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All you wanted to know about Hybrid Orbitals…

… but were afraid to ask


How I learned to stop worrying and not caring that much about hybridization.

The math behind orbital hybridization is fairly simple as I’ll try to show below, but first let me give my praise once again to the formidable Linus Pauling, whose creation of this model built a bridge between quantum mechanics and chemistry; I often say Pauling was the first Quantum Chemist (Gilbert N. Lewis’ fans, please settle down). Hybrid orbitals are therefore a way to create a basis that better suits the geometry formed by the bonds around a given atom and not the result of a process in which atomic orbitals transform themselves for better sterical fitting, or like I’ve said before, the C atom in CH4 is sp3 hybridized because CH4 is tetrahedral and not the other way around. Jack Simmons put it better in his book:

2017-08-09 20.29.45

Taken from “Quantum Mechanics in Chemistry” by Jack Simmons

The atomic orbitals we all know and love are the set of solutions to the Schrödinger equation for the Hydrogen atom and more generally they are solutions to the hydrogen-like atoms for which the value of Z in the potential term of the Hamiltonian changes according to each element’s atomic number.

Since the Hamiltonian, and any other quantum mechanical operator for that matter, is a Hermitian operator, any given linear combination of wave functions that are solutions to it, will also be an acceptable solution. Therefore, since the 2s and 2p valence orbitals of Carbon do not point towards the edges of a tetrahedron they don’t offer a suitable basis for explaining the geometry of methane; even more so these atomic orbitals are not degenerate and there is no reason to assume all C-H bonds in methane aren’t equal. However we can come up with a linear combination of them that might and at the same time will be a solution to the Schrödinger equation of the hydrogen-like atom.

Ok, so we need four degenerate orbitals which we’ll name ζi and formulate them as linear combinations of the C atom valence orbitals:

ζ1a12s + b12px + c12py + d12pz

ζ2a22s + b22px + c22py + d22pz

ζ3a32s + b32px + c32py + d32pz

ζ4a42s + b42px + c42py + d42pz

to comply with equivalency lets set a1 = a2 = a3 = a4 and normalize them:

a12 + a22 + a32 + a42 = 1  ∴  ai = 1/√4

Lets take ζ1 to be directed along the z axis so b1 = c1 = 0

ζ= 1/√4(2s) + d12pz

since ζ1 must be normalized the sum of the squares of the coefficients is equal to 1:

1/4 + d12 = 1;

d1 = √3/2

Therefore the first hybrid orbital looks like:

ζ1 = 1/√4(2s) +√3/2(2pz)

We now set the second hybrid orbital on the xz plane, therefore c2 = 0

ζ2 = 1/√4(2s) + b22px + d22pz

since these hybrid orbitals must comply with all the conditions of atomic orbitals they should also be orthonormal:

ζ1|ζ2〉 = δ1,2 = 0

1/4 + d2√3/2 = 0

d2 = –1/2√3

our second hybrid orbital is almost complete, we are only missing the value of b2:

ζ2 = 1/√4(2s) +b22px +-1/2√3(2pz)

again we make use of the normalization condition:

1/4 + b22 + 1/12 = 1;  b2 = √2/√3

Finally, our second hybrid orbital takes the following form:

ζ2 = 1/√4(2s) +√2/√3(2px) –1/√12(2pz)

The procedure to obtain the remaining two hybrid orbitals is the same but I’d like to stop here and analyze the relative direction ζ1 and ζ2 take from each other. To that end, we take the angular part of the hydrogen-like atomic orbitals involved in the linear combinations we just found. Let us remember the canonical form of atomic orbitals and explicitly show the spherical harmonic functions to which the  2s, 2px, and 2pz atomic orbitals correspond:

ψ2s = (1/4π)½R(r)

ψ2px = (3/4π)½sinθcosφR(r)

ψ2pz = (3/4π)½cosθR(r)

we substitute these in ζ2 and factorize R(r) and 1/√(4π)

ζ2 = (R(r)/√(4π))[1/√4 + √2 sinθcosφ –√3/√12cosθ]

We differentiate ζ2 respect to θ, and set it to zero to find the maximum value of θ respect to the z axis we get the angle between the first to hybrid orbitals ζ1 and ζ2 (remember that ζ1 is projected entirely over the z axis)

dζ2/dθ = (R(r)/√(4π))[√2 cosθ –√3/√12sinθ] = 0

sinθ/cosθ = tanθ = -√8

θ = -70.53°,

but since θ is measured from the z axis towards the xy plane this result is equivalent to the complementary angle 180.0° – 70.53° = 109.47° which is exactly the angle between the C-H bonds in methane we all know! and we didn’t need to invoke the unpairing of electrons in full orbitals, their promotion of any electron into empty orbitals nor the ‘reorganization‘ of said orbitals into new ones. Orbital hybridization is nothing but a mathematical tool to find a set of orbitals which comply with the experimental observation and that is the important thing here!

