# Density Functional Theory¶

#### Basics¶

• A function takes a number and returns another number
• An operator takes a function and returns another function
• A functional takes a function and returns a number

Rather than putting effort into solving for $$\Psi$$ which has far too many variables to be practical, we can make a few inferences and approximations that essentially allow us to calculate the electron density over a grid and allow us to determine what the resulting wavefunction and energy will be

The Hamiltonian is a parameter that can be used to give \Psi

$\widehat{H }=\widehat{T}_n+\widehat{T}_e+\widehat{V}_{nn}+\widehat{V}_{ne}+\widehat{V}_{ee}$

Through some assumptions made, we can translate this to be in terms of electron density ($$\rho$$)

$E[\rho]=T[\rho]+E_{ext}[\rho]+E_{coul}[\rho]+E_{xc}[\rho]$
• The nightmare of solving for this exchange-correlation interaction becomes a functional of the electron density

#### LDA - local (spin) density approximation¶

• $$V_{xc}$$ is defined as only depending on the local values of the electron density
• Is good for periodic systems but calculated bond strength and electron correlation to be too big
• E.g. SVWN, VWN5
$E_x^{LDA}[\rho]=−c_x\int{\rho^{\frac{4}{3}}(\hat{r})d\hat{r}}$

#### GGA - generalised gradient approximation¶

• $$V_{xc}$$ is also includes the first derivative of $$\rho$$
• Is better for molecules
• Builds upon LDA
• E.g. Exchange: PW86, B88, BP88, HCTH
• E.g. Correlation: LYP, PW91, BLYP
$E_x^{D88}[\rho]=E_x^{LDA}[\rho]−\beta\rho^{\frac{1}{3}} \frac{x^2}{1+6\beta x sin h^{−1} x′}$

Parameters = β, 1+6β

#### Meta-GGA¶

• Also includes second derivatives for better accuracy
• Not good for all molecules due to limited training set for determination of parameters
• E.g. M06-L, TPSS

#### Hybrid¶

• Mixes in HF exchange with GGA
• Most popular functionals
• E.g. B97/2,MPW1K
• E.g. B3LYP - 3 parameters; a, b and c
• Hybrid DFT mixes DFT with other post-HF methods to try and combine more concepts in to better account for correlation energy
$E_{xc}^{B3}=(1−a) E_x^{LDA}[\rho]+a_x^{HF}+b\Delta E_x^{B88}[\rho]+(1−c) E_c^{LDA}[\rho]+c\Delta E_c^{GGA}[\rho]$

Parameters = a, b, c

#### Hybrid-Meta-GGA¶

• E.g. M05-2X, M06-2x, MPWB1K
• Meta-GGA Hybrid

### Running DFT¶

#### Precautions¶

• Different methods and basis sets can yield highly different results
• It is important to know the errors associated with the particular choices of computations
• This is doubly important when looking at someone else’s results
• DO NOT TAKE ANYTHING AT FACE VALUE

#### Strengths and Weaknesses of DFT¶

• Strengths
• Low computational cost
• Accurate for structures and thermochemistry
• The density is conceptually simpler than $$\Psi$$
• Weaknesses
• Can fail in spectacular and unexpected ways
• There isn’t a systematic way of improving results
• Multidimensional integrals can be problematic

#### HF vs DFT¶

• DFT is not approximate, it is exact
• Everything we do however is a functional of $$\rho$$ which means that the density has to be accurate
• Hohenberg-Kohn proved that the functional of $$\rho$$ must exist
• There is no definition as to what the functional should look like
• We know $$f[\rho]$$ exists, we just don’t know what it is

#### Contrast¶

• HF is an approximate theory that solves the relevant equations exactly
• DFT is an exact theory that solves the relevant equations approximately (since we don’t know$$f[\rho]$$)
• DFT is not variation due to all the additions, however exact DFT is