Linear Combination Of Plane Waves Research Articles

The Weyl expansion for the scalar and vector spherical wave functions

The Weyl expansion technique, also known as the angular spectrum expansion, expresses an outgoing spherical wave as a linear combination of plane waves. The scalar spherical waves are the solutions of the homogeneous Helmholtz equation and therefore have direct relation to the scalar multipole fields. This paper gives the Weyl expansion of multipole fields, scalar and vector, of any degree and order for spherical wave functions. The expressions are given in closed form for the scalar, , and vector, , , multipole fields, evaluated across a plane orthogonal to any given direction. In the case of scalar spherical multipoles, the spherical gradient operator has been used, while for the vector spherical multipoles, the vector spherical wave operator has been constructed.

Plane wave elastography: a frequency-domain ultrasound shear wave elastography approach

In this paper, we propose plane wave elastography (PWE), a novel ultrasound shear wave elastography (SWE) approach. Currently, commercial methods for SWE rely on directional filtering based on the prior knowledge of the wave propagation direction, to remove complicated wave patterns formed due to reflection and refraction. The result is a set of decomposed directional waves that are separately analyzed to construct shear modulus fields that are then combined through compounding. Instead, PWE relies on a rigorous representation of the wave propagation using the frequency-domain scalar wave equation to automatically select appropriate propagation directions and simultaneously reconstruct shear modulus fields. Specifically, assuming a homogeneous, isotropic, incompressible, linear-elastic medium, we represent the solution of the wave equation using a linear combination of plane waves propagating in arbitrary directions. Given this closed-form solution, we formulate the SWE problem as a nonlinear least-squares optimization problem which can be solved very efficiently. Through numerous phantom studies, we show that PWE can handle complicated waveforms without prior filtering and is competitive with state-of-the-art that requires prior filtering based on the knowledge of propagation directions.

Benchmarking the performance of plane-wave vs. localized orbital basis set methods in DFT modeling of metal surface: a case study for Fe-(110)

Reproducing electronic structure of extended metallic systems is computationally demanding with the cost efficiency of this approach heavily dependent on both the density functional and the basis function used to approximate the electronic orbitals. It is well known that the generalized gradient approximation functional (GGA) is the most suitable and reliable approach for the description of metallic systems. As for the basis functions, two approaches dominate: the linear combination of localized basis functions (LB) such as Gaussian functions and the linear combination of plane waves (PW). Both have their own advantages and disadvantages, that may impact the efficiency and accuracy of the simulations. In this work, we use the VASP and the CRYSTAL14 suites of codes that employ plane waves and localized Gaussian basis sets, respectively, to establish a benchmark on their computational efficiency for the modeling of metal surfaces. The PW basis technique requires that the entire simulation box including the vacuum space be filled with plane waves which reduces the computational efficiency and limits the vacuum space. For its part, the LB method is based on atomic localized orbitals and does not require vacuum to model surfaces. Therefore, for calculations that require relatively large vacuum thickness such as modeling of adsorption, the LB method might be superior in terms of computational expense while providing the comparable accuracy.

Direct inversion of the iterative subspace with contracted planewave basis functions.

Ways to reduce the computational cost of periodic electronic structure calculations by using basis functions corresponding to linear combinations of planewaves have been examined recently. These contracted planewave (CPW) basis functions correspond to Fourier series representations of atom-centered basis functions, and thus provide access to some beneficial properties of planewave (PW) and localized basis functions. This study reports the development and assessment of a direct inversion of the iterative subspace (DIIS) method that employs unique properties of CPW basis functions to efficiently converge electronic wavefunctions. This method relies on access to a PW-based representation of the electronic structure to provide a means of efficiently evaluating matrix-vector products involving the application of the Fock matrix to the occupied molecular orbitals. These matrix-vector products are transformed into a form permitting the use of direct diagonalization techniques and DIIS methods typically employed with atom-centered basis sets. The abilities of this method are assessed through periodic Hartree-Fock calculations of a range of molecules and solid-state systems. The results show that the method reported in this study is approximately five times faster than CPW-based calculations in which the entire Fock matrix is calculated. This method is also found to be weakly dependent upon the size of the basis set, thus permitting the use of larger CPW basis sets to increase variational flexibility with a minor impact on computational performance. © 2018 Wiley Periodicals, Inc.

