# XC Functionals and How to Tame Them¶

Density functional theory (DFT) is the quantum chemical approximation central to ADF, BAND, and Quantum ESPRESSO. Within DFT there exist many different variants, approximations, and concepts and selecting the most appropriate method for a given problem can be difficult at times. This page aims to provide you with an overview of the functionals available in the Amsterdam Modeling Suite as well as a guide on their respective strengths and weaknesses.

## Introduction¶

In short terms DFT is a reformulation of quantum mechanics. The combination of the Hohenberg-Kohn theorem and the Levy-Lieb principle allows one to express the total energy as a functional of the particle density $$\rho$$ (instead of a many-body wavefunction). The Kohn-Sham formalism then turns DFT into an effective mean-field approach (in terms of a single Slater determinant) and thereby makes this theory applicable to real systems like molecules and materials. The resulting eigenvalue equation for the KS orbitals $$\{\phi_{i}\}$$

$\left(-\frac{1}{2}\nabla^{2} + v_{\mathrm{ext}} + v_{\mathrm{H}} + \frac{\delta E_{\mathrm{XC}}[\rho]}{\delta\rho}\right)\phi_{i} = \varepsilon_{i}\phi_{i}$

can be readily cast into a numerical eigenvalue problem which allows for very efficient implementations in computer programs. In the above expression $$v_{\mathrm{H}}$$ and $$v_{\mathrm{ext}}$$ denote the classical Coulomb term and the (usually electrostatic) effects of the nuclear potential, respectively. The so-called exchange-correlation (XC) functional $$E_{\mathrm{XC}}$$ is introduced by KS as the terms describing all quantum mechanical many-body interactions. The exact XC functional is, however, unknown and its specific form needs to be approximated in practical applications.

A plethora of different XC approximations has been suggested in the DFT community and a large variety of the most successful and popular XC functionals can be used to perform DFT calculations with ADF, BAND, and Quantum ESPRESSO. Their different strengths and weaknesses for the description of different physical properties and classes of quantum chemical systems shall be discussed in the following guide.

## Classes of Functionals¶

For many practical applications $$E_{\mathrm{XC}}$$ is split into terms for exchange $$E_{\mathrm{X}}$$ and correlation interactions $$E_{\mathrm{C}}$$, based on their respective scaling behaviors in terms of the density. Like $$E_{\mathrm{XC}}$$, neither $$E_{\mathrm{X}}$$ nor $$E_{\mathrm{C}}$$ are known exactly. However, many of their respective properties are. The most important property is thereby the behavior of $$E_{\mathrm{X}}[\rho]$$ for the so-called homogeneous electron gas (HEG), a model system simply defined by $$\rho(\mathbf{r}) = const.$$ Applying these HEG results locally leads to the following exchange functional:

$E_{\mathrm{X}}^{LDA}[\rho] = - \frac{3}{4}\left(\frac{3}{\pi}\right)^{1/3}\!\!\int\!\rho(\mathbf{r})^{4/3}\,\mathrm{d}\mathbf{r}$

Together with a corresponding correlation term parameterized on quantum Monte Carlo results, this yields the class of local density approximation (LDA) functionals, which in their general form read as:

$E_{\mathrm{XC}}^{LDA}[\rho] = \int\!\varepsilon_{\mathrm{XC}}^{LDA}(\rho(\mathbf{r}))\,\mathrm{d}\mathbf{r}$

Note, that integrating the rather involved function $$\varepsilon_{\mathrm{C}}^{LDA}(\rho(\mathbf{r}))$$ as well as all subsequently functionals typically require numerical grids. LDA functionals work relatively well given their simplicity and the usage of a local integral kernel like $$\varepsilon_{\mathrm{XC}}^{LDA}$$ also allows for relatively efficient implementations.

The concept of a local integral kernel is retained in the next class functionals, which introduce also a dependency on the magnitude of the local gradient of the electron density. However, attempts to derive a first principles formulation for $$\varepsilon_{\mathrm{X}}(\rho(\mathbf{r}),\,|\nabla\!\rho(\mathbf{r})|)$$ turned out unstable and more empirical forms for such kernel functions were used therefore. These make up the class of generalized gradient approximations (GGA):

$E_{\mathrm{XC}}^{GGA}[\rho] = \int\!\varepsilon_{\mathrm{XC}}^{GGA}(\rho(\mathbf{r}),\,|\nabla\!\rho(\mathbf{r})|)\,\mathrm{d}\mathbf{r}$

