Exchange Correlation Functional In Dft

15a In this initial publication an exchange as well as a correlation functional was proposed, both pure density functionals, often used in conjunction. 15b I don't know much about it's usefulness, since I prefer another quite robust variation of it: PBE0, which is a hybrid functional. Into the XC functional.! Exchange in DFT! Approximation to the exchange-correlation interaction is The n contains the charge density.

ADS! Download JAMB CBT Software Now for FREE!

Towards A New Exchange-Correlation Density Functional for More Accurate Band Gap Predictions.

Abstract

Density-Functional Theory (DFT) offers a simplification to electronic structure prob- lems by using the electron density instead of the wave-function.

Unlike the wave- function which is a function of 3N variables (excluding spin) for an N -electron system, the density depends only on three variables, irrespective of the number of electrons in the system.

While DFT, in principle, gives an accurate description of ground-state properties, practical applications of DFT are based on approximations to the so-called exchange-correlation (xc) potential.

The exchange-correlation po- tential describes the effects of the Pauli exclusion principle and the electron-electron Coulomb repulsion beyond a purely electrostatic interaction of the electrons.

A common description of exchange-correlation functional is the so-called local den- sity approximation (LDA) which locally substitutes the exchange-correlation energy density of an inhomogeneous system by that of an electron gas evaluated at that local density.

While many ground state properties (such as lattice constants and bulk moduli) are well described in the LDA, the band gap is underestimated by as much as 50% in LDA compared to experiments.

In this thesis, we focus on the development of an exchange-correlation functional with adjustable parameters which can give more accurate band gap energies.

This functional is based on the xc potential derived in 1988 from a tight-binding approx- imation by Hanke and Sham (HS).

Our contribution consists in expressing the HS potential in terms of the electron density and its gradient.

Table Of Contents

Abstract i

Acknowledgments ii

Dedication iv

Contents vi

  • Introduction 1
    • Successes and Failures of DFT….2
    • The Band Gap Problem…………..3
    • Atomic Units……………….. 4
  • Theoretical Background 6
    • Electronic Structure Problem………… 6
    • Wave-Function Based Methods………………. 9
      • The Hartree-Fock Formalism…………… 9
      • Correlated Methods Beyond Hartree-Fock………… 13
    • Density-Functional Thoery…………. 13
      • The Thomas-Fermi (TF) Model…………. 14
      • The Hohenberg-Kohn (HK) Theorem…..15
    • The Kohn-Sham (KS) Scheme……. 17
    • Interpretation of Kohn-Sham energies………20
    • Failure of DFT for Band Gap Energies……..20
  • Past Work on Correcting DFT Band Gap Energies….22
  • Functional Development 25
    • Exchange-Correlation Functionals………………. 25
      • The Local-Density Approximation (LDA)……….. 26
      • The Generalized-Gradient Approximation (GGA)…. 27
      • Meta-GGA (mGGA)……………… 28
      • Hybrid Schemes………………….. 29
    • A New Density Functional…………. 30
      • Xα-Method……………….30
      • The Corresponding xc Energy Exc……… 32
    • Methods 35
      • Basis Sets……………………………………. 35
        • Plane Waves…………. 36
        • Pseudo-potentials……………….. 37
      • Methodology………………………… 40
    • Results, Discussion and Conclusion 42
      • Results…………… 42
      • Discussion…………………… 48
      • Conclusion and Perspective……………… 48
    • Hanke-Sham (HS) xc Potential 49
      • Green’s Function and Self-Energy Operator…………. 49
      • A Tight-Binding (TB) Model for the Self-Energy (Σxc)……….. 51
      • A vxc for Insulators and Semiconductors………….. 54

Bibliography 57

Introduction

Background Of Study

Density-Functional Theory (DFT) is one of the most popular and successful Quan- tum Mechanics (QM) approach for large systems.

It is a widely used methods for “ab initio” calculations of the structure of atoms, molecules, crystals, surfaces and their interactions.

