Week of October 28, 2021
It's been nearly three months since my last blog entry. This is how things begin in the confessional. . . Let's jump back in.
If you were to ask most physics professors today when you should use quantum mechanics and when you should use classical mechanics, in their top five list will be the idea that quantum mechanics applies to very very small systems like atoms and molecules and that classical mechanics only applies to very very large systems like you, me, your house, or planets and stars.
Classical physics appears to work very well for big systems and quantum mechanics appears to work well for very small systems. Here, I want to drive home the fact that when size doesn't matter we find connections between classical and quantum mechanics.
I once presented a poster at a conference showing photos of a balloon inflated to different sizes. On the balloon were several dots I drew with a marker. No matter how large the balloon became, the geometric symmetry of the dots remained the same. The dots formed a triangle, and the shape of the triangle didn't change. Only the size of the triangle changed.
Imagine a picture of the Earth with an outline of the continents, no matter what size picture you have -- whether a tiny picture on your phone or a giant picture on a billboard, you would immediately recognize it as a photo of the Earth because of the shape of the continents. That's because the shapes of things in a system don't change when its size changes.
More to the point of this entire blog, here is a lovely animation.
In the animation are four small blue spheres attached to the surface of a larger transparent sphere. The size of the transparent sphere changes, but the tetrahedron whose vertices are formed by the small blue spheres doesn't change shape. It's size changes, but it remains a tetrahedron. The same is true of the sphere itself. If the blue spheres were electrons, for any size of the sphere the electrostatic configuration is formed with electrons at the vertices of a regular tetrahedron. Size doesn't matter.
When Size Doesn't Matter
When size doesn't matter, classical physics -- classical electrostatics -- reveals a very rich landscape of features consistent with numerous known physical quantities related to real atoms found throughout the periodic table of elements. This fact would have been warmly welcomed in the early decades of the 1900s when chemists Lewis and Langmuir were developing their static models of atoms (the cubic atom) that eventually gave rise to very useful concepts in chemistry like covalent bonds, the octet rule, and so on. But calculations from quantum mechanics had already revealed features consistent with known properties at the time. This is why I tend to believe that the rapid successes of quantum mechanics led to the demise of any further thinking in classical physics similar to that pursued by chemists of the day.
When size doesn't matter, it's the geometric symmetry of electron configurations of electrons no longer moving that dominate such things as electron shell-filling as I discussed in a previous blog.
To understand why geometric symmetry is so important, it may be instructive to take a look at how other scientists have looked at the problem before. Typically, one evaluates the energy solution of the Thomson problem for each N-electron case and compares it with neighboring energy solutions. For example, compare the energy for N=11 with the energy for N=12, and for N=12 with the energy for N=13, and so on in the figure below. The problem with this approach is that when an electron is added to a system, the number of electrons in the system and the geometric symmetry of the system are changed.
If we didn't impose the fixed size of the sphere in the Thomson problem, the size of the system would also change as is the case in my work involving electrons in a dielectric sphere -- the size of the electron system increases with N while the size of the dielectric sphere doesn't change. So there are three things that change in the approach taken by other researchers that affect the total energy of the system: size, electron number, and geometric symmetry. When researchers held the size fixed by using the Thomson problem, they didn't hold the electron number fixed to evaluate how geometric symmetry affects the energy. The reason I thought about this was because I knew the geometric symmetry for N=11 is very poor and the geometric symmetry for N=12 is very good (this was well-known among mathematicians and published in the 1980s) -- so why doesn't such large changes in geometric symmetry exhibit itself in the energy evaluation when we compare neighboring energy solutions? In fact, a plot of the energy solutions of the Thomson problem as N increases, looks like they it be fit with a very smooth mathematical function. We know this cannot be the case though, because to-date there is no known general solution of the Thomson problem! If we could fit the data in the figure below with a smooth mathematical function, we could predict the energy of any N-electron system. But we cannot and are so far restricted to calculating numerical approximations of each N-electron energy.
Absolutely nothing in the figure below appears to accounts for the stark difference in geometric symmetry between the N=11 and N=12 configurations. I've used data from the Thomson Problem Wikipedia page so you can check the plotted data for yourself. As you can see, there is nothing interesting here whatsoever, and most researchers simply dismissed the idea that there is any connection between the Thomson problem and atomic structure.
Numerical energy solutions of Thomson Problem using data from Wikipedia. [REF]
There are many published articles primarily concerned with the quantum mechanical evaluation of systems similar to the Thomson problem that yield visually obvious features that account for the underlying geometric symmetry differences. A great example from among these articles is this 1999 article by Bednarek ("Many Electron artificial atoms", Phys. Rev. B, 59(20) 13037 1999) that shows several plots of data of interest to novice readers. It's downloadable for free from Researchgate so click through and download it. Have a look at how uniformly distributed the energy levels are in the classical data compared to the rich non-uniformities found in the quantum mechanical solutions. In this article, the authors argue that the Thomson problem is the classical analog that exactly describes the quantum mechanical system they evaluate. They concluded no connection between the classical calculations and real atoms by stating that "no shell-filling effects take place."
