In my review of the new biography of Enrico Fermi I alluded to one of Fermi’s most notable qualities – his uncanny ability to reach rapid conclusions to tough problems based on order of magnitude, back of the envelope calculations. This method of approximation has since come to be known as the Fermi method, and problems which can especially benefit from applying it are called Fermi problems.
It struck me that chemistry is an especially fertile ground for applying the Fermi method, and in fact many chemists probably use the technique unconsciously in their daily work without explicitly realizing it. For understanding why this so, it’s worth taking a look at some of the details of the method and the kinds of problems to which it can be fruitfully applied.
At the heart of the Fermi method is a way to make educated guesses about different factors and quantities that could affect the answer to a problem. Usually when you are looking at complicated problems not just in physics or chemistry but in psychology or economics for that matter, much of complex problem solving involves examining different factors that could influence the magnitude and nature of the solution. For instance, say you were calculating the trajectory of a bomb dropped from an airplane. In that case you would consider parameters like the velocity of the plane, the velocity of the bomb, air resistance, the weight of the bomb, the angle at which it was dropped etc. If you were trying to gauge the impact of a certain economic proposal on the economy, you would consider the market and demographic to which the proposal was applied, the presence or absence of existing elements which could interact positively or negatively with the proposed policy, rates of inflation, potential changes in the prices of certain goods relevant to the policy etc. The first part of the Fermi method simply involves writing down such factors and making sure you have a more or less comprehensive list.
The second part of the Fermi method consists of making educated guesses for each of these factors. The crucial aspect of this part is that you don’t need to make highly accurate predictions for each factor to the fourth or fifth decimal place. In fact it was precisely this approach that made Fermi such a novelty in his time; it was because physicists could calculate quantities to four decimal places that they were often tempted to do this. Fermi showed that they didn’t have to, and in some sense he weaned them away from this temptation. The fact of the matter is you don’t always need a high degree of accuracy to reach actionable, semi-qualitative conclusions; you just need to know some rough numbers and get the answer right to an order of magnitude. That was the key insight from Fermi’s technique.
Now, before I proceed and discuss how these two aspects of the Fermi method may apply to chemistry, it’s worth noting that there are of course several examples in which an order of magnitude answer is simply not good enough. A famous example concerns the very Manhattan Project of which Fermi was such a valued member. In the early phases of the project when General Leslie Groves was picked as head of the project, he quizzed the scientists in Chicago about how much fissile material they would need. When they said that at that point all they give him was an answer correct to an order of magnitude, he was indignant and pointed out that that would be tantamount to ordering a wedding cake and not knowing whether to order enough cake for ten people or one person or a hundred people.
Notwithstanding such specific cases though, it’s clear that there are in fact several example of general problems which can benefit from Fermi’s technique. Chemistry in fact is a poster child for both the key aspects of the method illustrated above. Many problems in chemistry involve estimating the various kind of forces – electrostatic, hydrophobic, hydrogen bonds, Van der Waals – influencing the interaction of one molecule with another. For instance when a drug molecule is interacting with a protein, all these factors play an important role. Sometimes they synergize with each other and sometimes they oppose each other. Using the Fermi method then, you would first simply make sure you are listing all of them as comprehensively as possible. The goal is to come up with a total number resulting from all these contributions that would crucially provide you with the strength or free energy of interaction between the drug and the protein; a quantity measured in units of kcal/mol.
This part is where the method is especially useful. When you are trying to come up with numbers for each of these forces, it’s valuable simply to know some ranges; you don’t need to know the answers to three decimal places. For instance, you know that hydrogen bonds can contribute 2-5 kcal/mol, electrostatic interactions usually add 1-2 kcal/mol, and all the hydrophobic interactions will add a few kcal/mol to the mix. There are some trickier estimates such as those for the entropy of interaction, but there are also approximations for these. Sum up these interactions and you can come up with a reasonable estimate for the free energy of binding. The job becomes easier when what you are interested in are differences and not absolute values. For instance you may be given a list of small molecules and asked to rank these in order of their free energies. In those cases you just have to look at differences: for instance, if one molecule is forming an extra hydrogen bond and the other isn’t, you can say that the first one is better by about 2-3 kcal/mol. You can also use your knowledge of experimental measurements for calibrating your estimates, another trait which Fermi supremely exemplified.
This then is the Fermi method of approximate guesses in action. One of the reasons it’s far more prevalent in chemistry than physics is because unlike physics, in chemistry it’s usually not even possible to calculate numbers to very high accuracy. Therefore unlike some physicists, chemists would not even be tempted to attempt to do this and would have already resigned themselves (if you will) to making do with approximate solutions. Today the Fermi method is incorporated in both the minds of seasoned working chemists as well as in computer programs which try to automate the process. Both the seasoned chemist and the computer program try to first list all the interactions between molecules and then try to estimate the strengths of each interaction based on rough numbers, adding up to a final value.
The method does not work all the time since every interaction is modeled, so it may potentially miss some important real life component. But it works well enough for chemists and computers to employ it in a variety of useful tasks, from narrowing down the set of drug molecules that have to be made to prioritizing molecules for new materials and energy applications. Enrico Fermi’s ghost lives on in test tubes, computers, fume hoods and spectrometers, more than even his wide-ranging mind could have imagined.