# Category Archives: Mathematics

## No, seriously, why can’t orbitals be observed?

The concept of electronic orbital has become such a useful and engraved tool in understanding chemical structure and reactivity that it has almost become one of those things whose original meaning has been lost and replaced for a utilitarian concept, one which is not bad in itself but that may lead to some wrong conclusions when certain fundamental facts are overlooked.

Last week a wrote -what I thought was- a humorous post on this topic because a couple of weeks ago a viewpoint in JPC-A was published by Pham and Gordon on the possibility of observing molecular orbitals through microscopy methods, which elicited a ‘seriously? again?‘ reaction from me, since I distinctly remember the Nature article by Zuo from the year 2000 when I just had entered graduate school. The article is titled “direct observation of d-orbital holes.” We discussed this paper in class and the discussion it prompted was very interesting at various levels: for starters, the allegedly observed d-orbital was strikingly similar to a dz2, which we had learned in class (thanks, prof. Carlos Amador!) that is actually a linear combination of d(z2-x2) and d(z2-y2) orbitals, a mathematical -lets say- trick to conform to spectroscopic observations.

Pham and Gordon are pretty clear in their first paragraph: “The wave function amplitude Ψ*Ψ is interpreted as the probability density. All observable atomic or molecular properties are determined by the probability and a corresponding quantum mechanical operator, not by the wave function itself. Wave functions, even exact wave functions, are not observables.” There is even another problem, about which I wrote a post long time ago: orbitals are non-unique, this means that I could get a set of orbitals by solving the Schrödinger equation for any given molecule and then perform a unit transformation on them (such as renormalizing them, re-orthonormalizing them to get a localized version, or even hybridizing them) and the electronic density derived from them would be the same! In quantum mechanical terms this means that the probability density associated with the wave function internal product, Ψ*Ψ, is not changed upon unit transformations; why then would a specific version be “observed” under a microscope? As Pham and Gordon state more eloquently it has to do with the Density of States (DOS) rather than with the orbitals. Furthermore, an orbital, or more precisely a spinorbital, is conveniently (in math terms) separated into a radial, an angular and a spin component R(r)Ylm(θ,φ)σ(α,β) with the angular part given by the spherical harmonic functions Ylm(θ,φ), which in turn -when plotted in spherical coordinates- create the famous lobes we all chemists know and love. Zuo’s observation claim was based on the resemblance of the observed density to the angular part of an atomic orbital. Another thing, orbitals have phases, no experimental observation claims to have resolved those.

Now, I may be entering a dangerous comparison but, can you observe a 2? If you say you just did, well, that “2” is just a symbol used to represent a quantity: two, the cardinality of a set containing two elements. You might as well depict such quantity as “II” or “⋅⋅” but still cannot observe “a two”. (If any mathematician is reading this, please, be gentle.) I know a number and a function are different, sorry if I’m just rambling here and overextending a metaphor.

Pretending to having observed an orbital through direct experimental methods is to neglect the Born interpretation of the wave function, Heisenberg’s uncertainty principle and even Schrödinger’s cat! (I know, I know, Schrödinger came up with this gedankenexperiment in order to refute the Copenhagen interpretation of quantum mechanics, but it seems like after all the cat is still not out of the box!)

So, the take home message from the viewpoint in JPC is that molecular properties are defined by the expected values of a given wave function for a specific quantum mechanical operator of the property under investigation and not from the wave function itself. Wave functions are not observables and although some imaging techniques seem to accomplish a formidable task the physical impossibility hints to a misinterpretation of facts.

I think I’ll write more about this in a future post but for now, my take home message is to keep in mind that orbitals are wave functions and therefore are not more observable (as in imaging) than a partition function is in statistical mechanics.

## Dealing with Spin Contamination

Most organic chemistry deals with closed shell calculations, but every once in a while you want to calculate carbenes, free radicals or radical transition states coming from a homolytic bond break, which means your structure is now open shell.

