… but were afraid to ask
How I learned to stop worrying and not caring that much about hybridization.
The math behind orbital hybridization is fairly simple as I’ll try to show below, but first let me give my praise once again to the formidable Linus Pauling, whose creation of this model built a bridge between quantum mechanics and chemistry; I often say Pauling was the first Quantum Chemist (Gilbert N. Lewis’ fans, please settle down). Hybrid orbitals are therefore a way to create a basis that better suits the geometry formed by the bonds around a given atom and not the result of a process in which atomic orbitals transform themselves for better sterical fitting, or like I’ve said before, the C atom in CH4 is sp3 hybridized because CH4 is tetrahedral and not the other way around. Jack Simmons put it better in his book:
The atomic orbitals we all know and love are the set of solutions to the Schrödinger equation for the Hydrogen atom and more generally they are solutions to the hydrogen-like atoms for which the value of Z in the potential term of the Hamiltonian changes according to each element’s atomic number.
Since the Hamiltonian, and any other quantum mechanical operator for that matter, is a Hermitian operator, any given linear combination of wave functions that are solutions to it, will also be an acceptable solution. Therefore, since the 2s and 2p valence orbitals of Carbon do not point towards the edges of a tetrahedron they don’t offer a suitable basis for explaining the geometry of methane; even more so these atomic orbitals are not degenerate and there is no reason to assume all C-H bonds in methane aren’t equal. However we can come up with a linear combination of them that might and at the same time will be a solution to the Schrödinger equation of the hydrogen-like atom.
Ok, so we need four degenerate orbitals which we’ll name ζi and formulate them as linear combinations of the C atom valence orbitals:
ζ1= a12s + b12px + c12py + d12pz
ζ2= a22s + b22px + c22py + d22pz
ζ3= a32s + b32px + c32py + d32pz
ζ4= a42s + b42px + c42py + d42pz
to comply with equivalency lets set a1 = a2 = a3 = a4 and normalize them:
a12 + a22 + a32 + a42 = 1 ∴ ai = 1/√4
Lets take ζ1 to be directed along the z axis so b1 = c1 = 0
ζ1 = 1/√4(2s) + d12pz
since ζ1 must be normalized the sum of the squares of the coefficients is equal to 1:
1/4 + d12 = 1;
d1 = √3/2
Therefore the first hybrid orbital looks like:
ζ1 = 1/√4(2s) +√3/2(2pz)
We now set the second hybrid orbital on the xz plane, therefore c2 = 0
ζ2 = 1/√4(2s) + b22px + d22pz
since these hybrid orbitals must comply with all the conditions of atomic orbitals they should also be orthonormal:
〈ζ1|ζ2〉 = δ1,2 = 0
1/4 + d2√3/2 = 0
d2 = –1/2√3
our second hybrid orbital is almost complete, we are only missing the value of b2:
ζ2 = 1/√4(2s) +b22px +-1/2√3(2pz)
again we make use of the normalization condition:
1/4 + b22 + 1/12 = 1; b2 = √2/√3
Finally, our second hybrid orbital takes the following form:
ζ2 = 1/√4(2s) +√2/√3(2px) –1/√12(2pz)
The procedure to obtain the remaining two hybrid orbitals is the same but I’d like to stop here and analyze the relative direction ζ1 and ζ2 take from each other. To that end, we take the angular part of the hydrogen-like atomic orbitals involved in the linear combinations we just found. Let us remember the canonical form of atomic orbitals and explicitly show the spherical harmonic functions to which the 2s, 2px, and 2pz atomic orbitals correspond:
ψ2s = (1/4π)½R(r)
ψ2px = (3/4π)½sinθcosφR(r)
ψ2pz = (3/4π)½cosθR(r)
we substitute these in ζ2 and factorize R(r) and 1/√(4π)
ζ2 = (R(r)/√(4π))[1/√4 + √2 sinθcosφ –√3/√12cosθ]
We differentiate ζ2 respect to θ, and set it to zero to find the maximum value of θ respect to the z axis we get the angle between the first to hybrid orbitals ζ1 and ζ2 (remember that ζ1 is projected entirely over the z axis)
dζ2/dθ = (R(r)/√(4π))[√2 cosθ –√3/√12sinθ] = 0
sinθ/cosθ = tanθ = -√8
θ = -70.53°,
but since θ is measured from the z axis towards the xy plane this result is equivalent to the complementary angle 180.0° – 70.53° = 109.47° which is exactly the angle between the C-H bonds in methane we all know! and we didn’t need to invoke the unpairing of electrons in full orbitals, their promotion of any electron into empty orbitals nor the ‘reorganization‘ of said orbitals into new ones. Orbital hybridization is nothing but a mathematical tool to find a set of orbitals which comply with the experimental observation and that is the important thing here!
