How to calculate Fukui indices

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
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



About joaquinbarroso

Computational and theoretical chemist in his early forties, in love with life, science, baseball, and literature. Science literacy makes us responsible citizens, it is therefore a scientific duty to talk, write, and engage with the general public; as Feynman said, if you find science boring, your learning from the wrong teacher. "Make like a molecule and react!"

Posted on July 26, 2010, in Computational Chemistry, Theoretical Chemistry, White papers and tagged , , , . Bookmark the permalink. 115 Comments.

  1. I am trying to calculate some fukui indices as well but am having some practical problems. Do you know how I could calculate AIM using Gaussian09W? Thanks

    • Hi Kenny,

      I’ve been searching for the same thing without any success. I don’t think G09 supports AIM anymore. There are some freewares out there to calculate AIM and some which require licenses and then again from these some are free to academics. If I come across one I will let you know.



  2. Hi!

    I would be also interested in AIM freeware. I want to calculate partial charges within this theory, but due lack of software i had to leave my work undone.

  3. Dear sir
    Few days ago i have raised a question regarding the PES scan of 2-amino-5-nitropyrimidine.i would like to do 3D PES scan calculation.can u teel me how to do and how to visualize that using gaussview3.0.thanks in advance.

    • Sorry for such a long delay

      I strongly suggest you look into the post on PES scan in this blog, as well as on the Gaussian manual available at their website (look under keywords SCAN and OPT, in this last case look for the ModRedundant option) Although quite frankly I don’t know what angles can you possibly scan on 2-amino-5nitropyrimidine (both C-N bonds?) And didn’t I answer your question before? Never mind, I get confused sometimes.
      I hope this helps and as usual thanks for reading and posting your questions!
      Have a nice day

  4. Dear sir, I want caliculate Fukui indices for furan and dihydrofuran, how to caliculate? we have Gaussian09 software.

    Thanking you

    Yours sincerely
    seshi reddy surasani
    research scholar
    dept of chemistry

    • Hi Seshi

      You just have to follow the instructions located on this very same post! Please let me know if they are understandable ok?

      Thanks for reading!

      • Hi Professor,
        i’m very happy to visite and read your posts, i don’t understand how to prepare Gaussian for fukui calculation, i know the instructions for optimization (DFT B3lyb 631G(dp) for example ) but what i can do to make fukui indices results, and how to find the population in input file. help me plz and great thanks to you

    • Dear Seshi,

      Could you please share with me how to calculate fukui indices using gaussian09 package?

      many thanks

      Bijan Mondal
      Research Scholar

      • Well, I think you can read this post and do it very easily. You have to just be careful in your selection of charge and multiplicity on each part of the calculations and you should be able to do it very easily.

        Have a nice day!

  5. What is your opinion on the use of pop=mk charges and pop=chelpg charges?

    • to calculate Fukui indices I mean

      • Wow! Very delayed reply. Sorry about that, Benny.

        The important thing is the pop part. You want a method that is reliable of course, but since you are taking differences between two different populations then you should be able to get similar, at least qualitatively similar, results when using different population mehtods.

        Have a nice day

  6. Hola, Quisiera saber si usted sabe cómo utilizar AIMALL para cálculos en donde se utiliza LANL2DZ. Cuando calculo con este programa (desde mi .fch de G03) me da error. No se cuales son las palabras claves que debo colocar en el imput de gaussian para que al calcular con AIMAII (AMQB) tenga toda la información. Este problema no lo tengo con cálculos de G03 que no llevan LANL2DZ. Muchas gracia!

  7. Dr Anbial Sierraalta

    Hola Clara
    En Gaussian no es posible usar pseudopotenciales y el keyword AIM ya que las densidades electronicas que se obtienen cuando se utilizan pseudopotenciales presentan problemas. El Dr Jerzy Ciosłowski, que fue el enacragado de desarrollar esta parte, puso una condicion interna en el programa para que no funcionara cuando se utilizan psudopotenciales

  8. Thank you very much. Very useful.

  9. hi Joaquin

    thank you so much for all the topics they can be life saving

    can you please give us the reference for computing fukui function using the atomic populations instead of the atomic charges.