Closed shell systems are characterized by having doubly occupied molecular orbitals, that is to say the calculation is ‘restricted’: Two electrons with opposite spin occupy the same orbital. In open shell systems, unrestricted calculations have a complete set of orbitals for the electrons with alpha spin and another set for those with beta spin. Spin contamination arises from the fact that wavefunctions obtained from unrestricted calculations are no longer eigenfunctions of the total spin operator <S^2>. In other words, one obtains an artificial mixture of spin states; up until now we’re dealing only with single reference methods. With each step of the SCF procedure the value of <S^2> is calculated and compared to s(s+1) where s is half the number of unpaired electrons (0.75 for a radical and 2.0 for triplets, and so on); if a large deviation between these two numbers is found, the then calculation stops.

Gaussian includes an annihilation step during SCF to reduce the amount of spin contamination but it’s not 100% reliable. Spin contaminated wavefunctions aren’t reliable and lead to errors in geometries, energies and population analyses.

One solution to overcome spin contamination is using Restricted Open Shell calculations (ROHF, ROMP2, etc.) for which singly occupied orbitals is used for the unpaired electrons and doubly occupied ones for the rest. These calculations are far more expensive than the unrestricted ones and energies for the unpaired electrons (the interesting ones) are unreliable, specially spin polarization is lost since dynamical correlation is hardly accounted for. The IOP(5/14=2) in Gaussian uses the annihilated wavefunction for the population analysis if acceptable but since Mulliken’s method is not reliable either I don’t advice it anyway.

The case of DFT is different since rho.alpha and rho.beta can be separated (similarly to the case of unrestricted ab initio calculations), but the fact that both densities are built of Kohn-Sham orbitals and not true canonical orbitals, compensates the contamination somehow. That is not to say that it never shows up in DFT calculations but it is usually less severe, of course for the case of hybrid functional the more HF exchange is included the more important spin contamination may become.

So, in short, for spin contaminated wavefunctions you want to change from restricted to unrestricted and if that doesn’t work then move to Restricted Open Shell; if using DFT you can use the same scheme and also try changing from hybrid to pure orbitals at the cost of CPU time. There is a last option which is using spin projection methods but I’ll discuss that in a following post.

## Atoms in Molecules (QTAIM) – Flash lesson

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.

`filename.wfn`

(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.

Figure 1

Select your WFN/WFX file on which the calculation is to be run. (Figure 2)

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).

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).

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.

## Article in ‘Ciencia y Desarrollo’ (Science and Development)

Here is a link to an article I was invited to write by my good old friend, Dr. Eddie López-Honorato from CINVESTAV – Saltillo; Mexico, for the latest issue of the journal ‘Ciencia y Desarrollo’ (Science and Development) to which he was a guest editor. ‘Ciencia y Desarrollo’ is a popular science magazine edited by the National Council for Science and Technology (CONACyT) of which I’ve blogged before.

This magazine is intended for people interested in science with a general knowledge of it but not necessarily specialized in any field. With that in mind, I decided to write about the power of computational chemistry in predicting some phenomena while shedding light in certain aspects of chemistry that are not that readily available through experiments. The article is titled ‘Chemistry without flasks: Simulating chemical reactions‘. The link will take you to the magazine’s website which is in Spanish, as is the article itself, and only to the first page; so, below I translated the piece for anyone who could be interested in reading it (Hope I’m not infringing any copyright laws!). Don’t forget to also check out Dr. López-Honorato’s blog on nuclear energy research and the development of materials for nuclear waste containment! Encourage him to blog more often by liking and following his blog.

Typically we think of a chemist as a scientist who, dressed in a white robe and protected with safety glasses and latex gloves, busily working within a laboratory, surrounded by measurement equipment, glassware and bottles with colored substances; pours one substance onto other substance, transforming them into new substances while noting that the chemical reaction occurs through color changes, heat release , perhaps gas, and occasionally even an explosion.
Thus chemistry, the study of the material processing involves active experimentation to accomplish chemical reactions subsequently confirmed, although indirectly, that the changes have been conducted in the microscopic world, moreover, in the molecular and atomic world. The chemist plans these changes based on the knowledge he has of the chemical properties of the substances of which he started and, like any other substance, are due to its molecular structure, i.e., the spatial arrangement of the atoms that form it.

Under this archetypal image just posed, then it’s at least funny to think that there is a branch of chemistry named Theoretical Chemistry.

