# 10.5.6 Electron Localization Function

Formulated by Becke and Edgecombe,46 the electron localization function (ELF),

 $\text{ELF}=\Bigg{[}1+\Bigg{(}\frac{\tau_{\sigma}-\frac{(\nabla\rho_{\sigma})^{% 2}}{4\rho_{\sigma}}}{\frac{3}{5}(6\pi^{2})^{2/3}(\rho_{\sigma}^{5/3})}\Bigg{)}% ^{2}\Bigg{]}^{-1}$ (10.18)

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.270 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.492 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
$end  Please refer also to elf_methane.in the$QC/samples directory, which uses the newer \$plots format.