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Orbital Contributions to Excited States


This is a guest post by our very own Gustavo “Gus” Mondragón whose work centers around the study of excited states chemistry of photosynthetic pigments.

When you’re calculating excited states (no matter the method you’re using, TD-DFT, CI-S(D), EOM-CCS(D)) the analysis of the orbital contributions to electronic transitions poses a challenge. In this post, I’m gonna guide you through the CI-singles excited states calculation and the analysis of the electronic transitions.

I’ll use adenine molecule for this post. After doing the corresponding geometry optimization by the method of your choice, you can do the excited states calculation. For this, I’ll use two methods: CI-Singles and TD-DFT.

The route section for the CI-Singles calculation looks as follows:

%chk=adenine.chk
%nprocshared=8
%mem=1Gb

#p CIS(NStates=10,singlets)/6-31G(d,p) geom=check guess=read scrf=(cpcm,solvent=water)

adenine excited states with CI-Singles method

0 1
--blank line--

I use the same geometry from the optimization step, and I request only for 10 singlet excited states. The CPCP implicit solvation model (solvent=water) is requested. If you want to do TD-DFT, the route section should look as follows:

%chk=adenine.chk
%nprocshared=8
%mem=1Gb

#p FUNCTIONAL/6-31G(d,p) TD(NStates=10,singlets) geom=check guess=read scrf=(cpcm,solvent=water)

adenine excited states with CI-Singles method

0 1
--blank line--

Where FUNCTIONAL is the DFT exchange-correlation functional of your choice. Here I strictly not recommend using B3LYP, but CAM-B3LYP is a noble choice to start.

Both calculations give to us the excited states information: excitation energy, oscillator strength (as f value), excitation wavelength and multiplicity:

Excitation energies and oscillator strengths:

 Excited State   1:      Singlet-A      6.3258 eV  196.00 nm  f=0.4830  <S**2>=0.000
      11 -> 39        -0.00130
      11 -> 42        -0.00129
      11 -> 43         0.00104
      11 -> 44        -0.00256
      11 -> 48         0.00129
      11 -> 49         0.00307
      11 -> 52        -0.00181
      11 -> 53         0.00100
      11 -> 57        -0.00167
      11 -> 59         0.00152
      11 -> 65         0.00177

The data below corresponds to all the electron transitions involved in this excited state. I have to cut all the electron transitions because there are a lot of them for all excited states. If you have done excited states calculations before, you realize that the HOMO-LUMO transition is always an important one, but not the only one to be considered. Here is when we calculate the Natural Transition Orbitals (NTO), by these orbitals we can analyze the electron transitions.

For the example, I’ll show you first the HOMO-LUMO transition in the first excited state of adenine. It appears in the long list as follows:

35 -> 36         0.65024

The 0.65024 value corresponds to the transition amplitude, but it doesn’t mean anything for excited state analysis. We must calculate the NTOs of an excited state from a new Gaussian input file, requesting from the checkpoint file we used to calculate excited states. The file looks as follows:

%Oldchk=adenine.chk
%chk=adNTO1.chk
%nproc=8
%mem=1Gb

#p SP geom=allcheck guess=(read,only) density=(Check,Transition=1) pop=(minimal,NTO,SaveNTO)

I want to say some important things right here for this last file. See that no level of theory is needed, all the calculation data is requested from the checkpoint file “adenine.chk”, and saved into the new checkpoint file “adNTO1.chk”, we must use the previous calculated density and specify the transition of interest, it means the excited state we want to analyze. As we don’t need to specify charge, multiplicity or even the comment line, this file finishes really fast.

After doing this last calculation, we use the new checkpoint file “adNTO1.chk” and we format it:

formchk -3 adNTO1.chk adNTO1.fchk

If we open this formatted checkpoint file with GaussView, chemcraft or the visualizer you want, we will see something interesting by watching he MOs diagram, as follows:

We can realize that frontier orbitals shows the same value of 0.88135, which means the real transition contribution to the first excited state. As these orbitals are contributing the most, we can plot them by using the cubegen routine:

cubegen 0 mo=homo adNTO1.fchk adHOMO.cub 0 h

This last command line is for plotting the equivalent as the HOMO orbital. If we want to plot he LUMO, just change the “homo” keyword for “lumo”, it doesn’t matter if it is written with capital letters or not.

