Meeting the Snob Factor

The best journals employ a snob factor as a first cut to limit the deluge of submitted manuscripts that go out for peer review.  The editor uses the "desk reject" for potential papers that don't look interesting.  Then, the reviewers are asked to evaluate a manuscript's significance to the field prior commenting on the technical details.  These two layers of subjective assessment can doom a manuscript, relegating it to a polite rejection: the work might be technically correct, but it is not of broad enough interest.

One such journal is Optics Letters, published by the Optical Society of America.  Though it is eclipsed by the new OSA journal Optica in its impact, it is still a highly selective and respectable publication.  Recently, we beat the odds by receiving an acceptance letter (subject to minor revision) along with the initial reviews.  The preprint of the paper can be viewed at https://arxiv.org/pdf/1809.01216.pdf

While the paper is based on some esoteric principles, it provides the experimentalist with a recipe for adding one state to the simple model commonly used in the field to correct for the infinite number of states that are omitted for bovious practical reasons.  This magical state is a proxy for those infinite numbers of states that are ignored.  The figure shows a plot corresponding to the uncorrected model (left) and the corrected one (right).  The nice smooth green background and the sharp red along the diagonal is the signature of success.  We thought it cool and useful that such a proxy state could fix a problem that has been plaguing nonlinear-optical measurements for decades.  For once, the editor and reviewers agree.

Here is a summary of the reviews:

Reviewer 1:

The manuscript represents an important advance in the calculation of nonlinear susceptibilities because it presents for the first time a method for dealing with the difficult continuum states present in realistic models of molecules. Ignoring these states leads, as the authors identify, to large errors in the calculations while, perhaps surprisingly, a single proxy state allows one to eliminate these errors to a large degree. This proxy state is not just a mathematical fudge, it is defined through physically measurable quantities. I therefore strongly recommend publication. 

Reviewer 2:

This is an interesting work discussing corrections to polarizability and hyperpolarizability calculations for limited state models that can be made using a single proxy state.  The conclusions are well supported by the calculations and this will find significant interest in its community.  

One of my strangest papers

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Regardless of whether or not its conclusions turn out to be true, this is a very interesting thought provoking paper...

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 "We are pleased to inform you that your manuscript has been accepted for publication as a Regular Article in Physical Review A."  Even after decades of research, these words never cease to brighten my spirits.

I admit that a recent paper by Shoresh Shafei and me, "The paradox of the many-state catastrophe of fundamental limits and the three-state conjecture," is a bit odd.  We submitted it to Physical Review A, the premier physics journal, with the attitude, "heck, what have we got to loose by submitting it to a journal that will most likely reject it?"  Papers that are so far off the beaten path usually don't fare well.  That's why it is so satisfying that it got accepted so enthusiastically with no resistance.

The paper was reviewed by two individuals, who both liked it.  In the words Reviewer 1, "I really enjoyed reading this manuscript, which is well written and reads well. It is the last (for the time being I guess) of a long series of papers on the subject by the main author, who knows the subject perfectly well. In spite of some length, and a tendency to repeat concepts which have been already made clear, I must say, again, that I find the manuscript agreeable." The reviewer is right that this paper might be the end of one particular line of work, at least for the time being, which seeks to understand certain fundamental issues in what has become an applied research field.  The reviewer was also right that the paper was a bit wordy.  I have developed this bad habit from a continuous misunderstanding of our work by many of my colleagues.  Perhaps this wordiness served its purpose.

The second reviewer, opened his/her review with, "Regardless of whether or not its conclusions turn out to be true, this is a very interesting thought provoking paper on fundamental limits of nonlinear susceptibilities.  It should be useful for those seeking to design optimal nonlinear optical materials.  It is well thought out and has computer-modeling evidence to support it."  Reviewer 2 captures my thoughts on the paper.  The work brings up ideas that challenge our past results and leads us into unknown territory.  This type of work is not so common these days in mainstream fields such as mine.

The reviews go on to suggest changes, which we diligently implemented, leading to the paper being accepted for publication.  A copy of the pre-edited version can be found on the physics archives.  We will post the final version on the archives when the paper appears for publication.

So what's so strange about this research?  An answer requires a short introduction.