Fast solution of Schrödinger’s equation using linear combinations of plane waves

A method for solving Schrödinger’s equation that combines the Rayleigh quotient iteration (RQI) with the conjugate residual (CR) method is described. It is assumed that the eigenfunctions of the Hamiltonian operator Hˆ are expressed as linear combinations of plane waves, which allows the efficient computation of the product H⋅x via fast Fourier transform, where H is the representation of Hˆ in the plane-wave basis and x an arbitrary vector. The algorithm carries out two kinds of iterations: outer-loop RQI iterations (very few) and inner-loop CR iterations (many more). A very small number of orthogonalizations of the set of eigenvectors is required by the proposed method, just one per RQI iteration, which is one of its key advantages because, for large enough systems, orthogonalization is the costliest component of the calculation. The procedure is much faster than matrix diagonalization, and its cost with respect to the size of the plane-wave basis (NPW) has an effective scaling ranging from linear O(NPW) to quadratic O(NPW2).

All-electron mixed basisGWcalculations ofTiO2and ZnO crystals

In transition metal oxide systems, there exists a serious discrepancy between the theoretical quasiparticle energies and the experimental photoemission energies. To improve the accuracy of electronic structure calculations for these systems, we use the all-electron mixed basis $\mathit{GW}$ method, in which single-particle wave functions are accurately described by the linear combinations of plane waves and atomic orbitals. We adopt the full $\ensuremath{\omega}$ integration to evaluate the correlation part of the self-energy and compare the results with those obtained by plasmon pole models. We present the quasiparticle energies and band gap of titanium dioxide (${\mathrm{TiO}}_{2}$) and zinc oxide (ZnO) within the one-shot $\mathit{GW}$ approximation. The results are in reasonable agreement with experimental data in the case of ${\mathrm{TiO}}_{2}$ but underestimated by about 0.6--1.4 eV from experimental data in the case of ZnO, although our results are comparable to previous one-shot $\mathit{GW}$ calculations. We also explain a new approach to perform $\ensuremath{\omega}$ integration very efficiently and accurately.

Fast solution of elliptic partial differential equations using linear combinations of plane waves.

Given an arbitrary elliptic partial differential equation (PDE), a procedure for obtaining its solution is proposed based on the method of Ritz: the solution is written as a linear combination of plane waves and the coefficients are obtained by variational minimization. The PDE to be solved is cast as a system of linear equations Ax=b, where the matrix A is not sparse, which prevents the straightforward application of standard iterative methods in order to solve it. This sparseness problem can be circumvented by means of a recursive bisection approach based on the fast Fourier transform, which makes it possible to implement fast versions of some stationary iterative methods (such as Gauss-Seidel) consuming O(NlogN) memory and executing an iteration in O(Nlog(2)N) time, N being the number of plane waves used. In a similar way, fast versions of Krylov subspace methods and multigrid methods can also be implemented. These procedures are tested on Poisson's equation expressed in adaptive coordinates. It is found that the best results are obtained with the GMRES method using a multigrid preconditioner with Gauss-Seidel relaxation steps.

TOMBO: All-electron mixed-basis approach to condensed matter physics

TOMBO is a computer code for calculating the electronic structure of systems that consist both of core and valence electrons and nuclei, based on density-functional theory. It is based on an all-electron mixed-basis approach, in which the Kohn–Sham (KS) wave function is expressed by a linear combination of plane-waves and atomic-orbitals. This approach can describe both spatially localized and extended orbitals, which enables us to perform all-electron calculations with high accuracy from isolated clusters to periodic crystals. The present paper describes a theory of the all-electron mixed-basis approach, as well as input variables and benchmark tests in TOMBO. The algorithm for accelerating the computational time that is needed to solve the KS equation is also presented.

Plane wave approximation in linear elasticity

We consider the approximation of solutions of the time-harmonic linear elastic wave equation by linear combinations of plane waves. We prove algebraic orders of convergence both with respect to the dimension of the approximating space and to the diameter of the domain. The error is measured in Sobolev norms and the constants in the estimates explicitly depend on the problem wavenumber. The obtained estimates can be used in the h- and p-convergence analysis of wave-based finite element schemes.

