# Category Archives: Fukui

## Population Analysis in the Excited State with Gaussian

To calculate what the bonding properties of a molecule are in a particular excited state we can run any population analysis following the root of interest. This straightforward procedure takes two consecutive calculations since you don’t necessarily know before hand which excited state is the one of interest.

The regular Time Dependent Density Functional Theory (TD-DFT) calculation input with Gaussian 16 looks as follows (G09 works pretty much the same), let us assume we’ve already optimized the geometry of a given molecule:

```%OldChk=filename.chk
%nprocshared=16
%chk=filename_ES.chk

Title Card Required

0 1
--blank line--```

This input file retrieves the geometry and wavefunction from a previous calculation from filename.chk and doesn’t write anything new into it (that is what %OldChk=filename.chk means) and creates a new checkpoint where the excited states are calculated (%chk=filename_ES.chk)

In the output you search for the transition which peeks your interest; most often than not you’ll be interested in the one with the highest oscillator strength, f. The oscillator strength is a dimensionless number that represents the ratio of the observed, integrated, absorption coefficient to that calculated for a single electron in a three-dimensional harmonic potential [Harris & Bertolucci, Symmetry and Spectroscopy]; in other words, it is related to the probability of that transition to occur, and therefore it takes values from 0.0 to 1.0 (for single photon absorption processes.)

The output of this calculation looks as follows, the value of f for every excitation is reported together with its energy and the orbital transitions which comprise it.

``` Excitation energies and oscillator strengths:

Excited State   1:      Singlet-A      3.1085 eV  398.86 nm  f=0.0043  <S**2>=0.000
56 -> 59        -0.11230
58 -> 59         0.69339
This state for optimization and/or second-order correction.
Total Energy, E(TD-HF/TD-DFT) =  -1187.56377917
Copying the excited state density for this state as the 1-particle RhoCI density.

Excited State   2:      Singlet-A      4.0827 eV  303.68 nm  f=0.0016  <S**2>=0.000
52 -> 59         0.46689
52 -> 64        -0.20488
53 -> 59         0.19693
54 -> 59         0.40414
54 -> 64        -0.16261
...
...
Excited State   8:      Singlet-A      5.2345 eV  236.86 nm  f=0.8063  <S**2>=0.000
52 -> 60         0.17162
53 -> 59         0.47226
53 -> 60        -0.11771
54 -> 59        -0.27658
54 -> 60        -0.22006
55 -> 59         0.20496
56 -> 59         0.15029
```

Now we’ve selected excited state #8 because it has the largest value of f from the lot, we use the following input to read in the geometry from the old checkpoint file and we generate a new one in case we need it for something else. The input file for doing all this looks as follows (I’ve selected as usual the Natural Bond Orbital population analysis):