You must realize that the Natural Transition Orbitals are quite different from Molecular Orbitals. For visual comparisson, I’ve printed also the molecular orbitals, given from the optimization and from excited states calculations, without calculating NTOs:

These are the molecular frontier orbitals, plotted with Chimera with 0.02 as the isovalue for both phase spaces:

The frontier NTOs look qualitatively the same, but that’s not necessarily always the case:

If we analyze these NTOs on a hole-electron model, the HOMO refers to the hole space and the LUMO refers to the electron space.

Maybe both orbitals look the same, but both frontier orbitals are quite different between them, and these last orbitals are the ones implied on first excited state of adenine. The electron transition will be reported as follows:

If I can do a graphic summary for this topic, it will be the next one:

NTOs analysis is useful no matter if you calculate excited states by using CIS(D), EOM-CCS(D), TD-DFT, CASSCF, or any of the excited states method of your election. These NTOs are useful for population analysis in excited states, but these calculations require another software, MultiWFN is an open-source code that allows you to do this analysis, and another one is called TheoDORE, which we’ll cover in a later post.

Natural Transition Orbitals (NTOs) Gaussian


The canonical molecular orbital depiction of an electronic transition is often a messy business in terms of a ‘chemical‘ interpretation of ‘which electrons‘ go from ‘which occupied orbitals‘ to ‘which virtual orbitals‘.

Natural Transition Orbitals provide a more intuitive picture of the orbitals, whether mixed or not, involved in any hole-particle excitation. This transformation is particularly useful when working with the excited states of molecules with extensively delocalized chromophores or multiple chromophoric sites. The elegance of the NTO method relies on its simplicity: separate unitary transformations are performed on the occupied and on the virtual set of orbitals in order to get a localized picture of the transition density matrix.

[1] R. L. Martin, J. Chem. Phys., 2003, DOI:10.1063/1.1558471.

In Gaussian09:
After running a TD-DFT calculation with the keyword TD(Nstates=n) (where n = number of states to be requested) we need to take that result and launch a new calculation for the NTOs but lets take it one step at a time. As an example here’s phenylalanine which was already optimized to a minimum at the B3LYP/6-31G(d,p) level of theory. If we take that geometry and launch a new calculation with the TD(Nstates=40) in the route section we obtain the UV-Vis spectra and the output looks like this (only the first three states are shown):

Excitation energies and oscillator strengths:

Excited State 1: Singlet-A 5.3875 eV 230.13 nm f=0.0015 <S**2>=0.000
42 -> 46 0.17123
42 -> 47 0.12277
43 -> 46 -0.40383
44 -> 45 0.50838
44 -> 47 0.11008
This state for optimization and/or second-order correction.
Total Energy, E(TD-HF/TD-KS) = -554.614073682
Copying the excited state density for this state as the 1-particle RhoCI density.

Excited State 2: Singlet-A 5.5137 eV 224.86 nm f=0.0138 <S**2>=0.000
41 -> 45 -0.20800
41 -> 47 0.24015
42 -> 45 0.32656
42 -> 46 0.10906
42 -> 47 -0.24401
43 -> 45 0.20598
43 -> 47 -0.14839
44 -> 45 -0.15344
44 -> 47 0.34182

Excited State 3: Singlet-A 5.9254 eV 209.24 nm f=0.0042 <S**2>=0.000
41 -> 45 0.11844
41 -> 47 -0.12539
42 -> 45 -0.10401
42 -> 47 0.16068
43 -> 45 -0.27532
43 -> 46 -0.11640
43 -> 47 0.16780
44 -> 45 -0.18555
44 -> 46 -0.29184
44 -> 47 0.43124

The oscillator strength is listed on each Excited State as “f” and it is a measure of the probability of that excitation to occur. If we look at the third one for this phenylalanine we see f=0.0042, a very low probability, but aside from that the following list shows what orbital transitions compose that excitation and with what energy, so the first line indicates a transition from orbital 41 (HOMO-3) to orbital 45 (LUMO); there are 10 such transitions composing that excitation, visualizing them all with canonical orbitals is not an intuitive picture, so lets try the NTO approach, we’re going to take excitation #10 for phenylalanine as an example just because it has a higher oscillation strength:

%chk=Excited State 10: Singlet-A 7.1048 eV 174.51 nm f=0.3651 <S**2>=0.000
41 -> 45 0.35347
41 -> 47 0.34685
42 -> 45 0.10215
42 -> 46 0.17248
42 -> 47 0.13523
43 -> 45 -0.26596
43 -> 47 -0.22995
44 -> 46 0.23277