There is a quantity called the hyperpolarizability, which quantifies the strength of interaction between light and materials.  For those of you interested in a more in-depth explanation, please check out the tutorial on NLOsource.com.

The concept of a hyperpolarizability was originally applied to molecules, but it also applies to quantum dots, multiple quantum wells, quantum wires -- pretty much anything.  Since practical devices are based on it, making it as big as possible is often the goal.  Since the hyperpolarizability is fundamental to nonlinear light-matter interactions, it can give insights into basic science.

Given the importance of the hyperpolarizability, I calculated its fundamental limits back in 1999 and published the results in Physics Review Letters.  Aside from a few early emails expressing mild interest, the work remained largely un-noticed until a critical comment was penned by Champagne and Kirtman that appeared a few years later in PRL along with my response.  The process of writing my response got me thinking again about limits, which gave me ideas that led to a series of papers that both vindicated my approach but raised additional questions.

Any quantum system is represented by a spectrum of states, each having a characteristic energy.  Based on intuition, I guessed that at the limit, only three states contribute to the hyperpolarizability.  This was later called the three-level ansatz (the German for a guess).  There were still too many parameters remaining, and if they could have arbitrary values, there would be no limit.  Next, I used the sum rules, which relate these parameters to each other, to further simplify the equations.  The sum rules are neat because they come directly from the Schrodinger Equation without any approximations; and, they must hold for any system.

The combination of sum rules and the three-level ansatz lead to a limit, which turned out not to be a single number, but a function of the number of electrons in the system, N, and the energy of the first excited state E10.  This too made lots of sense because the limit must depend on the size of the system, which is related to N and E10.  The hyperpolarizability is like an area.  It is nonsensical to ask for the limit of area, but determining the largest possible area as a function of perimeter leads to insights about geometry.

Luckily, I had just edited a book on nonlinear optical materials, which contained a tabulation of all the molecules that had been measured before the book appeared in print.  A plot of a comparison of these molecules with the limit showed that they all obeyed the theory.  So far so good.  However, the best molecules were  a factor of 30 below the limit.  Molecules are hard to make, and they come in many shapes and flavors.  This gap suggested that there may be whole classes of molecules that are yet to be discovered that could fill the void.  Alternatively, it might be that no stable molecules with the required structure exist.  This set off an explosion of work in my group that was funded by the National Science Foundation for almost a decade, and still going strong.

To make a long story short, it is possible to "make" all sorts of quantum systems as theoretical models.  We can be like gods, holding the nuclei in positions that they would normally refuse to occupy in the real world.  This allows us to see how electrons would behave in all sorts of weird configurations.  We designed "electromagnetic bottles" that coax the electrons into highly peculiar orbits by simply adjusting a couple parameters.  No matter that the parameters we need might exceed the energy capacity of the world over the next decade.  We can know the result.  We even allowed electrons to interact with each other in the most bizarre ways that would make them blush.  Quantum mechanics and electromagnetic theory can predict the behavior of nature to unprecedented accuracies of many decimal places, so we can be confident that our musings correspond to a reality, however unpractical.

These investigations found that the best systems, independent of the approach in making them, yield a hyperpolarizability of 0.70899, where the limit is 1.  In every case, when the quantum system is at this extreme, we find that it is described by only three states.  Somehow, the three-level asnatz is always obeyed.  We also did what are called Monte Carlo studies, where instead of calculating the hyperpolarizability from the Schrodinger Equation, we determine all the parameters by randomly picking them under the constraint that they obey the sum rules.  Trying millions of runs, the largest values that we got  were 1, consistent with the prediction of our limit theory, and the three-level ansatz continued to be verified when the hyperpolarizability was 1.

All quantum systems must obey the sum rules, but, these equations are obeyed by more general systems than are described by the Schrodinger Equation.  We therefore hypothesized that the values between 0.7089 and 1 were the domain of exotic Hamiltonians governing phenomena that have not yet been discovered. We are busily pursuing this idea, but more on that later.

Aside from this curious 30% gap, the theory seems to correctly predict an upper bound and the calculations verify the three-level ansatz.  This state of affairs left many questions unanswered, but things looked to be self consistent.

