Real-space Grid Research Articles

An efficient implementation of a QM–MM method in SIESTA

We present the major features of a new implementation of a QM–MM method that uses the DFT code Siesta to treat the quantum mechanical subsystem and the AMBER force field to deal with the classical part. The computation of the electrostatic interaction has been completely revamped to treat periodic boundary conditions exactly, using a real-space grid that encompasses the whole system. Additionally, we present a new parallelization of the Siesta grid operations that provides near-perfect load balancing for all the relevant operations and achieves a much better scalability, which is important for efficient massive QM–MM calculations in which the grid can potentially be very large.

Basis set effects on the hyperpolarizability of CHCl3: Gaussian-type orbitals, numerical basis sets and real-space grids

Calculations of the hyperpolarizability are typically much more difficult to converge with basis set size than the linear polarizability. In order to understand these convergence issues and hence obtain accurate ab initio values, we compare calculations of the static hyperpolarizability of the gas-phase chloroform molecule (CHCl(3)) using three different kinds of basis sets: Gaussian-type orbitals, numerical basis sets, and real-space grids. Although all of these methods can yield similar results, surprisingly large, diffuse basis sets are needed to achieve convergence to comparable values. These results are interpreted in terms of local polarizability and hyperpolarizability densities. We find that the hyperpolarizability is very sensitive to the molecular structure, and we also assess the significance of vibrational contributions and frequency dispersion.

Förster Coupling in Nanoparticle Excitonic Circuits

Exciton transport in semiconductor nanoparticles underlies recent experiments on electrically controlled nanostructures and proposals for new artificial light-harvesting systems. In this work, we develop a novel method for the numerical evaluation of the Forster matrix element, based on a three-dimensional real space grid and the self-consistent solution of the mesoscopic exciton in a macroscopic dielectric environment. This method enables the study of the role of the nanoparticle shape, spatially varying dielectric environments, and externally applied electric fields. Depending on the orientation of the transition dipole, the Forster coupling is shown to be either increased or decreased as a function of the nanoparticle shape and of the properties of the dielectric environment. In the presence of an electric field, we investigate the relation between excitonic binding and confinement effects. We also study a type II core-shell quantum dot where electron and hole are spatially separated due to a particular configuration of the bandstructure.

Strategies for h-Adaptive Refinement for a Finite Element Treatment of Harmonic Oscillator Schrödinger Eigenproblem

A Schrödinger eigenvalue problem is solved for the 2D quantum simple harmonic oscillator using a finite element discretization of real space within which elements are adaptively spatially refined. We compare two competing methods of adaptively discretizing the real-space grid on which computations are performed without modifying the standard polynomial basis-set traditionally used in finite element interpolations; namely, (i) an application of the Kelly error estimator, and (ii) a refinement based on the local potential level. When the performance of these methods are compared to standard uniform global refinement, we find that they significantly improve the total time spent in the eigensolver.

Theoretical approximations to X-ray absorption spectroscopy of liquid water and ice

We review methods to compute X-ray absorption spectra (XAS) with special focus on the transition-potential approach of Triguero et al. [Phys. Rev. B 58 (1998) 8097] and its application to calculations on water in condensed phase. We discuss the absolute energy scale, functional dependence, broadening versus sampling of intra- and intermolecular vibrational modes, treatment of the continuum, cluster size convergence as well as compare with periodic calculations and with experiment; periodic and cluster model calculations are found to agree very closely in the relevant near-edge region although neither reproduces the pre-edge and main-edge features in the experimental spectra of thin ice films. The real-space grid representation of the wave function in the periodic calculations allows a more extended energy range to be described and we find satisfactory agreement with experiment for higher energy continuum resonances. Two proposed alternative approaches using either the potential from a full core-hole (FCH) or the full core-hole with an excited electron in the lowest state (XCH) are shown to lead to spectra that deviate significantly from experiment.