What is theoretical chemistry?
Theoretical chemistry is a kind of bridge between chemistry and physics; using laws and equations that govern the subatomic world, to calculate the molecular structure of a substance, more specifically calculate the distribution of electrons surrounding the molecule forming a cloud, which interact with the electron cloud of another molecule to form a new substance. It is based on the knowledge of the electron density cloud or we can understand and predict the chemical properties of any substance. We can then define theoretical chemistry as the set of physical theories that describe the distribution and properties of the electron cloud belonging to a molecule, in this particular mathematical description we call electronic structure and this is the starting point for descriptions and chemical predictions.

What is it good for?
Through theoretical chemistry we can find answers to fundamental questions about the structure of matter. Consider a molecule of water, which has the chemical formula H2O. This formula implies that there are two hydrogen atoms attached to an oxygen atom But what spacial structure does a water molecule have? The simplest geometry it could take would be a linear structure, in which the angle formed by the three atoms is 180 °. However, the water molecule has an angle of 109 °, far from a linear structure. In Figure 2 we can see the result of the calculation of the electronic structure of H2O, it observed that the electron cloud that exists on the oxygen atom also has a place in space and thus push the hydrogen atoms bringing them together instead of allowing them to take a  more comfortable conformation.

Figure 2. Oxygen remaining electrons (red cloud around the oxygen atom) that are not chemically push the hydrogen atoms towards each other.

The industrial area currently impacted by the application of theoretical chemistry is the pharmacist, as they generate a new drug involves significant investment in financial and human resources, so predicting the properties of a molecule with pharmacological activity before synthesizing is highly attractive. Therefore it has been generated within the theoretical chemistry field, otherwise known as branch Rational Drug Design.
Drugs acting on our organism when active molecules interact directly with the various proteins which are distributed in the tissue cells. If the structure of the protein is known and we attack is known also a drug which acts on it, then we can design similar drugs having greater efficacy in the treatment of diseases. But it is not only fit one molecule to another, but to calculate the energy of interaction, the energy of dissolution and the probability that this interaction can be observed experimentally (Figure 3). The calculation of the interaction energy between the drug and the protein tells how strongly attract each other, a weak attraction drug will result in a low efficiency, while a greater attraction involve a more effective drug.

How do you calculate a molecule?

All matter exists in the universe is made of atoms, which in turn are composed of a nucleus of protons and neutrons surrounded by a cloud of electrons. When two or more atoms combine to form a molecule combining do their electron clouds and how do these combinations are best described by the equations of quantum mechanics, the branch of physics that describes the behavior of the subatomic world. However, due to its complexity, the equations of quantum mechanics can only be accurately resolved in the simplest cases such as the hydrogen atom, which consists of a single electron orbiting a proton. We must therefore resort to a range of methods and approaches to tackle cases of chemical interest and even biological.
For years the only available computers could solve the approximate equations for small molecules, no later than thirty atoms, which which can be interesting, but not entirely useful. Today modern supercomputing equipment (which may amount to up to tens or even hundreds of powerful computers connected together to work cooperatively) allow us to make models with hundreds of atoms molecules such as proteins or DNA fragments.
While the software available to perform these calculations is developed continuously for the last thirty years has been the progress in the design of computer systems able to perform thousands of operations per second the cornerstone that has made the theoretical chemistry a predictive tool commonly used. Today the branch known as Molecular Dynamics, which studies the interactions between molecules over time, has benefited from the development of the latest game consoles, as their processors, known as graphics processing units (GPUs , for its acronym in English) are able to perform calculations in parallel: Many of the images seen in our video games are actually calculated, not animated, this means that the console must calculate how to answer each item on the screen According to each stimulus we introduce. Conversely, if the images were animated, the answers would be always the same and the game would become unrealistic. Each game event should be calculated almost immediately to maintain its fluidity and emotion, in such a way that these GPUs have to be able to perform several mathematical operations simultaneously.
Traditionally molecular dynamics is based on the equations of classical physics, which only see the time evolution of molecules like solid objects collide, hundreds of molecules floating in water or other solvent. With the advent of GPUs can include dynamic calculating the electronic structure so we can peek into biological processes such as DNA replication or the passage of nutrients through a channel protein embedded in the membrane of a cell.
Conclusions
Since the fundamental understanding of the distribution of electrons in a molecule, its structure and properties to rational drug design, new materials based on molecular modeling theoretical chemistry is a powerful tool which is constantly progressing. The development of computer systems increasingly powerful detail allows us to meet the electronic processes involved in a chemical reaction while we can predict the real-time progress of molecular transformations. All this brings us ever closer to the dream of modern alchemists: transform matter to obtain substances with properties designed to pleasure.
In the nineteenth century, the American philosopher Ralph Waldo Emerson, wrote: “Chemistry was born from the dream of the alchemists to turn cheap metals into gold. By failing to do so, they have accomplished much more important things. ” And yes. Today we delve into the innermost secrets of nature not only to understand how it works but also to modify its operation on our behalf.