Each set of NTOs for each transition must be calculated separately. First, copy you filename.chk file from the TD-DFT result to a new one and name it after the Nth state of interest as shown below (state 10 in this case). NOTE: In the route section, replace N with the number of the excitation of interest according to the results in filename.log. Run separately for each transition your interested in:

#chk=state10.chk

#p B3LYP/6-31G(d,p) Geom=AllCheck Guess=(Read,Only) Density=(Check,Transition=N) Pop=(Minimal,NTO,SaveNTO)

0 1
--blank line--

By requesting SaveNTO, the canonical orbitals in the state10.chk file are replaced with the NTOs for the 10th excitation, this makes it easier to plot since most visualizers just plot whatever set of orbitals they read in the chk file but if they find the canonical MOs then one would need to do some re-processing of them. This is much more straightforward.

Now we format our chk files into fchk with the formchk utility:

formchk -3 filename.chk filename.fchk
formchk -3 state10.chk state10.fchk

If we open filename.fchk (the file where the original TD-DFT calculation is located) with GaussView we can plot all orbitals involved in excited state number ten, those would be seven orbitals from 41 (HOMO-3) to 47 (LUMO+2) as shown in figure 1.

Figure 1. Canonical orbitals involved in the 10th excited state according to the TD-DFT calculation

If we now open state10.fchk we see that the numbers at the side of the orbitals are not their energy but their occupation number particular to this state of interest, so we only need to plot those with highest occupations, in our example those are orbitals 44 and 45 (HOMO and LUMO) which have occupations = 0.81186; you may include 43 and 46 (HOMO-1 and LUMO+1, respectively) for a much more complete description (occupations = 0.18223) but we’re still dealing with 4 orbitals instead of 7.

Figure 2. Natural Transition Orbitals for Phenylalanine. Orbital 44 (particle) and Orbital 45 (hole) exhibit the largest occupations for Excited State No. 10

The NTO transition 44 -> 45 is far easier to conceptualize than all the 10 combinations given in the canonical basis from the direct TD-DFT calculation. TD-DFT provides us with the correct transitions, NTOs just paint us a picture more readily available to the chemist mindset.

NOTE: for G09 revC and above, the %OldChk option is available, I haven’t personally tried it but using it to specify where the excitations are located and then write the NTOs of interest into a new chk file in the following way, thus eliminating the need of copying the original chk file for each state:

%OldChk=filename.chk
%chk=stateN.chk

NTOs are based on the Natural Hybrid orbitals vision by Löwdin and others, and it is said to be so straightforward that it has been re-discovered from time to time. Be that as it may, the NTO visualization provides a much clearer vision of the excitations occurring during a TD calculation.

Thanks for reading, stay home and stay safe during these harsh days everyone. Please share, rate and comment this and other posts.

A new paper on the Weak Link Approach


Chemically actuating a molecule is a very cool thing to do and the Weak Link Approach (WLA) allows us to do precisely that through the reversible coordination of one or various organometallic centers to a longer ligand that opens or closes a macrocyclic cavity. All this leads to an allosteric effect so important in biological instances available in inorganic molecules. Once again, the Mirkin group at Nortwestern University in Evanston, Illinois, has given me the opportunity to contribute with the calculations to the energetic properties of these actuators as well as their electronic properties for their use as molecular scavengers or selective capsules for various purposes such as drug delivery agents.

As in the previous WLA work (full paper), the NBODel procedure was used at the B97D/LANL2DZ level of theory, only this time the macrocycle consisted of two organometallic centers and for the first time the asymmetric opening of the cavity was achieved, as observed by NMR. With the given fragments, all possibilities shown in scheme 1 were obtained. The calculated bond energies for the Pt – S bonds are around 60 – 70 kcal/mol whereas for the Pt – Cl bonds the values are closer to 90 kcal/mol. This allows for a selective opening of the cavity which can then be closed by removing the chlorine atoms with the help of silver salts.

wla

For the case of complex mixture 4a, 4b, and 4c, the thermochemistry calculations show they are all basically isoenergetic with differences in the thousandths of kcal/mol. The possibilities for the groups in the weakly bonded ligands are enormous; currently, there is work being done about substituting those phenyl rings for calix[4]arenes in order to have a macrucyclic capsule made by macrocylic capusules.

Thanks to Andrea D’Aquino for taking me into her project, for all the stimulating discussions and her great ideas for expanding WLA into new avenues; I’m sure she’ll succeed in surprising us with more possibilities for these allosteric macrocycles.

