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Condensed question formulation:

Is there an experimental method to directly visualise the 3D form of a MO wave function, or at least the electron density associated with it?

Full statement:

Molecular orbital theory is a crowning achievement of modern Chemistry: it is by far the most accurate description of chemical bonding we possess and accounts for the vast majority of related phenomena.

An array of evidence exists to confirm predictions from MO theory. Ab initio computational methods such as Hartree-Fock are able to produce a very accurate description of orbital coefficients and energies, which match experimental data for the energies (via photoelectron spectroscopy etc.) or predicted reaction mechanisms/FMO interactions (in kinetic runs).

But do we have any experimental evidence for the shape of MOs?

I'd presume the most feasible way of observing some measure related to the MO wave function $\Psi$ would be to follow a probabilistic approach and try imaging the electron density assumed to be given by $|\Psi|^2$.

Of course, X-ray diffraction first springs to mind; the notorious issue with that being the phase problem. In my superficial understanding: due to the loss of phase information, in going from the XRD pattern to the electron density isosurface (via ab initio methods), essentially we model some initial guess phase and try to fit it to the intensity data.

Since usually the location of nuclei is of far greater practical interest, the formalisms developed to model the phases (e.g. Hansen-Coppens) tend to focus on charge densities localised on each atom (though they are not necessarily of spherical symmetry). This is useful in actually getting the form of the molecule, but produces a picture different to the expected electronic occupation of delocalised MOs.

Is it possible to instead produce the delocalised picture if our ab initio phase guess is based on a RHF calculation? But even if possible, does this have any more legitimacy than the localised picture? I.e. it seems, as far as XRD is concerned, these are equally possible interpretations of the same intensity data where we just forced some phase information as convenient.

Sorry for the long description, this is an in depth presentation of my thoughts on the issue thus far. Any experimental technique/fundamental aspect of MO behaviour I'm missing would be appreciated. But the bottom line is, as stated in the beginning, can we experimentally visualise the shape of MOs or at least their modulus squared?

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  • $\begingroup$ There are two problems here: 1) Molecular orbitals coming from a one-electron model of the wavefunction, i.e. they are essentially an approximation. 2) Wavefunction, just like the molecular orbitals are not observables. In your question, you said that 2) is not an issue, as you accept electron densities, too, which are observables. Still, we have a problem with 1). While MOs are not real physical objects, a possible solution to your problem is to find physical properties that well approximated in a simple orbital picture / one MO Note: HF is far from an accurate description of chemistry. $\endgroup$
    – Greg
    Nov 24 '20 at 17:02
  • $\begingroup$ chemistry.stackexchange.com/questions/5376/… $\endgroup$
    – Mithoron
    Nov 24 '20 at 21:29
  • $\begingroup$ chemistry.stackexchange.com/questions/57784/… $\endgroup$
    – Mithoron
    Nov 24 '20 at 21:35
  • $\begingroup$ Thanks for the suggestions and the quite helpful sign towards the right direction! Though not exactly the same question, the second link brought STM into my attention and eventually helped me find this paper on STM for individual MO imaging -which is the sort of technique I was asking about. $\endgroup$
    – aureolin
    Nov 26 '20 at 10:26
  • $\begingroup$ > Molecular orbital theory is a crowning achievement of modern Chemistry This grand value judgement is matter of opinion, but ok. > it is by far the most accurate description of chemical bonding we possess well, that's just false. > Ab initio computational methods such as Hartree-Fock are able to produce a very accurate description of orbital coefficients and energies, which match experimental data for the energies (via photoelectron spectroscopy etc.) or predicted reaction mechanisms/FMO interactions (in kinetic runs). $\endgroup$
    – Lorents
    Nov 26 '20 at 12:59
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The very question you pose is addressed thoroughly in this open access work:
https://www.nature.com/articles/ncomms9287 .
The short answer to your question is that the electron density can be mapped using a technique akin to diffraction as described in their work. You mentioned a distinction between the electron density and the orbitals, and the orbital is not itself observable; but as you suggested, the electron density can be mapped.

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  • $\begingroup$ Thank you very much, this is outstanding! I was not familiar with the ARPES technique, which seems to provide the exact answer I was looking for. Two quick additional questions (for anyone with better understanding in these methods than my half an hour earlier ignorant self): (a) is the angle-resolved part of ARPES which makes it not suffer from the phase problem of XRD? (b) do any further techniques as suitable as ARPES or STM (mentioned elsewhere in comments) for single MO imaging exist? $\endgroup$
    – aureolin
    Nov 26 '20 at 10:30

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