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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.

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.


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==

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’.




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!

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