gap, and Percolation Threshold in Disordered Bilayer Graphene

The inhomogenous real-space electronic structure of gapless and gapped disordered bilayer graphene is calculated in the presence of quenched charge impurities. For gapped bilayer graphene, we find that for current experimental conditions the amplitude of the fluctuations of the screened disorder potential is of the order of (or often larger than) the intrinsic gap Δ induced by the application of a perpendicular electric field. We calculate the crossover chemical potential Δ(cr), separating the insulating regime from a percolative regime in which less than half of the area of the bilayer graphene sample is insulating. We find that most of the current experiments are in the percolative regime with Δ(cr)≪Δ. The huge suppression of Δ(cr) compared with Δ provides a possible explanation for the large difference between the theoretical band gap Δ and the experimentally extracted transport gap.

Emerging local Kondo screening and spatial coherence in the heavy-fermion metal YbRh2Si2

The entanglement of quantum states is both a central concept in fundamental physics and a potential tool for realizing advanced materials and applications. The quantum superpositions underlying entanglement are at the heart of the intricate interplay of localized spin states and itinerant electronic states that gives rise to the Kondo effect in certain dilute magnetic alloys. In systems where the density of localized spin states is sufficiently high, they can no longer be treated as non-interacting; if they form a dense periodic array, a Kondo lattice may be established. Such a Kondo lattice gives rise to the emergence of charge carriers with enhanced effective masses, but the precise nature of the coherent Kondo state responsible for the generation of these heavy fermions remains highly debated. Here we use atomic-resolution tunnelling spectroscopy to investigate the low-energy excitations of a generic Kondo lattice system, YbRh(2)Si(2). We find that the hybridization of the conduction electrons with the localized 4f electrons results in a decrease in the tunnelling conductance at the Fermi energy. In addition, we observe unambiguously the crystal-field excitations of the Yb(3+) ions. A strongly temperature-dependent peak in the tunnelling conductance is attributed to the Fano resonance resulting from tunnelling into the coherent heavy-fermion states that emerge at low temperature. Taken together, these features reveal how quantum coherence develops in heavy 4f-electron Kondo lattices. Our results demonstrate the efficiency of real-space electronic structure imaging for the investigation of strong electronic correlations, specifically with respect to coherence phenomena, phase coexistence and quantum criticality.

Real-space electronic structure calculations with full-potential all-electron precision for transition metals

We have developed an efficient computational scheme utilizing the real-space finite-difference formalism and the projector augmented-wave (PAW) method to perform precise first-principles electronic-structure simulations based on the density-functional theory for systems containing transition metals with a modest computational effort. By combining the advantages of the time-saving double-grid technique and the Fourier-filtering procedure for the projectors of pseudopotentials, we can overcome the egg box effect in the computations even for first-row elements and transition metals, which is a problem of the real-space finite-difference formalism. In order to demonstrate the potential power in terms of precision and applicability of the present scheme, we have carried out simulations to examine several bulk properties and structural energy differences between different bulk phases of transition metals and have obtained excellent agreement with the results of other precise first-principles methods such as a plane-wave-based PAW method and an all-electron full-potential linearized augmented plane-wave (FLAPW) method.

A momentum-dependent perspective on quasiparticle interference in Bi2Sr2CaCu2O8+δ

Scanning tunnelling spectroscopy and angle-resolved photoemission spectroscopy are complementary probes, and yet the results of recent studies using these techniques on quasiparticle excitations in the copper oxide superconductors seem to be contradictory. In fact, there is no contradiction. Angle-resolved photoemission spectroscopy (ARPES) probes the momentum-space electronic structure of materials and provides invaluable information about the high-temperature superconducting cuprates1. Likewise, scanning tunnelling spectroscopy (STS) reveals the cuprates’ real-space inhomogeneous electronic structure. Recently, researchers using STS have exploited quasiparticle interference (QPI)—wave-like electrons that scatter off impurities to produce periodic interference patterns—to infer properties of the quasiparticles in momentum space. Surprisingly, some interference peaks in Bi2Sr2CaCu2O8+δ (Bi-2212) are absent beyond the antiferromagnetic zone boundary, implying the dominance of a particular scattering process2. Here, we show that ARPES detects no evidence of quasiparticle extinction: quasiparticle-like peaks are measured everywhere on the Fermi surface, evolving smoothly across the antiferromagnetic zone boundary. This apparent contradiction stems from differences in the nature of single-particle (ARPES) and two-particle (STS) processes underlying these probes. Using a simple model, we demonstrate extinction of QPI without implying the loss of quasiparticles beyond the antiferromagnetic zone boundary.

Curvature effects on optical response of Si nanocrystals inSiO2having interface silicon suboxides

Several models of oxygenated and hydrogenated surfaces of Si quantum wells and Si nanocrystals of variable shapes have been constructed in order to assess curvature effects on energy gaps due to the three Si-suboxides. Si-suboxides in partially oxydized models of nanocrystals of spherical shapes (or quantum dots) are shown to reduce energy gaps compared to the hydrogenated nanocrystals, consistent with previous results in the literature. This trend is shown to be reversed when planar interfaces are formed in Si-NC inside SiO2 thin films, as in Si quantum wells. At planar interfaces (or surfaces) the electronic charge density is shown to become extended and to distribute among the Si-suboxides, thus generating along these planar interfaces extended, or delocalized, states. This delocalization of the electronic states then increase the energy gap compared to equivalent hydrogenated interfaces (or surfaces). Determination of geometric effects of curvature on the band gaps are based on density functional theory (DFT) using the real-space numerical approach implemented in the ``Pseudopotential Algorithm for Real-Space Electronic Structure'' (PARSEC) program. A method for measuring interface effects due to Si-suboxides in Si-NC in SiO2 is also suggested: by comparing photoluminescence (PL) between as-grown and annealed Si-NC samples. The DFT calculations suggest that blueshift of more than 0.2 eV of the PL should be observed in as-grown samples having Si-suboxides at their planar interfaces, in comparison to annealed samples above 950 degree Celcius that have fewer or no Si-suboxides once annealed, providing that the bulk of Si-NC remains unaltered.