To summarize, you can take any number of orbitals and build any linear combination you want, in order to comply with the observed geometry. Furthermore, no matter what hybridization scheme you follow, you still take the entire orbital, you cannot take half of it because they are basis functions. That is why you should never believe that any atom exhibits something like an sp2.5 hybridization just because their bond angles lie between 109 and 120°. Take a vector v = xi+yj+zk, even if you specify it to be v = 1/2i that means x = 1/2, not that you took half of the unit vector i, and it doesn’t mean you took nothing of j and k but rather than y = z = 0.
This was a very lengthy post so please let me know if you read it all the way through by commenting, liking, or sharing. Thanks for reading.
Guillermo Caballero, a graduate student from this lab, has written this two-part post on the nuances to be considered when searching for transition states in the theoretical assessment of reaction mechanisms. He’s been quite successful in getting beautiful energy profiles for organic reaction mechanisms, some of which have even explained why some reactions do not occur! A paper in Tetrahedron has just been accepted but we’ll talk about it in another post. I wanted Guillermo to share his insight into this hard practice of computational chemistry so he wrote the following post. Enjoy!
Yes, finding a transition state (TS) can be one of the most challenging tasks in computational chemistry, it requires both a good choice of keywords in your route section and all of your chemical intuition as well. Herein I give you some good tricks when you have to find a transition state using Gaussian 09 Rev. D1
METHOD 1. The first option you should try is to use the opt=qst2 keyword. With this method you provide the structures of your reagents and your products, then the program uses the quadratic synchronous transit algorithm to find a possible transition state structure and then optimize it to a first order saddle point. Here is an example of the input file.
link 0 --blank line-- #p b3lyp/6-31G(d,p) opt=qst2 geom=connectivity freq=noraman --blank line-- Charge Multiplicity Coordinates of reagents --blank line-- Charge Multiplicity Coordinates of products --blank line---
It is mandatory that the numbering must be the same in the reagents and the products otherwise the calculation will crash. To verify that the label for a given atom is the same in reagents and products you can go to Edit, then Connection. This opens a new window were you can manually modify the numbering scheme. I suggest you to work in a split window in gaussview so you can see at the same time your reagents and products.
The keyword freq=noraman is used to calculate the frequencies for your optimized structure, it is important because for a TS you must only observe one imaginary frequency, if not, then that is not a TS and you have to use another method. It also occurs that despite you find a first order saddle point, the imaginary frequency does not correspond to the bond forming or bond breaking in your TS, thus, you should use another method. I will give you advice later in the text for when this happens. When you use the noraman in this keyword you are not calculating the Raman frequencies, which for the purpose of a TS is unnecessary and saves computing time. Frequency analysis MUST be performed AT THE VERY SAME LEVEL OF THEORY at which the optimization is performed.
The main advantage for using the qst2 option is that if your calculation is going to crash, it generally crashes at the beginning, in the moment of guessing your transition state structure. Once the program have a guess, it starts the optimization. I suggest you to ask the algorithm to calculate the force constants once, this generally improves on the convergence, it will take slightly more time depending on the size of your structure but it pays off. The keyword in the route section is opt=(qst2,calcfc). Indeed, I hardly encourage you to use the calcfc keyword in any optimization you want to run.
