Long-Range-Corrected Density Functionals

Most common density functionals cannot correctly describe the distance dependence of charge-transfer excitation energies since the non-local Hartree-Fock exchange is not 100% included. One way to solve this problem is to preserve the form of common GGAs and hybrid functionals at short range while applying 100% Hartree-Fock exchange at long range. This can be done with a partitioning of the electron-electron Coulomb potential into long- and short-range components [192, 193]:

(1)

where erf and erfc are the error function and complementary error function. The first term on the right in Eq. (1) is short range and decays to zero on a length scale of 1/ω , while the second term is long-range.

The DFT exchange-correlation energy for a hybrid functional is normally expressed as

(2)

where E_C is the correlation energy, E_X is the DFT exchange energy, and is the Hartree-Fock exchange energy with C_HF the weight of Hartree-Fock exchange. The two components of the range-separated Coulomb operator are weighted separately, to give a generic long-range-corrected (LRC) energy that can be expressed as

(3)

where SR and LR indicate that these components of the functional are evaluated using either the short-range (SR) or the long-range (LR) component of the Coulomb operator. The first three terms on the right are the same as in the uncorrected functional.

The short-range Hartree-Fock exchange is replaced, either entirely or in part, with a modified version of the DFT exchange functional in which the error function is applied to truncate it, either by the method of Iikura et al. [68] or that of Henderson et al. [70]. With the introduction of the short-range corrected functional, the DFT exchange-correlation energy can be expressed in general as

(4)

The coefficients are not independent, since as a consequence of Eq. (1),

(5)

Thus, if then there is no short-range Hartree-Fock exchange and only long-range Hartree-Fock exchange. The same is true for the DFT exchange. This form is useful, however, for writing the functional as the original functional (first three terms) plus a correction term (last four terms).