fig1

Figure 1. Jacob’s ladder of density functional approximations to the exchange-correlation energy of a many-electron system. This energy is approximated as an integral over three-dimensional position space of a function of one or more arguments. The argument added on each rung is listed to the left of that rung: the electron density n as a function of position r, its gradient, the positive orbital kinetic energy density τ, etc. The exact exchange information comes from the one-particle density matrix ρ (r, r') constructed from the occupied Kohn-Sham orbitals or one-electron wavefunctions, which is also the input for proper self-interaction corrections. The proper self-interaction corrections are now added to the hybrid functionals on the fourth rung. The random phase approximation or RPA, on the highest rung, employs also the unoccupied Kohn-Shan orbitals. Each of the most widely-used density-functional approximations sits on one of the rungs.