Real-space Grid Research Articles

Configuration interaction singles based on the real-space numerical grid method: Kohn-Sham versus Hartree-Fock orbitals.

We developed a program code of configuration interaction singles (CIS) based on a numerical grid method. We used Kohn-Sham (KS) as well as Hartree-Fock (HF) orbitals as a reference configuration and Lagrange-sinc functions as a basis set. Our calculations show that KS-CIS is more cost-effective and more accurate than HF-CIS. The former is due to the fact that the non-local HF exchange potential greatly reduces the sparsity of the Hamiltonian matrix in grid-based methods. The latter is because the energy gaps between KS occupied and virtual orbitals are already closer to vertical excitation energies and thus KS-CIS needs small corrections, whereas HF results in much larger energy gaps and more diffuse virtual orbitals. KS-CIS using the Lagrange-sinc basis set also shows a better or a similar accuracy to smaller orbital space compared to the standard HF-CIS using Gaussian basis sets. In particular, KS orbitals from an exact exchange potential by the Krieger-Li-Iafrate approximation lead to more accurate excitation energies than those from conventional (semi-) local exchange-correlation potentials.

High order forces and nonlocal operators in a Kohn-Sham Hamiltonian.

Real space pseudopotentials have a number of advantages in solving for the electronic structure of materials. These advantages include ease of implementation, implementation on highly parallel systems, and great flexibility for describing partially periodic systems. One limitation of this approach, shared by other electronic structure methods, is the slow convergence of interatomic forces when compared to total energies. For real space methods, this requires a fine grid to converge a solution of the Kohn-Sham problem, which is accompanied by concurrent increase in memory and additional matrix-vector multiplications. Here we introduce a method to expedite the computation of interatomic forces by employing a high order integration technique. We demonstrate the usefulness of this technique by calculating accurate bond lengths and vibrational frequencies for molecules and nanocrystals without using fine real space grids.

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.

Dirac physics in flakes of artificial graphene in magnetic fields

Artificial graphene is a recently realized, man-made nanosystem that exhibits graphene-like physics in a tunable setup. The system can be created by, e.g., positioning molecules in a triangular lattice on a metal surface. Here, we model finite flakes of artificial graphene on a real-space grid and calculate their single-electron properties as a function of the flake size and the strength of an external magnetic field. Our calculations reveal the gradual formation of Dirac cones as well as a self-similar Hofstadter butterfly as the flake size is increased. Moreover, the density of states qualitatively agrees with the experimental data with and without the magnetic field.

Optimal control of charge with local gates in quantum-dot lattices

Semiconductor quantum dots are among the leading candidates for next-generation nanoscale devices due to their tunable size, shape, and low energy consumption. Here we apply quantum optimal control theory to coherently manipulate the single-electron charge distribution in quantum-dot lattices of various sizes. In particular, we show that to control the charge distribution it is sufficient to optimize the gate voltage acting on a single quantum dot in the lattice. We generally find yields around 99% in the picosecond time scale when using realistic models for the quantum-dot lattices on a real-space grid. We analyze and discuss both the limitations of the model regarding the gate parameters as well as the potential of the scheme for applications as quantum-dot cellular automata.

Periodic boundary conditions for QM/MM calculations: Ewald summation for extended Gaussian basis sets

An implementation of Ewald summation for use in mixed quantum mechanics/molecular mechanics (QM/MM) calculations is presented, which builds upon previous work by others that was limited to semi-empirical electronic structure for the QM region. Unlike previous work, our implementation describes the wave function's periodic images using "ChElPG" atomic charges, which are determined by fitting to the QM electrostatic potential evaluated on a real-space grid. This implementation is stable even for large Gaussian basis sets with diffuse exponents, and is thus appropriate when the QM region is described by a correlated wave function. Derivatives of the ChElPG charges with respect to the QM density matrix are a potentially serious bottleneck in this approach, so we introduce a ChElPG algorithm based on atom-centered Lebedev grids. The ChElPG charges thus obtained exhibit good rotational invariance even for sparse grids, enabling significant cost savings. Detailed analysis of the optimal choice of user-selected Ewald parameters, as well as timing breakdowns, is presented.

