Quantum mechanics is a proverbially and quintessentially challenging topic. It is the poster-child of difficulty in chemistry. Setting up the discussion for the four quantum numbers is fraught with potential and confusion—so much so there are those who may choose to gloss or skip over a topic that shows the beauty of nature in its most fundamental form.
Quantum mechanics has rules based on the unfamiliar mathematics of waves. But where to start with general chemistry students? When stripped, quantum mechanics has to do with quantities, counting. And what are we counting? Orbitals: the space around atoms where exist electrons. This method leaves the explanation of the spatial (size, shape, direction), though necessary, until later, and reduces the spatial of the orbital to a simple cube.
The embedded static images below are screenshots from the attached Supporting Information high-resolution PowerPoint presentation, which I welcome you to open at this time. I invite you to click along with the animated presentation. The animated PowerPoint slide presentation begins with a recap of orbitals: from whence they are created and the rules of counting (slides 2-3). Next is the set-up for the rest of the presentation (slides 4-8), specifically collapsing of the conventional orbital shapes to uniform cubes. Cubes were chosen for their uniformity and the ease of visual "stacking" of a three-dimensional structure on a two-dimensional slide.
Counting up Orbitals: In subshells. What this module hopes to do is to create pattern recognition. The Catch-22 of pattern recognition is that it is difficult for those who do not recognize the pattern. Though a pattern is not made of one thing, we begin here with an s-orbital. To start, I emphasize to the students that one item does not make a pattern. It primes them for pattern recognition while relieving them of the burden of premature and futile effort. Continuing, we build the pattern, from s to p to d orbitals. Again and again, emphasizing the obvious; "It's a square". This is somewhat absurd, but it pulls them into the game, and has them search for more patterns coming up. Before you reveal f-orbitals, you can ask students if they see the pattern, and congratulate the insightful.
Counting up Orbitals: In shells. The next slide expands on the previous slide to see the building up of shells and subshells. The principal quantum number, n, is introduced and connected to: a) the number of types, n = Σ types of orbitals in shell, and b) n2 = total orbitals in shell. "Because it is a square!" First "Ah-ha moment".
Orbitals in shells: Stacking them up. The next slide recapitulates the previous but makes the expansion of the type and number of orbitals more obvious by putting them in increasing order from bottom to top, both spatial (distance from the nucleus) and energetic. Repetition in an expanded (literally in this case) context reinforces all of the concepts covered. N.B.: There is no way to graphically represent on the same axis both distance and energy trends in proper proportion, but in this introductory simplification it is necessary to do to keep from being bogged down. N.B. = nota bene, "note well".
Labeling ℓ and mℓ: Having set up the square of orbitals and their energetics, we can reveal the beauty of quantum numbers. By wave function rules unnamed, the values of mℓ need to be symmetrical around zero. The s-orbitals can be symmetrical only by being at zero. Again, we have a pattern of 'no pattern' with only s-orbitals. Setting a diagonal line of zeros sets up the filling in of the rest of the pattern. The second "Ah-ha moment" occurs where recognition arrives and confusion melts. And what naturally falls out is that conventional formula: mℓ goes from +ℓ to -ℓ by 1.
Slight digression: The "Ah-ha moment" in its natural state. If you wish to illustrate to someone an "Ah-ha moment", this is the place to do it. Yet, it only works if the student (or instructor for that matter) has struggled with the concept. Though I do not have the language to describe analytically what happens, I have seen it enough times to know it is special. I cannot say that the reliability of this is "Ah-ha moment" is unique, I can only say that I have never seen it occur so reliably. The symmetry and flow of it is beautiful.
Orbitals in shells: Many electron energies As a wrap-up and re-recapitulation, the next slide has a mechanical engineering-style blow-up diagram which shows the orbitals when symmetry is broken between subshells. The energetic ordering is aligned so it is clear that 1s < 2s < 3s < 4s, etc…and 4s < 4p < 4d < 4f. The blow-up angle and proportion is such that the energies 4s ~ 3d and 6s ~ 5d ~ 4f.
Finally, here is the "mike drop" moment. Tell the students, in their minds' eyes, which at this point should be attuned to the model, to straighten each elbow of p, d and f subshells. Now, double the length of all subshell sets (to account for two electrons, ms = +1/2 and -1/2), and finally, flip it over, bottom to top. There is the Periodic Table.
Thus, the most iconically "Chemistry" image is now connected to orbitals without mention of the shape of orbitals. A pattern in properties (which is how the Periodic Table was derived) comes from a pattern in structure which begets a pattern in energy. It is no wonder that we find patterns beautiful; they are in nature block-by-block.
"Counting orbitals: A beautiful way to teaching quantum numbers and hybrids", Joseph Lomax, ACS Spring 2019 National Meeting in Orlando, FL, March 31–April 4, 2019, CHED 1974, Sci Mix invited poster 1 April 2019, Oral Presentation 4 April 2019.