It is nowadays routinely applied for calculating e.g., the binding energy of molecules in chemistry and the band structure of solids in physics.

First application relevant for fields traditionally considered more distant from quantum mechanics, such as biology and mineralogy are beginning to appear.

Superconduc- tivity, atoms in the focus of strong laser pulses, relativistic effects in heavy elements and in atomic nuclei, classical liquids, and magnetic properties of alloys have been studied with DFT.

Bibliography

Hanke, L.J. Sham Phys. Rev. B 38, 13361 (1988)

Capelle, A bird’s-eye view of density-functional theory, http://arxiv.org/abs/cond-mat/0211443.

Fermi E (1926) Z Phys 36:902.

Dirac PAM (1930) Proc Cam Phil Soc 26:376.

Baroni and R. Resta, Phys. Rev B 33, 7017 (1986).

Hybertson and S. Louie, Phys. Rev. B 34, 5390 (1986).

Join Over 3,500 000+ Readers Online Now!

=>FOLLOW US ON INSTAGRAM | FACEBOOK & TWITTER FOR LATEST UPDATES

ADS: KNOCK-OFF DIABETES IN JUST 60 DAYS! - ORDER YOURS HERE

COPYRIGHT WARNING! Contents on this website may not be republished, reproduced, redistributed either in whole or in part without due permission or acknowledgement. All contents are protected by DMCA.The content on this site is posted with good intentions. If you own this content & believe your copyright was violated or infringed, make sure you contact us at [[email protected]] to file a complaint and actions will be taken immediately.

RECOMMENDED!

Libxc is a library of exchange-correlation and kinetic energy functionals for density-functional theory. The original aim was to provide a portable, well tested and reliable set of these functionals to be used by all the codes of the European Theoretical Spectroscopy Facility (ETSF), but the library has since grown to be used in several other types of codes as well; see below for a partial list.

Libxc is written in C, but it also comes with Fortran and Python bindings. It is released under the MPL license (v. 2.0). Contributions are welcome. Bug reports and patches should be submitted over gitlab.

To cite Libxc, the current reference is

The previous reference to the library was

In Libxc you can find various types of functionals: LDA, GGA, and meta-GGA (mGGA) functionals. LDAs, GGAs, and meta-GGAs depend on local information, in the sense that the value of the density functional part of the energy density at a given point depends only on the values of the density, the gradient of the density, and the kinetic energy density and/or the density laplacian, respectively, at the given point:

$$E^mathrm{LDA}_mathrm{xc} = E^mathrm{LDA}_mathrm{xc}[n(vec{r})],$$

$$E^mathrm{GGA}_mathrm{xc} = E^mathrm{GGA}_{xc}[n(vec{r}), vec{nabla}n(vec{r})],$$

$$E^mathrm{mGGA}_mathrm{xc} = E^mathrm{mGGA}_mathrm{xc}[n(vec{r}), vec{nabla}n(vec{r}), nabla^2 n(vec{r}), tau(vec{r})].$$

Libxc is designed to evaluate this energy density and its derivatives in a correct fashion. Because several functionals are complicated in form, Libxc is based on the use of computer algebra and automatic code generation to enable the generation of bug-free code. Libxc can calculate both the functional itself, as well as its first through fourth derivatives, satisfying even the stringest requirements for applications.

Global hybrid (GH) and range-separated hybrid (RSH) functionals are also supported by Libxc:$$E^mathrm{GH}_mathrm{xc} = c_x E^mathrm{EXX} + E^mathrm{DFT}_mathrm{xc}[n(vec{r}), dots],$$

What Is Exchange Correlation Functional

$$E^mathrm{RSH}_mathrm{xc} = c_mathrm{sr} E^mathrm{EXX}_mathrm{sr} + c_mathrm{lr} E^mathrm{EXX}_mathrm{lr} + E^mathrm{DFT}_mathrm{xc}[n(vec{r}), dots].$$

Exchange

For these functionals, Libxc only handles the local part (as above); the evaluation of the exact exchange components must be done in the calling program. Libxc, however, does contain all the information necessary to perform the calculations (fraction of exact exchange, range separation parameter(s)).

