Lecture 6: The Atomic Orbital and Quantum Numbers

Lecture 6: The Atomic Orbital and Quantum Numbers Summary One of the main limitations of the Bohr model is that it can only describe an atom with a single electron. To find the properties of multiple electrons in an atom, it is necessary to solve the Schrodinger equation, (K + V )ψ = Eψ, where K is kinetic energy, V is potential energy ( − 1/r for an electron in an atom), E is the total magnitude of energy, and ψ is the wavefunction, or orbital of the electron. The solution to the Schrodinger equation for an electron in a hydrogen atom has three separate components: ψ = R(r)P (θ)F (ψ). The full form of the solution form has three big implications: 1. Electrons don’t really orbit around atoms, at least in the sense that planets orbit around the sun. The solution to the Schrodinger equation yields a wavefunction, which when squared, gives an expression |ψ|2 = 1for the probability distribution of the electron’s location in relation to the nucleus. Further, though each electron has some radius it is most likely to be found, there is also a chance that it is much closer to or much further from the nucleus. For example, while Bohr model electrons with n=1 live at exactly 0.529 Å from the nucleus, the wavefunction for a real electron in the hydrogen atom has occupational probability peaked at 0.529 Å and smeared out on either side of that value. As the principle quantum number increases, the number of nodes—forbidden regions—increases, giving distinct bands around the atom where the electron is likely to be. 2. Four quantum numbers are necessary to fully describe a particular solution. The principle quantum number, n is primarily responsible for the energy level of the electron. Just like in the Bohr model, n can take any integer value from 1 to infinity. As n gets larger, the total energy gets less negative (bigger). An electron is equivalently referred to as being in the nth energy level and the nth shell; the remaining quantum numbers describe subshells. The angular momentum quantum number, l, is primarily responsible for the shape of the orbital. The values that l can take on depend on which shell the electron is in: the range goes from 0 to (n-1). The magnetic quantum number, ml , is responsible for the orientation of the orbital in space. Its values depend on both n and l, and the range of ml goes from (-l) to (l). Finally, the spin quantum number, ms can have one of two values: +1/2 or -1/2, which is equivalently referred to as up or down. 3. Electrons that solve the Schrodinger equation are distinct from Bohr electrons. While there is only one electron per shell in the Bohr model, the Schrodinger equation gives degenerate solutions: as n increases, there are more and more orbital levels. Further, the Bohr model gives a fixed allowed radius for the electron to live, while the Schrodinger equation gives a distribution of probability that an electron is at a given distance from the center (and some forbidden zones!). Finally, though the Bohr model could only describe the behavior of one-electron atoms, the Schrodinger equation is much more broad: it can describe the electronic structure of an atom with any number of electrons. We also discussed how the values for the l quantum number get the letter names s, p, d, and f corresponding to l = 0, 1, 2, 3, respectively. If we say an electron is in the 2s orbital, then we know it corresponds to the quantum numbers n = 2, l = 0, ml = 0, and spin could be up or down. Other examples of sets of quantum numbers and the corresponding orbitals were covered and these hydrogen orbitals were shown as the building blocks of chemistry. That’s because to describe an atom and its properties, we fill these orbitals from lowest energy up, with the number of electrons Z in the atom. This occupation follows a general trend that is the topic of the next lecture, but in this one we covered a very important rule: the Pauli Exclusion Principle. This rule states that no two electrons can have the same set of quantum numbers, meaning that two electrons and only two electrons can occupy any given orbital. Why this matters Quantum weirdness is one of the most important reasons why nanotechnology has gotten so much attention recently. Now, there has been quite a bit of press about nanotechnology over the last two decades, but one of my all-time favorite articles appeared long ago in the U.K. Telegraph: “How Super-Cows and Nanotechnology will Make Ice Cream Healthy.” I always remember that piece because of the way they define nanotechnology: they describe a company as “experimenting with nanotechnology, or the science of invisibly tiny things.” I suppose I like this quote because of how ridiculous it is, since first of all there are many different realms of science and engineering associated with tiny things we cannot see (nanotechnology is one of them, but how about nuclear physics with its quarks and gluons, or solar wind with its hot plasma particles that reach our earth only to be deflected by our planet’s magnetic field, or dark