Subject: CCL:orbitals and reality

From: Alan Shusterman <alan.shusterman[at]reed.edu>
Date: Thu, 20 Jan 2005 10:22:17 -0800

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

---

From: Ignacio Nebot-Gil <Ignacio.Nebot_at_uv.es>
Date: Thu, 20 Jan 2005 18:53:27 +0100

Begin forwarded message:

> From: Ignacio Nebot-Gil <Ignacio.Nebot_at_uv.es>
> Date: 20 de enero de 2005 17:32:12 GMT+01:00
> To: "Jorge Seminario" <jorge.seminario_at_chemail.tamu.edu>
> Subject: Re: CCL:orbitals and reality
>
> Dear all
>
> I do'nt agree.
>
> Orbital, following McWeeny and Sutcliffe definition (Methods of  
> Molecular Quantum Mechanics) is the wavefunction obtained from a  
> monoelectronic hamiltonian operator, actual or effective.
>
> If the hamiltonian operator is actually monoelectronic, no other  
> electrons are present. If it is effective, then, the electron  
> interaction is averaged in some way, as in the majority of systems  
> (all but H atom and H2+ molecule and some other ions). Then, orbital  
> is only the result of an approximate calculation, although it can be  
> used to establish models and to interpret experimental facts, as many  
> chemists do.
>
> But my main objection is that orbital is a wavefunction (no physical  
> meaning according to the Bohr interpretation of the wavefunction) and  
> electron density is related not with the orbitals but with its SQUARE  
> (probabilistic meaning, Bohr again). Hence, the difference between the  
> experimental electron density of N2 and N2+ cannot be related with the  
> HOMO, but, in any case, with its square.
> Even that will not be right, because the electron density difference  
> is experimental, and then the ACTUAL electron interaction is involved.  
> This interaction is not taken in consideration in the orbital concept  
> (see definition above). In other words, if we calculate the N2 HOMO,  
> even using an infinite basis set, and we obtain its square, the  
> result, which can be considered as the exact HOMO, will not be  
> identical to the electron density difference between N2 and N2+. The  
> electron interactions taken into account in these two situations are  
> DIFFERENT.
>
> Ignacio Nebot-Gil
>
> On 20/01/2005, at 11:36, Jorge Seminario wrote:
>
>> Dear All,
>> I understand the problem is whether MO's have any physical
>> reality or not to be observable.  Certainly, the goal of any model we  
>> create
>> to explain a feature of nature is to explain such feature the best it  
>> can;
>> needless to say, some models are better than others.
>> Let's assume, as many agree, that the molecular densities can be  
>> measured,
>> so we can get rho(r) at any point r for the neutral of N2
>> and also for its vertical cation, N2+.
>> Let me define the difference of these two densities as the
>> "experimental density of the HOMO" (just a definition).
>> This density difference corresponds to a one-electron density,
>> it certainly corresponds to the ejected electron,
>> and therefore has correspondence to a one-electron wavefunction;
>> this should be trivial to accept (well, to same extent...)
>> Now, if you read the papers of my dear friends Levy and Perdew
>> several years ago, this experimental HOMO should correspond "exactly"
>> to the HOMO in DFT (provided you have the "exact" functional, any way
>> it will be a good approximation otherwise).
>> Thus, my point is that if we can observe one-electron densities of
>> molecules, we certainly can observe the HOMO (at least indirectly).
>> For sure, we can have another long discussion to decide
>> which of the theoretical HOMOs is the best but that is another  
>> problem.
>> Cordially
>> Jorge
>> Jorge M. Seminario
>> Professor  of Chemical Engineering and
>> Professor of Electrical Engineering
>> Texas A&M University
>> 3122 TAMU
>> College Station, TX 77843-3122
>> Phone and Fax 979-845-3301
>> seminario:at:tamu.edu
>> http://cheweb.tamu.edu/faculty/seminario/
>>
>>
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>>
>>
>>
>
> Prof. Ignacio Nebot-Gil
>
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