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All you wanted to know about Hybrid Orbitals…

… but were afraid to ask




2017-08-09 20.29.45

Taken from “Quantum Mechanics in Chemistry” by Jack Simmons




ζ1a12s + b12px + c12py + d12pz

ζ2a22s + b22px + c22py + d22pz

ζ3a32s + b32px + c32py + d32pz

ζ4a42s + b42px + c42py + d42pz




ζ= 1/√4(2s) + d12pz


1/4 + d12 = 1;

d1 = √3/2

Therefore the first hybrid orbital looks like:





ζ1|ζ2〉 = δ1,2 = 0

1/4 + d2√3/2 = 0

d2 = –1/2√3



again we make use of the normalization condition:


Finally, our second hybrid orbital takes the following form:



ψ2s = (1/4π)½R(r)

ψ2px = (3/4π)½sinθcosφR(r)

ψ2pz = (3/4π)½cosθR(r)





sinθ/cosθ = tanθ = -√8

θ = -70.53°,




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




Chemistry without flasks?

What is theoretical chemistry?

What is it good for?

How do you calculate a molecule?


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.


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.


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.


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.



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!


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.

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