A Guided Stochastic Energy-Domain Formulation of the Second Order Møller-Plesset Perturbation Theory.

We develop an alternative formulation in the energy-domain to calculate the second order Møller-Plesset (MP2) perturbation energies. The approach is based on repeatedly choosing four random energies using a nonseparable guiding function, filtering four random orbitals at these energies, and averaging the resulting Coulomb matrix elements to obtain a statistical estimate of the MP2 correlation energy. In contrast to our time-domain formulation, the present approach is useful for both quantum chemistry and real-space/plane wave basis sets. The scaling of the MP2 calculation is roughly linear with system size, providing a useful tool to study dispersion energies in large systems. This is demonstrated on a structure of 64 fullerenes within the SZ basis as well as on silicon nanocrystals using real-space grids.

Self-interaction corrected density functional calculations of molecular Rydberg states

A method is presented for calculating the wave function and energy of Rydberg excited states of molecules. A good estimate of the Rydberg state orbital is obtained using ground state density functional theory including Perdew-Zunger self-interaction correction and an optimized effective potential. The total energy of the excited molecule is obtained using the Delta Self-Consistent Field method where an electron is removed from the highest occupied orbital and placed in the Rydberg orbital. Results are presented for the first few Rydberg states of NH3, H2O, H2CO, C2H4, and N(CH3)3. The mean absolute error in the energy of the 33 molecular Rydberg states presented here is 0.18 eV. The orbitals are represented on a real space grid, avoiding the dependence on diffuse atomic basis sets. As in standard density functional theory calculations, the computational effort scales as NM(2) where N is the number of orbitals and M is the number of grid points included in the calculation. Due to the slow scaling of the computational effort with system size and the high level of parallelism in the real space grid approach, the method presented here makes it possible to estimate Rydberg electron binding energy in large molecules.

Multi-Thousand-Atom DFT Simulation of Li-Ion Transfer through the Boundary between the Solid–Electrolyte Interface and Liquid Electrolyte in a Li-Ion Battery

Microscopic mechanisms of the Li-ion transfer through the boundary between the solid–electrolyte interface (SEI) formed on the graphite anode and liquid electrolyte in the Li-ion battery are investigate theoretically. A simulation system (about 2400 atoms) for the boundary is modeled using dilithium ethylene dicarbonate (Li2EDC), ethylene carbonate (EC), and LiPF6 for the SEI, solvent, and salt, respectively. After inserting Li-ions in the Li2EDC region, we perform the first-principles molecular dynamics simulation for 4.8 ps using the divide-and-conquer-type real-space grid DFT. Enhanced stability of the Li-ions at the boundary where EDC2– and EC contact with each other is thereby found in the runs without salt, which acts to impede the Li-ion transfer through the boundary. It is also found that inclusion of 1.0 M LiPF6 salt in the liquid EC weakens such impedance effect significantly. Physical reasons for those phenomena are explained in combination with separate DFT calculations.

Expeditious Stochastic Approach for MP2 Energies in Large Electronic Systems

A fast stochastic method for calculating the second order Møller-Plesset (MP2) correction to the correlation energy of large systems of electrons is presented. The approach is based on reducing the exact summation over occupied and unoccupied states to a time-dependent trace formula amenable to stochastic sampling. We demonstrate the abilities of the method to treat systems with thousands of electrons using hydrogen passivated silicon spherical nanocrystals represented on a real space grid, much beyond the capabilities of present day MP2 implementations.