    • Hi Zineb

      Thanks for reading the blog and for your kind words. The Fukui functions are always calculated with the atomic population, never with the charges. I’ll look for the reference for you but I just wanted to make clear that the use of the atomic population is always the way to go.

      Hope this helps!

  10. Abdel-Illah Djelloul Smir

    Dear Professor, I ask you a stone pertmet information about the calculation of indices fukui. In fact, for the molecule H2C = C (CN) (Morpholyno), I try to calculate indices of fukui two doubly bonded carbon. I, unwanted DFT/B3LYP/6-31G the *. I managed easily optimize the neutral molecule and quite difficult to negatively charged molecule. However, I can not optimize the positively charged molecule!
    The question I ask the technical point, how to use the optimization of the neutral molecule to optimize the positively charged molecule.
    Thank you for your help

    • Hi Smir!

      You should NOT optimize the +1e or the -1e structures! only the reference state. This reference state is usually neutral but you could also want to analyze the Fukui indexes for an ionic species.
      So, in short. If you already optimized the neutral structure, take that structure and change the charge and multiplicity in your new +1e and -1e calculations but do NOT optimize the structure, just calculate the atomic populations with the method of your preference.

      I hope this helps!

  11. djelloul smir

    Hello dear Professor, I am very sorry for this late reply. Also, I want to thank you very much for your help I was able to solve my problem with Fukui indices.

  12. djelloul smir

    Please can I ask you another question. In fact, I want to know what criteria I have to rely to explain the fragility of a link leading to it to break the bridge of view of quantum chemistry.

  13. hi Joaquin
    should i use the URHF calculations (not RHF) when doing calculation of the N-1 (charge =1, multiplicity= 2) or N+1species (charge =11, multiplicity= 2) ?

    best regards

    • Yes, Saied, you are correct (it says charge=11 in the N+1 case, I guess you meant charge=-1, right?), but you have to be careful because Gaussian will print the population analysis for the alpha and then for the beta population; I’m pretty sure somewhere later on on the log file you’ll find the total population (alpha+beta) but if you can’t find it just do the addition yourself.

      I hope this helps! Have a nice day!

  14. Thanks Joaquin, i get from your blog a very useful informations. i am using the pop=NBO analysis to get the natural charges then using either GaussView or chemcraft to get the natural charges to be used in the Fukui calculations, is it ok?

    great thanks

  15. According to Chandra and Nguyen, Int. J. Mol. Sci. 2002,3,310-323, Mendez and Gazquez J. Ame. Chem. Soc. 116, 9298 (1994), and others, the terms Electrophilicity and Nucleophilicity for your definition of the Condensed Fukui Index should be reversed.

    • With all due respect, they should not be reversed, they are correct! The language might be misleading, though, I’ll concede that. I am fully aware of those references you cite and in those it says that f(subk)^+ corresponds to the susceptibility of atom k to nucleophillic attack (which is proportional if not equivalent to the electrophilicity of the same atom k as I describe herein). The same also holds for f(subk)^- (susceptibility towards electrophilic attack on atom k ~ nucleophilicity of atom k). I will try to clarify the matter in a future post.

      That aside, I’m honored to see your name on my blog. I admire the work of your foundation and of course your own. It would be a privilege for me to ever collaborate with the work performed there.

      Thanks for your input. Best wishes.

  16. Dear Joaquin, thank you very much for your blog and your attention.
    Could you please let me know how one calculate polarizability in gaussian output?

    • Hello Ali,

      Thank you very much for your kind words. I just wrote a post addressing your question. However the post is still a work in progress. Sorry about that but I want to watch some Olympic action 🙂

      I hope it helps

  17. Hello doctor, I optimized a molecule to a level of theory B3LYP / 6-31G (d) 0 and singlet charged, have not had problems but I wanted to calculate Fukui changing the charge +1 and singlet but I get error, and when I do from gaussianview +1 I get that to be double multiplicity. just calculate it as one energy. and i get
    “Serveral Error 2070” I would appreciate if you could help me.
    How could calculate the Fukui function.

