# 10.5.6 Electron Localization Function

(June 30, 2021)

Formulated by Becke and Edgecombe, the electron localization function (ELF),

 $\text{ELF}=\Bigg{[}1+\Bigg{(}\frac{\tau_{\sigma}-\frac{(\nabla\rho_{\sigma})^{% 2}}{4\rho_{\sigma}}}{\frac{3}{5}(6\pi^{2})^{2/3}(\rho_{\sigma}^{5/3})}\Bigg{)}% ^{2}\Bigg{]}^{-1}$ (10.17)

is a measure of electron localization. It is derived from the Hartree-Fock conditional pair probability and can reveal information about bonding and shell structure. The function $\text{ELF}(\mathbf{r})$ has values that lie between 0 and 1, with $\text{ELF}=1$ representing perfect localization and $\text{ELF}=1/2$ representing electron-gas-like pair probability. To generate ELF plots with Q-Chem, set the PLOT_ELF $rem variable to TRUE. For closed-shell systems, only the $\alpha$-spin ELF is calculated whereas for open-shell systems (spin-unrestricted calculations), both $\alpha$- and $\beta$-spin ELFs are computed. The following example illustrates the calculation of the ELF for a water molecule. Example 10.15 A job that evaluates the ELF for H${}_{2}$O on a 50 x 50 x 50 grid. The output is in a cube file called elf_alpha.0.cube. $molecule
0 1
O    -4.5320698567   0.2524215916      0.0130780103
H    -3.5641829319   0.2173288989     -0.0173259969
H    -4.8109190521  -0.4489616171     -0.5945943692
$end$rem
JOBTYPE           opt
METHOD            b3lyp
BASIS             6-31g*
PLOT_ELF          true
MAKE_CUBE_FILES   true
$end$plots
water
50 -7 7
50 -4 4
50 -4 4
0 1 0 0
0
$end  View output Please refer also to elf_methane.in the$QC/samples directory, which uses the newer \$plots format.