The full paper is published in Inorganic Chemistry from the ACS (DOI: 10.1021/acs.inorgchem.7b02745). Thanks for reading and -if you made it this far into the post- happy new year!

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.

Tribology – New paper in JPC A


Tribology isn’t exactly an area with which us chemists are most familiar, yet chemistry has a great impact on this branch of physics of high industrial importance. Tribology is basically the science which studies the causes and consequences of friction between surfaces. 

The plastic bag industry requires the use of chemical additives to reduce the electrostatic adherence between sheets of plastic. My good old friend Dr. Armando Gama has studied through Dissipative Particle Dynamics (DPD) coarse-grained simulations the friction coefficients of having two slightly different molecules: erukamide and behenamide, which only differ in the presence of a double bond between carbon atoms 12 and 13 (Fig1).

picture1

Fig 1

In order to study the electronic aspects that give rise to different tribological effects in these very similar molecules, four chains of each kind were bounded to a frozen graphene surface (four bonds apart to prevent steric crowding) and were optimized at the B97D/6-31G(d,p) level of theory.

 

 

Double bonds in erukamide pile together through pi-pi stacking interactions (Fig2) which are absent in behenamide which is why these last ones are able to slide better between each other (Fig3). Interaction energies calculated for the inner chains at the same level of theory are 44.21 and 34.46 kcal/mol for erukamide and behenamide, respectively. As per the suggestion of a referee we extended the calculations to a 2D system by placing seven molecules on graphene, which once again was kept at the optimized geometry of its isolated state, at four bonds of separation in order to prevent steric crowding (Fig 4).

picture4

Fig 4

This calculations clearly represent a limit case with a high density covering of the surface, but they nevertheless reflect the observed trend that behenamide works better than erukamide in reducing the static friction coefficient between sheets.

The paper is now available at JPC-A. Thanks to Dr. Gama for this great opportunity to work with his team, I know it wont be the last.

A new chemist is graduated


It is with great pleasure that I announce the graduation of another member of our research group: Luis Enrique “Kike” Aguilar defended his BSc thesis yesterday and is now counting the days left for the Autumn when he’ll move to the Netherlands for a masters in computational chemistry.

Luis Enrique, Kike, calculated the interaction energies of 144 different inclusion complexes where calix and thia-calix[n]arenes were once again the chosen hosts (36 of them) and two drugs for the treatment of chronic myeloid leukemia (CML), namely Sorafenib and Bosutinib, were the guests.

The publication of the corresponding article in which we once again were fortunate enough to count with the collaboration of Dr. Rodrigo Galindo from Utah University in the molecular dynamics section, is still pending but we’re confident enough that it wont take much longer until it’s out there.

Kike is a very diligent student with great learning skills, I’m sure he’ll succeed in any enterprise he sets himself off.  Congratulations, Kike! Thanks for being a part of our research but more importantly for being a part of our community.


    

New paper in Journal of Chemical Theory and Computation


Happy new year to all my readers!

Having a new paper published is always a matter of happiness for this computational chemist but this time I’m excedingly excited about anouncing the publishing of a paper in the Journal of Chemical Theory and Computation, which is my highest ranked publication so far! It also establishes the consolidation of our research group at CCIQS as a solid and competitive group within the field of theoretical and computational chemistry. The title of our paper is “In Silico design of monomolecular drug carriers for the tyrosine kinase inhibitor drug Imatinib based on calix- and thiacalix[n]arene host molecules. A DFT and Molecular Dynamics study“.

In this article we aimed towards finding a suitable (thia-) calix[n]arene based drug delivery agent for the drug Imatinib (Gleevec by Novartis), which is a broadly used powerful Tyrosine Kinase III inhibitor used in the treatment of Chronic Myeloid Leukaemia and, to a lesser extent, Gastrointestinal Stromal Tumors; although Imatinib (IMB) exhibits a bioavailability close to 90% most of it is excreted, becomes bound to serum proteins or gets accumulated in other tissues such as the heart causing several undesired side effects which ultimately limit its use. By using a molecular capsule we can increase the molecular weight of the drug thus increasing its retention, and at the same time we can prevent Imatinib to bind, in its active form, to undesired proteins.