The same can be said about dispersion corrections: several functionals are available in Libxc that were parametrized with either semiclassical dispersion corrections à la Grimme, or various van der Waals functionals; neither of these can be evaluated with the local density information provided to Libxc, and must be handled by the calling program. The necessary parameters for VV10-type correlation kernels are, however, provided by Libxc as part of the functional definition.

Exchange Correlation Functional In Dft

At the moment, we are aware of Libxc being used in the following codes (in alphabetical order):

  • Abinit - a software suite to calculate the optical, mechanical, vibrational, and other observable properties of materials
  • ACE-Molecule - a quantum chemistry package based on a real-space numerical grid
  • ADF - a density functional theory program for molecules and condensed matter
  • APE - a computer package designed to generate and test norm-conserving pseudopotentials within density functional theory
  • AtomPAW - a program for generating projector augmented wave functions
  • BAGEL - a parallel electronic-structure program
  • BigDFT - a fast, precise, and flexible density functional theory code for ab-initio atomistic simulation
  • CP2K - a program to perform atomistic and molecular simulations of solid state, liquid, molecular, and biological systems
  • DFT-FE - a massively parallel real-space code for first principles based materials modelling using Kohn-Sham density functional theory
  • DP - a linear response time-dependent density functional theory code with a plane wave basis set
  • Chronus Quantum - a computational chemistry software package focused on explicitly time-dependent and post-SCF methods
  • Elk - an all-electron full-potential linearised augmented-plane wave code
  • entos - a software package for Gaussian-basis ab initio molecular dynamics calculations on molecular and condensed-phase chemical reactions and other processes
  • ERKALE - a DFT/HF molecular electronic structure code based on Gaussian-type orbitals
  • exciting - a full-potential all-electron density-functional-theory package implementing the families of linearized augmented planewave methods
  • FHI-AIMS - an efficient, accurate, all-electron, full-potential electronic structure code package for computational molecular and materials science
  • GAMESS (US) - a general ab initio quantum chemistry package
  • GPAW - a density-functional theory Python code based on the projector-augmented wave method
  • HelFEM - Finite element methods for electronic structure calculations on small systems
  • Horton - Python development platform for electronic structure methods
  • JDFTx - plane-wave code designed for joint density functional theory
  • MADNESS - a multiwave adaptive numerical grid program for electroni
  • MOLGW - many-body perturbation theory for atoms, molecules, and clusters
  • Molpro - a comprehensive system of ab initio programs for advanced molecular electronic structure calculations
  • MRCC - a suite of ab initio and density functional quantum chemistry programs for high-accuracy electronic structure calculations
  • Octopus - a program aimed at the ab initio virtual experimentation on a hopefully ever-increasing range of system types
  • ORCA - ab initio quantum chemistry program that contains modern electronic structure methods
  • PROFESS - orbital-free density functional theory implementation to simulate condensed matter and molecules
  • Psi4 - an open-source suite of ab initio quantum chemistry programs designed for efficient, high-accuracy simulations of molecular properties
  • PySCF - Python-based Simulations of Chemistry Framework
  • QuantumATK - code including pseudopotential-based density functional theory methods with LCAO and plane-wave basis sets in one framework
  • Quantum Espresso - an integrated suite of open source computer codes for electronic-structure calculations and materials modeling at the nanoscale
  • Turbomole - a program package for electronic structure calculations
  • WIEN2k - program for electronic structure calculations of solids using density functional theory based on the full-potential (linearized) augmented plane-wave + local orbitals method
  • Yambo - a program that implements many-body perturbation theory methods such as GW and BSE and time-dependent density functional theory