METHOD 2. If method 1 does not work, my next advice is to use the opt=ts keyword. For this method, the coordinates in your input file are those for the TS structure. Here is an example of the input file.
link 0 --blank line-- #p b3lyp/6-31G(d,p) opt=ts geom=connectivity freq=noraman --blank line-- Charge Multiplicity Coordinates of TS --blank line--
The question that arises here is how should I get the coordinates for my TS? Well, honestly this is not a trivial task, here is where you use all the chemistry you know. For example, you can start with the coordinates of your reagents and manually get them closer. If you are forming a bond whose length is to be 1.5Å, then I suggest you to have that length in 1.6Å in your TS. Sometimes this becomes trial and error but the most accurate your TS structure is, based on your chemical knowledge, the easiest to find your TS will be. As another example, if you want to find a TS for a [1,5]-sigmatropic reaction a good TS structure will be putting the hydrogen atom that migrates in the middle point through the way. I have to insist, this method hardly depends on your imagination to elucidate a TS and on your chemistry background.
Most of the time when you use the opt=ts keyword the calculations crashes because of an error in the number of eigenvalues, you can avoid it adding noeigen to the route section; here is an example of the input file, I encourage you to use this method.
link 0 --blank line-- #p b3lyp/6-31G(d,p) opt=(ts,noeigen,calcfc) geom=connectivity freq=noraman --blank line-- Charge Multiplicity Coordinates of TS --blank line--
If you have problems in the optimization steps I suggest you to ask the algorithm to calculate the force constants in every step of the optimization opt=(ts,noeigen,calcall) this is quite a harsh method because will consume long computing time but works well for small molecules and for complicated TSs to find.
Another ‘tricky’ way to get your coordinates for your TS is to run the qst2 calculation, then if it fails, take the second- or the third-step coordinates and used them as a ‘pre-optimized’ set of coordinates for this method.
By the way, here is another useful trick. If you are evaluating a group of TSs, let’s say, if you are varying a functional group among the group, focus on finding the TS for the simplest case, then use this optimized TS as a template where you add the moieties and use this this method. This works pretty well.
For this post we’ll leave it up to here and post the rest of Guillermo’s tricks and advice on finding TS structures next week when we’ll also discuss the use of IRC calculations and some considerations on energy corrections when plotting the full energy profile. In the mean time please take the time to rate, like and share this and other posts.
Thanks for reading!
Some atomic properties such as an atomic charge are isotropic, but every now and then some derivations of them become anisotropic, for example the plotting of the Molecular Electrostatic Potential (MEP) on the electron density surface can exhibit some anisotropic behavior; quantifying it can be a bit challenging.
It is well known that halogen atoms such as Chlorine can form so-called halogen-bonds of the type R-Cl-R in crystals with a near perfect 180° angle. This finding has lead to the discovery of σ-holes in halogens. σ-Holes are electrophilic portions of the anisotropic electrostatic potential in an otherwise nucleophilic atom. Recently, Guillermo “Memo” Caballero and I calculated the MEP for a series of trichloromethyl-containing compounds at the MP2/cc-pVQZ level of theory and the mapping shows evidence of such σ-holes as seen in Figure1. Those small blue portions on an otherwise red atom indicate that some electron density is missing in that position, which by the way is located at 180° away from the carbon atom.
But having the picture is not enough. We want to quantify just how strong are those σ-holes to effectively attract a nucleophile and perhaps perform some chemistry on the C-Cl bond. That’s when we resorted to MultiWFN, a Multifunctional Wavefunction Analyzer developed by Tian Lu (卢天) at the Beijing Kein Research Center for Natural Sciences. You can check the project leader list of publications here. Among many other capabilities, MultiWFN is able to print details about properties along a surface.