The three-level ansatz is the key.  It is a guess that always seems to hold, but has never been rigorously proven.  Our attempts to prove the three-level ansatz basically boiled down to showing that when an M-level model is reduced to an (M-1)-level model, the hyperpolarizability gets larger.  Since the two-level model was previously proven to be unphysical, by induction, the three-level model would remain standing as the model that yields the maximum.

Shoresh took a different approach.  He started with a four-level model and varied the parameters using sliders in a popular program called Mathematica.  Like an audiophile adjusting the levels on an equalizer, he watched how the hyperpolarizability ebbed and waned with various choices of parameters.  He demanded that the equations obey the sum rules, but did not restrict the number of states. To his surprise, he found that when the second and third states become degenerate (of the same energy), the hyperpolarizability is 1.28, breaking the limit.  He continued to add states and found a pattern that for a system with M states, if all the states save the ground state and highest-energy state are degenerate, the hyperpolarizability is bigger than 1 and gets bigger and bigger as more states are added.  For a system with an infinite number of states, the hyperpolarizability becomes infinite.  This behavior, we dubbed, the many-state catastrophe.

The many-state catastrophe has many implications.  First, it invalidates the three-level ansatz by counterexample; the larges hyperpolarizability is not given by a three-state system.  Secondly, it shows that there is no limit. However, both of these results run counter to all observations.  As we have seen over and over, there is an observed limit and three-states always dominate the response at the limit.  Granted, we have sampled less than 1,000,000,000 systems, so perhaps we have missed the cases that invalidate our theory.

Having an infinitely-degenerate system that leads to infinite hyperpolarizability is clearly unphysical.  It appears that the three-level ansatz, though quite simple, somehow acts to restrict the space of all possible quantum systems that obey the sum rules to the ones that are physical.  How it can possibly do this blows my mind.

The three-level ansatz has problems because it leads to a 30% overestimate to what is observed, but that's pretty close for a guess.  There are also other mathematical issues with the theory that I will not explain here; but nevertheless, the theory appears to be highly predicative and has been successfully applied to many studies.  For example, the theory has found a new paradigm for making better molecules. 

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In a sense, we have been spending lots of time coloring, but can't tell what fraction of the page we've covered.

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Given that the many-state catastrophe invalidates the three-level ansatz, but is found to always hold brings up the possibility that it is true for all real systems, but unprovable unless we can find a different general constraint that limits systems to being real ones.  Such an additional constraint would then allow us to prove the three-level ansatz as its consequence.  However, given the generality of the problem, finding such a constraint may not be possible.  We have been testing various classes of quantum systems, so in some ways, we have been coloring in the regions corresponding to real systems.  If we can somehow show that we have colored in the whole region corresponding to all possible REAL systems, and the three-level ansatz always holds, that would also constitute a proof.  In a sense, we have been spending lots of time coloring, but can't tell what fraction of the page we've covered.

Our work stirs up mathematical inconsistencies and proposes ideas that are unproven and might even be wrong; but the  implications are tantalizing.  Mathematicians most likely can point to defects that invalidate much of what we have done, while technologist may complain that we have not really made a practical advance.  The fact that nature seems to behave according to the predictions of our theory is an indication that we are at least on the right path and the fact that the theory may not be derivable by deduction from known physics is thrilling.

In summary, the many-state catastrophe leads us to propose that the three-level ansatz is the correct constraint to enforce nature's will by restricting the sum rules to the realm of the real world.  If the three-level ansatz, which has so far been observed to be correct in the real world is not provable, then it may be a fundamental principle.  The chances of this being the case is highly improbably, but the quest for trying to find the proof will undoubtedly uncover many treasures.

I agree with the reviewer that this stuff is interesting even if it is wrong.

A correspondence on the intrinisic hyperpolarizability

I often get correspondence from scientists form all over the world. One such arrived a couple days ago asking about the intrinsic hyperpolarizability and why it is a useful quantity for comparing molecules. Below is the original message and my response:

Email to me:

Dear Sir,

I have a doubt regarding beta-intrinsic value. Which molecule is of greater practical importance, having a greater beta-intrisic value or a greater beta-value? If molecule has greater beta-intrinsic and lesser beta-value as compared to its related counterpart can it be regarded as a better molecule for practical applicability?
Thanks.

