Real-space Electronic Structure Calculations Research Articles

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.

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.

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.

Implementation of ultrasoft pseudopotentials in large-scale grid-based electronic structure calculations

An implementation of Vanderbilt ultrasoft pseudopotentials in real-space grid-based electronic structure calculations is presented. Efficient utilization of these pseudopotentials requires the use of different grids for i wave functions, ii charge density, and iii sharply peaked operators within the atomic core radii. High-order interpolation between the various grids is important for accuracy, as is high-order discretization for the differential operators. However, efficiency is also of paramount importance, especially when parallelizing over hundreds or thousands of processors. We describe algorithms and procedures used to achieve an effective implementation in the real-space multigrid code, and provide test results for first-row diatomics, bulk transition metals, and energy-conserving quantum molecular dynamics of water. The code parallelizes efficiently over several thousands of processors on modern parallel supercomputers, such as the Cray XT3 and XT4.

Real-space electronic-structure calculations with a time-saving double-grid technique

We present a set of efficient techniques in first-principles electronic-structure calculations utilizing the real-space finite-difference method. These techniques greatly reduce the overhead for performing integrals that involve norm-conserving pseudopotentials, solving Poisson equations, and treating models which have specific periodicities, while keeping a high degree of accuracy. Since real-space methods are inherently local, they have a lot of advantages in applicability and flexibility compared with the conventional plane-wave approach, and promise to be well suited for large and accurate {\it ab initio} calculations. In order to demonstrate the potential power of these techniques, we present several applications for electronic structure calculations of atoms, molecules and a helical nanotube.

Compensation mechanism for N acceptors in ZnO

We present a mechanism for the compensation of N acceptors in ZnO through real-space multigrid electronic structure calculations within the local-density-functional approximation. We find that at low N doping levels using a normal ${\mathrm{N}}_{2}$ source, O vacancies are the main compensating donors for N acceptors, while N acceptors are compensated via the formation of defect complexes with Zn antisites at high doping levels. When an active plasma ${\mathrm{N}}_{2}$ gas is used to increase the N solubility, N acceptors are still greatly compensated by ${\mathrm{N}}_{2}$ molecules at oxygen sites and N-acceptor--${\mathrm{N}}_{2}$ complexes, explaining the difficulty in achieving low-resistivity p-type ZnO.

Mechanism for the Enhanced Diffusion of Charged Oxygen Ions inSiO2

Based on real-space multigrid electronic structure calculations, we find that a double Si-O-Si bridge structure is the lowest energy configuration of interstitial oxygen ions (O(-) and O(2-)) in SiO2, where two additional Si-O bonds are formed with almost no interaction between the interstitial and host O atoms, while the peroxy linkage is the most stable structure for neutral interstitial O. We propose a diffusion mechanism of interstitial O ions generated from molecular O2 under UV radiation, and find extremely low energy barriers of 0.11--0.27 eV for migration in the form of the double-bridge structure, in good agreement with enhanced oxidation experiments.

Real-space electronic structure calculations of charged clusters and defects in semiconductors using a multigrid method

We present an efficient real-space multigrid method for first-principles electronic structure calculations, based on the pseudopotential method within the local-density-functional approximation. The Poisson and Kohn-Sham equations are accurately discretized by a higher-order finite difference method, and solved efficiently by a multigrid technique, which uses different relaxations for different sets of real-space grids. Testing various systems, we find the convergence to be nearly independent of the number of real-space grids. We demonstrate that our method is very useful for charged clusters and defects in localized bulk systems.

Adaptive coordinate, real-space electronic structure calculations on parallel computers

We present a method for electronic structure calculations that retains all of the advantages of real space, but addresses the major weakness of a regular grid, i.e., its inability to treat some regions of space with more resolution than others. The computations are carried out on a regular mesh in curvilinear space, which allows natural and efficient decomposition on parallel computers, and effective use of iterative numerical methods. A novel feature is the use of error analysis to optimize the curvilinear grid for highly inhomogeneous electronic distributions. We report accurate all-electron calculations for H 2, O, and O 2.

Structural transitions in metal clusters

The methods for first-principles calculation of the structure and dynamics of clusters have now progressed to a point where clusters containing ca. 50 non-transition-metal atoms can be studied. As a paradigm, we studied the energetics of structural transformations in 13- and 55-atom Al clusters, which can assume both perfect icosahedral and cuboctahedral structures. Using the Car–Parrinello (quantum molecular dynamics) formalism, we found Al13 has a unique structure, a slightly distorted icosahedron, but Al55 has several inequivalent but energetically nearly degenerate structures. The degeneracy in Al55 is due to the short range of the effective interatomic interactions in a metallic cluster and should lead to floppiness at finite temperatures. A new accurate procedure for calculating ionization potentials (Ei) and electron affinities (Eea) within the Car–Parrinello formalism was developed and applied to Al clusters. Unfortunately, it appears that at least for some clusters, most notably for Al55, the Ei and Eea are very similar for different structural models of this cluster. A formulation and the first tests of a new multigrid-based method for real space electronic structure calculations are briefly described. This method should make possible calculations similar to the above for clusters containing transition metal and/or first-row atoms in the fairly near future.