In order to work with MultiWFN you need to generate a *.wfn file, if you have a previous Gaussian calculation for which you want to analyze their surface you can run a guess=only calculation in order to extract the wavefunction from the checkpoint file. Here is a dummy of the input for such calculation
%chk=oldfile.chk # output=wfn geom=check guess=(only,read) density=current --blank-- Title Card --blank-- 0 1 --blank-- filename.wfn --blank--
In our case, having a post-Hartree-Fock calculation, the use of density=current is mandatory to get the MP2 density matrix and not just the HF one. Running this calculation will generate the file filename.wfn which is now used with MultiWFN. When starting MultiWFN you get to see a terminal window like the one below in which you are asked to input the path of your wfn file:
After loading it you will get the following window with the various options available. Type 12// (these two slashes are mandatory) to get the quantitative analysis of molecular surface option.
Then you will be asked to define some elements of that surface (we used the default options 0)
The following screenshot shows the results section in which several maxima and minima of electrostatic potential were found (7 and 11 in our case); a star is placed on the side of the global maximum. The value of the MEP at those points is given in Hartrees, eV and kcal/mol which I personally hate because there isn’t such a thing as a mole of ‘potentials’ (same argument as giving an orbital’s energy in kcal/mol, moles of what? orbitals? Personally, I don’t like it even if its valid).
Their visualizer is activated through the option 0 and although it is far from pretty it is quite good enough to find the numbers corresponding to maxima and minima of the MEP on the isodensity surface. If we look for the maxima then we find for our example (CHCl3) that a maximum is located in front of each Cl atom in a straight line from the C atom. Now we get to put a number on the mapped isosurface provided by Gaussian or even import the file into Chimera.
We are still working our way around MultiWFN, I hope we can find the batch option, it would be most useful. In the mean time, Guillermo and I will continue to search for σ-holes in chlorinated reagents. Thanks to Guillermo for his ongoing work in this and other topics within the realm of organic reactivity.
Have you any suggestions or ideas to work with MultiWFN? Please share them!
In a nutshell, computational chemistry models are about depicting, reproducing and predicting the electronic-based molecular reality. I had this conversation with my students last week and at some point I drew a parallel between them and art in terms of how such reality is approached.
Semi empirical methods
Prehistoric wall paintings depict a coarse aspect of reality without any detail but nevertheless we can draw some conclusions from the images. In the most sophisticated of these images, the cave paintings in Altamira, we can discern a bison, or could it be a bull? but definitely not a giraffe nor a whale, most in the same way Hückel´s method provides an ad hoc picture of π electron density without any regard of the σ portion of the electron density or the conformational possibilities (s-cis and s-trans 1,3-butadiene have the same Hückel description).
More sophisticated semi-empirical Hamiltonians like PM3 or PM6 have better parametrizations and hence yield better results. We are still replacing a lot of information for experimental or adjusted parameters but we still cannot truly adopt it as truthful. Take this pre-medieval painting of one of the first Kings of England, Aelred the Unready. It is, by today standards, a good children´s drawing and not a royal portrait, we now see more detail and can discern many more features yielding a better description of a human figure than those found in Altamira or Egypt.
HF is the simplest of ab initio methods, meaning that no experimental results or adjustable parameters are introduced. Even more so, from the HF equations for a multi-electron system that complies with Pauli’s exclusion principle the exchange operator arises as a new quantum feature of matter with no classical analogue. Still, there are some shortcomings. Correlation energy is disregarded and most results vary according to the basis set employed. Take the impressionist movement, specially in France: In Monet´s Lady with Umbrella we have a more complicated composition, we observe many more features and although we have a better description of color composition some details, like her face, remain obscure. The impressionists are characterized by their broad strokes, the thicker the strokes the harder it is to observe details similar to what happens in HF when we change from a small to a large basis set, respectively.
CI (Configurations Interaction)
Extension of HF to a multi-reference method yields better results. In CI we take the original guess wavefunction -as expressed through a Slater Determinant- and extend it with one or many more wavefunctions; thus a linear combination of Slater Determinants gives rise to a broader description of the ground state because other electronic configurations are involved to include more details like the ionic and covalent pictures (configurations). The more terms we include the more real the results feel. If we take classical figurative paintings we have a similar result; most of these paintings are constituted of many elements and the more realistically each element is captured the more real the whole composition looks even if some are just merely indicated.
