What methods exist for us to measure the position and momentum of atoms that make up molecules? In this paper,

Localization of an atom by homodyne measurement. A. M. Herkommer et al. Quantum Semiclass. Opt. 8 no. 1, p. 189 (1996) (paywalled).

the authors are able to localize atoms using homodyne measurement. Would it be too naive to consider we can measure the position of atoms that make up molecules? I know that the LCAO (Linear Combination of Atomic Orbitals) is a widely used method to approximate molecular orbitals. This leads me to think, whether or not we can use homodyne measurements to measure the position of atoms in a molecule. If homodyne measurements don't work, what other ways are there to gain information about the position wavefunction of molecules for let's say, H2?
Example I'm considering:
I'm thinking that the position wavefunction for H2 would be its molecular orbitals. If we estimated its molecular wavefunction as a linear combination of two atomic H orbitals that are in close proximity to each other, then... can't there be some way of applying homodyne measurement techniques to gain information about the position wavefunction of the two H atoms?
 A: The paper you reference describes measurement of the atom's center-of-mass position using light. This is definitely possible, though it is fundamentally limited to resolutions of a large-ish fraction of the light's wavelength (at about 400 mn) at most. This method could in principle be extended to a small molecule like H2 (though I suspect you'd need them to be in their rotational ground state, so the nuclei's orientation would be completely delocalized), but the resolution makes it completely impossible to resolve sub-molecular detail: the nuclei are at most ~1 Å apart, so a few hundred times smaller than your resolution.
To access the spatial structure of a molecule, then, you need finer tools - and, specifically, a much smaller wavelength. The typical solutions to this problem are to use either x-rays (which are routinely employed in x-ray crystallography to read off crystal structures from diffraction patterns) or fast electrons (which produce beautiful images in electron microscopes). Both of these can be used for single molecules, but as you might expect there is quite a bit of work to get useful images from molecules that are just flying around instead of sitting in an orderly crystal.
One big problem is alignment. If you want to, say, measure the internuclear distance in H2, then you need to know the molecule is pointing in some specific direction relative to your apparatus. For this you need to cool your molecules to their rotational ground state in a supersonic gas jet, then shine a strong(ish) laser pulse at your molecules to give them a kick, and wait one rotational period for them to align again. (This technique is known as impulsive alignment.)
Having aligned your molecules, you now have your choice of x-rays and electrons.


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*To use x-rays for this kind of application, you need a very short, intense and coherent burst of x rays to bring out a diffraction pattern form gas-phase molecules. These are available most notably from free-electron lasers. (You could also use high harmonic generation as a light source, with the very relative disadvantage of having to work with two table-top gas jets.) Both sources produce pulses which are short enough to fit inside a rotational revival and - even better - permit time-resolved studies of nuclear motion inside a molecule. Not only can you read out the internuclear distances from the diffraction patterns, but you can watch this distance oscillate after the molecule is excited in some way. For a good reference, try

X-ray diffraction from isolated and strongly aligned gas-phase molecules with a free-electron laser. J. Küpper et al. Submitted to PRL. arXiv:1307.4577.


*Using electrons is hard from a conventional approach: if you tried to shower your gas-phase molecules in an electron beam strong enough to detect interference, you'd completely blast your samples. The clever way around this is to perform electron diffraction using an electron that comes from the molecule itself: it is removed by a strong laser field and then re-accelerated towards the molecule by the same laser. It then recollides at energies of 100 eV or more, which makes for very small wavelengths and thus high spatial resolution. In the collision a number of things can happen:


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*There is a finite chance that the electron will recombine to the neutral ground state, and it will release its energy as a short burst of XUV radiation. This is High Harmonic Generation, and you can look in the spectrum of the generated harmonics of your driving field to obtain a ton of information about your molecule. You might try, for example, Xibin Zhou's PhD thesis for an introduction.

*The electron can simply diffract off of the molecule. This is called Laser Induced Electron Diffraction and it permits highly sensitive imaging of both the nuclear positions as well as the electronic orbital (yes! including its phase!) from which the electron was ionized. These slides by Eric Charron provide a nice hand-waving introduction; for a fuller reference see

Imaging ultrafast molecular dynamics with laser-induced electron diffraction. C. I. Blaga et al. Nature 483, pp. 194–197 (2012). Slightly dubious pdf is available.


*You can also get holography going: you can observe interference between the diffraction pattern and the original ionized wavepacket. This offers a wealth of information about the phase of the diffracted electron beam, and therefore more information about the molecule. One place to go for this is

Time-resolved holography with photoelectrons. Y. Huismans et al. Science 331 no. 6013 (2011), pp. 61-64. HAL e-print.

... and I think I'll stop here. As you can see, this is a very active field of study and new applications will most likely appear within the next decade or two. Free-electron lasers are only really getting started as a trusty research tool, though they're unwieldy as they require an accelerator facility. HHG and its related setups need tabletop lasers so they fit inside a lab (but the tables are pretty big) and are still growing fast. All in all, keep tuned!