Functionals improving over GGAs would naturally involve the second derivatives of $$\rho$$ like the density Laplacian $$\Delta\rho$$. However, accurately representing this quantity on a numerical grid often requires very dense grid points and is thus associated with higher computational costs. A simpler alternative is the usage of the kinetic energy density $$\tau(\mathbf{r})=\sum_{i}|\nabla\phi_{i}(\mathbf{r})|^{2}$$, which results in meta-GGA functionals:

$E_{\mathrm{XC}}^{mGGA}[\rho] = \int\!\varepsilon_{\mathrm{XC}}^{mGGA}(\rho(\mathbf{r}),\,|\nabla\!\rho(\mathbf{r})|,\,\tau(\mathbf{r}))\,\mathrm{d}\mathbf{r}$

Meta-GGA functionals introduce an explicit dependency on the KS orbitals. While this puts most of their practical implementations formally outside of the KS formalism, this is of little consequence in practical applications.

The forth class of DFT functionals push the usage of KS orbitals further. Hybrid XC functionals come with an additional scaled exchange term originating from Hartree-Fock theory:

$\begin{split}E_{\mathrm{XC}}^{hybrid}[\rho] &= E_{\mathrm{XC}}^{GGA/mGGA}[\rho] + E_{\mathrm{EXX}}[\{\phi_{i}\}]\\ &= \int\!\varepsilon_{\mathrm{XC}}^{GGA/mGGA}(\rho(\mathbf{r}),\,|\nabla\!\rho(\mathbf{r})|,\,\tau(\mathbf{r}))\,\mathrm{d}\mathbf{r} +\sum\limits_{ab}\!\iint\frac{\phi_{a}(\mathbf{r})\phi_{b}(\mathbf{r})\phi_{b}(\mathbf{r}^{\prime})\phi_{a}(\mathbf{r}^{\prime})}{|\mathbf{r}-\mathbf{r}^{\prime}|}\,\mathrm{d}\mathbf{r}\mathrm{d}\mathbf{r}^{\prime}\end{split}$

Adding $$E_{\mathrm{EXX}}$$ has similar implications as $$\tau$$, however, due to their nonlocal form hybrid DFT methods are associated with much higher computational costs. Nonetheless, the nonlocal exchange term often provides more accurate results or is even essential for achieving qualitatively correct results in some cases.

The last and even more demanding group of XC functionals include also dependencies on unoccupied (virtual) KS orbitals and usually involve a nonlocal correlation term. Examples of such approximations are random phase approximations (RPA) typically used in solid states calculations or double hybrid functionals which include an additional correlation term from many-body perturbation theory.

$E_{\mathrm{XC}}^{doubleHybrid}[\rho] = E_{\mathrm{XC}}^{GGA/mGGA}[\rho] + E_{\mathrm{EXX}}[\{\phi_{i}\}] + E_{\mathrm{MP2}}[\{\phi_{i}\}]$

## Overview of Available Functionals¶

Type Term Availability Citation Remarks
VWN LDA X & C ADF
PW92 LDA X & C ADF

## Properties – Accuracy of Different Types of Functionals¶

### Structures¶

LDA
Due to an often fortituous error cancellation, LDA methods perform surprisingly well for geometries. The accuracy for bond lengths can even exceed that of Hartree-Fock or MP2 calculations, especially in the case of systems involving transition metal atoms. While accurate, LDA methods are known to underestimate bond lengths slightly.
GGA
GGA methods typically use $$\varepsilon_{\mathrm{X}}^{GGA} = \varepsilon_{\mathrm{X}}^{LDA}(\rho(\mathbf{r}))\cdot f_{\mathrm{X}}^{GGA}(\rho(\mathbf{r}),\,|\nabla\!\rho(\mathbf{r})|)$$ as kernel for their exchange part. With $$f_{\mathrm{X}}^{GGA}\ge 1$$, the exchange energy becomes more negative (more stabilizing) for corrugated electron densities. As a result, structures optimized with a GGA method tend to exhibit somewhat elongated bond lengths and are less accurate than corresponding LDA results.
meta-GGA
Because of their more flexible functional form, meta-GGA methods usually improve over GGA functionals and can in many cases also expected to be at least on a par with LDA results.
Hybrid DFT
The accuracy of hybrid DFT approximations for structural parameters is quite system-dependent. While hybrid DFT methods usually provide an accurate description of compounds consisting of main-group elements, they become unreliable for systems including transition metals, with near-degenerate ground states or featuring vanishing band gaps.