Mixed Si-Ge nanoparticle quantum dots: a density functional theory study

We present a systematic study of small mixed composition Si and Ge nanoparticle quantum dots performed using density functional theory (DFT) and time-dependent density functional theory (TDDFT) with real space grids and norm-conserving pseudopotentials. Quantum dot models are obtained by distributing Si and Ge atoms within basic models containing a specified number of atomic sites, tetrahedrally-coordinated about a central site, with H atoms added to terminate the surfaces. We consider models containing up to 35 semiconductor atoms, and a number of subsets of the total number of possible configurations (e.g. alternating elements in successive shells). Following calculation of structural and energetic properties, the optical absorption spectra are determined using time-dependent DFT within Casida’s linear-response approach. With respect to Si, the inclusion of Ge moves spectral weight to lower energies. We observe that the relative fractions of Si and Ge have stronger effects on the absorption spectra than the structural distribution of those atoms within the nanoparticle, which tends to have quite small effects.

Umbrella integration in two or more reaction coordinates

Umbrella integration is a method to analyze umbrella sampling simulations by calculating and integrating the mean force. Here, the method is extended to multidimensional reaction coordinates. Approximation of the probability distribution obtained from sampling by a multivariate normal distribution allows to calculate the mean force from the average and the covariance matrix of the reaction coordinate. Integration schemes of the free-energy gradient field are discussed. Integration on a real-space grid is compared to expansion of the gradient in a series of analytic functions (such as a Fourier analysis), which can be integrated, and the expansion of the gradient only at the window means in a series of analytic functions. The Fourier analysis was found particularly useful for periodic reaction coordinates, such as torsion angles. An expression is provided to calculate the Hessian of the free energy with respect to the reaction coordinates from sampling data. The utility of the method is demonstrated at the example of the free-energy surface of the alanine dipeptide in vacuum calculated with respect to the backbone torsion angles Phi and Psi. Relevance of the Jacobian term for non-Cartesian reaction coordinates is discussed.

Ab initio molecular simulations with numeric atom-centered orbitals

We describe a complete set of algorithms for ab initio molecular simulations based on numerically tabulated atom-centered orbitals (NAOs) to capture a wide range of molecular and materials properties from quantum-mechanical first principles. The full algorithmic framework described here is embodied in the Fritz Haber Institute “ab initio molecular simulations” (FHI-aims) computer program package. Its comprehensive description should be relevant to any other first-principles implementation based on NAOs. The focus here is on density-functional theory (DFT) in the local and semilocal (generalized gradient) approximations, but an extension to hybrid functionals, Hartree–Fock theory, and MP2/GW electron self-energies for total energies and excited states is possible within the same underlying algorithms. An all-electron/full-potential treatment that is both computationally efficient and accurate is achieved for periodic and cluster geometries on equal footing, including relaxation and ab initio molecular dynamics. We demonstrate the construction of transferable, hierarchical basis sets, allowing the calculation to range from qualitative tight-binding like accuracy to meV-level total energy convergence with the basis set. Since all basis functions are strictly localized, the otherwise computationally dominant grid-based operations scale as O(N) with system size N. Together with a scalar-relativistic treatment, the basis sets provide access to all elements from light to heavy. Both low-communication parallelization of all real-space grid based algorithms and a ScaLapack-based, customized handling of the linear algebra for all matrix operations are possible, guaranteeing efficient scaling (CPU time and memory) up to massively parallel computer systems with thousands of CPUs.

Application of time-dependent density-functional theory to molecules and nanostructures

We present ab initio time-dependent density-functional calculations for the optical properties of molecules, atomic clusters, functionalized carbon nanotubes, and metal-nanotube heterostructures. Our calculations are carried out in the framework of a real-space higher-order finite difference method combined with the pseudopotential approximation. In this method, the Kohn–Sham equations for electronic states are solved self-consistently on a real-space three-dimensional Cartesian grid without the use of explicit basis functions. The time-dependent density-functional linear response formalism is applied to calculate the excited-state properties of the water molecule, analyze the optical spectra of potassium atoms and clusters adsorbed on graphene and carbon nanotubes, study the assembly of organic molecules to carbon nanotubes and compute the Stokes shifts in hydrogenated silicon clusters. The results of our calculations show that the time-dependent density-functional computational approach is flexible and can be successfully applied to a variety of different physical problems.

