# Blog Archives

## All you wanted to know about Hybrid Orbitals…

#### … but were afraid to ask

#### or

#### 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 CH_{4} is sp^{3} hybridized because CH_{4} 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 *ζ _{i}* and formulate them as linear combinations of the C atom valence orbitals:

*ζ _{1}*=

*a*+

_{1}2s*b*+

_{1}2p_{x}*c*+

_{1}2p_{y}*d*

_{1}2p_{z}*ζ _{2}*=

*a*+

_{2}2s*b*+

_{2}2p_{x}*c*+

_{2}2p_{y}*d*

_{2}2p_{z}*ζ _{3}*=

*a*+

_{3}2s*b*+

_{3}2p_{x}*c*+

_{3}2p_{y}*d*

_{3}2p_{z}*ζ _{4}*=

*a*+

_{4}2s*b*+

_{4}2p_{x}*c*+

_{4}2p_{y}*d*

_{4}2p_{z}to comply with equivalency lets set *a _{1}* =

*a*=

_{2}*a*=

_{3}*a*and normalize them:

_{4}*a _{1}*

*+*

^{2}*a*

_{2}*+*

^{2}*a*

_{3}*+*

^{2}*a*

_{4}*= 1 ∴*

^{2}*a*= 1/√4

_{i}Lets take *ζ _{1}* to be directed along the

*z*axis so

*b*=

_{1}*c*= 0

_{1}*ζ _{1 }*= 1/√4(

*2s*) +

*d*

_{1}2p_{z}since *ζ _{1}* must be normalized the sum of the squares of the coefficients is equal to 1:

^{1}/_{4} + *d _{1}^{2}* = 1;

*d _{1}* =

^{√3}/

_{2}

Therefore the first hybrid orbital looks like:

*ζ _{1}* =

^{1}/

_{√4}(

*2s*) +

^{√3}/

_{2}(

*2p*)

_{z}We now set the second hybrid orbital on the xz plane, therefore *c _{2}* = 0

*ζ _{2}* =

^{1}/

_{√4}(

*2s*) +

*b*+

_{2}2p_{x}*d*

_{2}2p_{z}since these hybrid orbitals must comply with all the conditions of atomic orbitals they should also be orthonormal:

〈*ζ _{1}*|

*ζ*〉 = δ

_{2}_{1,2}= 0

^{1}/_{4} + *d _{2}*

^{√3}/

_{2}= 0

*d _{2}* = –

^{1}/

_{2√3}

our second hybrid orbital is almost complete, we are only missing the value of *b _{2}*:

*ζ _{2}* =

^{1}/

_{√4}(

*2s*) +

*b*+-

_{2}2p_{x}^{1}/

_{2√3}(

*2p*)

_{z}again we make use of the normalization condition:

^{1}/_{4} + *b _{2}^{2}* +

^{1}/

_{12}= 1;

*b*=

_{2}^{√2}/

_{√3}

Finally, our second hybrid orbital takes the following form:

*ζ _{2}* =

^{1}/

_{√4}(

*2s*) +

^{√2}/

_{√3}(

*2p*) –

_{x}^{1}/

_{√12}(

*2p*)

_{z}The procedure to obtain the remaining two hybrid orbitals is the same but I’d like to stop here and analyze the relative direction *ζ _{1}* and

*ζ*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:

_{2}ψ* _{2s}* = (1/4π)

^{½}

*R*(

*r*)

ψ* _{2px}* = (3/4π)

^{½}sinθcosφ

*R*(

*r*)

ψ* _{2pz}* = (3/4π)

^{½}cosθ

*R*(

*r*)

we substitute these in *ζ _{2}* and factorize R(r) and

^{1}/

_{√(4π)}

*ζ _{2}* = (

^{R(r)}/

_{√(4π)})[

^{1}/

_{√4}+ √2 sinθcosφ –

^{√3}/

_{√12}cosθ]

We differentiate *ζ _{2}* 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

*ζ*and

_{1}*ζ*(remember that

_{2}*ζ*is projected entirely over the

_{1}*z*axis)

d*ζ _{2}*/dθ = (

^{R(r)}/

_{√(4π)})[√2 cosθ –

^{√3}/

_{√12}sinθ] = 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 *sp ^{2.5}* hybridization just because their bond angles lie between 109 and 120°. Take a vector

*v*= x

*i*+y

*j*+z

*k*, even if you specify it to be

*v*= 1/2

*i*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.

## 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.

*Chemistry without flasks?*

*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 =*a*x+*b*y+*c*z 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.

## Knots, fishing and the origin of the universe

Most awful post title ever, I know, but maybe I’m still hooked on prof. Schaefer’s conference from two weeks ago.

I went fishing on Sunday and although my luck was better this time (I caught four fish!) I spent a great deal of time tying hooks, untangling my line from others or even from my own. Whenever the knot became too complicated to solve I just cut the line and tied a new hook or floater. At some point I was wishing there was a tool that could help me to untie those nasty knots and make better ones, I would have settled at least for a recipe! That tool/algorithm exists, of course, and it’s called topology; and within this branch there is a whole area devoted to knots (knots theory.) Of course in topology a knot has no ends, that is, they consist of single loops. This is one of those math areas which found little use during the time of their development but that in time became the framework for complex physical theories such as quantum gravity or string theory, these theories account for the wacky title, of course.

Within topology we come accross graph theory too, which is an everyday chemist’s tool although most of us are unaware of it. 2d representations of chemistry structure are graphs, dots joined by edges. If you look at an old text, the 2D representation of norbornane looks like two fused squares with a methylene in the middle of the common edge. This representation is topologically correct but geometrically incorrect. more complicated molecules were just drawn into texts.

In chemistry, although molecular symmetry is described by group theory (and this in turn connects molecular structure to its quantum properties,) many computational chemistry efforts are conducted on topology and graph theory. For lack of a better example think of SciFinder’s molecule builder tool: in it you can draw a molecule (or a piece of it) disregarding everything you know about structural chemistry, hybridization, the VSEPR model, Bent rules, and so on, and still SciFinder can find related structures to your query because all that it reads are labeled points (atoms) and edges (bonds); it understands the graph, not the symmetry arising from geometry, let alone the molecule. Another example of graphs theory applied to chemoinformatics are those softwares that take a IUPAC name and yield the structure (the graph) or viceversa; what the algorithms do is interpreting or generating graphs once a set of rules were provided.

Among graphs there is a particular kind that is called planar graphs; these can be presented in such a way that no edges overlap each other. There is an online game with which I came across a few years ago and I’m still addicted to it, its name is planarity and it can be found here (NSFW). Molecules are planar graphs but their non-overlaping-edges representation is hardly of any help since their chemical properties rely on their 3D structure.

Now, if I was to set my mind to evil, could we think of people as dots or connectors and their relationships/story-lines as edges and ultimately come up with an algorithm for untangling a lie? It would require a lot of data (the edges) if we were to untangle a lie made by others, but what if we want to weave a life of lies? we know what vertexes are around us and up to some extent the edges between connectors close to us; therefore we could draw bogus edges (lies) provided we could come up with a planar graph in which no two bogus edges overlap. That could be a planar graph plotted on top of a non-necessarily planar one. Definitely unethical but nonetheless feasible from my point of view.

Maybe I should just stick to untie knots in my fishing line next Sunday.