CCSD(T) full-CI, CASPT2
In Edwards Much’s the scream, we might think we have lost some information again and went back to impressionism but we know this is actually an expressionist painting; we can now not only observe details of the figurative portion of the image but Munch has captured his subject´s fear in the form of distorsions on the subjective reality. In this way, CCSD(T), full-CI and CASPT2 methods provide a description of the ground as well as the excited states which -in experimental reality- are only accessed through a perturbation of the elecron density by electromagnetic radiation. Something resembling radiation has perturbated the subject in The Scream rendering him frightened and wondering how to return to his ground state or if such thing will be even possible.
Density Functional Methods
At least due to its widespread use, DFT has risen as the preferred method. One of the reasons behind its success is the reduced computing time when compared to previous ab initio methods. So DFT is pretty much like photography, in which reality is captured in full but only apparently after selecting a given lens, an exposition, a filter, shutter speed and the occasional Photoshop for correcting issues such as aliasing. In photography, as in DFT, all details concerning the procedure or method for capturing an uncanny reproduction of reality must be stated in every case for reproduction purposes.
Now, in the end it all comes down to Magritte’s Pipe. Ceci n’est pas une pipe -or, ‘this is not a pipe’- reminds us that painting as with modeling we don’t get reality but rather a depiction of it. In this famous painting we look at an image that in our heads resembles that of a pipe but we cannot grab it, fill it with tobacco and smoke it.
The image above is a digital file, which translated becomes a scaled reproduction of an image painted by Magritte in which we see the 2D projection of the image of an object that reminds us of a pipe. In fact, the real name of this work is The Treachery of Images, definitely quite an epistemology problem on perception and knowledge but before I get too metaphysical I should finish this post.
Can you find where cubism or surrealism should be placed? with MPn methods, perhaps?
As far as population analysis methods goes, the Quantum Theory of Atoms in Molecules (QTAIM) a.k.a Atoms in Molecules (AIM) has become a popular option for defining atomic properties in molecular systems, however, its calculation is a bit tricky and maybe not as straightforward as Mulliken’s or NBO.
Personally I find AIM a philosophical question since, after the introduction of the molecule concept by Stanislao Cannizzaro in 1860 (although previously developed by Amadeo Avogadro who was dead at the time of the Karlsruhe congress), the questions of whether or not an atom retains its identity when bound to others? where does an atom end and the next begins? What are the connections between atoms in a molecule? are truly interesting and far deeper than we usually consider because it takes a big mental leap to think about how matter is organized to give rise to substances. Particularly I’m very interested with the concept of a Molecular Graph which in turn is concerned with the way we “draw lines” to form conceptual molecules. Perhaps in a different post we can go into the detail of the method, which is based in the Laplacian operator of the electron density, but today, I just want to collect the basic steps in getting the most basic AIM answers for any given molecule. Recently, my good friend Pezhman Zarabadi-Poor and I have used rather extensively the following procedure. We hope to have a couple of manuscripts published later on. Therefore, I’ve asked Pezhman to write a sort of guest post on how to run AIMALL, which is our selected program for the integration algorithm.
The first thing we need is a WFN or WFX file, which contains the wavefunction in a Fortran unformatted file on which the Laplacian integration is to be performed. This is achieved in Gaussian09 by incluiding the keyword output=wfn or output=wfx in the route section and adding a name for this file at the bottom line of the input file, e.g.
(NOTE: WFX is an eXtended version of WFN; particularly necessary when using pseudopotentials or ECP’s)
Analyzing this file requires the use of a third party software such as AIMALL suite of programs, of which the standard version is free of charge upon registration to their website.
OpenAIMStudio (the accompanying graphical interface) and select the AIMQB program from the run menu as shown in figure 1.