Kind regards,
Sincerely,
So-and-so

My response:

Dear Dr. So-and-so,

The intrinsic hyperpolarizability is used to understand the origin of the nonlinear response of a molecule. Making a molecule larger will yield a bigger value of beta; but, the intrinsic hyperpolarizability tells you how effective it is given its size. This kind of understanding can lead to the rational design of better molecules by first identifying ones that have a large intrinsic hyperpolarizability and then making them larger using the same "shape" or theme.

Having a molecule with a large hyperpolarizability in itself is not technological significant because that property alone will not necessarily make it useful in a device. It needs to be incorporated into a material with a large bulk response and then needs to be formed into a device component that is photochemically stable, etc. Thus, a molecule with large beta is not of technological interest without lots of other work to determine other properties; and, a small intrinsic beta makes it less interesting from the point of view of science.

A large beta molecule may be of interest to others if it has other unique properties, such as an ability to attach it to microdots to enhance local electric fields, or if it acts a charge sensitizer in a polymer, etc.

Best,
Mark G. Kuzyk

In the near future, I plan to write a description of our research aimed at the non-expert so (s)he can gain an appreciation of our work, which is based on trying to understand complex properties of a system by looking at large-scale patterns. Stay tuned.

Research summary on COMPLEX MOLECULES MADE SIMPLE

A review of our work that will appear in Physical Review

COMPLEX MOLECULES MADE SIMPLE

Organic molecules are versatile and can be custom tailored for a large variety of applications that span such diverse fields as cell microscopy, cancer therapy, computing technology, or high speed communications, to cite just a few. With this high degree of flexibility comes complications. Even when a system can be reasonably approximated by two excited states, at least 7 parameters are required to predict important properties such as the nonlinear-optical response. In the present work, a combination of sum rules and symmetry constraints has been shown to allow the problem to be reduced to 3 parameters, which can be determined using two simple measurements -- the linear absorption spectrum and a measurement of the hyperpolarizability at just one one wavelength using hyper Rayleigh Scattering. This approach has been applied to the complex molecule AF455, and shown to accurately predict not only the correct shape of the two-photon absorption spectrum, but also its absolute magnitude. Here is a rare instance where two simple measurements accurately predict all the linear and nonlinear optical properties of a molecule. With complexity simplified, new paradigms for making better materials may follow.

Simple Scaling and too much to do in too little time

This was a good week. Nathan successfully defended his dissertation with flying colors and our manuscript for Advanced Materials, a high impact journal, was accepted. Our paper is a comment on another paper that previously appeared in Advanced Materials.

One central theme of our work that uses fundamental limits and sum rules to understand the nonlinear-optical response is the idea of scale invariance. The Schrodinger Equation has the property that the shape of the wavefunction does not change when the width of the potential energy function is decreased by a scaling factor b and the depth of the well is simultaneously increased by b squared. Under such a transformation, the wavefunction is compressed by the factor b, but otherwise, the shape remains the same. We call this simple scaling.

The nonlinear-optical quantity of interest to many applications is called the hyperpolarizability. Making the hyperpolarizability as large as possible is an ongoing area of intense research activity. To nobody's surprise, a larger molecule will generally have a larger hyperpolarizability. To better understand what makes a material tick, we have defined a quantity called the intrinsic hyperpolarizability, which is simply the ratio of the hyperpolarizability of a quantum system, divided by the fundamental limit. Interestingly, the intrinsic hyperpolarizability is invariant under simple scaling, so large and small molecules that are related to each other by simple scaling will have the same intrinsic hyperpolarizability.

In examining all the molecules that had been studied for nonlinear-optical applications over a 3 decade period, we found that the large range of hyperpolarizability values could mostly be accounted for by simple scaling. Thus, researchers were making molecules larger and larger but the best intrinsic hyperpolarizabilities remained static at about 0.03 - suggesting that it would be possible to make a factor of 30 improvement; but to get there would undoubtedly requrie a major paradigm shift.

