CCL:orbitals and reality

 My 2 cents (devalued by years of inflation)...
I read both Nature papers, the original research and the news perspective. I confess that I couldn't understand the experimental technique as described in either paper. Therefore, I cannot describe the experimental technique in terms of quantum theory.
So I wrote the authors of both papers for more information. Both of them were kind enough to respond to my note and I hope that we will continue to exchange emails because I have more to learn. I have put their comments below and added my own thoughts about orbitals at the end. 
 #1. Response from David Villeneuve:
 "I don't want to leave you unhappy. We were being somewhat provocative to
 show an orbital wave function, knowing that introductory quantum mechanics
 courses say that you cannot do this. However it is known that one can
 measure a quantum state if the system can be repeatedly prepared in the same
 state (see A Royer, Foundations of Physics 19, 3 (1989)). We use an
 interference technique of the same electron with itself to record the phase
 of the wave function.
 The other issue is whether a set of single-electron wave functions can
 represent the N body quantum system. The answer is no, but quantum chemists
 do it anyway. It is the best that we can do. There is lots of work in
 molecular structure that depends on these approximations, such as GAUSSIAN
 ab initio calculations. SO even if orbitals do not exist, they are
 extensively used.
 I learned after we published this, that the HOMO of N2 is not that well
 known. See Maksic and Vianello, J Phys Chem 106, 6515 (2002), and Stowasser
 and Hoffman, J Am Chem Soc 121, 3414 (1999) for conflicting calculations.
 Finally, if you are still unhappy, you can just think that we measured a set
 of transition dipole matrix elements from some state to a set of continuum
 plane waves."
 #2. Response from Henrik Stapelfeldt:
 "It is well established that a wave function or an orbital is not directly
 observable for a single quantum system. However, a set of measurements of
 observables (which are real measurable quantities) on a sufficiently large
 number of identically prepared systems may contain enough information to
 determine the wave function (or more generally the density matrix) of the
 system considered   up to an overall phase.
This approach has been taken in a large number of works over the past decade experimentally as well as theoretically. For instance, using quantum tomography the density matrix, or equivalently the Wigner function, has been reconstructed 
 for vibrational states of molecules, nonclassical states of light, trapped
 ions, atomic beams and dissociating molecules. I will be happy to send you
 references if you are not familiar with these works.
 In the recent work of Itatani et al. in Nature the molecular orbital is
 reconstructed from a large number of measurements of laser induced high
 harmonic spectra (the observables) using a mathematical tomography routine
essentially identical to the one used in medicine CT. This is what I tried to 
 explain in my News & Views article in Nature."
 #3. My current thoughts on these emails and orbitals:
One of the authors admitted that the "measuring an orbital" claim was deliberately provocative. He also said that one could view the experiment as only providing information about matrix elements. Unfortunately, I don't understand the latter, so I can't say whether he is withdrawing his claim of "measuring an orbital". Perhaps he still means this? Perhaps not.
Both of the authors pointed to other publications that describe experimental techniques for observing wave functions. I have not read these publications yet and I don't know what they will tell me. I suspect (but this is only a suspicion) that the maximal claim will be one can get information about the *full* molecular wave function. The minimal claim may turn out to be quite a bit less.
Supposing that one can get information about *full* wave functions, is it correct to claim that one can "measure orbitals"? (I believe that this is Sengen Sun's question.)
This poses interesting questions about what we mean by measurement and reality. I don't want to get entangled with these deep issues. I will be simple-minded and assume experimental measurements are real.
Some of the emails that I have read so far in this discussion make the following argument: if an experiment generates data Y that looks like a familiar entity X, then it is reasonable to say that Y tells us something useful about X. Other emails have gone farther: since Y comes
from an experiment ("Y is real" !), it is inappropriate to object to "Y 
 tells us about X" on merely theoretical grounds.
There is something to both of these arguments, but it is important to recognize their limitations. "Y tells us about X" is an interpretation of experimental data. It is a mental decision that comes after the experiment and not an intrinsic property of these data. If we decide to challenge this interpretation (and this can be done on a variety of grounds, including theoretical notions about X), we are not jeopardizing the status of Y as experimental data. We are only contesting the interpretation of Y.
Also, we have to remember that the interpretation "Y tells us about X", even though it looks reasonable, may be incorrect. Y may be correlated with our ideas about X, but there may be good reasons for thinking that X cannot cause Y.
Let's take a familiar example. Koopmans theorem equates experimental ionization energies with orbital energies (when the orbitals are defined in a certain way). Many experiments have shown that there is rough, sometimes excellent, agreement between measured energies (Y) and orbital energies (X). At this point, we might be tempted to say "Y tells us about X". Are we correct in doing this?
I think there is general agreement that Y (experimental ionization energies) do *not* tell us about X (orbital energies) even though this is an attractive (and frequently used) interpretation. First, we can object to this interpretation on purely theoretical grounds: 1) according to quantum theory, orbitals are not real, so when a molecule is ionized, we are not "really" changing the occupancy of one orbital and leaving the other orbitals unchanged, and 2) quantum theory provides an alternative description of ionization in terms of molecular states. The latter description is a) quantitatively more successful than the "orbital energy" interpretation, and b) is internally consistent with the rest of quantum theory.
If one stubbornly insisted that ionization energies tell us about orbital energies, I would have to ask, "since the task of theory is to explain experiment, do you plan to revise quantum theory so that it predicts orbital energies in accord with experiment?" It is interesting that this hasn't happened. Chemists are largely satisfied with the current quantum theory (it isn't broken) even when they find the siren-call of orbitals irresistable. This is an example of the compartmentalized thinking that we all engage in; different parts of my mind happily and simultaneously cling to contradictory points of view.
Coming back to the newest experiment in Nature (which I still don't understand): I don't think it can actually be measuring something about orbitals because quantum theory says 1) orbitals are not real, and 2) the theory always provides a superior orbital-free way to describe the experiment. I wish I could supply the latter, but I can't (yet).
If you feel that my inability to provide an orbital-free explanation proves that the "experiment tells us about orbitals", then I would say you have two challenges ahead of you. 1) Since current quantum theory says orbitals are not real, come up with a new quantum theory in which orbitals are real and have measurable properties. 2) Since your new theory will say that orbitals are real, accept the theoretical challenge of bringing your theory into line with experiment, i.e., modify it so that it correctly reproduces experimentally measured orbital properties.
If you are still caught in the middle -- not ready to give up the current theory, but you find the apparent similarity/correlation of experimental data and orbital shapes appealing, inspiring, provocative, and qualitatively useful -- I can sympathize. 
 Alan
 --
 Alan Shusterman
 Chemistry Department
 Reed College
 Portland, OR 97202-8199
 503-517-7699
 http://academic.reed.edu/chemistry/alan/
 "The Way you can go isn't the real Way." Lao Tzu