The static-exchange electron-water pseudopotential, in conjunction with a polarizable water model: A new Hamiltonian for hydrated-electron simulations

Previously, Turi and Borgis [J. Chem. Phys. 117, 6186 (2002)] parametrized an electron-water interaction potential, intended for use in simulations of hydrated electrons, by considering H(2)O(-) in the "static exchange" (essentially, frozen-core Hartree-Fock) approximation, then applying an approximate Phillips-Kleinman procedure to construct a one-electron pseudopotential representing the electron-water interaction. To date, this pseudopotential has been used exclusively in conjunction with a simple point charge water model that is parametrized for bulk water and yields poor results for small, neutral water clusters. Here, we extend upon the work of Turi and Borgis by reparametrizing the electron-water pseudopotential for use with the AMOEBA water model, which performs well for neutral clusters. The result is a one-electron model Hamiltonian for (H(2)O)(n)(-), in which the one-electron wave function polarizes the water molecules, and vice versa, in a fully self-consistent fashion. The new model is fully variational and analytic energy gradients are available. We have implemented the new model using a modified Davidson algorithm to compute eigenstates, with the unpaired electron represented on a real-space grid. Comparison to ab initio electronic structure calculations for (H(2)O)(n)(-) cluster isomers ranging from n=2 to n=35 reveals that the new model is significantly more accurate than the Turi-Borgis model, for both relative isomer energies and for vertical electron detachment energies. Electron-water polarization interactions are found to be much more significant for cavity states of the unpaired electron than for surface states.

Real-space pseudopotential method for first principles calculations of general periodic and partially periodic systems

We present a real-space method for electronic-structure calculations of systems with general full or partial periodicity. The method is based on the self-consistent solution of the Kohn-Sham equations, using first principles pseudopotentials, on a uniform three-dimensional non-Cartesian grid. Its efficacy derives from the introduction of a new generalized high-order finite-difference method that avoids the numerical evaluation of mixed derivative terms and results in a simple yet accurate finite difference operator. Our method is further extended to systems where periodicity is enforced only along some directions (e.g., surfaces), by setting up the correct electrostatic boundary conditions and by properly accounting for the ion-electron and ion-ion interactions. Our method enjoys the main advantages of real-space grid techniques over traditional plane-wave representations for density functional calculations, namely, improved scaling and easier implementation on parallel computers, as well as inherent immunity to spurious interactions brought about by artificial periodicity. We demonstrate its capabilities on bulk GaAs and Na for the fully periodic case and on a monolayer of Si-adsorbed polar nitrobenzene molecules for the partially periodic case.

Time-dependent density-functional theory in the projector augmented-wave method

We present the implementation of the time-dependent density-functional theory both in linear-response and in time-propagation formalisms using the projector augmented-wave method in real-space grids. The two technically very different methods are compared in the linear-response regime where we found perfect agreement in the calculated photoabsorption spectra. We discuss the strengths and weaknesses of the two methods as well as their convergence properties. We demonstrate different applications of the methods by calculating excitation energies and excited state Born-Oppenheimer potential surfaces for a set of atoms and molecules with the linear-response method and by calculating nonlinear emission spectra using the time-propagation method.