Select your WFN/WFX file on which the calculation is to be run. (Figure 2)
You can control several options for the integration of the Laplacian of the electron density as well as other features. If your molecules are simple enough, you may go through with a successful and meaningful calculation using the default settings. After the calculation is finished, several result files are obtained. We’ll work in this tutorial only with *.mpgviz (which contains information about the molecular graph, MG) and *.sum (which contains all of needed numerical data).
Visualization of the MG yields different kinds of critical points, such as: 1) Nuclear Attractor Critical Points (NACP); 2) Bond Critical Points (BCP); 3) Ring CP’s (RCP); and 4) Cage CP’s (CCP).
Of the above, BCP are the ones that indicate the presence of a chemical bond between two atoms, although this conclusion is not without controversy as pointed out by Foroutan-Njead in his paper: C. Foroutan-Nejad, S. Shahbazian and R. Marek, Chemistry – A European Journal, 2014, 20, 10140-10152. However, at a first approximation, BCP’s can help us to explore chemical interactions.
Now, let’s go back to visualizing those MGs (in our examples we’ve used methane and ethylene and acetylene). We open the corresponding *.mpgviz file in AIMStudio and export the image from the file menu and using the save as picture option (figure 3).
The labeled atoms are NACP’s while the green dots correspond to BCP’s. Multiplicity of a bond cannot be discerned within the MG; in order to find out whether a bond is a single, double or triple bond we have to look into the *.sum file, in which we’ll take a look at the bond orders between pairs of atoms in the section labeled “Diatomic Electron Pair Contributions and Delocalization Data” (Figure 4).
Delocalization indexes, DI’s, show the approximate number of electrons shared between two atoms. From the above examples we get the following DI(C,C) values: 1.93 for C2H4 and 2.87 for C2H2; on the other hand, DI(C,H) values are 0.98 for CH4, 0.97 in C2H4 and 0.96 in C2H2. These are our usual bond orders.
This is the first part of a crash tutorial on AIM, in my opinion this is the very basics anyone needs to get started with this interesting and widespread method. Thanks to all who asked about QTAIM, now you have your long answer.
Thanks a lot to my good friend Dr Pezhman Zarabadi-Poor for providing this contribution to the blog, we hope you all find it helpful. Please share and comment.
A bit outside the scope of this blog (maybe), but just too cool to overlook. Augmented reality in chemistry education.
This is a guest post from Samantha Morra of EdTechTeacher.org, an advertiser on FreeTech4Teachers.com.
Augmented Reality (AR) blurs the line between the physical and digital world. Using cues or triggers, apps and websites can “augment” the physical experience with digital content such as audio, video and simulations. There are many benefits to using AR in education such as giving students opportunities to interact with items in ways that spark inquiry, experimentation, and creativity. There are a quite a few apps and sites working on AR and its application in education.
There are 6 physical paper cubes printed with different symbols from the periodic table. It takes a while to cut out and put together the cubes, but it…
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How to calculate the Delta G of solvation? This is a question that I get a lot in this blog, so it is about time I wrote a (mini)post on it, and at the same time put an end to this posting drought which has lasted for quite a few months due to a lot of pending work with which I’ve had to catch up. Therefore, this is another post in the series of SCRF calculations that are so popular in this blog. For the other posts on this subjects remember to click here and here.
SMD is the keyword you want to use when performing a Self Consistent Reaction Field (SCRF) calculation with G09. This keyword was only made available in this last version of the program and it corresponds to Truhlar’s and coworkers solvation model which is recommended by Gaussian itself as the preferred model to calculate Delta G of solvation. The syntax used is the standard way used in any other Gaussian input files as follows:
# 'route section keywords' SCRF=SMD
Separately, we must either perform a gas phase calculation or use the DoVacuum keyword within the same SCRF input, and then take the energy difference between gas phase and solvated models.
# 'route section keywords' SCRF=(SMD,DoVacuum)
No solvation or cavity model should be defined since, by definition, SMD will use the IEFPCM model which is a synonym for PCM.