  18. Hello doctor, I optimized a molecule to a level of theory B3LYP / 6-31G (d) 0 and singlet charged, have not had problems but I wanted to calculate Fukui changing the charge +1 and singlet but I get error, and when I do from gaussianview +1 I get that to be double multiplicity. just calculate it as one energy. and i get
    “Serveral Error 2070″ I would appreciate if you could help me.
    How could calculate the Fukui function.

    • Hello Luis,

      all the instructions are there in the post you just have to read them carefully. When you change the charge of your molecule you must also change the multiplicity of the wavefunction (in this case from a singlet in the neutral case to a doublet in the +1 state). You will have to do the same when you go from q = 0 to q = -1.
      Did you find this error after you changed the multiplicity? also, remember the +1 and -1 states should NOT be optimized or you will relax the density and what you want to observe is where it is more polarized at the moment of insertion (or extraction) of a single electron.

      I hope this helps. Have a nice day!

  19. Thank you doctor good load calculation = +1 and multiplicity = doublet works, now I wondered with Mulliken charges can be calculated Fukui indices ..
    fx +1, -1 fx, fx
    for attack electrophilic, nucleophilic, and radical I think there are four strategies to get the Fukui function. Is there to do some extra treatment Mulliken charges, for Fukui indexes?
    Yang and Mortier Method
    fx-= [qx (N)-qx (N-1)]
    fx + = [qx (N +1)-qx (N)]

    • If you know about four different approaches, please share the references here with us!!!
      The problem with Mulliken charges is that these are basis set dependent, this means you will not obtain the same numbers when you change from 3-21G to 6-31G(d,p) to 6-311G, etc. In other words they are not consistent and therefore they are not reliable. You can use those charges but due to the nature of their calculation you wont get good results. I suggest using a different population method (natural, Hirschfeld, etc.) Use Mulliken only if you want to compare many many molecules and you dont have large computing resources.

      I hope this helps, have a nice day!

  20. Thanks for helping me, I’m a student of Pure, of Peru and am interested in computational chemistry and there is not much information arrives to your blog and use a translator to write, I’ll send you the link of these three strategies.

    Have some manual as determine reactivity using other methods of population analysis.

  21. I am very happy to find a blog of quantum chemistry and a master like you can guide me and share about these issues that are very exciting.

    • Muchas gracias por tus amables palabras. Si bien te recomiendo que estudies inglés, puedes poner tus comentarios en español para que a ambos nos sea más fácil entendernos.

      ¡Saludos hasta Perú!

  22. Buenas doctor Joaquin ahora tengo un problema con el potencial quimico y la dureza en la parte de la cadena coloco esto pero no logro hallar estos valores.. # opt b3lyp/6-31++g(d,p). Agradeceria mucho me de alguna pautas para resolver este problema.

    • Ninguno de esos parámetros son calculados directamente con Gaussian. Necesitas revisar las definiciones originales de ambas cantidades y calcular con Gaussian las diferencias finitas. El route section que estás utilizando únicamente pide una optimización de geometría.


  23. Doctor Buenos Dias he colocado la cadena
    Pop=Reg donde obtengo los orbitales homo y lumo
    5 virtuales y 5 Ocupados
    reemplazando directamente aqui en esta formula puedo hallar el potencial quimico.

    lo mismo sucederia para la dureza. Esto es una aproximacion por diferencias finitas y Koopmans.
    Es correcto lo que estoy haciendo.

  24. Good Morning Doctor I have placed the chain
    Pop = Reg where I get HOMO and LUMO orbitals
    5 and 5 virtual Occupied
    replacing directly here in this formula I can find the chemical potential.
    PotQuimico = (1/2) (EL + EH)

    The same would apply for hardness. This is a finite difference approximation and Koopmans.
    Is it correct what I’m doing.

  25. Thanks’ll put your name in my article for your help … I wondered units values ​​are in Hartree orbitals or kJ / mol …?

    • Thanks a lot! please send a copy of your article when you have it 🙂
      The energy of the orbitals in a Gaussian file is in Hartrees (also referred to as atomic units a.u.); it is my opinion that it is conceptually wrong to convert them to KJ/mol or kCal/mol (A mole of what? a mole of orbitals? makes little sense). I suggest reporting them in a.u. or eV (1 hartree = 27.2107 eV)

      I hope this helps!