I couldn't reconcile the idea that there are stark geometric symmetry differences between neighboring N-electron systems, but they didn't appear in the energy data. Why had no one found a connection?! I sank my teeth into changes in geometry and figured out a way to hold onto all the electrons in the system.
Hold the Electrons, Pass the Geometric Symmetry
Over the course of one weekend in February 2005, I think, I realized the calculations made by others were muddied because they included contributions from both a change in the number of electrons and a change in the geometric symmetry. So I thought about how to compare the energy of neighboring N-electron systems without changing the number of electrons in the system. I walked into this problem with some inane experience in which I knew that an electron placed at the center of a dielectric sphere was in a location farthest away from all the other electrons in the Thomson problem (at a common distance from the origin) as they can be other than to be outside the dielectric sphere entirely. This strategy was part of my original guess procedure to find the lowest possible energy of the system. That is, putting an electron at a location in the sphere as far from the others as possible seemed like an intuitive location to place each new electron added to the system. However, I quickly learned these charge-centered configurations are never lower in energy than the configuration with all electrons at the same distance from the origin. This is very much different from the two-dimensional case I called the snowflake problem in a previous blog entry.
I wanted to compare the sets of energy data I already had that included charge-centered solutions of the Thomson problem with Thomson energy solutions with all electrons at the same distance from the origin. The "eureka moment" was when I looked at just three points of data on my screen and saw that the energy difference between the case for N=1 and N=2 Thomson problem solutions in a dielectric sphere was much smaller than the energy difference between the case of N=2 and N=3. This was when I realized that I had compared energies while only changing the geometric symmetry of the system AND there it was -- the first 'energy jump' that I believed to be why the first row of the periodic table has just two atoms and the second row begins with an atom having three electrons. The energy jump indicates the end of the first electron shell.
The open circles are energy differences of the Thomson problem treated in free space (no dielectric sphere). The solid circles are energy differences of the Thomson problem treated inside a dielectric sphere. Note that their distributions are not suitable to fit with a smooth function.This is from my work.
Being a bit skeptical, as scientists must be, I double checked my calculations to be sure the data were correct and that the energy jump between N=2 and N=3 wasn't the result of a silly mistake. It wasn't. I was, however, worried that the electrostatics framework involving the dielectric interface I constructed to perform the evaluations may have been in error. Ultimately, the comprehensive electrostatics interactions framework I constructed isn't needed here, because the bare Thomson problem of electrons in free space yields the same features. The difference is that the dielectric framework makes the energy variations much more pronounced as you can see in the figure above for up to 7 electrons (x-axis) where solid circles are data for the dielectric system and open circles are for the Thomson problem in free space.
That Monday morning I compared the energy differences for the first seven Thomson problem solutions and found an energy jump responsible for the orbital shape change that occurs between the element with 4 electrons and the element with 5 electrons as you can see in the figure above and schematically in the figure below. This jump doesn't indicate the beginning of a new electron shell, though. It coincides with the beginning of a new electron orbital. On the periodic table, this happens between the spherical-shaped s-orbital and the dumbbell-shaped p-orbital. So, just as the energy differences in the figure above result from the geometric symmetry shape change between the 4-electron solution of the Thomson problem, and the 5-electron solution of the Thomson problem by moving one electron from the center of the system to the surface of the Thomson sphere as shown in the figure below (red electron), the shape of the electron orbitals in real atoms also change (from spherical s-orbitals to dumbbell shaped p-orbitals). In fact, this is the first instance of an orbital shape-change in the periodic table. I have some ideas about how and why this might happen in the context of the Thomson problem that I'll discuss in the next blog entry.
What is an "orbital"? An orbital describes the SHAPE of the region in the space around an atom's nucleus in which an electron is most likely to be found. The SHAPE. The geometric symmetry.
I had evaluated energy differences from only changing the geometry shape of electron configurations, so seeing results consistent with a change in the shape of electron orbitals came as no surprise to me. I was, however, beside myself knowing something no one else in the world knew. It was time to deliver the news to my doctoral advisor. Not only was I overly confident as only a graduate science student can be, but I was armed with nothing more than 7 points of data -- and the printer decided not to work that day, so I had all of 7 data points gloriously sketched by hand on a sheet of scratch paper to show my advisor. That's an entirely different story I'll take up in a later blog entry.