To summarize, you can take any number of orbitals and build any linear combination you want, in order to comply with the observed geometry. Furthermore, no matter what hybridization scheme you follow, you still take the entire orbital, you cannot take half of it because they are basis functions. That is why you should never believe that any atom exhibits something like an sp2.5 hybridization just because their bond angles lie between 109 and 120°. Take a vector v = xi+yj+zk, even if you specify it to be v = 1/2i that means x = 1/2, not that you took half of the unit vector i, and it doesn’t mean you took nothing of j and k but rather than y = z = 0.

This was a very lengthy post so please let me know if you read it all the way through by commenting, liking, or sharing. Thanks for reading.


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. 

Rank your QM knowledge according to Pauli’s Exclusion Principle

Pauli’s Exclusion Principle is a paramount concept in Quantum Mechanics which has implications from statistical mechanics to quantum chemistry, consequently, there are many different statements to summarize it depending on the forum. I occasionally joke with my students about how we learnt it in kindergarten an how we state it now at the end of our computational chemistry course.

So, are you a toddler or high up there with W. Pauli predicting the existence of sub-atomic particles at CERN? Which statement of Pauli’s Exclusion Principle sounds more familiar to you?

QM Evolutionary tree!

QM Evolutionary tree!

LOL just feeling a little humorous this morning!

New paper in JPC-A

As we approach to the end of another year, and with that the time where my office becomes covered with post-it notes so as to find my way back into work after the holidays, we celebrate another paper published! This time at the Journal of Physical Chemistry A as a follow up to this other paper published last year on JPC-C. Back then we reported the development of a selective sensor for Hg(II); this sensor consisted on 1-amino-8-naphthol-3,6-disulphonic acid (H-Acid) covalently bound to a modified silica SBA-15 surface. H-Acid is fluorescent and we took advantage of the fact that, when in the presence of Hg(II) in aqueous media, its fluorescence is quenched but not with other ions, even with closely related ions such as Zn(II) and Cd(II). In this new report we delve into the electronic reasons behind the quenching process by calculating the most important electronic transitions with the framework laid by the Time Dependent Density Functional Theory (TD-DFT) at the PBE0/cc-pVQZ level of theory (we also included an electron core potential on the heavy metal atoms in order to decrease the time of each calculation). One of the things I personally liked about this work is the combination of different techniques that were used to assess the photochemical phenomenon at hand; some of those techniques included calculation of various bond orders (Mayer, Fuzzy, Wiberg, delocalization indexes), time dependent DFT and charge transfer delocalizations. Although we calculated all these various different descriptors to account for changes in the electronic structure of the ligand which lead to the fluorescence quenching, only delocalization indexes as calculated with QTAIM were used to draw conclusion, while the rest are collected in the SI section.


Thanks a lot to my good friend and collaborator Dr. Pezhman Zarabadi-Poor for all his work, interest and insight into the rationalization of this phenomenon. This is our second paper published together. By the way, if any of you readers is aware of a way to finance a postdoc stay for Pezhman here at our lab, please send us a message because right now funding is scarce and we’d love to keep bringing you many more interesting papers.

For our research group this was the fourth paper published during 2014. We can only hope (and work hard) to have at least five next year without compromising their quality. I’m setting the goal to be 6 papers; we’ll see in a year if we delivered or not.

I’d like to also take this opportunity to thank all the readers of this little blog of mine for your visits and your live demonstrations of appreciation at various local and global meetings such as the ACS meeting in San Francisco and WATOC14 in Chile, it means a lot to me to know that the things I write are read; if I were to make any New Year’s resolutions it would be to reply quicker to questions posted because if you took the time to write I should take the time to reply.

I wish you all the best for 2015 in and out of the lab!

XIth Mexican Reunion on Theoretical Physical Chemistry

For over a decade these meetings have gathered theoretical chemists every year to share and comment their current work and to also give students the opportunity to interact with experienced researchers, some of which in turn were even students of Prof. Robert Parr, Prof. Richard Bader or Prof. Per Olov Löwdin. This year the Mexican Meeting on Theoretical Physical Chemistry took place last weekend in Toluca, where CCIQS is located. You can find links to this and previous meetings here. We participated with a poster which is presented below (in Spanish, sorry) about our current research on the development of calixarenes and tia-calixarenes as drug carriers. In this particular case, we presented our study with the drug IMATINIB (Gleevec as branded by Novartis), a powerful tyrosinkynase inhibitor widely employed in the treatment of Leukaemia.

The International Journal of Quantum Chemistry is dedicating an issue to this reunion. As always, this meeting posed a great opportunity to reconnect with old friends, teachers, and colleagues as well as to make new acquaintances; my favourite session is still the beer session after all the seminars! Kudos to María Eugenia “Maru”  Sandoval-Salinas for this poster and the positive response it generated.

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