  26. Hi Still working Good Doctor knows you as it is called in the software they
    use Linux for computational chemistry and can be downloaded ……. have
    some link is that there are certain advantages when using linux …

  27. In a relatively recent article (J. Phys. Chem. A 2011, 115, 4738–4742) Fukui functions were used to distinguish between HAT and PCET processes.
    Question remains in what to use for LUMO and HOMO when you have reaction with radicals.
    Tnx for a really nice site!

  28. Dear Joaquin,
    Thanks for your excellent and helpful blog. I’m an organic chemist and deal with the calculation of the fukui functions for pericyclic reactions. Using the usual method and Mullican charges I got some negative values for FFs. Then, I tend to use a method based on the Chemical Physics Letters 304 1999. 405–413 of Contreras et al using eqs. 15 and 16 of this paper, to calculate the FFs by just a single point run. I was succeeded to reproduce the FFs of water and methylamin in the paper by STO-3G basis set but for higher basis sets such as 6-311+G I can’t reproduce the FFs reported in the paper. I used the commands POP=FULL IOP=(3/33=3 or 1) in rout section to print the orbital coeff. and utilized the OVERLAP and DENSITY MATRIX and also the orbital coefficients according to the formula 15 and 16 of the paper. Could you please tell me is it correct and why this method works for STO-3G only.

    Have a nice day,

    • Hi Mahshid,

      Sorry for the delay of my reply. I’ve read that paper by Contreras but I don’t remember the details. Did they use Gaussian? If they didn’t then maybe it has to do something with the way the basis sets are defined in Gaussian (the contraction scheme). Since the STO3G uses Slater orbitals these are the same under any definition. Are your results too different from those by Contreras?

      I hope this helps

  29. Sai vikrama Vummaleti

    Dear Joaquin, very useful blog. Thanks a lot.

  30. Dear doctor,
    I have problems with the fukui + and – indices. All of them are negative and my calculations are correct. I obtain this functions since the Hirchfield charges.

    Can you help me?

    • The problem is that you are using charges and not population numbers.
      If you are calculating Hirschfeld charges (or any kind of charges, for that matter) you need to first calculate the population from the basic equation: pop(sub_i) = Atomic Number (sub_i) – Hirschfeld charge

      e.g. for a Carbon atom with HC = 0.9 its population number would be:
      Pop(sub_C) = 6.0 – 0.9 = 5.1
      for an Oxygen atom with HC = -0.9
      Pop(sub_O) = 8.0 – (-0.9) = 6.9

      Those are the numbers to be used in the Fukui functions.

      Have a nice day!

      • Puede recomendar alguna literatura?

        Chandra and Nguyen, Int. J. Mol. Sci. 2002,3,310-323, Mendez and Gazquez J. Ame. Chem. Soc. 116, 9298 (1994)

  31. Hello, I also think about calculating Fukui indexes, since my molecules are of biological relevance. As most of your readers I use Gaussian09. From your post I’ve learned that it is possible and quite easy to calculate those indexes.
    However, there are still two unanswered questions:
    1. for ions: is it bette to use UHF (UB3LYP, etc….) or ROHF (ROB3LYP)?
    2. which of the population analyses should we performed?The choice is pretty big, unfortunately…


  32. @Radek
    I ve been using Fukui indices for a couple of years now and I found that the best choice of population analysis is the Natural population analysis NPA which allow to avoid the negative values of the FI.

  33. Morning dr.

    I am PhD student from Malaysia. May i know, how to calculate the cation molecule.For your information, my structure is cation (+1). Should i put charge +2 for electrophilic attack and charge 0 for nucleophilic attack.Hopefully Dr can help me.tq

    • Yes Jimy, you are correct. Almost 🙂

      The example given is for a neutral molecule but the method can be extended to ions. Hence, if your reference or molecule under study is a cation then you need to calculate a single point for the +2 cation (in order to get f- nucleophile character of each atom) and then for the neutral species with charge=0 (in order to get f+, electrophile character of each atom). So you are correct but you should switch definitions.
      + and – refer to adding an electron and removing an electron respectively, not to the acquired charge of the hypothetical neutral reference state.