```%oldchk=a_ES.chk
%nprocshared=16
%chk=a_nbo.chk

Title Card Required

0 1

\$NBO BOAO BNDIDX E2PERT \$END

--blank line--

```

The flags at the bottom request the calculation of Wiberg Bond Indexes (BNDIDX) as well as Bond Order in the Atomic Orbital basis (BOAO) and a second order perturbation theory for the electronic delocalization (E2PERT). Now we can compare the population analysis between ground and the 8th excited state; check figure 1 and notice the differences in Wiberg’s bond order for this complex made of two molecules and one Na+ cation.

In this example we can observe that in the ground state we have a neutral and a negative molecule together with a Na+ cation, but when we analyze the population in the 8th excited state both molecules acquire a similar charge, ca. 0.46e, which means that some of the electron density has been transferred from the negative one to the neutral molecule, forming an Electron Donor-Acceptor complex (EDA) in the excited state.

This procedure can be extended to any other kind of population analysis and their derived combination, e.g. one could calculate their condensed fukui functions in the Nth excited state; but beware! These calculations yield vertical excitations, should the excited state of interest have a minimum we can first optimize the ES geometry and then perform the population analysis on said geometry; just add the opt keyword to perform both jobs in one go, but bear in mind that the NBO population analysis is performed before and after the optimization process so look for the tables and values closer to the end of the output file.

In the case of open shell systems the procedure is the same but one should be extremely careful in searching for the total population analysis since the output file contains this table for the alpha and beta populations separately as well as the added values for the total number of electrons.

## The HOMO-LUMO Gap in Open Shell Calculations. Meaningful or meaningless?

The HOMO – LUMO orbitals are central to the Frontier Molecular Orbital (FMO) Theory devised by Kenichi Fukui back in the fifties. The central tenet of the FMO theory resides on the idea that most of chemical reactivity is dominated by the interaction between these orbitals in an electron donor-acceptor pair, in which the most readily available electrons of the former arise from the HOMO and will land at the LUMO in the latter. The energy difference between the HOMO and LUMO of any chemical species, known as the HOMO-LUMO gap, is a very useful quantity for describing and understanding the photochemistry and photophysics of organic molecules since most of the electronic transitions in the UV-Vis region are dominated by the electron transfer between these two frontier orbitals.

But when we talk about Frontier Orbitals we’re usually referring to their doubly occupied version; in the case of open shell calculations the electron density with α spin is separate from the one with β spin, therefore giving rise to two separate sets of singly occupied orbitals and those in turn have a α-HOMO/LUMO and β-HOMO/LUMO, although SOMO (Singly Occupied Molecular Orbital) is the preferred nomenclature. Most people will then dismiss the HOMO/LUMO question for open shell systems as meaningless because ultimately we are dealing with two different sets of molecular orbitals. Usually the approach is to work backwards when investigating the optical transitions of a, say, organic radical, e.g. by calculating the transitions with such methods like TD-DFT (Time Dependent DFT) and look to the main orbital components of each within the set of α and β densities.

To the people who have asked me this question I strongly suggest to first try Restricted Open calculations, RODFT, which pair all electrons and treat them with identical orbitals and treat the unpaired ones independently. As a consequence, RO calculations and Unrestricted calculations vary due to variational freedom. RO calculations could yield wavefunctions with small to large values of spin contamination, so beware. Or just go straight to TDDFT calculations with hybrid orbitals which include a somewhat large percentage of HF exchange and polarized basis sets, but to always compare results to experimental values, if available, since DFT based calculations are Kohn-Sham orbitals which are defined for non-interacting electrons so the energy can be biased. Performing CI or CASSCF calculations is almost always prohibitive for systems of chemical interest but of course they would be the way to go.

## Python scripts for calculating Fukui Indexes

One of the most popular posts in this blog has to do with calculating Fukui indexes, however, when dealing with a large number of molecules, our described methodology can become cumbersome since it requires to manually extract the population analysis from two or three different output files and then performing the arithmetic on them separately with a spreadsheet or something.

Our new team member Ricardo Loaiza has written a python script that takes the three aforementioned files and yields a .csv file with the calculated Fukui indexes, and it even points out which of the atoms exhibit the largest values so if you have a large molecule you don’t have to manually check for them. We have also a batch version which takes all the files in any given directory and performs the Fukui calculations for each, provided it can find file triads with the naming requirements described below.

Output files must be named filename.log (the N electrons reference state), filename_plus.log (the state with N+1 electrons) and filename_minus.log (the N-1 electrons state). Another restriction is that so far these scripts only work with NBO population analysis as provided by the NBO3.1 program available in the various versions of Gaussian. I imagine the listing is similar in NBO5.x and NBO6.x and so it should work if you do the population analysis with them.

The syntax for the single molecule version is:

`python fukui.py filename.log filename_minus.log filename_plus.log`

For the batch version is:

`./fukuiPorLote.sh`

(Por Lote means In Batch in Spanish.)

These scripts are available via GitHub. We hope you find them useful, and you do please let us know whether here at the comments section or at our GitHub site.