Efficient and accurate solver of the three-dimensional screened and unscreened Poisson's equation with generic boundary conditions

We present an explicit solver of the three-dimensional screened and unscreened Poisson's equation, which combines accuracy, computational efficiency, and versatility. The solver, based on a mixed plane-wave/interpolating scaling function representation, can deal with any kind of periodicity (along one, two, or three spatial axes) as well as with fully isolated boundary conditions. It can seamlessly accommodate a finite screening length, non-orthorhombic lattices, and charged systems. This approach is particularly advantageous because convergence is attained by simply refining the real space grid, namely without any adjustable parameter. At the same time, the numerical method features O(NlogN) scaling of the computational cost (N being the number of grid points) very much like plane-wave methods. The methodology, validated on model systems, is tailored for leading-edge computer simulations of materials (including ab initio electronic structure computations), but it might as well be beneficial for other research domains.

Parallel finite element density functional computations exploiting grid refinement and subspace recycling

In this communication computational methods that facilitate finite element analysis of density functional computations are developed. They are: (i) h–adaptive grid refinement techniques that reduce the total number of degrees of freedom in the real space grid while improving on the approximate resolution of the wanted solution; and (ii) subspace recycling of the approximate solution in self-consistent cycles with the aim of improving the performance of the generalized eigenproblem solver. These techniques are shown to give a convincing speed-up in the computation process by alleviating the overhead normally associated with computing systems with many degrees-of-freedom.

Plasmon excitations in sodium atomic planes: A time-dependent density functional theory study

The collective electronic excitation in planar sodium clusters is studied by time-dependent density functional theory calculations. The formation and development of the resonances in photoabsorption spectra are investigated in terms of the shape and size of the two-dimensional (2D) systems. The nature of these resonances is revealed by the frequency-resolved induced charge densities present on a real-space grid. For long double chains, the excitation is similar to that in long single atomic chains, showing longitudinal modes, end and central transverse modes. However, for 2D planes consisting of (n × n) atoms with n being up to 16, new 2D characteristic modes emerge regardless of the symmetries considered. For in-plane excitations, besides the equivalent end mode, mixed modes with contrary polarity occur. The relation between the frequency of the primary modes and the system size is similar to the case of a 2D electron gas but with a correction due to the realistic atomic structure. For excitations perpendicular to the plane there are corner, side center, bulk center, and circuit modes. Our calculation reveals the importance of dimensionality for plasmon excitation and how it evolves from 1D to 2D.

Linear scaling algorithm of real-space density functional theory of electrons with correlated overlapping domains

The real-space grid based implementation of the Kohn–Sham density functional theory of electrons using the finite difference method for derivatives of variables, has attractive features of parallelizability and applicability to various boundary conditions in addition to universality in target materials. Following the divide-and-conquer strategy, we propose a linear scaling algorithm of it by advancing the algorithm in [F. Shimojo et al., Comput. Phys. Comm. 167 (2005) 151]. In the Kohn–Sham-type equation for a domain, we introduce (i) the density-template potential for density continuity with simple stepwise weight functions and (ii) the embedding potential to take into account all the quantum correlation effects with other overlapping domains in addition to the classical effects of ionic and electronic Coulomb potentials. We thereby realize reasonably high accuracies in atomic forces with relatively small numbers of buffer ions irrespective of the electronic characters of materials. The timing tests on parallel machines demonstrate the linear scaling of the code with little communication time between the domains.

Reference electronic structure calculations in one dimension

Large strongly correlated systems provide a challenge to modern electronic structure methods, because standard density functionals usually fail and traditional quantum chemical approaches are too demanding. The density-matrix renormalization group method, an extremely powerful tool for solving such systems, has recently been extended to handle long-range interactions on real-space grids, but is most efficient in one dimension where it can provide essentially arbitrary accuracy. Such 1d systems therefore provide a theoretical laboratory for studying strong correlation and developing density functional approximations to handle strong correlation, if they mimic three-dimensional reality sufficiently closely. We demonstrate that this is the case, and provide reference data for exact and standard approximate methods, for future use in this area.