Novel basis functions for the partition of unity boundary element method for Helmholtz problems

SUMMARYThe BEM is a popular technique for wave scattering problems given its inherent ability to deal with infinite domains. In the last decade, the partition of unity BEM, in which the approximation space is enriched with a linear combination of plane waves, has been developed; this significantly reduces the number of DOFs required per wavelength. It has been shown that the element ends are more susceptible to errors in the approximation than the mid‐element regions. In this paper, the authors propose that this is due to the use of a collocation approach in combination with a reduced order of continuity in the Lagrangian shape function component of the basis functions. It is demonstrated, using numerical examples, that choosing trigonometric shape functions, rather than classical polynomial shape functions (quadratic in this case), provides accuracy benefits. Collocation schemes are investigated; it is found that the somewhat arbitrary choice of collocating at equally spaced points about the surface of a scatterer is better than schemes based on the roots of polynomials or consideration of the Fock domain. Copyright © 2012 John Wiley & Sons, Ltd.

Plane wave approximation of homogeneous Helmholtz solutions

In this paper, we study the approximation of solutions of the homogeneous Helmholtz equation Δu + ω2u = 0 by linear combinations of plane waves with different directions. We combine approximation estimates for homogeneous Helmholtz solutions by generalized harmonic polynomials, obtained from Vekua’s theory, with estimates for the approximation of generalized harmonic polynomials by plane waves. The latter is the focus of this paper. We establish best approximation error estimates in Sobolev norms, which are explicit in terms of the degree of the generalized polynomial to be approximated, the domain size, and the number of plane waves used in the approximations.

Variational solution of the Schrödinger equation using plane waves in adaptive coordinates: The radial case

A new method for solving the Schrödinger equation is proposed, based on the following details. First, a map u=u(r) from Cartesian coordinates r to a new coordinate system u is chosen. Second, the solution (orbital) psi(r) is written in terms of a function U depending on u so that psi(r)=/J(u)/(-1/2)U(u), where /J(u)/ is the Jacobian determinant of the map. Third, U is expressed as a linear combination of plane waves in the u coordinate, U(u)= sum (k)c(k)e(ik x u). Finally, the coefficients c(k) are variationally optimized to obtain the best energy, using a generalization of an algorithm originally developed for the Coulomb potential [J. M. Perez-Jorda, Phys. Rev. B 58, 1230 (1998)]. The method is tested for the radial Schrödinger equation in the hydrogen atom, resulting in micro-Hartree accuracy or better for the energy of ns and np orbitals (with n up to 5) using expansions of moderate length.

Intrinsic point defects and electron-energetic characteristics of hexagonal monoxide of beryllium

Using the zone method of a linear combination of plane waves for the supercell model, there have been studied energy and electron states of beryllium monoxide BeO with intrinsic point defects: the vacancy in the sub-lattice of beryllium + interstitial atom of Be {VBe + Bei}. The most stable among the considered configurations of defects are the vacancies and interstitial atoms of beryllium Be i o occupying octahedral positions at R(VBe-Be i o ) ∼ 3.15–3.35 A distances from each other. Depending on the configuration of defects, the forbidden gap of beryllium oxide can change by ∼0.4–2.4eV.

Vibration Analysis of Continuous Bodies Using Analytic Solutions Discretized on Boundaries : A Basic Study on Two-Dimensional Problems Described by Helmholtz's Equation

We discuss a new method of analyzing two-dimensional eigenvalue problems described by Helmholtz's equation. A general solution is assumed as a linear combination of plane waves which have the same wave number and travel in different directions. Nodal points are located only on the boundary of a convex domain which has an arbitary shape. A set of equations similar to finite-element equations can be derived from the general solutions. Eigenvalues obtained by this method are very close to the exact values. This method is effective in simplifying of the programming and in shortening computational time.

Vibration Analysis of Continuous Bodies with the Use of Analytic Solutions Discretized on Boundaries. A Basic Study on Two-Dimensional Problems Described by Helmholtz's Equation.