## The Chuck Norris of chemistry

It is widely known by now, the existence of a list called “The Chuck Norris facts” in which macho attributes of this eighties redneck action hero are exacerbated for the sake of humor. The list includes such amusing facts like:

• “Chuck Norris doesn’t eat honey, he chews bees”
• “When Chuck Norris does a pushup, he’s pushing the Earth down”
• “Chuck Norris counted to infinity; twice!”
• “There is no evolution, only a list of creatures Chuck Norris allows to live”

This last one is funny also because Chuck Norris is a Born-Again-Christian who doesn’t believe in evolution. The list is very funny although the original site has become plagued of not so good ones thanks to uninspired people with web access.

A not so old list, and definitely funnier for us people in the science business, is “The Carl Friederich Gauss list of facts“, which includes gems like:

• “Gauss can divide by zero” (funny although a bit obvious, right? well this is warm up)
• “Gauss didn’t discover the normal distribution, nature conformed to his will”
• “Gauss can write an irrational number as the ratio of two integers”
• “Gauss doesn’t look for roots of equations, they come to him”
• “Gauss knows the topological difference between a doughnut and a coffee mug”
• “Parallel lines meet where Gauss tells them to”.

All these facts imply one thing: impossibilities being allowed to one paradigmatic character for humor’s sake. What could be considered an impossibility in chemistry by now and who could be the one to bear Norris’ fame? Who could be deemed as the Chuck Norris of chemistry?
The impossibility of synthesizing noble gas compounds comes to my mind as the historical impossibility in modern chemistry most imprinted in chemists minds since its written in Pauling’s textbook and is supported by Lewis’ theory; yet Bartlett achieved their synthesis during the 60’s! Chemistry is a science which generates it’s own study matter and as such, impossibilities become challenges. What are the current challenges in chemistry? what is the direction our science is taking or even worse that it should be taking?

So here is my first attempt at emulating the list of facts in the chemistry field and my chosen one is Roald Hoffmann!

• Roald Hoffmann can make a C atom hybridize d orbitals into its valence shell
• Roald Hoffmann drinks AlLiH4 aqueous cocktails
• Roald Hoffmann can stabilize a tertiary carbanion and a primary carbocation
• Roald Hoffmann can analytically solve the Schrödinger equation for H2 and beyond (of course)
• Roald Hoffmann denatures a protein by looking at it and refolds it at will
• Roald Hoffmann always gets a 100% yield
• Le’Chatellier’s principle first asks for Hoffmann’s permission
• Roald Hoffmann once broke the Periodic Table with a roundhouse kick
• Roald Hoffmann can make a molecule stop vibrating at absolute zero; it’s called fear!
• Born-Oppenheimer’s approximation is a consequence of nuclei being too frightened to move in the presence of Roald Hoffmann. Electrons? they are just trying to escape
• Roald Hoffmann’s blood is a stronger acid than SbF5

A pretty lame attempt I admit. Who is your favorite chemist in history and why? Try to come up with your own Chuck Norris of Chemistry list and we’ll share it here in this site.

As usual thanks for  reading (yeah! the whole three of you)

## It’s that time of the year again… The Nobel Prizes

Around early October the scientific community -or at least part of it- starts getting excited about what could be considered the most prestigious award a scientist could ever achieve: The Nobel Prize.

