## Evaluating exchange-correlation energy Some typically applied method is proposed: - LDA: local density approximation The exchange-correlation energy density at each point is assumed to be the same as in the homogeneous electron gas with that density. >[!Note] > For this LDA exchange part can be computed as an analytical form, correlation may be obtained by Monte Carlo. > The exchange energy density is $E_{x}=c\int \rho^{4/3} \mathrm{d}r$. - GGA: generalized gradient approximation GGA includes the gradient of density and typically constructed in the form of a correlation term added to a LDA. $E_{xc}^{GGA} = E_{xc}^{LDA} + \Delta E \nabla \rho f(\rho)$ >[!Note] >For GGA, even at large gradients, it is also valid and is able to present desired properties. - meta-GGA: Includes additional kinetic energy ($\sum(\nabla \psi_{i})^{2}$), may trade different chemical bonds more accurately. - Hybrid functionals: It includes fraction of exact HF exchange energy calculated from KS molecular orbitals, and more orbitals in most cases. $E_{xc} = (1-a)E_{xc}^{DFT}+aE_{x}^{HF}$ >[!Note] >The term $E_{x}^{HF}$ is only an exchange energy. >A successful one of this type is $E_{xc}^{B3LYP}$, which is an combination of multiple results. >[!Notice] > For lattice constant in solids. LDA overestimates, GGA underestimates. For absorption and binding energy. LDA over binds, GGA under binds. >[!Info] >See more on: >https://en.wikipedia.org/wiki/Hybrid_functional >https://www.cup.uni-muenchen.de/ch/compchem/energy/dft1.html