We suggested 36 different calix and thia-calix[n]arenes (CX) as possible candidates; IMB-CX complexes were manually docked and then optimized at the B97D/6-31G(d,p) level of theory; Stephan Grimme’s B97D functional was selected for its inclusion of dispersion terms, so important in describing π-π interactions. Intermolecular interaction energies were calculated under the Natural Bond Order approximation; a stable complex was needed but a too stable complex would never deliver its drug payload! This brings us to the next part of the study. A monomolecular drug delivery agent must be able to form a stable complex with the drug but it must also be able to release it. Molecular Dynamics simulations (+100 ns) and umbrella sampling methods were used to analyse the release of the drug into the aqueous media.

Optimized geometries for all complexes under study (B97D/6-31G*)

Optimized geometries for the 20 most stable complexes under study (B97D/6-31G*)

Potential Mean Force profiles for the four most stable complexes for position N1 and  N2 from the QM simulations are shown below (Red, complexes in the N1 position, blue, N2 position). These plots, derived from the MD simulations  give us an idea of the final destination of the drug respect of the calixarene carrier. In the next image, the three preferred structures (rotaxane-like; inside; released) for the final outcome of the delivery process are shown. The stability of the complexes was also assessed by calculating the values of ΔG binding through the use of the Poisson equations.

PMF for the most stable compounds

PMF for the most stable compounds

General MD simulation final structures

General MD simulation final structures

Thanks to my co-authors Maria Eugenia Sandoval-Salinas and Dr. Rodrigo Galindo-Murillo for their enormous contributions to this work; without their hard work and commitment to the project this paper wouldn’t have been possible.

First graduated student!


Today is truly a landmark in our lab because on this day, María Eugenia “Maru” Sandoval-Salinas has defended her thesis and has thus obtained her B. Sc. in Chemistry. She is the first student under my supervision to achieve this goal, and I hope it won’t be long until we get some more, although now the bar has been set quite high. For the time being, Maru is pursuing a career in the pharmaceutical industry but has every intention of coming back to the lab for her Masters degree; she has a reserved spot here with us at CCIQS.

Hard work pays off - We wait for you to come back for your Masters Degree!

Hard work pays off – We wait for you to come back for your Masters Degree!

Maru’s thesis deals mainly, but not exclusively, with calculating the interaction energies of calix- and thia-calix[n]arenes with the tyrosine kinase inhibitor Imatinib, which is widely used in the treatment of Chronic Myeloid Leukemia (CML), in order to rationally design a drug delivery agent for this drug. Her work is (a huge) part of an article currently under revision that I only wish had been published before her defense. Still, we await for that paper to be published in the next few weeks.

Throughout her stay at our lab, Maru was a dedicated student willing to learn new skills every time. As she replied today to one of the questions: “it’s not so much how many calculations I got right, but how many I got wrong!“. I find deep meaning in this sentence, perhaps deep enough as to consider it an aphorism, because indeed the more we try the more we fail, and the more we fail the more we learn and the closer we get to success.

Congratulations, Maru! I personally thank you for all the hard work invested in your thesis, all the long hours in front of the computer and your disposition to learn and work during the last 1.5 years. I’m certain you’ll find success in any venture you undertake; and I’m certain of it because you never stop trying.

Taking the Oath after being unanimously approved

Taking the Oath after being unanimously approved

Natural Bond Orbitals (NBO) Visualization with Chemcraft


It’s been a long time since I last posted something and so many things have happened in our research group! I should catch up with them in short but times have just been quite hectic.

I’m glad to publicly thank Prof. Frank Weinhold’s gesture to include this blog in the bibliography section of the new NBO6.0 website under the NBO-Related Websites tab.

Here is a contribution from Igor Marques at the University of Aveiro in Portugal (Group Website); the original text can be found as a comment in the original NBO Visualization post but it is pretty much the same thing you can find in this post. Here is a link to Chemcraft’s website. Thanks for sharing this, Igor!

=> Examples provided by Igor Marques used Chemcraft Version 1.7, build 365 <=

In the Gaussian input, with the NBORead option included under the population keyword, we should include the PLOT option as illustrated below. The gfoldprint keyword will print the basis set to the output file in the old G03 format. Some visualization programs require a certain format of the basis set to be printed to the output file in order to plot orbitals and other surfaces like the electron density; therefore, if you want to play safe, use gfoldprint, gfprint and gfinput in the same line. gfprint will print the basis set as a list but in the new G09 format, whereas gfinput will print the basis set using Gaussian’s own input format. (The used level of theory and number of shared processors are shown as illustrations only; also the Opt keyword is not fundamental to the visualization of the NBO’s)

%chk=filename.chk
%nprocshared=8

#P b3lyp/6-311++g** Opt pop=(full,nboread) gfoldprint 
filename

0 1
molecular coordinates
$NBO BNDIDX PLOT $END

this will generate files from *.31 to *.41
For the visualization of NBOs, you’ll need FILE.31 and FILE.37. Open FILE.31 from chemcraft. It will automatically detect FILE.37 (if in the same directory).