The three categories that interest me the most are: Chemistry, Physics and Literature. I’m not saying I don’t care for the other three (well, maybe the one in economy is way out of my league to grasp) but these three are the ones that always arouse my curiosity. This year laureates have really had me excited! For starters, in chronological order of announcement, Geim and Novoselov seem to be quite younger than the average recipient (52 and 36 years old, respectively). But so is the field for what they got it since the first paper these two scientists from the University of Manchester published on the topic is only about six years old. Discovery of Graphene and most importantly the characterization and understanding of its properties is one of the most promising areas in materials sciences since graphene exhibits very interesting electronic as well as structural behaviors. Nobel prizes are always controversial, but we have to admit that although graphite has been around us for ages, these two England-based Russian scientist have kicked off a promising area of science that will no doubt contribute to further technological developments we can only begin to imagine.

On the other hand, the Nobel Prize in Chemistry was awarded to Heck, Negishi and Suzuki for their work on Pd (palladium) catalyzed coupling reactions. What I liked the most about this prize is that a few years ago I published alongside Dr. David Morales-Morales from the National Autonomous University of Mexico, a paper in J. Molecular Cat. A., in which we performed  a systematic study of a phosphane-free Heck reaction for a series of Pd catalysts with the general formula [ArFNH]PdCl2 (ArFNH = Fluorinated or polyfluorinated aniline). In this paper theoretical calculations were used to assess the relationship between the substitution pattern in fluorinated anilines upon the catalyst’s eficiency, a sort of small quantum-QSAR. Another thing that got me (and a bunch of other chemists) excited was the fact that this year the Nobel Prize in Chemistry went to people working in old fashioned synthetic chemistry, so to speak. Recently a long list of researchers working on the field of BIO-chemistry were awarded the prestigious prize, which comes to no surprise since the development of the Human Genome Project has, and will continue to have, a huge impact in biotechnology. Be that as it may, good for Heck, Suzuki and Negishi and the Pd-catalyzed-carbon-carbon-bond-forming-reactions!

About my initial remark: For reasons I don’t know (I wont subscribe to any of the existing urban-legend-level hypothesis) there is no Nobel Prize in Mathematics, although a lot of mathematicians have been awarded the Nobel Prize in Economical Sciences. For mathematicians the Fields Medal would be the equivalent of a Nobel Prize. However, the Fields Medal is only awarded every four years. Four years ago, this captivating character named Grigori “Grisha” Perelman was awarded the Fields Medal for solving what the Clay Institute in Massachusetts deemed one of the problems of the millennium: The Poincare Conjecture. What is so noteworthy is that Grisha (diminutive for Grigori in Russian) rejected the medal as well as the million dollars awarded by the Clay Institute for solving it. He also rejected a position at Princeton University. His lack of faith in any institution was also reflected in his work, since he did not publish his solution to Poincare’s conjecture in any peer reviewed journal but instead uploaded it on-line and alerted some notorious mathematicians he had worked with in the past about it. Secluded in his St. Petersburg apartment, this remarkable fourty year old, Rasputin-looking-genius, mathematician keeps rejecting not only all fame, money and glory but human contact altogether. It is said that at some point Sir Isaac Newton did the same thing. I guess great minds do think alike.

## Teaching QSAR and QSPR at UAEMex

Teaching has never been my cup of  tea. Karl Friederich Gauss said “Good students do not need a teacher and bad students, well, why do they want one?” I once read this quote somewhere, and although I don’t know if he actually said it or not, there is some truth to it. It is known that Gauss didn’t like teaching, still spent most of his life doing it. Anyway, teaching is important and it has to be done!