      I know it can be confusing and I have it drawn on my office window so I never screw up 🙂

      I hope this helps

  34. Euhhh êtes vous certain de ce que vous nous écrivez ??

  35. shailendra trivedi

    I found blog interesting.can you provide step by step process to calculate fukui function by taking simple example.

    • shailendra trivedi

      Hi Dr,
      Since I am Ph.D. scholar . I am working in computational chemistry & verymuch interested in fukui function but got confused.
      Plz provide detail step by step working to calculate ff by taking example.

  36. charges from pop=NBO or pop=MK did not matching the results from scientific papers
    how i can exactly calculate the q(N) q(n+1) q(N-1)

    • It specifically says in the post to not use charges but populations instead! Of course you can’t reproduce results from others, you are off by a factor equal to the nuclear charge in every atom. Maybe the use of q in my equations is misleading but they don’t mean charge they mean population.
      I hope this helps

  37. Ricardo Miranda

    If you can help me understand some things about Fukui index to nucleophiles and electrophiles. I can do the calculation using the protocol below:

    B3LYP/6-31G++(d,p) gfoldprint iop (3/36 = -1) = pop (full, esp)

    I do this calculation in three stages:

    1 – neutral molecule – (charge 0 and multiplicity 1)
    2 – Cation – (charge +1 and multiplicity 2)
    3 – anion – (charge -1 and multiplicity 2)

    With the generation of molecular orbitals HOMO and LUMO, i make a difference:

    Nucleophilic (f +) = HOMO (neutral molecule) – HOMO (cation)
    Electrophilic (f-) = HOMO (anion) – HOMO (neutral molecule)

    And to analyze the most character nucleophile just check atom which showed the highest values of f (+). Please tell me if I’m applying the protocol correctly. Help me a lot, thank you very much.

  38. Favero Reisdorfer Paula

    Dear Prof.
    Your post is very helpful to understand the Fukui indices. I do not know if you still answers questions and yet I would like to elaborate one.
    Natural values used populational NBO and B3LYP / 6-31G ++ (d, p), I got negative values of Fukui indexes.
    There are sites and papers such as, J. Melin, P. W. Ayers, J V Ortiz, J. Phys. Chem. A, 2007, 111, 10017, [164], that accept the validity of negative Fukui functions.

    Is there validity in these Fukui negative values?

    thank you

  39. Favero Reisdorfer Paula

    Dear Prof.
    I use natural populational NBO and B3LYP / 6-31G ++ (d, p) to calculate the Fukui índices and I got negative values.
    There are sites and papers such as, J. Melin, P. W. Ayers, J V Ortiz, J. Phys. Chem. A, 2007, 111, 10017, [164], that accept the validity of negative Fukui functions.

    Is there validity in these Fukui negative values?

    thank you

    • Yes there is! You are getting atoms depleted of electron density after the change of charge. I like to interpret them as disfavorable sites for molecular attacks.

      I hope this helps

  40. Dear prof,

    First I would like to appreciate the useful information that you provide in this website. I have still ambiguity about Which value from output file we have to select to put in as PA(N+1), PA(N), and PA(N-1) to calculate Fukui Functions?
    I personally believe that ,It is in “Summary of Natural Population Analysis” that has core,valence,Rydberg and we should get the total value and subtract it from natural charges of each atom. however, someone in linkedin explained “You can use the *Total* column in the NPA summary section of the output for the population as is.” Could you please confirm which approach is correct?


  41. Dear Prof,

    I calculated the fukui indices (f+) for a sulfonyl chloride and got the highest values not for the expected sulfur atom but for the two adjacent oxygens. Somehow, this makes sense since I would expect that the electron will be accepted by the sulfur but will be then located on the oxygens after all due to the electronegativity. However, isn’t that a problematic case in sense of the interpretation: f+=electrophilicity? The oxygens of sulfonylchlorides are surely not electrophilic. Thank you in advance for your answer.

