Exchange-correlation

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.

- The quality of exchange-correlation functionls is assessed by comparing the performance with experiments or high-level wave function based calculations.

- uniform electron gas $\approx$ slowly varying densities $\approx$ small reduced density gradients

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.

- 2016 - IPAM - Back to Putting the Theory Back in Density Functional Theory
- http://www.ipam.ucla.edu/abstract/?tid=13996&pcode=DFT2016

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:

- Assume that $E_{xc}[n] = 0$, meaning that we are solving the Kohn-Sham equations like doing the Hartree-Fock calculations without the self-interaction correction. This was tried in 1960s, but not good. Lattice constants of solids and bond lengths of molecules are too large by quite a lot (10~20%) using this method. It makes the binding energy of molecules or solids very small and sometimes zero. So atoms barely bind to form molecules and solids.
- $E_{xc}[n] = \text{Nature's glue of atoms and solids}$
- Kohn-Sham 1965: LDA. Walter didn't expect LDA much better than $E_{xc}[n] = 0$. But it turned out much better and more realistic.

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

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

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.

- exact for a uniform density
- correct for density that varies slowly over space (slowly varying system)
- $\frac{\vert \nabla n \vert}{n} \ll k_F = (3 \pi n)^{1/3} \text{ (local Fermi wave vector)}$
- $\frac{\vert \nabla n \vert}{n} \ll k_s \sim n^{1/6} \text{ (Thoma-Fermi screening length)}$

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

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

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

- $r_s$ is the local Seitz radius: $n = \dfrac{3}{4} \pi r_s^3 = \dfrac{k_F^3}{3\pi^3}$
- $\zeta$ is the relative spin polarization: $\zeta = \dfrac{n_\uparrow - n_\downarrow}{n}$
- $t$ is the a dimensionless density gradient: $t = \dfrac{\vert \nabla n \vert}{2 \phi k_s n}$
- $\phi (\zeta)$ is a spin-scaling factor: $\phi (\zeta) = \frac{(1+\zeta)^{2/3} + (1-\zeta)^{2/3}}{2}$
- $k_s$ is the Thomas-Fermi screening wave number: $k_s = \sqrt{\frac{4k_F}{\pi a_0}}$
- $a_0 = \frac{\hbar^2}{me^2}$

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

- $\epsilon_{x}^{uniform}(n(\bold{r})) = - \dfrac{3e^2k_F}{4\pi}$
- $F_x(s)$ is the enhancement factor: $F_x(s) = 1 + \kappa - \dfrac{\kappa}{1 + \frac{\mu s^2}{\kappa} }$
- $s$ is a dimensionless density gradient: $s = \dfrac{\vert \nabla n \vert}{2 k_F n}$
- $\kappa$ is 0.804 in PBE

- The PBEsol is optimized for bulk properties with a new $\mu$ parameter while keeping $\kappa$ the same as PBE.
- PBEsol cohesive energies are known to be inaccurate, because of the poor performance for the single atom.

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.

- Minnesota functionals such as M06-L, M06 and M06-2X are known to be more sensitive to the integration grid than other functionals.

- 量子化学研究中切换泛函应当注意的问题 - Sobereva
- 对有机体系，M06-2X做几何优化也是比B3LYP略微好一点点的，特别是存在分子内弱相互作用的时候（当然，对B3LYP可以通过加DFT-D校正弥补这点），更何况M06-2X的计算耗时终究还是和B3LYP在一个档次。
- 通常来说对M06-2X加不加D3影响很小。
- 简谈量子化学计算中DFT泛函的选择 - Sobereva
- 计算有机体系热力学数据（反应能、构型能量差、能垒等）：M06-2X（加上DFT-D3校正往往更好）
- 其它情况或模棱两可的时候首选B3LYP-D3(BJ)，若发现其结果不理想，可尝试M06-2X、wB97XD、MN15。当然，如果能接受更大计算量，最好尝试双杂化泛函。
- M06-2X等明尼苏达系列泛函对DFT积分格点精度要求比普通泛函高很多，有时候因为格点不够精细导致几何优化和SCF不容易收敛、势能面上出现皱纹。提高积分格点精度（如Gaussian09里用int=ultrafine。极个别时候需要int=superfine）可解决，但会明显增加耗时。
- M06-2X的一个鲜明优点是有机体系反应能、异构化能、势垒等热力学量有关方面的计算精度颇好，有时候甚至接近双杂化泛函的精度，这也跟拟合参数时引入了这类问题直接相关。
- 算有机体系的热化学性质、结构优化等方面，在非双杂化泛函里，M06-2X是目前几乎最顶尖的。
- 如果是比较大的有机体系，特别是体系比较柔的情况，虽然M06-2X精度确实很好，但是由于本身泛函形式复杂而耗时比B3LYP高，再加上有时候需要用很高质量积分格点才能解决其几何优化难收敛、有细小虚频之类麻烦事，所以从实际角度，我更推荐用B3LYP-D3(BJ)做优化和振动分析，这样又便宜又好，而能量计算精度更好的M06-2X只在计算能量时候再用。

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.