Turbo charging time-dependent density-functional theory with Lanczos chains

We introduce a new implementation of time-dependent density-functional theory which allows the entire spectrum of a molecule or extended system to be computed with a numerical effort comparable to that of a single standard ground-state calculation. This method is particularly well suited for large systems and/or large basis sets, such as plane waves or real-space grids. By using a superoperator formulation of linearized time-dependent density-functional theory, we first represent the dynamical polarizability of an interacting-electron system as an off-diagonal matrix element of the resolvent of the Liouvillian superoperator. One-electron operators and density matrices are treated using a representation borrowed from time-independent density-functional perturbation theory, which permits us to avoid the calculation of unoccupied Kohn-Sham orbitals. The resolvent of the Liouvillian is evaluated through a newly developed algorithm based on the nonsymmetric Lanczos method. Each step of the Lanczos recursion essentially requires twice as many operations as a single step of the iterative diagonalization of the unperturbed Kohn-Sham Hamiltonian. Suitable extrapolation of the Lanczos coefficients allows for a dramatic reduction of the number of Lanczos steps necessary to obtain well converged spectra, bringing such number down to hundreds (or a few thousands, at worst) in typical plane-wave pseudopotential applications. The resulting numerical workload is only a few times larger than that needed by a ground-state Kohn-Sham calculation for a same system. Our method is demonstrated with the calculation of the spectra of benzene, C(60) fullerene, and of chlorophyll a.

Calculating scattering matrices by wave function matching

AbstractThe conductance of nanoscale structures can be conveniently related to their scattering properties expressed in terms of transmission and reflection coefficients. Wave function matching (WFM) is a transparent technique for calculating transmission and reflection matrices for any Hamiltonian that can be represented in tight‐binding form. A first‐principles Kohn–Sham Hamiltonian represented on a localized orbital basis or on a real space grid has such a form. WFM is based upon direct matching of the scattering‐region wave function to the Bloch modes of ideal leads used to probe the scattering region. The purpose of this paper is to give a pedagogical introduction to WFM and present some illustrative examples of its use in practice. We briefly discuss WFM for calculating the conductance of atomic wires, using a real space grid implementation. A tight‐binding muffin‐tin orbital implementation very suitable for studying spin‐dependent transport in layered magnetic materials is illustrated by looking at spin‐dependent transmission through ideal and disordered interfaces. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Auger recombination and carrier multiplication in embedded silicon and germanium nanocrystals

For Si and Ge nanocrystals (NCs) embedded in wide band-gap matrices, the Auger recombination and carrier multiplication (CM) lifetimes are computed exactly in a three-dimensional real space grid using empirical pseudopotential wave functions. Our results in support of recent experimental data offer other predictions. We extract simple Auger constants valid for NCs. We show that both Si and Ge NCs can benefit from photovoltaic efficiency improvement via CM due to the fact that under an optical excitation exceeding twice the band-gap energy, the electrons gain lion's share from the total excess energy and can cause a CM. We predict that CM becomes especially efficient for hot electrons with an excess energy of about $1\phantom{\rule{0.3em}{0ex}}\mathrm{eV}$ above the CM threshold.

Divide-and-conquer density functional theory on hierarchical real-space grids: Parallel implementation and applications

A linear-scaling algorithm based on a divide-and-conquer (DC) scheme has been designed to perform large-scale molecular-dynamics (MD) simulations, in which interatomic forces are computed quantum mechanically in the framework of the density functional theory (DFT). Electronic wave functions are represented on a real-space grid, which is augmented with a coarse multigrid to accelerate the convergence of iterative solutions and with adaptive fine grids around atoms to accurately calculate ionic pseudopotentials. Spatial decomposition is employed to implement the hierarchical-grid DC-DFT algorithm on massively parallel computers. The largest benchmark tests include $11.8\ifmmode\times\else\texttimes\fi{}{10}^{6}$-atom ($1.04\ifmmode\times\else\texttimes\fi{}{10}^{12}$ electronic degrees of freedom) calculation on 131 072 IBM BlueGene/L processors. The DC-DFT algorithm has well-defined parameters to control the data locality, with which the solutions converge rapidly. Also, the total energy is well conserved during the MD simulation. We perform first-principles MD simulations based on the DC-DFT algorithm, in which large system sizes bring in excellent agreement with x-ray scattering measurements for the pair-distribution function of liquid Rb and allow the description of low-frequency vibrational modes of graphene. The band gap of a CdSe nanorod calculated by the DC-DFT algorithm agrees well with the available conventional DFT results. With the DC-DFT algorithm, the band gap is calculated for larger system sizes until the result reaches the asymptotic value.

