# Interpreting electron localization function (ELF) results

Studying the interaction of a boron-nitride nanostructure with lead, we calculated the wave function using Gaussian software (single point energy with basis-set mix 6-311G++/lanl2dz and hseh1pbe functional) and did a topological analysis using the Multiwfn software.

The properties analyzed in the topological study were the critical points/basins in the electron density (via QTAIM), the Laplacian of the electron density and the electron localization function (ELF). Images with some of the results are below.

From the QTAIM analysis we got the the lead make a bond with one nitrogen atom at the nanostructure surface. This is due to the interbasin path together with the bond critical point between both atoms.

The electron localization function is shown in Fig. 2. According to its definition1, ELF took values among 0 and 1. The upper limit ELF=1, correspond to perfect localization and ELF=1/2 correspond to electron-gas like probability.

My doubts is: after looking to Fig. 2, if a bond is formed between lead and nitrogen atoms, we shouldn't expect more red regions between both atoms?

Fig. 1. Electron density. Bold deep blue lines: interbasin paths. Brown: (3,-3) nuclear critical point (NCP). Blue: (3,-1) bond critical point (BCP).

Fig. 2. Electron localization function.

Definition of ELF: $$ELF = {\left( {1 + \chi _\sigma ^2} \right)^{ - 1}}$$

$${\chi _\sigma } = {D_\sigma }/D_\sigma ^0$$

$${D_\sigma } = {\tau _\sigma } - \frac{1}{4}\frac{{{{\left( {\nabla {\rho _\sigma }} \right)}^2}}}{{{\rho _\sigma }}}$$

$$D_\sigma ^0 = \frac{3}{5}{\left( {6{\pi ^2}} \right)^{2/3}}\rho _\sigma ^{5/3}$$

$${\tau _\sigma } = \sum\limits_i^\sigma {{{\left| {\nabla {\psi _i}} \right|}^2}}$$

Where: $$\tau _\sigma$$ is the kinetic energy for spin $$\sigma$$, and $$\rho$$ is the electron density.

1. A. D. Becke and K. E. Edgecombe. A simple measure of electron localization in atomic and molecular systems. J. Chem. Phys. 92, 5397 (1990) (DOI: 10.1063/1.458517)
• Could you share your geometric structure?
– Jack
Nov 11 '20 at 21:05
• @Jack, just the structure (the Gaussian input)?
– Camps
Nov 11 '20 at 21:44
• Like the POSCAR of VASP.
– Jack
Nov 12 '20 at 6:57
• If ELF is 0.5 around center and homogeneous distribution exists you can say its like metallic bond but if electron localization occur in between two atoms and concentrated with ELF value higher that 0.5 covalent bond starts forming, and if one atom has higher ELF around 1 and other has atoms none or less than 0.5 , total electron transfer occur and ionic bond forms. Jul 4 at 4:45
• This could be a ionic bond. In that case the ELF function value can be very low at the AIM bond-critical-point. Did you evaluate the local kinetic and potential energy densities at the bond critical point ? They can help to identify the character of the bond. It's also a good idea to look at 3D isosurface plots of the ELF function to see if it forms localized basins between the atoms at isovalues > 0.5. Another approach would be a calculation of a similar system with a well known bond of Pb-N to get a "standard" for comparison. Jul 4 at 8:02