# Category Archives: Quantum Mechanics

## 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 *dz ^{2}*, which we had learned in class (thanks, prof. Carlos Amador!) that is actually a linear combination of

*d(z*and

^{2}-x^{2})*d(z*orbitals, a mathematical -lets say- trick to conform to spectroscopic observations.

^{2}-y^{2})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*)*Y ^{l}_{m}*(

*θ*,

*φ*)

*σ*(

*α*,

*β*) with the angular part given by the spherical harmonic functions

*Y*(

^{l}_{m}*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.

## Maru Sandoval M.Sc. – Our First Graduate Thesis

It is with great pride that I’d like to announce that for the first time we have a Masters Student graduated from this Comp.Chem. lab: María Eugenia “*Maru*” Sandoval-Salinas has finished her graduate studies and just last Friday defended her thesis admirably earning not only the degree of **Masters of Science in Chemistry** but doing so **with the highest honors** given by the National Autonomous University of Mexico.

Maru’s thesis is for many reasons a landmark in this lab not only because it is the first graduate thesis published from our lab but also the first document on our work about the study of Photosynthesis, a long sought after endeavor now closer to publication. It must also be said that Maru came to this lab when she was an undergraduate student five years ago when I just recently joined UNAM as a researcher fresh out of a postdoc stay. After getting her B.Sc. degree and publishing an article in JCTC (**DOI: 10.1021/ct4004178**) she now is about to publish more papers that I’m sure will be as highly ranked as the previous one. Thus, Maru was a pioneer in our lab giving it a vote of confidence when we had little to nothing to show for; thanks to her hard work and confidence, along with that of the students who have followed her, we managed to succeed as a consolidated research group in the field of computational chemistry.

More specifically, her thesis centered around finding a mechanism for the excitonic transference between pigments (bacteriochlorophyl-a, BChl-a) in the Fenna-Matthews-Olson (FMO) complex, a protein trimer with seven BChl-a molecules in each monomer, located between the antenna complex and the reaction center in green sulfur bacteria. Among the possible mechanisms explored were Förster’s theory, a modification to Marcus’ theory and finally we explored the possibility of Singlet Fission occurring between adjacent molecules with the help of Dr. David Casanova from the Basque Country University where Maru took a short research stay last autumn. Since nature doesn’t conform to any specific mechanism -specially in a complex arrangement such as the FMO- then it could be possible that a combination of the above might also occur but lets just wait for the papers to be published to discuss it. Calculations were performed through the TD-DFT and the C-DFT formalisms using G09 and Q-Chem; comparing experimental data in CH3OH (SMD implicit calculations with the SVWN5 functional) were undertaken previously for selection of the level of theory.

Now, after two original theses written and successfully defended, an article published in JCTC and more in process, at least five posters, a couple of oral presentations and countless hours at her desk, Maru will go pursuit a PhD abroad where I’m sure she will exceed anyone’s expectations with her work, drive, dedication and scientific curiosity. Thank you, Maru, for all your hard work and trust when this lab needed it the most, we wish you the best for you earn it. You will surely be missed.

## The Local Bond Order, LBO (Barroso et al. 2004)

I don’t know why I haven’t written about the Local Bond Order (LBO) before! And a few days ago when I thought about it my immediate reaction was to shy away from it since it would constitute a blatant self-promotion attempt; but hell! this is my blog! A place I’ve created for my blatant self-promotion! So without further ado, I hereby present to you one of my own original contributions to Theoretical Chemistry.

During the course of my graduate years I grew interested in weakly bonded inorganic systems, namely those with secondary interactions in bidentate ligands such as xanthates, dithiocarboxylates, dithiocarbamates and so on. Description of the resulting geometries around the central metallic atom involved the invocation of secondary interactions defined purely by geometrical parameters (Alcock, 1972) in which these were defined as present if the interatomic distance was longer than the sum of their covalent radii and yet smaller than the sum of their van der Waals radii. This definition is subject to a lot of constrictions such as the accuracy of the measurement, which in turn is related to the quality of the monocrystal used in the X-ray difraction experiment; the used definition of covalent radii (Pauling, Bondi, etc.); and most importantly, it doesn’t shed light on the roles of crystal packing, intermolecular contacts, and the energetics of the interaction.

This is why in 2004 we developed a simple yet useful definition of bond order which could account for a single molecule in vacuo the strength and relevance of the secondary interaction, relative to the well defined covalent bonds.