Therefore as part of my duties as researcher at CCIQS I will have to teach a class at the Faculty of Chemistry of the Mexico State’s Autonomous University (UAEMex). Obviously they want you to teach a class on a subject you are an expert on; I could teach organic chemistry for sure, despite the fact that I haven’t touched a flask in years. My colleague, Dr. Fernando Cortés-Guzmán and I seem to be two of the very few theoretical chemists around so it is up to us to teach all classes within the range of theoretical chemistry, computational chemistry and their applications. This year someone, I still need to find out who, came up with the idea that an interesting application would be QSAR which of course is a very relevant model for drug discovery. Thus, starting today, I will be the first teacher of this subject at UAEMex’s Chemistry Faculty. Although to be quite frank, I think I would have felt better teaching calculus or differential equations, since those already have a syllabus. On the other hand, those subject wouldn’t get me in touch with students in their final years who are the ones to be attracted as potential students for my incipient research group. It has been interesting so far, building the syllabus from scratch; finding all the topics that are worth covering in a semester as well as a proper way to illustrate and teach them. It will be a work in progress all the time and I intend to expand it somehow beyond the classroom; my first thought was to record all the lessons for a podcast. I’m still not sure how to include this blog into the equation or if I should open a new one for the class but I guess I’ll figure it out along the way. I’m not an expert on QSAR or QSPR but I know a good deal about it, mostly because of Dr. Dragos Horvath whom I met in Romania years ago. Perhaps I could persuade him of leaving Strasbourg for a couple of weeks and giving a few lectures.

Wish me luck, or maybe I should say: “wish my students luck”!

## Basis sets

In this new post I will address some issues regarding the correct use of the terminology used about basis sets in ab initio calculations.

One of the keys to achieve good results in ab initio calculations is to wisely select a basis set; however this requires some previous insight about the specific model to use, the system (molecule/properties) to be calculated and the computational resources at hand. Most of the basis sets available today remain in our codes due to historical reasons more than to their real practical use. We know the Schrödinger equation is not analytically solvable for an interestingly big enough molecule, so the Hartree-Fock (HF) approach approximates its solution in terms of MOs but these MOs have to be constructed of smaller functions, ideally AOs but even these are constructed as linear combinations of simpler, linearly independent, mutually orthogonal functions which we call Basis Sets.

INTRODUCTION

For true beginners: Imagine the 3D vector space as you know it. The position vector corresponding to any point in this space can be deconstructed in three different vectors: R =ax+by+cz In this case x, y and z would be our basis vectors which comply with the following rules: A) They are linearly independent; none of them can be expressed in terms of the others. B) They are orthogonal; their pairwise scalar product is zero. C) Their pairwise vector product yields the remaining one with its sign defined by the range three tensor epsilon. In a vector space with more than three dimensions we can always find a basis which has the same properties described above with which we are able to uniquely define any other vector belonging to this hypothetical space. In the case of Quantum Mechanics we are dealing with function spaces (since our entity of interest is the Wavefunction of a quantum system) instead of vector ones, so what we look for are basis functions that allow us to generate any other function belonging to this space.

SLATER TYPE ORBITALS (STO’s)

This is one of those examples that survive for historical reasons. Its value relies on the fact that is a good first start to obtain the properties of small systems. EXPAND

minimal basis: This term refers to the fact of using a single STO for each occupied atomic orbital present in the molecule.

double zeta basis: Here each STO is replaced by two STO’s wich differ in their zeta values. This improves the description of each orbital at some computational cost.

SPLIT VALENCE

A single STO is used to describe core orbitals (a minimal core basis set) while two or more are used to describe the valence orbitals.

DIFFUSION AND POLARIZATION FUNCTIONS

PLANE WAVES

A plane wave is a wave of constant frequency whose wavefronts are described as infinite parallel planes. When dealing with -tranlational- symmetrical systems (such as crystals) the total wavefunction can be deconstructed as a combination of plane waves. This kind of basis set is suitable for Periodic Boundary Conditions (PBC) computations if a suitable code is available for it, since plane wave solutions converge slowly. Softwares such as CRYSTAL make use of plane wave solution to find the electronic properties of crystaline solids.

As usual I hope this post is of help. Please rate or comment on this post just to know we are working on the right path!

REFERENCES

## Wheel? I think knot!

Once again an awful title. This post follows my previous one on graphs and chemistry, and it addresses an old idea which I have shared in the past with many patient people willing to listen to my ramblings.

It is a common conception/place to state that the wheel was the invention that made mankind spring from its more hominid ancestors into the incipient species that would eventually become homo sapiens; that it was the wheel, like no other prehistoric invention or discovery, what made mankind to rise from its primitive stage. I’ve always believed that even if the wheel was fundamental in the development of mankind, man first had to build tools to make wheels out of something; otherwise they would have been just a good theoretical conception.