A theoretical model for electron transfer in ion–atom collisions: Calculations for the collision of a proton with an argon atom

We have developed a theoretical model of ion–atom collisions based on the time-dependent density-functional theory. We solve the time-dependent Kohn–Sham equation for electrons employing the real-space and real-time method, while the ion dynamics are described in classical mechanics by the Ehrenfest method. Taking advantage of the real-space grid method, we introduce the “coordinate space translation” technique to allow one to focus on a certain space of interest. Benchmark calculations are given for collisions between proton and argon over a wide range of impact energy. Electron transfer total cross sections showed a fairly good agreement with available experimental data.

Linear density response function in the projector augmented wave method: Applications to solids, surfaces, and interfaces

We present an implementation of the linear density response function within the projector-augmented wave (PAW) method with applications to the linear optical and dielectric properties of both solids, surfaces, and interfaces. The response function is represented in plane waves while the single-particle eigenstates can be expanded on a real space grid or in atomic orbital basis for increased efficiency. The exchange-correlation kernel is treated at the level of the adiabatic local density approximation (ALDA) and crystal local field effects are included. The calculated static and dynamical dielectric functions of Si, C, SiC, AlP and GaAs compare well with previous calculations. While optical properties of semiconductors, in particular excitonic effects, are generally not well described by ALDA, we obtain excellent agreement with experiments for the surface loss function of the Mg(0001) surface with plasmon energies deviating by less than 0.2 eV. Finally, we apply the method to study the influence of substrates on the plasmon excitations in graphene. On SiC(0001), the long wavelength $\pi$ plasmons are significantly damped although their energies remain almost unaltered. On Al(111) the $\pi$ plasmon is completely quenched due to the coupling to the metal surface plasmon.

Coordinate space translation technique for simulation of electronic process in the ion–atom collision

Recently we developed a theoretical model of ion-atom collisions, which was made on the basis of a time-dependent density functional theory description of the electron dynamics and a classical treatment of the heavy particle motion. Taking advantage of the real-space grid method, we introduce a "coordinate space translation" technique to allow one to focus on a certain space of interest such as the region around the projectile or the target. Benchmark calculations are given for collisions between proton and oxygen over a wide range of impact energy. To extract the probability of charge transfer, the formulation of Lüdde and Dreizler [J. Phys. B 16, 3973 (1983)] has been generalized to ensemble-averaging application in the particular case of O((3)P). Charge transfer total cross sections are calculated, showing fairly good agreements between experimental data and present theoretical results.

Calculation of transmission probability by solving an eigenvalue problem

The electron transmission probability in nanodevices is calculated by solving an eigenvalueproblem. The eigenvalues are the transmission probabilities and the number of nonzeroeigenvalues is equal to the number of open quantum transmission eigenchannels. Thenumber of open eigenchannels is typically a few dozen at most, thus the computational costamounts to the calculation of a few outer eigenvalues of a complex Hermitian matrix(the transmission matrix). The method is implemented on a real space grid basisproviding an alternative to localized atomic orbital based quantum transportcalculations. Numerical examples are presented to illustrate the efficiency of the method.

TOWARDS DFT CALCULATIONS OF METAL CLUSTERS IN QUANTUM FLUID MATRICES

This paper reports progress on the simulation of metallic clusters embedded in a quantum fluid matrix such as 4 He . In previous work we have reported progress developing a real-space density functional method. The core of the method is a diffusion algorithm that extracts the low-lying eigenfunctions of the Kohn-Sham Hamiltonian by propagating the wave functions (which are represented on a real-space grid) in imaginary time. Due to the diffusion character of the kinetic energy operator in imaginary time, algorithms developed so far are at most fourth order in the time-step. The first part of this paper discusses further progress, in particular we show that for a grid based algorithm, imaginary time propagation of any even order can be devised on the basis of multi-product splitting. The new propagation method is particularly suited for modern parallel computing environments. The second part of this paper addresses a yet unsolved problem, namely a consistent description of the interaction between helium atoms and a metallic cluster that can bridge the whole range from a single atom to a metal. Using a combination of DFT calculations to determine the response of the valence electrons, and phenomenological acounts of Pauli repulsion and short-ranged correlations that are poorly described in DFT, we show how such an interaction can be derived.