A MIA enhanced linear scaling approach to the computation of the exchange-correlation terms in DFT/LDA

doi:10.1016/j.theochem.2003.08.011

Journal of Molecular Structure: THEOCHEM

Volumes 666–667, December 2003, Pages 41-50

https://doi.org/10.1016/j.theochem.2003.08.011 Get rights and content

Abstract

The Multiplicative Integral Approximation is applied in the linear scaling local density approximation density functional theory. Our method is a modified version of the algorithm of Stratmann et al. [Chem. Phys. Lett. 257 (1996) 213]. We suggest an alternative shell pair based selection scheme for the identification of non-negligible terms in the expression of charge density and exchange-correlation contribution of Kohn–Sham matrix together with an iterative update procedure for these quantities. These modifications enable us to implement the Multiplicative Integral Approach, which further reduces the computational cost. The linear scaling behaviour as well as the CPU time reduction is demonstrated on a test system containing up to 350 water molecules.

Introduction

Although the Kohn–Sham formulation of density functional theory (DFT) opened new perspectives in the theoretical treatment of spatially extended systems, the computations for large clusters and biological macromolecules remained unfeasible. This problem induced an intensive development of computer algorithms and resulted in several methods that require a computational effort proportional to the system size (linear scaling methods). Instead of summarizing them, we refer here only to two recent review articles written by Goedecker [2] and Wu and Jayanthi [3].

Besides the treatment of Coulomb interactions and diagonalization of the Kohn–Sham matrix, the numerical integration needed for the calculation of the exchange part of the Kohn–Sham matrix is the most time-consuming part of a DFT calculation. Stratmann et al. [1] suggested in 1996 a grid point driven linear scaling method which selects the basis functions (significant basis functions) giving non-negligible contribution to the numerical integration during the exchange build-up procedure. Most recently, Challacombe [4] suggested a new hierarchical cubature for the numerical integration of the exchange-correlation matrix. He uses an entirely Cartesian grid and a k-dimensional binary search tree data structure that fits well to the large variability of electron density both in range and in magnitude.

The multiplicative integral approach (MIA) was originally developed for the linear scaling Hartree–Fock treatment of large molecules [7] and was successfully applied in many cases [8]. In this paper, we present an application of the MIA in the linear scaling DFT. Our method is based on the algorithm of Stratmann et al. [1] and suggests an alternative shell pair based selection scheme for the identification of non-negligible terms in the expression of charge density and exchange-correlation contribution of Kohn–Sham matrix. This modification enable us to implement the MIA, which further reduces the computational cost. The linear scaling behaviour as well as the CPU time reduction is demonstrated on a test system containing up to 350 water molecules.

Section snippets

Exchange-correlation in the direct Kohn–Sham scheme

In the Kohn–Sham theory [9], based on the Hohenberg–Kohn theorems [10], the electron density is determined by the one-electron equation $F ̂ ψ_{k} =ε_{k} ψ_{k}$ in the form of $ρ(r)= ∑ k ψ_{k}^{∗} (r)ψ_{k} (r).$ The Kohn–Sham operator F̂ is the following $F ̂ = t ̂ +v(r)+ J ̂ + F ̂^{xc},$ where t̂ is the kinetic energy, $v(r)$ is the external potential $J ̂ =∫ ρ(r ′) | r − r ′| d r ′$ is the Coulomb part of the electron–electron interaction and $F ̂^{xc}$ is the exchange-correlation potential. Expanding the one-electron functions on a M dimensional basis set {χ_μ}_μ=1^M $ψ_{k} ($

A dominantly linear scaling method for the computation of F^xc matrix

The numerical integration makes the build up of the exchange part of the Kohn–Sham matrix in DFT LDA very time consuming. There are two problematic steps in this procedure. The first being the calculation of the electron density at every grid point and the second being the numerical integration in the expression of the exchange-correlation contribution to the Kohn–Sham matrix elements. Stratmann et al. [1] suggested a modification of the Becke [11] weighting scheme that reduces the

The multiplicative integral approximation in DFT

In MIA [7], the product of two basis functions $χ_{μ_{I}} (r − R_{I})χ_{ν_{J}} (r − R_{J})$ centered, respectively, at $R_{I}$ and $R_{J}$ is approximated by $χ_{μ_{I}} (r − R_{I})χ_{ν_{J}} (r − R_{J})≈ ∑ α=1 N_{aux} C_{α}^{μ_{I}ν_{J}} χ_{μ_{I},α} (r − R_{μ})= ∑ α=1 N_{aux} C_{α}^{μ_{I}ν_{J}} χ_{μ_{I}} (r − R_{I})σ_{α_{I}} (r − R_{I}),$ where {σ_{α_I}}_{α_I=1,N_aux} is an auxiliary basis set containing N_aux basis functions, centered on the same atom as χ_{μ_I}. The exponents used in this basis set can be chosen to be the same for every shells of a molecule. In practice, an uncontracted P shell with exponent 1.0 and a M shell (S, P and D shells

Implementation of DFT/MIA

The DFT/MIA method was implemented in program package BRABO [12]. In this section, we describe the algorithm of our method. The accuracy of error estimation in MIA (Eq. (8)) is also discussed. Although the method described above can be applied for any exchange-correlation functional, we have chosen the Slater exchange [5] and VWN correlation [6] for testing purposes. Water clusters [13] were chosen as a test system to demonstrate the scaling properties of the method as well as the CPU time

Acknowledgements

This work was supported by the Flemish-Hungarian Scientific and Technological Joint Fund (TeT B-2/01 and BIL01/72). One of us (F.B.) is indebted to the National Scientific Research Fund Hungary (OTKA M36803) and the Hungarian Ministry of Education (MU-00094/2002). B.R. acknowledges the Flemish governmental institution IWT for a predoctoral grant. This research was supported by the University of Antwerp under Grant GOA-BOF-UA No. 23.

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A MIA enhanced linear scaling approach to the computation of the exchange-correlation terms in DFT/LDA

Abstract

Introduction

Section snippets

Exchange-correlation in the direct Kohn–Sham scheme

A dominantly linear scaling method for the computation of Fxc matrix

The multiplicative integral approximation in DFT

Implementation of DFT/MIA

Acknowledgements

Chem. Phys. Lett.

Phys. Rep.

Phys. Rev. A

J. Am. Chem. Soc.

Rev. Mod. Phys.

J. Chem. Phys.

Phys. Rev.

A dominantly linear scaling method for the computation of F^xc matrix