As opposed to the previous versions of Gaussian, the output energy already contains all corrections, this is why we must take the difference between both values (remember to calculate them both at the same level of theory if calculated separately!). Nevertheless, when using the SMD keyword we get a separate line, just below the energy, stating the SMD-CDS non electrostatic value in kCal/mol.
The radii were also defined in the original paper by Truhlar; I’m not sure if using the keyword RADII with any of its options yields a different result or if it even ends in an error. Its worth the try!
Some calculation variations are not available when using SMD, such as Dis (calculation of the solute-solvent dispersion interaction energy), Rep (solute-solvent repulsion interaction energy) and Cav (inclusion of the solute cavitation energy in the total energy). I guess the reason for this might be that the SMD model is highly parametrized.
Have you found any issue with any item listed above? Pleases share your thoughts in the comments section below. As usual I hope this post was useful and that you all rate it, like it and comment.
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 😉
Calculating both Polarizability and the Hyperpolarizability in Gaussian is actually very easy and straightforward. However, interpreting the results requires a deeper understanding of the underlying physics of such phenomena. Herein I will try to describe the most common procedures for calculating both quantities in Gaussian09 and the way to interpret the results; if possible I will also try to address some of the most usual problems associated with their calculation.
The dipole moment of a molecule changes when is placed under a static electric field, and this change can be calculated as
pe = pe,0 + α:E + (1/2) β:EE + … (1)
where pe,0 is the dipole moment in the absence of an electric field; α is a second rank tensor called the polarizability tensor and β is the first in an infinite series of dipole hiperpolarizabilities. The molecular potential energy changes as well with the influence of an external field in the following way
U = U0 – pe.E – (1/2) α:EE – (1/6) β:EEE – … (2)
Route Section Keyword: Polar
This keyword requests calculation of the polarizability and, if available, hyperpolarizability for the molecule under study. This keyword is both available for DFT and HF methods. Hyperpolarizabilities are NOT available for methods that lack analytic derivatives, for example CCSD(T), QCISD, MP4 and other post Hartree-Fock methods.
Frequency dependent polarizabilities may be calculated by including CPHF=RdFreq in the route section and then specifying the frequency (expressed in Hartrees!!!) to which the calculation should be performed, after the molecule specification preceded by a blank line. Example:
#HF/6-31G(d) Polar CPHF=RdFreq Title Section Charge Multiplicity Molecular coordinates ==blank line== 0.15
In this example 0.15 is the frequency in Hartrees to which the calculation is to be performed. By default the output file will also include the static calculation, that is, ω = 0.0. Below you can find an example of the output when the CPHF=RdFreq is employed (taken from Gaussian’s website) Notice that the second section is performed at ω = 0.1 Ha
SCF Polarizability for W= 0.000000: 1 2 3 1 0.482729D+01 2 0.000000D+00 0.112001D+02 3 0.000000D+00 0.000000D+00 0.165696D+02 Isotropic polarizability for W= 0.000000 10.87 Bohr**3. SCF Polarizability for W= 0.100000: 1 2 3 1 0.491893D+01 2 0.000000D+00 0.115663D+02 3 0.000000D+00 0.000000D+00 0.171826D+02 Isotropic polarizability for W= 0.100000 11.22 Bohr**3.
You may have noticed now that the polarizabilities are expressed in volume units (Bohr^3) and the reason is the following:
Consider the simplest case of an atom with nuclear charge Q, radius r, and subjected to an electric field, E, which creates a force QE, and displaces the nucleus by a distance d. According to Gauss’ law this latter force is given by:
(dQ^2)/(4πεr^3) = QE (Hey! WordPress! I could really use an equation editor in here!)
if the polarizability is defined by Qd/E then we can rearrange the previous equation and yield
α = 4πεr^3 which in atomic units yields volume units, r^3, since 4πε = 1. This is why polarizabilities are usually referred to as ‘polarizability volumes’.
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The use of double zeta quality basis sets is paramount but it also makes these calculations more time consuming. Polarization functions on the basis set functions are a requirement for good results.
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