Exchange-correlation

Exchange-correlation functionals

From the Kohn-Sham theory, one can predict ground state energy and ground state electron density of a many electron system in principle exactly by solving self-consistent one electron equations. It is much faster to solve a one electron Schrodinger equation than many electron Schrodinger equation. The real many body methods are potentially more accurate than density functional approximations, but computationally more expensive, especially for a large system. In practice, the exchange correlation functional must be approximated.

Tips

LDA

In the local density approximation (LDA), the localized density can be treated as a uniform electron gas, or equivalently that the density is a slowly varying function. For a spin-unpolarized, a local-density approximation for the exchange-correlation energy is written as:

$$E_{xc}^{LDA}[n] = \int f_{xc} (n) d\bold{r} = \int n(\bold{r}) \epsilon_{xc}^{uniform} (n(\bold{r})) d\bold{r}$$

where $n(\bold{r})$ is the electron density and $\epsilon_{xc}^{uniform} (n(\bold{r}))$ is the exchange-correlation energy per particle a homogeneous electron gas of charge density $n(\bold{r})$ ($\epsilon$: energy density). - LDA gives good geometries, but it overbinds molecules.

Perdew's lecture about LDA in 2016

It is very important to understand the exact theory which is the key to develop the approximations. These approximations are the engines that make DFT work in practice. LDA is the first level approximation of exchange-correlation functional which originates from the foundation paper (1965) from Kohn and Sham who contributed to the exact theory of DFT. LDA is also the foundation of other two levels approximations (GGA and meta-GGA). We don't know the analytical form of $E_{xc}[n]$, but we know some constrains that the exact $E_{xc}[n]$ have. These constrains give the direction of how to approximate $E_{xc}[n]$. There is also an empirical approach to construct $E_{xc}[n]$ by fitting properties (like atomization energy) of big molecular databases.

Perdew prefers building $E_{xc}[n]$ from exact constrains and appropriate norms (system we know the energy exactly). We can construct our functional by making exact or near-exact energy for that system. The uniform electron gas is an example of that the appropriate norm and used for constructing LDA.

$E_{xc}[n]$ is relatively a small part of the total energy of typical atoms or molecules, but also an important part for the following reasons:

$$\underbrace{E_{xc}^{LDA}[n]} _ {\text{functional of } n(\bold{r})} = \int d\bold{r} n(\bold{r}) \underbrace{\epsilon_{xc}^{unif}(n(\bold{r}))}_{\text{function of n at }\bold{r}}$$

$$\epsilon_{xc}^{unif}(n(\bold{r})) = \text{xc energy per particle of an electron gas of uniform density n}$$

This is probably the simplest non-zero functional of density one can make. It is motivated by the Thomas-Fermi approximation which was a similar approximation for non-interacting kinetic energy in 1920s.

Perdew didn't know the work of Kohn-Sham 1965 until early 1970s when people realized that LDA was a very good for solid state physics because LDA gave realistic properties: - lattice constant for solids - phonon frequencies - surface energies for metals.

Why is LDA good? - Langreth and Perdew 1975 - Gunnarsson and Lundqvist 1976: more readable and easier to understand - LDA not only matches the uniform electron gas, but also satisfies "hidden exact constrains" valid any physical system $v(\bold{r})$.

One of the exact constrain for $E_{xc}[n]$ is that

$$ \begin{aligned} E_{xc}[n] &= \frac{1}{2} \int d \bold{r} n(\bold{r}) \int d n(\bold{r}^\prime) \dfrac{n_{xc}(\bold{r}, \bold{r}^\prime)}{\vert \bold{r}^\prime - \bold{r} \vert} \\ n_{xc}(\bold{r}, \bold{r}^\prime) &= \text{density at point } \bold{r}^\prime \text{of the xc hole surrounding an electron at } \bold{r} \end{aligned} $$

GGA

General Gradient approximation (GGA) which depends on not only the electron density but also the derivatives of the density, is a way to consider non-uniform electron gas which is not included in LDA. $$ E_{xc}^{GGA}[n] = \int f_{xc} (n, \nabla n) d\bold{r} $$ The exchange part: $$ E_{x}^{GGA}[n] = E_{x}^{LDA}[n] F_x(s) = \int n(\bold{r}) \epsilon_{xc}^{uniform} (n(\bold{r})) F_x(s) d\bold{r} $$

PBE paper

PBE is very popular because it gives a satisfactorily description of both isolated and periodic systems. - The exchange-correlation energy must be approximated under the DFT framework. - Comparison between LDA and GGA - Previous developments of GGA and remaining issues - General form of $E_{c}^{GGA}$ and requirements for building $E_{c}^{GGA}$ - General form of $E_{x}^{GGA}$ and conditions for building $E_{x}^{GGA}$ - Improvements of this new GGA functional

The general form for the correlation energy within GGA: $$E_{c}^{GGA} [n_\uparrow, n_\downarrow] = \int n(\bold{r}) [\underbrace{\epsilon_c^{uniform}(r_s, \zeta)} _ {\epsilon_c^{LDA}} + \underbrace{H(r_s, \zeta, t)} _ \text{the gradient contribution}] d\bold{r}$$

The exchange energy within GGA: $$E_{x}^{GGA} = \int n(\bold{r}) \underbrace{\epsilon_{x}^{uniform}(n(\bold{r}))} _ {\epsilon_{x}^{LDA}} \underbrace{F_x(s)} _ {\text{enhancement factor}} d\bold{r}$$

PBEsol

Meta-GGA

Meta-GGA functionals are in general more sensitive to the integration grid than GGA functionals, and therefore they usually require a finer integration grid than the default of most popular programs.

Hybrid

The B3LYP functional suffers from a significant electron self-interaction error. The consequence is a strong tendency for B3LYP to overdelocalize the wave functions; this is the opposite of Hartree–Fock methods, which overlocalize the wave functions. Thus, in extended π-conjugated systems, B3LYP favors fully coplanar conformations—with B3LYP torsion potentials overestimating the energy barriers for interconversion between stable conformers—and spreads the HOMO–LUMO wave functions.

Chem. Mater. 2017, 29, 2, 477–478