Skip to content

Density Functional Theory


  • 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

Jacobs Ladder

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


  • 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


  • 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


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

Running DFT

What you need

  • Molecule geometry
  • Molecular charge
  • Spin multiplicity (2s+1)
  • Basis set
  • Exchange functional (S,B,B3 etc..)
  • Correlation functional (LYP, PW91 etc…)


  • 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

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


  • 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


  • 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

About DFT

  • DFT is good for determining geometries, but not so much for calculating energy
  • Totally fails for non-covalent interaction
  • Can have large errors for excitation energies
    • Fixes include CAM-B3LYP or TD-DFT