It seems a bit weird that there isn’t much information on this topic on the internet. Recently I’ve had to calculate some of these indices to explain an anomalous behaviour in lactones formation and out of curiosity I ran a small search on the net about how to calculate them. Most of the information I retrieved were papers dealing with calculated Fukui and condensed Fukui indices, but unless you count with electronic subscriptions to the corresponding journals you were left in the dark. Moreover, even if you get to read the paper they will tell you as much about how to calculate Fukui indices as they tell you about the Hartree-Fock procedure details.

Therefore here I post this information specifically under “how to calculate Fukui indices” so others might find it. It seems to me this blog is taking a rather educational turn which was not its original intention. Still, if I can atract people to read about my work while finding useful information I’m glad to do it.

Fukui indices are, in short, reactivity indices; they give us information about which atoms in a molecule have a larger tendency to either loose or accept an electron, which we chemist interpret as which are more prone to undergo a nucleophilic or an electrophilic attack, respectively. This in turn has to do with a molecule’s tendency of becoming polarized in the presence of an external field or upon the change of electron density. The key word here is electron density, the whole idea behind Fukui’s lies in the realm of conceptual DFT.

Fukui functions are defined as the functional derivative of the chemical potential respect to the external potential (the one produced by the nuclei) at a constant electron number. Since the chemical potential is defined as the derivative of the density functional respect to the electron density, fukui functions are also defined as the derivative of the electron density respect to the number of electrons at a constant potential, and this latter definition is what we want to work with because it means that we can calculate how the density changes at every point (since it is already different at every point r) when adding or removing an electron while keeping the potential constant (that is the position of the nuclei, in other words mantaining the molecular geometry). The name fukui function stems from the fact that these added/removed electrons go into the frontier or Fukui orbitals HOMO/LUMO, but the reality is that the definition was conceived by Yang and Parr (like almost everything in conceptual DFT).

To practically perform these calculations the finite differences method is employed and so the condensed to atom fukui index is obtained.

Electrophilicity of atom A in molecule M (of N electrons)
fA+ = PA(N+1)-PA(N)
Nucleophilicity of atom A in molecule M (of N electrons)
fA = PA(N)-PA(N-1)
Radical attack susceptibility of atom A in molecule M (of N electrons)
fA0 = 1/2[PA(N+1)-PA(N-1)]

where P stands for the population of atom A in molecule M. If you want to analyze the fukui function at an ionized species the procedure is the same but most people need to beware that N is the number of electrons of the original ion! (i.e. the species you are trying to analyze), sometimes it may be confusing if you are trying to analyze the nucleophilicity (f-) of an atom in a cationic species (M+).
The population analysis on the + or – system has to be performed at the same equilibrium geometry as the original molecule! If we optimize again the system then we are letting the system relax and therefore we loose information on the polarization of the electron density upon the change in number of electrons.

“Negative indices are meaningless and should be disregarded” This is a common statement that ever since the definition of Fukui indices has been regarded as true; however there have been a couple of recent publications (see below) that defy to some extent such notion.

Major drawback: Since we are ultimately dealing with occupation numbers on each atom these indices are very sensitive towards changes in basis sets and population analysis paradigm. I strongly recomend never to take those numbers as absolutes, only as comparative parameters within the same system! I also find useful to compute them using more than just one method and one level of theory in order to further confirm or even to dismiss the trends observed. Natural population analysis and AIM are much more robust than simple Mulliken PA.

Hopefully this will be helpful to people trying to calculate fukui functions and also to understand what they mean. Subscribe to this post for further updates (you never know). Rates and comments are always most welcome specially if you found this post interesting or useful. Cheers!

Further reading.-Computational Chemistry by Errol Lewars
-Bultinck et al. Negative Fukui functions: New insights based on electronegativity
equalization
J.Chem.Phys 118 (10) 4349-4356 (2003)
-Melin J, Ayers PW, Ortiz JV. Removing electrons can increase the electron density: a computational study of negative Fukui functions. J Phys Chem A. 2007 111(40):10017-9
-Bultnick & Cabo-Dorca Negative and Infinite Fukui Functions: The Role of Diagonal Dominance in the Hardness Matrix J. Mat. Chem. 34 (2003) 67-74

PLEASE CHECK THIS OTHER POST FOR A NICE SCRIPT OF OURS TO AUTOMATE THIS PROCEDURE. THANKS!
https://joaquinbarroso.com/2017/12/20/python-scripts-for-calculating-fukui-indexes/