Tools > Orbitals > Render molecular orbitals

select the NBOs of interest (whcih are in the same order of the output),

Adjust settings > OK

On the left side of the window, select the NBO of interest and then click on ‘show isosurface’. Adjust the remaining settings. To represent another orbital, click on ‘keep this surface’ and then select another orbital from the rendered set and follow the previous steps.

Some Considerations:

> It’s possible to open a formated checkpoint file, containing the NBOs, in chemcraft.
Gaussian input:

%Chk=filename.chk
%nprocshared=4
#P b3lyp/6-311++g** Opt pop=(full,nboread,savenbo) gfoldprint 
filename

0 1
molecular coordinates
$NBO BNDIDX $END

the procedure is identical, but it is only necessary to read the *fchk file and then render the desired orbitals.
However, two problems might arise:
a) Orbitals in the checkpoint are reordered, thus requiring some careful inspection of the output.
b) Sometimes, for a larger molecule, the checkpoint might not be properly saved and the Gaussian job (as previously reported – http://goo.gl/DrSgA ) will end with:

Failed in SchOr1 in NBStor.
Error termination via Lnk1e in /data/programs/g09/l607.exe at Wed Mar 6 15:27:33 2013.

****

As usual, thanks to all for reading/commenting/rating this and other posts in this blog!

The Local Bond Order, LBO (Barroso et al. 2004)


I don’t know why I haven’t written about the Local Bond Order (LBO) before! And a few days ago when I thought about it my immediate reaction was to shy away from it since it would constitute a blatant self-promotion attempt; but hell! this is my blog! A place I’ve created for my blatant self-promotion! So without further ado, I hereby present to you one of my own original contributions to Theoretical Chemistry.

During the course of my graduate years I grew interested in weakly bonded inorganic systems, namely those with secondary interactions in bidentate ligands such as xanthates, dithiocarboxylates, dithiocarbamates and so on. Description of the resulting geometries around the central metallic atom involved the invocation of secondary interactions defined purely by geometrical parameters (Alcock, 1972) in which these were defined as present if the interatomic distance was longer than the sum of their covalent radii and yet smaller than the sum of their van der Waals radii. This definition is subject to a lot of constrictions such as the accuracy of the measurement, which in turn is related to the quality of the monocrystal used in the X-ray difraction experiment; the used definition of covalent radii (Pauling, Bondi, etc.); and most importantly, it doesn’t shed light on the roles of crystal packing, intermolecular contacts, and the energetics of the interaction.

This is why in 2004 we developed a simple yet useful definition of bond order which could account for a single molecule in vacuo the strength and relevance of the secondary interaction, relative to the well defined covalent bonds.

Barroso-Flores, J. et al. Journal of Organometallic Chemistry 689 (2004) 2096–2102
http://dx.doi.org/10.1016/j.jorganchem.2004.03.035,

Let a Molecular Orbital be defined as a wavefunction ψi which in turn may be constructed by a linear combination of Atomic Orbitals (or atom centered basis set functions) φj

We define ζLBO in the following way, where we explicitly take into account a doubly occupied orbital (hence the multiplication by 2) and therefore we are assuming a closed shell configuration in the Restricted formalism.

The summation is carried over all the orbitals which belong to atom A1 and those of atom A2.
Simplifying we yield,

where Sjk is the overlap integral for the φj and φk functions.

By summing over all i MOs we have accomplished with this definition to project all the MO’s onto the space of those functions centered on atoms A1 and A2. This definition is purely quantum mechanical in nature and is independent from any geometric requirement of such interacting atoms (i.e. interatomic distance) thus can be used as a complement to the internuclear distance argument to assess the interaction between them. This definition also results very simple and easy to calculate for all you need are the coefficients to the LCAO expansion and the respective overlap integrals.

Unfortunately, the Local Bond Order hasn’t found much echo, partly due to the fact that it is hidden in a missapropriate journal. I hope someone finds it interesting and useful; if so, don’t forget to cite it appropriately 😉

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