Coarse-grained density functional theories for metallic alloys: Generalized coherent-potential approximations and charge-excess functional theory (original) (raw)

N-scaling algorithm for density-functional calculations of metals and insulators

Physical review, 1994

An algorithm for minimization of the density-functional energy is described that replaces the diagonalization of the Kohn-Sham Hamiltonian with block diagonalization into explicit occupied and partially occupied (in metals) subspaces and an implicit unoccupied subspace. The progress reported here represents an important step toward the simultaneous goals of linear scaling, controlled accuracy, efficiency, and transferability. The method is specifically designed to deal with localized, nonorthogonal basis sets to maximize transferability and state-by-state iteration to minimize any charge-sloshing instabilities. It allows the treatment of metals, which is important in itself, and also because the dynamics of "semiconducting" systems can result in metallic phases. The computational demands of the algorithm scale as the particle number, permitting applications to problems involving many inequivalcnt atoms. I. INTRODUCTION Meaningful simulation of the microscopic behavior of condensed-matter systems requires a reliable description of interatomic forces. More than 20 years of experience have shown that the local-density-functional approximation (LDA) can accurately predict structure and properties of many classes of materials, including crystals, interatomic compounds, surfaces, and small molecules. Consequently, LDA represents a general framework for the study of important material-science issues, e. g., crystal growth, defect properties, catalytic chemistry, and materials degradation and failure. Since the seminal paper of Car and Parrinello, ' a great deal of effort has been devoted to making LDA calculations more efficient. However, until very recently, all LDA algorithms scaled with N, where N is the number of inequivalent atoms in the system. The outlook for treating large systems changed with the publications of Yang and Baroni and Giannozzi, which showed that linear scaling with N is in principle attainable in LDA. Although there are more rigorous, and, in general, more accurate, methods than LDA for the solution of electronic structure problems, none clearly shows the promise of applicability to even modestly complex systems, since they all scale rapidly with system size.

Mathematical Analysis and Numerical Approximations of Density Functional Theory Models for Metallic Systems

2022

In this paper, we investigate the energy minimization model of the ensemble Kohn-Sham density functional theory for metallic systems, in which a pseudo-eigenvalue matrix and a general smearing approach are involved. We study the invariance and the existence of the minimizer of the energy functional. We propose an adaptive double step size strategy and the corresponding preconditioned conjugate gradient methods for solving the energy minimization model. Under some mild but reasonable assumptions, we prove the global convergence of our algorithms. Numerical experiments show that our algorithms are efficient, especially for large scale metallic systems. In particular, our algorithms produce convergent numerical approximations for some metallic systems, for which the traditional self-consistent field iterations fail to converge.

Mathematical foundations of the density functional theory dft. An efficient method for theoretical calculations in materials science

2013

Se presentan los fundamentos matematicos de la teoria funcional de la densidad DFT en este trabajo. Empezamos con el inicio de la mecanica cuantica de esta teoria, es decir, el modelo de Thomas-Fermi (TF), que utiliza la densidad de electrones n (R), una funcion de solo 3 coordenadas, como la unica variable fisica. A continuacion mostramos la fundacion formal de DFT, los teoremas de Hohenberg y Kohn, expresados en una teoria bien establecida representada por pruebas twoexcited. Le mostramos al final del articulo como Kohn y Sham (KS) idearon una aplicacion practica y trajeron DFT en los calculos de la corriente principal de la estructura electronica.

A new density functional method for electronic structure calculation of atoms and molecules

arXiv: Chemical Physics, 2019

This chapter concerns with the recent development of a new DFT methodology for accurate, reliable prediction of many-electron systems. Background, need for such a scheme, major difficulties encountered, as well as their potential remedies are discussed at some length. Within the realm of non relativistic Hohenberg-Kohn-Sham (HKS) DFT and making use of the familiar LCAO-MO principle, relevant KS eigenvalue problem is solved numerically. Unlike the commonly used atom-centered grid (ACG), here we employ a 3D cartesian coordinate grid (CCG) to build atom-centered localized basis set, electron density, as well as all the two-body potentials directly on grid. The Hartree potential is computed through a Fourier convolution technique via a decomposition in terms of short- and long-range interactions. Feasibility and viability of our proposed scheme is demonstrated for a series of chemical systems; first with homogeneous, local-density-approximated XC functionals followed by non-local, gradi...

