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EXCHANGE CORRELATION FUNCTIONAL IN DFT

16-09-2021 

 1. Comparison of exchange-correlation functionals: from LDA to GGA and beyond
    Martin Fuchs Fritz-Haber-Institut der MPG, Berlin, Germany
    Density-Functional Theory Calculations for Modeling Materials and
    Bio-Molecular Properties and Functions - A Hands-On.
 2. Therefore, if only one exchange-correlation functional should be chosen, 6
    functionals such as SCAN+rVV10, DFT-D3, LDA, SCAN, PBEsol, and DFT-D3(BJ)
    should be chosen. Cases of incorrect calculation of the ground structures.

 * What Is Exchange Correlation Functional
 * Exchange Correlation Functional In Dft

Next:Bloch's Theorem

Bloch's Theorem Up: The Many Body Problem Previous: Density Functional Theory
Contents The Exchange-Correlation Term. The Kohn-Sham equations in 2.30 are thus
far exact: no approximations have yet been made; we have simply mapped the fully
interacting system onto an auxiliary non-interacting system that yields the same
groundstate density.

Up:The Many Body Problem Previous:


WHAT IS EXCHANGE CORRELATION FUNCTIONAL

Density Functional TheoryContents


The Kohn-Sham equations in 2.30 are thus far exact: no approximations have yet
been made; we have simply mapped the fully interacting system onto an auxiliary
non-interacting system that yields the same groundstate density.

As mentioned earlier, the Kohn-Sham kinetic energy is not the true kinetic
energy; we may use this to define formally the exchange-correlation energy as




where and are the exact kinetic and electron-electron interaction energies
respectively. Physically, this term can be interpreted as containing the
contributions of detailed correlation and exchange to the system energy. The
definition above is such that it ensures that the Kohn-Sham formulation is
exact. However, the actual form of is not known; thus we must introduce
approximate functionals based upon the electron density to describe this term.
There are two common approximations (in various forms) in use: the local density
approximation (LDA) [47], and the generalised gradient approximation (GGA) [48].
The simplest approximation is the LDA: this assumes that the
exchange-correlation energy at a point is simply equal to the
exchange-correlation energy of a uniform electron gas that has the same density
at the point . Thus we can write




(2.33)


so that the exchange-correlation potential may be written





with




(2.35)


where in the last equation the assumption is that the exchange-correlation
energy is purely local. The most common parametrisation in use for is that of
Perdew and Zunger [49], which is based upon the quantum Monte Carlo calculations
of Ceperley and Alder [50] on homogeneous electron gases at various densities;
the parametrisations provide interpolation formulae linking these results.

The LDA ignores corrections to the exchange-correlation energy due to
inhomogeneities in the electron density about . It may seem surprising that this
is as successful as it is given the severe nature of the approximation in use;
to large extent, it appears [40] that this is due to the fact that the LDA
respects the sum rule, that is, that exactly one electron is excluded from the
immediate vicinity of a given electron at point . The LDA is known to overbind,
particularly in molecules. It is for this reason that in this study we have
neglected it in favour of the GGA.

The GGA attempts to incorporate the effects of inhomogeneities by including the
gradient of the electron density; as such it is a semi-local method. The GGA XC
functional can be written as




where is known as the enhancement factor. Unlike the LDA, there is no unique
form for the GGA, and indeed many possible variations are possible
[48,52,53,54], each corresponding to a different enhancement factor. The GGA
succeeds in reducing the effects of LDA overbinding [51], and is significantly
more successful when applied to molecules. In this work, the PW91 GGA due to
Perdew and Wang is used [48].
Next:Bloch's Theorem Up:The Many Body Problem Previous:Density Functional
TheoryContentsWeb Page Administrator2004-12-16

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:$$E^mathrm{GH}_mathrm{xc} = c_x E^mathrm{EXX} +
E^mathrm{DFT}_mathrm{xc}[n(vec{r}), dots],$$


EXCHANGE CORRELATION FUNCTIONAL IN DFT

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

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.

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



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