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.

dz2 is NOT a linear combination of d(z2-x2) and d(z2-y2). dz2 is d0, an eigenvalue of both L2 (eigenvalue 2) and Lz (eigenvalue 0). d(x2-y2) is a linear combination of d(z2-x2) and d(z2-y2).

Sorry, I misspoke! [kicking myself!] What I meant to say was that dz2 is not just “a mathematical trick to conform to spectroscopic observations” but is an eigenvalue of both L2 (eigenvalue 2) and Lz (eigenvalue 0). Scratch what I wrote regarding d(x2-y2).

You made me scratch my head on that one! But you’re right is a mathematical trick to get a full set of eigenvalues to a complete set of commuting observables.

Thanks for pointing it out.

I’m confused. The orbital concept I was familiar is “a high probability region of finding the electron.” In this way, the orbital is not the wave function itself, but the square of the wave function integrated into space (the wave function itself does not justify the famous lobes we all chemists know and love). In fact, I do not intend to get into the discussion of the observation of the cited articles, but, having said that, I do not think that the orbital is merely a mathematical representation, but that it has a real physical meaning.

Those lobes we all know and love have phases and nodal planes, those have no physical meaning, yet the squared of the orbital yields an electron density function which in turn does have an interpretation. The orbitals and lobes are wavefunctions hence non-observables.