Chemistry Labs

Theoretical and computational chemistry

Density functional theory (DFT)

How the electron density, instead of the many-electron wave function, can determine the energy of a molecule or solid.

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AdvancedThe Hohenberg–Kohn idea

The wave function of NN electrons depends on 3N3N spatial coordinates. Hohenberg and Kohn proved that the ground-state energy is fixed by the electron density ρ(r)\rho(\mathbf r), a function of only three coordinates.

E[ρ]=Ts[ρ]+∫vext(r) ρ(r) dr+J[ρ]+Exc[ρ]E[\rho] = T_s[\rho] + \int v_{\mathrm{ext}}(\mathbf r)\,\rho(\mathbf r)\,d\mathbf r + J[\rho] + E_{xc}[\rho]

Definition: Kohn–Sham scheme

Replace the interacting electrons by non-interacting ones that reproduce the same density, and solve one-electron equations self-consistently. All the difficult many-body effects are packed into the exchange–correlation functional Exc[ρ]E_{xc}[\rho], which is not known exactly and must be approximated.

3D point cloud of the electron density of the H₂ molecule (minimal LCAO model), with the bonding and antibonding orbitals as options.
Density is what DFT works with. Here it is built from a bonding orbital of two 1s1s functions (a simple model, not a DFT calculation): change the H–H distance and see the density pile up between the nuclei, or switch to the antibonding orbital, which has a node there.

Common approximations are the local density approximation (LDA), generalized gradient approximations (GGA, such as PBE), meta-GGAs and hybrid functionals that mix in exact exchange (such as B3LYP). Choosing a functional means choosing a trade-off between cost and accuracy.

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References

  • Inhomogeneous Electron Gas · P. Hohenberg, W. Kohn, 1964
  • Self-Consistent Equations Including Exchange and Correlation Effects · W. Kohn, L. J. Sham, 1965
  • Density-Functional Theory of Atoms and Molecules · R. G. Parr, W. Yang, 1989