matter?). Second of all, I like this quote because it shines a spotlight – albeit incorrectly – on one hold-up in nanotechnology development, which was not being able to observe what was happening. This changed once we could use electrons as a way to illuminate matter, as discussed in last chapter’s Why This Matters. The main point is that nanotechnology holds great promise. Not so much for the reasons that gave it fame in the early 2000’s (healthy ice cream being the least of it, we were supposed to have a space elevator by now!), but rather because of the chemistry that’s “under the hood” of nanotechnology. What does nanotech even mean, and why do we have a whole name for it? Let’s answer that with an example directly related to today’s lecture. We’ve been talking about orbitals of electrons in atoms and how the orbital is the probability cloud for the electron. Well what if we take that cloud, and we squeeze it? Or in other words, confine it so that it can’t take up the same volume as it typically would. There’s a name for that: quantum confinement. And when we confine quantum things, like in this case an electron in an atom, new properties emerge. Take color as an example. For a piece of bulk material, where “bulk” here just means it’s not nanoImages of bulk semiconductors, ice cream sized, I might have something that’s kind of opscoop, quantum dot © sources unknown. tically boring. It’s just a piece of gray material. But then I take out my nano-ice-cream-scooper and I make a little tiny chunk of the material. Suddenly, without changing anything except the size, I can tune the color to pretty much anything I want! The images above are excluded from our Creative Commons license. The fact that changing the size of this material allows us to tune a given property is quite incredible. Imagine if I were to take one of those strips of metal from your first Goodie Bag and break it into two pieces: you wouldn’t expect the color of the metal to change. But if you could break just a tiny piece of it off, like one with just 10-20 atoms in it, then that piece would change color. How weird is that? It’s because of this electron cloud running out of real estate when the amount of matter is small enough. When it’s nanoscale, the confined probability cloud changes the energy of the electron, which in turn changes the color of the material. We talked about electrons changing their energy levels in atoms as a way to either absorb or emit a photon corresponding to that energy shift. Well here we’ve got a new way to shift the starting and terminating levels themselves, simply by changing the size of the material. That’s an example of why nano is such a big deal. The ability to control the properties of a material is one of the foundations of our modern era. As we’ve already learned, understanding the periodic table of the elements and all of the differences between the behavior of these elements gave scientists and engineers knowledge of the basic ingredients that are used to build our world. Going nano is kind of like adding a whole other dimension to the periodic table: it’s like each element can now do new things, take on new properties, and be utilized in new applications. Why this employs Quantum computing has gotten a lot of attention recently, and for good reason. Over the past decade, we have reached the point where quantum bits, or qubits, can be made and controlled experimentally. The computers we use today manipulate information in the form of individual bits, which are the famous 1’s and 0’s of the last 50+ years. But in a quantum computer, the quantum mechanical phenomena that we’ve been learning about in this chapter are used to manipulate information. It’s qubits that hold the key since those are the keepers of quantum states, which have quantum numbers just like the electron states in an atom. When two qubits interact with one another we get much more than just 1’s and 0’s, instead we get all possible superpositions. It’s like being able to harness the probability cloud as a computing cloud where all possible answers are computed at once. Quantum computing is getting a lot bigger and a lot more exciting, with many companies predicting commercial quantum computers by 2025. The Why This Employs idea for this lecture is to take those quantum states we’ve just discussed and run with them. . . all the way to Google’s Artificial Intelligence Lab, or to the team at IBM Q, or to D-Wave, or to the folks at Intel working on their Qubit Chips, or to any of the 63 companies currently listed at Wikipedia under “Companies worldwide engaged in the development of quantum computing.” A list that is growing rapidly. And that doesn’t even include doing research on this topic in a lab here at MIT :) Example problems 1. Write the quantum numbers for each electron in a neutral carbon atom. Solutions 1. Note: the spin signs are selected arbitrarily, so it would be just as correct to flip them all. For n 1 1 2 2 2 2 l 0 0 0 0 1 1 m 0 0 0 0 -1 0 ms 1/2 -1/2 1/2 -1/2 1/2 1/2 the 2p electrons, they must have the same spin due to Hund’s rule.