Several years ago, I published a paper that showed how to calculate a related quantity called the intrinsic two-photon absorption (TPA) cross-section. More recently, Javier Perez-Moreno and I published a paper that introduced a rough rule of thumb for determining the intrinsic TPA cross-section - simply divide it by the square of the number of electrons. My earlier paper showed how to determine the number of electrons.

To my delight, my paper on TPA gets lots of citations, not because of what I believe is the beautiful physics of the work, but because scientists refer to my method of counting the number of effective electrons - a quite trivial (and approximate) procedure. To my horror, when comparing molecules, most researchers then go on to divide by the number of electrons rather than by the square of the number of electrons as suggested by Javier's work. This leads to a flawed comparison between molecules.

A recent paper in Advanced Materials reported on TPA cross section measurements of a new class of dendrimers, molecules with ever-branching pieces much like veins and arteries. They referred to my paper when calculating the number of electrons; but, as is usually the case, they divided by the number of electrons and found that the new dendrimer class was an order of magnitude better than the best existing dendrimers - an impressive improvement.

In our comment on this paper, we reanalyzed the data using the N squared rule and found that the new materials were in fact two orders of magnitude better. In addition, the dendrimers within each class, though of vastly differing sizes, all had approximately the same intrinsic TPA cross section. Thus, we were able to show that the authors had made an even more important discovery than they had realized. Usually, comments on a paper point out a negative flaw, leading to strong rebuttals and counter-rebuttals. In this case, all parties were winners.

Unfortunately, these small successes were overshadowed by a pile of work. After the Thanksgiving break, I am going to an NSF meeting in Hawaii (I hate to travel and I hate hot and humid places), where I will be reporting on the results of our projects. I need to prepare a glitzy poster as well as an oral presentation. This, on top of being hopelessly behind in preparing problem sets/solutions, grading, and catching up on lectures for my graduate mechanics class. In addition, I need to write a pile of recommendations and read a 350 page dissertation; the defense will take place early Monday morning.

Smack in the middle of this stressful week, after months of waiting for an estimate, a flooring contractor handed us an estimate and told us that he could get new floors in before Thanksgiving. Things moved fast, requiring us to immediately move large and heavy furniture back and forth between two rooms, which included taking down built-in cabinets and then replacing them, as well painting all the walls. After spending three solid days on manual labor (actually a satisfying break from work), my time pressures have become critical. I cringe at the accumulating piles of manuscripts waiting to be written and the papers that I need to review for journals.

So, how did I handle the stress? I squandered a couple hours writing about my frustrations on this blog. And now, back to work...

The good, the bad, and the nasty

Just a few minutes ago I was preparing my lectures for my graduate mechanics class. As I often do, I was once again distracted, thumbing through the latter pages of the textbook - marveling at the simple beauty of physics. Even classical physics is rich with phenomena, yet it is unified by simple ideas from the calculus of variations. I yearn to savor the physics that lies before me, but awaken to reality with every ping alerting me to a new email. Sadly, life is filled with trivial tasks that keep me from my passions.

The last few days brought some good and bad news. As I had anticipated, Nathans' revised manuscript was accepted for publication. And, Shoresh's and Mark's JOSA B paper, which I had previously reported as being highlighted on the Optical Society of America's website, was the second most downloaded JOSA B paper in September 2010. See http://www.opticsinfobase.org/josab/abstract.cfm?uri=josab-27-9-1849 for a free download of the paper.

The paper that I had submitted to the Journal of Chemical Computation and Theory with David Watkins, over which I waged a full out battle with the referees, was finally rejected. I had expected this outcome from the outset, but thought it worth a try. My intention was to expose a new audience to our work, but apparently, this was not to be. Below is the strongest criticism of our paper.

"Professor Kuzyk seems strongly taken with his own work but is not sufficiently mindful of the work by others. Perhaps the most egregious example deals with the issue of neglecting the effect of vibronic interactions on the static first hyperpolarizability. As justification he cites two of his own papers, but totally misses the extensive literature in this field showing that the vibronic effect cannot be ignored. There are, in fact, many instances where the vibronic term is comparable to, or larger than, the pure electronic contribution. Hence, the maximum value he uses (derived only by considering electrons) could easily be breached without invoking any exotic systems."