Shell-Correction and Orbital-Free Density-Functional Methods for Finite Systems

Recent Advances in Computational Chemistry, 2013

Orbital-free (OF) methods promise significant speed-up of computations based on density functional theory (DFT). In this field, the development of accurate kinetic-energy density functionals remains an open question. In this chapter we review the shell-correction method (SCM, commonly known as Strutinsky's averaging method) applied originally in nuclear physics and its more recent formulation in the context of DFT [Yannouleas and Landman, Phys. Rev. B 48, 8376 (1993)]. We demonstrate the DFT-SCM method through its earlier applications to condensed-matter finite systems, including metal clusters, fullerenes, and metal nanowires. The DFT-SCM incorporates quantum mechanical interference effects and thus offers an improvement compared to the use of Thomas-Fermi-type kinetic energy density functionals in OF-DFT.

Density Functional Theory Methods for Computing and Predicting Mechanical Properties

Over the past few decades, tremendous progress has been made in the development of computational methods for predicting the properties of materials. At the heart of this progress is density functional theory (DFT) [13, 17, 31, 39, 65], one of the most powerful and efficient computational modeling techniques for predicting electronic properties in chemistry, physics, and material science. Prior to the introduction of DFT in the 1960s [31, 39] the only obvious method for obtaining the electronic energies of materials required a direct solution of the many-body Schrödinger equation [62]. While the Schrödinger equation provides a rigorous path for predicting the electronic properties of any material system, analytical solutions for realistic systems having more than one interacting electron are out of reach. Moreover, since the Schrödinger equation is inherently a many-body formalism (3N spatial coordinates for N strongly interacting electrons), numerically accurate solutions of multi-electron systems are also impractical. Instead of the full 3N-dimensional Schrödinger equation, DFT recasts the electronic problem into a simpler yet mathematically equivalent 3-dimensional theory of non-interacting electrons (cf. Fig. 4.1). The exact form of this electron density, .D n.r//, hinges on the mathematical form of the exchange-correlation functional, E xc OEn.r/, which is crucial for providing accurate and efficient solutions to the many-body Schrödinger equation. Unfortunately, the exact form of the exchange-correlation functional is currently unknown, and all modern DFT functionals invoke various degrees of approximation.

Massively parallel density functional calculations for thousands of atoms: KKRnano

Physical Review B, 2012

Applications of existing precise electronic-structure methods based on density functional theory are typically limited to the treatment of about 1000 inequivalent atoms, which leaves unresolved many open questions in material science, e.g., on complex defects, interfaces, dislocations, and nanostructures. KKRnano is a new massively parallel linear scaling all-electron density functional algorithm in the framework of the Korringa-Kohn-Rostoker (KKR) Green's-function method. We conceptualized, developed, and optimized KKRnano for large-scale applications of many thousands of atoms without compromising on the precision of a full-potential all-electron method, i.e., it is a method without any shape approximation of the charge density or potential. A key element of the new method is the iterative solution of the sparse linear Dyson equation, which we parallelized atom by atom, across energy points in the complex plane and for each spin degree of freedom using the message passing interface standard, followed by a lower-level OpenMP parallelization. This hybrid four-level parallelization allows for an efficient use of up to 100 000 processors on the latest generation of supercomputers. The iterative solution of the Dyson equation is significantly accelerated, employing preconditioning techniques making use of coarse-graining principles expressed in a block-circulant preconditioner. In this paper, we will describe the important elements of this new algorithm, focusing on the parallelization and preconditioning and showing scaling results for NiPd alloys up to 8192 atoms and 65 536 processors. At the end, we present an order-N algorithm for large-scale simulations of metallic systems, making use of the nearsighted principle of the KKR Green's-function approach by introducing a truncation of the electron scattering to a local cluster of atoms, the size of which is determined by the requested accuracy. By exploiting this algorithm, we show linear scaling calculations of more than 16 000 NiPd atoms.

Density-density functionals and effective potentials in many-body electronic structure calculations

2008

We demonstrate the existence of different density-density functionals designed to retain selected properties of the many-body ground state in a non-interacting solution starting from the standard density functional theory ground state. We focus on diffusion quantum Monte Carlo applications that require trial wave functions with optimal Fermion nodes. The theory is extensible and can be used to understand current practices in several electronic structure methods within a generalized density functional framework. The theory justifies and stimulates the search of optimal empirical density functionals and effective potentials for accurate calculations of the properties of real materials, but also cautions on the limits of their applicability. The concepts are tested and validated with a near-analytic model.