    Best wishes,


    • Hello Adriano,

      Calculation of Fukui indices is always done under certain risks such as the one you have observed which is counterintuitive to your chemistry knowledge.
      First, I’d like to ask what level of theory are you using. I assume you are using the same basis set for all the atoms in the molecule, but maybe the polarization scheme you are using is not right.
      On the other hand, I’d like to ask what population analysis are you using. I hope it’s not Mulliken’s or that could be the source of the problem.
      The third observation is that even when your interpretation seems right it is also a bit ad hoc, since Fukui can’t tell you where the electron density will go after equilibrium but rather which are the atoms most perturbed by the inclusion of an extra electron.
      Last, did you optimize both structures? remember only the first one should be optimized (your reference state).

      I hope this helps!

      • Dear Prof.,

        thank you for your fast reply. I’m using m062x, 6-31+(d,p) and npa for the population analysis (for every atom the same). I optimized only the reference structure and used this optimized structure to calculate the population of the reference state and the N+1 species.
        Is it right to assume that the increase in electron density after a nucleophilic attack is somehow correlated to the AO-coefficients of the LUMO in the molecule referred to those found in the HOMO?

        thank you again and best wishes,


      • That is an interesting question, Adriano! To be honest with you I don’t know if such correlation exists but it would be a good idea to check for it since it sounds reasonable. However, I’m inclined to think it only works for calculations in which you don’t have polarizable atoms with too many basis functions or your correlation would be biased.
        It is worth considering. Let me know how it went!


  43. Dear sir, I want caliculate Fukui indices for flavone, how to caliculate? we have Gaussian09 software.

  44. Deepika Kathuria

    Can I use NBO charges for the calculation of NBO?

  45. Deepika Kathuria

    The next thing I would like to know for taking charges for N+1 and N-1 species do I have to optimise these species. Or directly i can optimise the molecule and then consider the single point state of N+1 and N-1 species?

    • Do not optimize the N-1 nor the N+1 states! The Fukui index is a response coefficient to the change in particles at a given state or PES. Optimizing this stated changes the PES and therefore your conclusions would have no correlation.

  46. Dear Professor,
    Is it possible to calculate Fukui indices for an anionic complex (charged negatively)?
    if, yes what would be the cationic and neutral states?

    Thanks in advance,

    Benoit De Lafort

    • Dear Benoit,

      It is perfectly possible. Now the anion (say it’s charge is -1) is your reference state now the f+ is calculated with -2 charge and the f- with zero charge.
      I hope this helps!

  47. Dear Professor,

    Thank you so much for your kind help.



  48. I quite like reading through a post that can make people
    think. Also, thank you for allowing for me to comment!

  49. Thanks for the description about Fukui indices. I came here as I read an article where Fucui indices were used to describe the reactivity. Hopefully, in future if Ill use them, your blog would help me.

  50. Hello Sir,
    I have optimised a cationic species having N electrons at the B3LYP/6-311G**(d, p) level of theory using Gaussian 09 program. The charge and multiplicity was 0 and 2 respectively and performed NBO analysis. We calculated Fukui function (fk+), to know the most electrophilic site in that cationic species. For doing so, we changed the charge = -1 and multiplicity = 1 for the N+1 system on the same optimised geometry. And finally, used the formula given below to calculate the fukui function:
    fk+ = qk(N+1)-qk(N)

    where qk is taken as the natural charge obtained on atom k after NBO analysis.
    For N+1 system
    Natural —
    Atom No Charge
    C 1 -0.08527
    C 2 0.15819

    For N system
    Atom No Charge
    C 1 -0.02604
    C 2 0.08278

    fk+= qk(N+1)-qk(N)

    F1+ = 6.08527-6.02604 = 0.05923
    F1+ = 5.84181-5.91722 = -0.07541
    This is how we had calculated fukui function. I need to know wether we have calculated it correct or not because I am really confused about qk.

    • Hello,

      There are some inconsistencies in your procedure.
      First, you claim that your molecule of interest is a cationic species but the charge you use is 0. This is a mistake, if you have a cationic species as a reference state (the N state) then you should use charge = 1 multiplicity = 2 (assuming this is an organic molecule); Therefore the N+1 state (to which you have added an electron) would have charge = 0 and multiplicity = 1 (This second calculation should be performed at the optimized geometry of the N state without further optimization).
      Now, the qk values in the equations written in the post are not charges but atomic populations, which only differ with charges in the value of Z the nuclear charge; from what I see you included Z = 6 to compensate.
      So, in short, the use of the fk expressions is appropriate but your numbers are incorrect due to the inconsistency of charges in the N and the N+1 states.