The reviewer is referring to the fact that I cited one of our papers that shows that vibronic states do not contribute substantially to the hyperpolarizability. Undoubtedly, in heated debates, the parties involved often speak past each other. To do my part, I am once again plowing through this literature to understand the issue. However, I find it is a bit annoying that the reviewer did not point out an error in our logic, but rather made sweeping statements. My guess is that it is certainly possible for real molecules to have vibronic contributions that are large compared with electronic excitations; but, I believe that when a quantum system is designed to have a hyperpolarizability near the fundamental limit, the vibronic contribution will never be as large. And since we are always concerned with the physics of a system near the limits, I believe that we are correct.

It is interesting that the reviewer states that the limit could easily be breached in the presence of vibronic states. Since the best molecules ever measured fall a factor of 30 short of the limit, I do not believe that the reviewer's confidence is justified.

Though this exchange with the reviewer was one of the nastier ones I have experienced, I find it all to be trivial in the larger scheme of things. Happiness is most abundant when I am absorbed in Physics. At some point, I will rewrite this manuscript, taking into account these comments, and will see how our results mesh with the body of scientific understanding. I take solace in the fact that people are reading our papers. I see this review not as a devastating blow, but as an opportunity for new investigations. The last 6 years of my research on fundamental limits were ignited by a comment that was published on my 2000 PRL paper. In the process of accumulating evidence to show the criticisms wrong, I made many discoveries and gained deep insights. These kinds of experiences should not be seen as defeat, but as an invigorating start to a new chapter of research.

We finally got it right (we hope)

It's been three years since Juefei Zhou finished his Ph.D. research that culminated in a nice piece of collaborative work with the group of Koen Clays of the University of Leuven in Belgium. The research used a combination of theory and experiments to determine all the parameters needed to predict the full wavelength dependence of the two-photon absorption cross section. The beauty of the approach is that the theory uses the Thomas Kuhn sum rules to significantly reduce the number of parameters required to describe a molecule. This reduced set of parameters was determined from two experiments - a linear absorption spectrum and the measurement of the hyperpolarizability at just one wavelength.

Given that nonlinear-optical quantum calculations are notoriously inaccurate; and, independent measurements (such as first and second hyperpolarizability measurements) often disagree, we were elated that our approach led to a global agreement between all quantities using just one small set of parameters. The only problem was that our theory was wrong. We had made a false assumption. Thus, our manuscript was placed on the back burner.

A year later, in 2009, I spent a summer in Belgium, and used a combination of symmetry arguments and sum rules to show that our equations turned out to be correct, but for very different reasons. As we were applying the final touches to the manuscript when I returned back to Pullman, Xavi found an error. After days of intense debate, we found a way to correct the mistake and submitted a revised manuscript to Physical Review A.

A very sharp reviewer caught what appeared to be a fatal error. Our symmetry arguments were correct, but they implied an additional condition that rendered our approach untenable. For the next 12 months, we were haunted by a model that was wrong yet seemed to fit the data perfectly well.

At the beginning of the summer of 2010, Shengting was getting frustrated with a laser that refused to work properly, and asked for a theoretical project as a diversion. I suggested that he learn group theory and apply it to fixing the model. While Shengting was making good progress in both learning group theory and developing a plan of attack to address our problems, Xavi arrived from Belgium for a six-week stay. During my trip to Budapest, they had found a solution, albeit with a few holes.

A day before I returned, Koen Clays arrived in Pullman, and spent some time discussing the problem with Xavi and Shengting. As a result, they got closer to a solution. When I got back to Pullman (a day late because of a missed connection in Amsterdam), the four of us met to discus the problem and the proposed fix. Xavi acted as the spokesperson and very animatedly described the approach on the blackboard. Within an hour, everything fell into place. Not only did the original mathematical form of the theory turn out to be true, the underlying physics was even more beautiful than we had imagined. This project has led to new ideas that will take us into novel areas of research that we hope will make a closer connection between our theory, which is a bit esoteric, and real molecules.

It was worth the wait. This paper will add a significant new paradigm to the body of knowledge that seeks to more deeply understand the nonlinear-optical response of complex molecules. The path of our research took us through exhilarating highs and unbearable lows. Hopefully, our models are finally right. If not, the self-correcting process of the scientific method will eventually lead us, or someone else, closer to the truth.