Barroso-Flores, J. et al.Journal of Organometallic Chemistry689 (2004) 2096–2102 http://dx.doi.org/10.1016/j.jorganchem.2004.03.035,

Let a Molecular Orbital be defined as a wavefunction *ψ*i which in turn may be constructed by a linear combination of Atomic Orbitals (or atom centered basis set functions) φj

We define *ζ*LBO in the following way, where we explicitly take into account a doubly occupied orbital (hence the multiplication by 2) and therefore we are assuming a closed shell configuration in the Restricted formalism.

The summation is carried over all the orbitals which belong to atom A1 and those of atom A2.

Simplifying we yield,

where *S*jk is the overlap integral for the *φ*j and *φ*k functions.

By summing over all *i* MOs we have accomplished with this definition to project all the MO’s onto the space of those functions centered on atoms A1 and A2. This definition is purely quantum mechanical in nature and is independent from any geometric requirement of such interacting atoms (i.e. interatomic distance) thus can be used as a complement to the internuclear distance argument to assess the interaction between them. This definition also results very simple and easy to calculate for all you need are the coefficients to the LCAO expansion and the respective overlap integrals.

Unfortunately, the Local Bond Order hasn’t found much echo, partly due to the fact that it is hidden in a missapropriate journal. I hope someone finds it interesting and useful; if so, don’t forget to cite it appropriately 😉

## Polarizability and Hyperpolarizability in Gaussian

Calculating both Polarizability and the Hyperpolarizability in Gaussian is actually very easy and straightforward. However, interpreting the results requires a deeper understanding of the underlying physics of such phenomena. Herein I will try to describe the most common procedures for calculating both quantities in Gaussian09 and the way to interpret the results; if possible I will also try to address some of the most usual problems associated with their calculation.

**Introduction**

The dipole moment of a molecule changes when is placed under a static electric field, and this change can be calculated as

*pe = pe,0 + α:E + (1/2) β:EE + …* (**1**)

where pe,0 is the dipole moment in the absence of an electric field; α is a second rank tensor called the polarizability tensor and β is the first in an infinite series of dipole hiperpolarizabilities. The molecular potential energy changes as well with the influence of an external field in the following way

*U = U0 – pe.E – (1/2) α:EE – (1/6) β:EEE –* … (**2**)

.

Route Section Keyword: **Polar**

This keyword requests calculation of the polarizability and, if available, hyperpolarizability for the molecule under study. This keyword is both available for DFT and HF methods. Hyperpolarizabilities are NOT available for methods that lack analytic derivatives, for example CCSD(T), QCISD, MP4 and other post Hartree-Fock methods.

Frequency dependent polarizabilities may be calculated by including CPHF=RdFreq in the route section and then specifying the frequency (**expressed in Hartrees**!!!) to which the calculation should be performed, after the molecule specification preceded by a blank line. Example:

#HF/6-31G(d) Polar CPHF=RdFreq Title Section Charge Multiplicity Molecular coordinates==blank line==0.15

In this example 0.15 is the frequency in Hartrees to which the calculation is to be performed. By default the output file will also include the static calculation, that is, ω = 0.0. Below you can find an example of the output when the CPHF=RdFreq is employed (taken from Gaussian’s website) Notice that the second section is performed at ω = 0.1 Ha

SCF Polarizability for W= 0.000000: 1 2 3 1 0.482729D+01 2 0.000000D+00 0.112001D+02 3 0.000000D+00 0.000000D+00 0.165696D+02 Isotropic polarizability for W= 0.000000 10.87 Bohr**3. SCF Polarizability for W= 0.100000: 1 2 3 1 0.491893D+01 2 0.000000D+00 0.115663D+02 3 0.000000D+00 0.000000D+00 0.171826D+02 Isotropic polarizability for W= 0.100000 11.22 Bohr**3.

You may have noticed now that the polarizabilities are expressed in volume units (Bohr^3) and the reason is the following:

Consider the simplest case of an atom with nuclear charge *Q*, radius *r*, and subjected to an electric field, *E*, which creates a force *QE*, and displaces the nucleus by a distance* d*. According to Gauss’ law this latter force is given by:

(dQ^2)/(4πεr^3) = QE (Hey! WordPress! I could really use an equation editor in here!)

if the polarizability is defined by Qd/E then we can rearrange the previous equation and yield

α = 4πεr^3 which in atomic units yields volume units, r^3, since 4πε = 1. This is why polarizabilities are usually referred to as ‘polarizability volumes’.

******THIS POST IS STILL IN PROGRESS. WILL COMPLETE IT IN SHORT. SORRY FOR ANY INCONVENIENCE******

The use of double zeta quality basis sets is paramount but it also makes these calculations more time consuming. Polarization functions on the basis set functions are a requirement for good results.

As usual, please rate/comment/share this post if you found it useful or if you think someone else might find it useful. Thanks for reading!

## 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.