An iterative semi-implicit scheme with robust damping

An efficient, iterative semi-implicit (SI) numerical method for the time integration of stiff wave systems is presented. Physics-based assumptions are used to derive a convergent iterative formulation of the SI scheme which enables the monitoring and control of the error introduced by the SI operator. This iteration essentially turns a semi-implicit method into a fully implicit method. Accuracy, rather than stability, determines the timestep. The scheme is second order accurate and shown to be equivalent to a simple preconditioning method. We show how the diffusion operators can be handled so as to yield the property of robust damping, i.e. dissipating the solution at all values of the parameter DΔt, where D is a diffusion operator and Δt the timestep. The overall scheme remains second order accurate even if the advection and diffusion operators do not commute. In the limit of no physical dissipation, and for a linear test wave problem, the method is shown to be symplectic. The method is tested on the problem of Kinetic Alfvén wave mediated magnetic reconnection. A Fourier (pseudo-spectral) representation is used. A two-field gyrofluid model is used and an efficacious k-space SI operator for this problem is demonstrated. CPU speed-up factors over a CFL-limited explicit algorithm ranging from ∼20 to several hundreds are obtained, while accurately capturing the results of an explicit integration. Possible extension of these results to a real-space (grid) discretization is discussed.

Calculation of Landégfactors for III-V nanowhisker quantum dots and comparison with experiment

The coupling between a magnetic field and the spin of an electron confined to a quantum dot is determined by the $g$ factor, which is strongly affected by the structure of the dot. Uncertainties in a dot's geometry and composition can obscure quantitative comparison of theory and experiment. Nanowhisker quantum dots (NWQDs) provide a well-controlled structure that is ideal for such comparison. We have performed detailed three-dimensional numerical calculations of the electronic properties of NWQDs consisting of an $\mathrm{In}\mathrm{P}∕\mathrm{In}\mathrm{As}∕\mathrm{In}\mathrm{P}$ quantum well embedded in a [111] oriented InAs nanowhisker. We have computed $g$ factors, confinement energies, and wave functions for valence and conduction states as a function of dot size. The calculations are in excellent agreement with experiments and differ markedly from $g$ factors obtained from extrapolation of bulk formulas, providing strong confirmation of the effect of angular momentum quenching. The closeness of our results to experiment enables us to identify critical well and barrier widths yielding $g\ensuremath{\approx}0$, which are important for technological applications. We also predict larger and more negative $g$ factors for $\stackrel{P\vec}{B}$ oriented along [111]. The calculations were carried out using eight-band strain-dependent $k∙p$ theory in the envelope-function approximation using a finite difference technique on a real-space grid.

Algorithms for defects in nanostructures

We illustrate recent progress in developing algorithms for solving the Kohn–Sham problem. Key ingredients of our algorithm include pseudopotentials implemented on a real space grid and the use of damped-Chebyshev polynomial filtered subspace iteration. This procedure allows one to predict electronic properties for many materials across the nano-regime, i.e., from atoms to nanocrystals of sufficient size to replicate bulk properties. We will illustrate this method for large silicon quantum dots doped with phosphorus defect.

Algorithms for the evolution of electronic properties in nanocrystals

We illustrate recent progress in developing algorithms for solving the Kohn–Sham problem. Key ingredients of our algorithm include pseudopotentials implemented on a real space grid and the use of damped-Chebyshev polynomial filtered subspace iteration. This procedure allows one to predict electronic properties for many materials across the nano-regime, i.e. from atoms to nanocrystals of sufficient size to replicate bulk properties. We will illustrate this method for large silicon quantum dots.