Sophistication versus Understanding

I attended the SPIE meeting in San Diego over the weekend, where I gave an invited talk about work done in collaboration with David Watkins of the Math Department. The meeting was in a small room with perhaps a couple dozen attendees, all experts in organic nonlinear optics. We had a good time exchanging ideas and throwing about some new thoughts.

As I have mentioned in previous posts, I find travel physically draining. So, spending eight hours en route to San Diego on Saturday and eight hours on Sunday to return home took its toll. While I usually work on weekends, I find it much more relaxing than sitting on an airplane. So, I started the week in an uncharacteristically bad mood.

To add insult to injury, I learned on Sunday and Monday that two of my papers were outright rejected -- not something that I commonly experience. In addition, a paper submitted by my collaborators, to which my contribution was relatively minor, was also rejected. Thus, in a span of three days I had more rejection than in a typical decade. I think that my darker than usual mood was warranted given the extraordinary circumstances. In fact, I entertained the notion of quitting the professional life of physics altogether.

After a good-night's sleep, my mood dramatically improved so on Tuesday morning, I wrote a levelheaded email to the two editors who had rejected my papers. In the process of composing these letters, I realized that science has turned into a big efficient machine, with creativity a reluctant causality.

For more than three decades, lots of people have been expending a great deal of effort to make better molecules. In parallel, computational methods are getting more sophisticated so that theoretical chemists can calculate the properties of ever-larger molecules, taking into account more subtle effects and getting more accurate results. Similarly, chemists have made a huge number of very complex structures, many of which are pieces of art, such as the ever-branching dendrimers.

My own work, which uses sum rules to understand the nonlinear optical response of quantum systems (which I started ten years ago), illustrates how simple but powerful ideas can arise even in a mature field. The basic question that I asked was if there was a fundamental limit to the nonlinear optical response. The answer was a resounding "yes." This limit is not based on practical considerations, but on very fundamental quantum mechanical principles that span the basis of chemical reactions of life and govern the flow of electrons in electronic circuits. If the fundamental theory of quantum mechanics were to be wrong, then the world would be alien to us. In fact, we probably wouldn't exist. In short, I feel that my fundamental limits calculations stand on solid ground.

I expected great admiration for my theory when I first presented a talk on the topic a decade ago. Instead, I got some very nasty comments to the effect that my work was an insult to all the hard-working chemists who were trying to make better molecules. Who was I to say that it was not possible to do any better? Though that sentiment did not reflect my intentions, mother nature DOES place limits on what is possible.

To put this into perspective, my calculation does not imply a single numerical limit, but rather a limit in the presence of a contraint. For example, to investigate the largest possible area is nonsensical. It makes more sense to determine the largest possible area given a fixed perimeter. Similarly, when studying molecules, it is more appropriate to determine the largest nonlinear response for a given molecular size. Since size is not well defined in a quantum system, we used the more abstract concept of scaling.

The bottom line is that after three decades of research, the best molecules fall short of the fundamental limit by a factor of thirty. While people have been making bigger molecules with a larger nonlinear-optical response, the intrinsic nonlinearity has not changed since the birth of the field. That's why one of the reviewer's comments was particularly annoying. (S)he was critical of our simple fundamental approach as passe in light of all the sophisticated and precise methods available. Ironically, our simple approach has been the only one that has led to an improvement in the intrinsic hyperpolarizability.

Being a scientist, my priority is to understand, not to participate in the frenzy of doing the most sophisticated calculations. I prefer to study broad principles that apply to all systems rather than seeking higher precision in more complex calculations that apply to specific molecules. My long-term goal is to build an understanding of the fundamental issues that identify universal properties of systems that approach the fundamental limit. And in this quest, my small effort continues.

I end this post with good news. In response to my emails, both editors reversed their decisions and are giving us the opportunity to submit a revised manuscript. While I am concerned that the trend of sophistication worship is wasteful, I take comfort in the fact that our group is supported to continue our work. Perhaps a time will come when I will become one of the dinosaurs that was left behind, but for now; I take great satisfaction in my research and the potential it has for making a lasting contribution to the body of science.