      I hope this helps. Have a nice day

  51. Hello Sir,
    If I am using charge = 1 and multiplicity = 2 for the reference cationic state (N), then link died and server message #2070 displayed on the screen. In gauss view, if we kept charge = 1 then system self took multiplicity = 1 and output file is generated with normal termination. I want to know this is correct or not, actually the results obtained are quite satisfactory. The NPA charges are calculated and fukui calculation displayed desired results.

  52. Thank you for instant reply.

  53. Good review! Is it possible to run a comparative Fukui indexes study between two molecules where just one atom changes (i.e acyl chloride vs. acyl fluoride) to explain and rationalize which one is more electrophilic?


    • Thank you Carlos!

      I was going to reply that I don’t think that is a formal comparison since the number of electrons is different, but then again you may compare population changes in the C atom bearing the halogen and see how it changes. So, yes, I think you can do it and in fact you have given me an idea I needed for my own work. Thanks!

      Have a nice day

  54. Hello,

    I am interested in calculating f+ for the phosphorus atom in a neutral molecule (using PCM solvation model) using Gaussian 09. I would like to ask for clarification on which value from the output file should be used for P(N+1) and P(N). For the anionic molecule I used charge = -1 and multiplicity = 2, and for the neutral molecule I used charge = 0 and multiplicity = 1 (using the same optimized neutral molecular geometry for both). NBO analysis was performed using the input keywords:
    #p b3lyp/6-311++g** scrf=(solvent=water) pop=nbo

    I am copying the section of the output files for “Summary of Natural Population Analysis” for the phosphorus atom in my molecule(s):
    Natural Population
    Natural ———————————————————-
    Atom No Charge Core Valence Rydberg Total
    P 1 2.33191 9.99851 2.54127 0.12831 12.66809

    Natural Population
    Natural ——————————————————–
    Atom No Charge Core Valence Rydberg Total
    P 1 1.17785 4.99911 1.25904 0.06400 6.32215

    Should the population values in the Fukui equation be taken from the final “Total” column, or should they be calculated as Nuclear Charge – “Natural Charge”?

    For the neutral molecule, the population values are the same either way.
    P(N) = 15 – 2.33191 = 12.66809

    But for the anionic molecule, the population value would be different depending on which way it’s calculated:
    P(N+1) = 6.32215 OR
    P(N+1) = 15 – 1.17785 = 13.82215

    Depending on which way I calculate P(N+1), the Fukui index is either positive or negative:
    f+ = P(N+1) – P(N) = 6.32215 – 12.66809 = -6.34594 OR
    f+ = 13.82215 – 12.66809 = 1.15406

    Please let me know which is the correct method. Thank you!

    • Dear Matt,
      Do try listing all “Summary of Natural Population Analysis” instances of anion radical in your output. Only 6.32 electrons (out of 16 for anion radical) means only alfa electrons are shown here.

      • Thanks for pointing that out! Previously, I had only done NBO calculations on neutral molecules. So I wasn’t aware that there were separate “Summary of Natural Population Analysis” sections for alpha and beta electrons in the output files for anions and cations.

    • Hello,

      thank you (all) for your explanations but to get back to the last question:

      “Should the population values in the Fukui equation be taken from the final “Total” column, or should they be calculated as Nuclear Charge – “Natural Charge”?”

      –> What is the correct answer?

      Thanks for your attention. I’m looking forward to your reply.

      • Hi Janek,

        Here is the relevant “Summary of Natural Population Analysis” from my output file that includes both alpha and beta electrons for the anion:

        Natural Population
        Natural ———————————————–
        Atom No Charge Core Valence Rydberg Total
        P 1 2.23131 9.99824 2.55968 0.21077 12.76869

        Thus, the population value for P for the anion is:
        P(N+1) = 15 – 2.23131 = 12.76869

        And then the Fukui index is:
        f+ = P(N+1) – P(N) = 12.76869 – 12.66809 = 0.1006

        In summary, the population values in the Fukui equation can be taken either from the final “Total” column, or they can be calculated as Nuclear Charge – “Natural Charge.” They should provide the exact same values as long as you’re using the correct “Summary of Natural Population Analysis” instances in the output files.

