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Density functional Theory

There are a couple of properties that make electron density so useful:

  1. It is a much more simple object, only depending on three coordinates (\(r_1\))instead of the coordinates of the basis functions
  2. It is much better defined at the nuclei, with the electron density approaching \(0\) rather than approaching \(\infin\)
  3. At large distances from the nuclei it also approaches \(0\)

If we re-write the electronic Schrodinger equation using density as a functional, we get:

\[ E[\rho]=\color{blue}T_s[\rho]+J[\rho]+E_{ne}[\rho]\color{red}+E_{XC}[\rho] \]

The blue terms are all known things that we can obtain from HF (\(T_s=\) Kinetic energy, \(J=\) Coulomb energy, \(E_{ne}=\) nuclei-electron interaction) , though the red component is the unknown term, called the “bin” of DFT.

It is made up of an exchange (electron swapping) and a correlation component (the electron dance/degenerate orbital swapping):

\[ E_{XC}[\rho]=E_X[\rho]+E_C[\rho] \]


Are made up of an exchange component and a correlation component

Exchange Correlation

Jacob’s Ladder

Tells us about the increasing accuracy with the increasing terms/components to the functionals

Level Type Components Examples
Level 5 Double Hybrid Level ⅔ + MP2 Correlation energy B2LYP
Level 4 Hybrid GGA
Hybrid meta-GGA
Level ⅔ + Exact HF exchange M06/M06-2X (meta)
M11 ( meta)
B3LYP ( non-meta)
ωB97X-D/V ( non-meta)
Level 3 Meta-GGA Level 2 + Gradient of electron’s kinetic energy \(\rho(r)''\) TPSS
Level 2 GGA Level 1 + Gradient of density \(\rho(r)'\) PBE
Level 1 Pure Functionals (LDA) Electron Density, \(\rho(r)\) LDA

From Phys.Chem.Chem.Phys., 2014, 16, 9904

  • M06-2X has the best overall performance for thermochemistry (3.21 kcal mol−1)
    • After M06-2X, the next best density functionals for thermochemistry are ωB97X-V and ωB97X-D, with RMSDs of 3.60 and 3.61 kcal mol−1, respectively
  • ωB97X-V has the best overall performance for noncovalent interactions (0.32 kcal mol−1)
    • After ωB97X-V, the next best density functionals for noncovalent interactions are M06-L and B97-D2, with RMSDs of 0.47 and 0.48 kcal mol−1, respectively.

From PhysChem Webinar

“ωB97XD does a generally good job for geometries of quite diverse systems. I’d suggest B3LYP-D3(BJ) or wB97XD these days for a simple DFT method.”

M06 Functionals (Minnesota)

Functional HF Exchange Function
M06 27% Thermodynamics and kinetics
M06-L 0% Transition metals and organometallics
M06-2X 54% Non-covalently bound interactions
M06-HF 100% Charge transfer complexes

Limitations of DFT

  • Electron correlation isn’t considered explicitly - it’s parameterised
    • Solved by Grimme’s addition of explicit empirical dispersion corrections (-D)
  • Electrons can see parts of themselves though self interaction errors (long range issue, effecting dissociation curves)
    • Solved by including the HF exchange (cannot include 100%, as the performance of the functional will be poor)


One approximation for long range corrected functionals, as it accounts for HF exchange at short distances with a different scaling factor (coefficient) to longer distances.

Can also have explicit dispersion corrections with -D

The Grid

The results of DFT calculations are complex analytical expressions that cannot be solved numerically. As a result, the system needs to be divided into sections and the density within each grid point calculated.

When to be Cautious!

  1. Reactions with drastic changes in chemical nature, such as ring formation, conjugation, isomerisations, free-radical polymerisations
  2. Excited states (including UV-Vis). Double hybrids can be useful but may still not be accurate
  3. Molecular systems with strong static correlation, such as transition metal complexes and Biradical reactions (\(\omega\)B97X-D can be useful here)
  4. Redox potentials of metal-based (inorganic) compounds and surfaces

Kohn-Sham Formalism

The KS formalism is based on the self consistent field method, which means that HF code can be used for DFT methods (convenient), however the orbitals produced don’t really make much sense. Koopman’s theorem doesn’t apply to DFT.