But even despite the fact that building tools was in itself a pretty damn good start, I strongly believe that mankind’s first groundbreaking invention were knots. For even a wheel was a bit useless until it was tied to something. From my perspective, the invention of the wheel was an event bound to happen since there are many round shaped things in nature: from the sun and the moon to some fruits and our own eyes. Achieving the mental maturity of taking a string (or a resembling equivalent of those days) and tie it, whether around itself or to something, was, in my opinion, the moment in which the opposable thumbs of mankind realized they could transform it’s surroundings. Furthermore, at that stage the mental maturity achieved made it possible for man to remember how to do it over again in a consistent way.

The book ‘2001 – a space odissey’ by A. C. Clarke, describes this process in the first chapter when a group of hominids bumps into the famous monolith. Their leader (i think his name was moonlight), under the spell of this strangely straight and flat thing takes two pieces of grass and ties them together without knowing or understanding what he is doing. I was pleased to read that I was not alone in that thought.

The concept of a knot keeps on amazing me given their variety and the different purposes they serve according to their properties. These were known to ancient sailors who have elevated the task of knot-making to a practical art form. The mathematical background behind them has served to lay one of today’s most fundamental (and controversial) theories about the composition of matter: string theory. Next time when you make the knot of your necktie think about this tedious, obnoxious little habit was based on something groundbreaking that truly makes us stand out from the rest of the species in the animal kingdom.

## Pauling hybridization model

Is the C atom in methane sp3 hybridized because it’s tetrahedral or is it tetrahedral because it’s sp3 hybridized? It’s funny how many students think to this date that the correct answer is the latter; specially those working in inorganic chemistry. I ignore the reason for such trend. What is true is that most chemistry teachers seem to have lost links to certain historical facts that have shaped our scientific discipline; most of those lay in the realm of physics, maybe that’s why.

What Linus Pauling, in a very clever way, stated was that once you have a set of eigenvectors (orbitals) of the atomic Hamiltonian any combination of them will also be an eigenvector (which is normal since one of the properties of Hermitian operators is that they are linear); so why not making a symmetry adapted one? Let’s take the valence hydrogenoid orbitals (hydrogenoid being the keyword here) and construct a linear combination of them, in such a way that the new set transforms under the irreducible representations of a given point group.  In the case of methane, the 2s and 2p orbitals comprise the valence set and their symmetry-adapted-linear-combination under the Td point group constitutes a set of new orbitals which now point into the vertexes of a tetrahedron. Funny things arise when we move to the next period of the table; it has been a controversy for a number of years the involvement of empty d orbitals in pentacoordinated P(V) compounds. Some claim that they lay too high in energy to be used in bond formation; while others claim that their involvement depends on the nature (electronegativity mainly) of the surrounding substituents.

In many peer reviewed papers authors are still making the mistake of actually assigning a type of hybridization to set of valence orbitals of an atom based on the bond angles around it. Furthermore, it is not uncommon to find claims of intermediate hybridizations when such angles have values in between those corresponding to the ideal polyhedron. Symmetry is real, orbitals are not; they are just a mathematical representation of the electron density distribution which allows us to construct mind images of a molecule.

Linus Pauling is one of my favourite scientific historical figures. Not only did he build a much needed at the time bridge between physics and chemistry but he also ventured into biochemistry (his model of an alpha-helix for the alanine olygopeptide became the foundation to Watson & Cricks later double  helix DNA model), X-ray diffractometry, and humanities (his efforts in reducing/banning the proliferation of nuclear weapons got him the Nobel Peace Prize long after he had already received the Nobel Price in Chemistry). He was a strong believer of ortho-molecular nutrition, suggesting that most illnesses can be related to some sort of malnutrition. Linus Pauling and his book On the Chemical Bond will remain a beacon in our profession for the generations to come.

Disclaimer: The question above, with which I opened this post, was taken from an old lecture by Dr. Raymundo Cea-Olivares at UNAM back in the days when I was an undergraduate student.