This paper discusses a new method of analysis of two-dimensional eigenvalue problems de-scribed by Helmholtz's equation. A general solution is assumed as a linear combination of plane waves which have the same wave number and travel in different directions. Nodal points are located only on the boundary of a convex domain which has an arbitrary shape. A set of equations similar to finite element equations can be derived from the general solutions. Eigenvalues obtained by this method are very close to the exact values. This method is effective in simplification of programming and in shortening of computational time.

Influence of a Mid-Ocean Ridge on Wind-Driven Barotropic Rossby Waves

Abstract An analytical study investigates the influence of a mid-ocean ridge in a barotropic ocean on the energy radiation by wind-driven, quasi-geostrophic, linear Rossby waves. The ocean basin is bounded on the east and west by meridional boundaries and is separated in its middle by a north-south ridge with zonal exponential depth profiles. The forcing function is oscillatory, either propagative or not, and represented by one or a linear combination of plane waves. The excitation frequencies range from a week to about six months (the chosen spin-down time for Rossby waves). High-frequency forcings (about 10-days period) excite basin-size topographic Rossby-modes propagating westward along the f/h contours. Low-frequency forcings (several-week period) excite Rossby-modes relative to each half basin, but modified by the bottom slope. At very low-frequencies (several-month period), Rossby wave reflection and bottom friction set up a boundary layer at the western boundary of the basin but no such sign signi...

Theory of the spin polarization of field-emitted electrons from nickel

We calculate the spin polarization of field-emitted electrons from nickel in the framework of the Stoner-Wohlfarth-Slater theory of band magnetism. As was first proposed by Hertz and Aoi for the case of tunneling, we find that the $s\ensuremath{-}d$ hybridization plays an important role and we give a more quantitative treatment of this effect. The crystal wave functions are described as linear combinations of plane waves and tight-binding $d$ wave functions. In a first approximation the contribution of the $d$ states to the emitted current is neglected; the problem of the matching of the wave functions at the crystal boundary can then be solved exactly and, using the band parameters of Zornberg, very large positive spin polarizations are obtained. Next, the contribution of the $d$ electrons is included in a semiphenomenological way and reasonable agreement with experiment is obtained without invoking any many-body effects. The dependence of the spin polarization on crystallographic direction is investigated. This makes it necessary to consider the case of high-index directions for which field emission may be dominated by surface scattering. The final results compare well with the preliminary measurements of Campagna et al.

Crystal symmetries of plane-wave-like functions I. The symmorphic space groups

Abstract A set of tables of coefficients for forming linear combinations of plane waves, both without and with spin, symmetrized with respect to all the necessary sub-groups of all 73 symmorphic space groups, is presented here. A symmetry analysis of all the Brillouin zone symmetry points of these 73 space groups is also included and serves as an index to the coefficient tables. A discussion of the method of calculation is included as well as a review of the theory which underlies the desirability of symmetrized functions, and the theory of the projection operator, by which these coefficients were calculated. Section 4, a guide to the use of the tables, may be read independently of the rest.

DBands in the Body-Centered Cubic Lattice

A model of a crystal, consisting of positive point charges neutralized by a uniform distribution of negative charge, is employed to study the form of the $d$ bands in a body-centered cubic lattice as a function of the lattice spacing. The wave functions are expressed as linear combinations of plane waves and the potential treated as a perturbation. It is shown that the perturbation series for the energy is a power series in $\mathrm{Za}$, where $Z$ is the atomic number and $a$ is the lattice parameter. The leading term in the series is of the order ${(\frac{1}{a})}^{2}$, and the coefficients of successive terms in the series decrease rapidly. The first three terms are evaluated for the states of predominantly $d$ symmetry at the center of the Brillouin zone, and the corner $H$.

The use of the repulsive potential in the quantum theory of solids

A new method is proposed for calculating the energy in certain special points of the Brillouin zone. The wave functions of valence and conduction electrons are given in the form of the linear combination of plane waves and the orthogonality condition of these functions to the wave functions of lower states is replaced by the repulsive potential. The practical application of this very simple method is illustrated on the energy spectrum of silicon in the centre of the Brillouin zone. It is proved that the results are comparable with some other methods, e. g. the orthogonalized plane-wave method.