    • Michael Ludeña

      yo tambien quiero saber eso,,
      he tratado de replicar los valores de un paper y obtengo valores iguales pero de signo difrente nose por que

      se puede realizar directamente
      f+= 1.1778 – 2.3319= – 1.15406

      esto es correcto?

      I also want to know that.
      I have tried to replicate the values of a paper and I get the same values but with a different sign, I don’t know why

      can be done directly
      f + = 1.1778 – 2.3319 = – 1.15406

      this is correct?

  55. Thank you all for pitching in! Indeed when we go to the radical calculation for whatever state, the NBO analysis is performed separately for the alpha and beta electrons and after that the program adds both results for a total count which is the one we usually want to look at.
    There is always a confusion about whether using charges or populations and as Matt pointed out the nuclear charge always cancels itself out so it really doesn’t matter as long as you ALWAYS do it the same way.

    Best regards to all!

  56. Hi,

    I have a question related to the spin multiplicity setting for Fukui calculations in Gaussian 16. If my neutral molecule is a triplet, what should be the correct spin states for its anionic and cationic counterparts? Naively, I would choose doublet for both, but I am not sure.


    • Hello Hieu,

      If your reference (neutral) molecule is an organic radical in the triplet state then that means you have two unpaired electrons; your anion and cation would then add or remove an electron respectively, leaving just one unpaired electron and thus the multiplicity for the N+1 and N-1 states should be indeed a doublet.
      If your molecule involves a transition metal then the problem becomes more complicated for it implies knowledge about the ligand (high/low field or spin).

      I hope this helps!

      • Thanks for the prompt reply!

        My system actually consists of transition metal atoms bounded by oxygen (oxo). According to what I know, O_oxo is a weak field ligand, therefore the system is likely to have high spin. Does it mean that I should pick quartet for both N-1 and N+1 states, or does it also depend on the orbital filling of the transition metal of interest? If the later is true, would you suggest testing both possibilities (doublet and quartet) and pick out the lowest energy spins for Fukui calculations?

  57. Why bother? If people want to investigate the reactivities of different parts of a molecule, they can check the shapes of HOMO and LUMO in a visualiser, eg GaussView, and then they’ll be able to estimate the reactivities of different atoms. I don’t understand the need to calculate the ‘Fukui indices’.

    • Interesting perspective. I guess the quick answer would be that canonical delocalized HOMO/LUMO distributions while informative to us comp.chemists, they are not very informative to synthetic chemists who want to draw curly arrow mechanisms from specific atoms or knowing which specific atoms will undergo what kind of reaction. I can think of some other ‘needs’ or implications but I’ll save them for a later post 🙂

  58. Dear Pr. Joaquine Barosso,

    I would like to ask you about the NPA population when doing the Fukui calculations. For the cationic and anionic state of a molecule, in the output we can see there some tables (alpha and beta) of charge for each atom. Let’s say for the anionic one, we can use the charges from the table given the total (in which the charge is -1) or from the table at the end in which also the charge is -1 (alpha). Please I need a to make this clear in my mind.

    Best regards,


  59. Dear Professor,
    I get confused about setting the multiplicity. I want to calculate Fukui indices for an anionic complex (charged negatively -2). Now the f+ is calculated with -3 charge and the f- with -1 charge. What would be the corresponding multiplicities ?

  60. Hi.
    Is there a any difference between Fukui Indices and Condensed Fukui Indices? If that’s the case, which of the two would be better for one to calculate?

    By the way, this post is great and very helpful! Thanks!

  61. Dear Prof. Jaoquin,

    I have gained a lot from all your posts I consulted on this blog.

    I cannot but comment on this one regarding fukui function.

    Thanks so much for creating time to help junior researchers in this field.
    I believe many senior computational chemistry researchers are benefiting as well. Lol….


  1. Pingback: Python scripts for calculating Fukui Indexes | Dr. Joaquin Barroso's Blog

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