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§ Phase iii · The Hydrogen Atom · Chapter 04

Reading orbitals

Before quantum mechanics had a name, alkali metals were already telling chemists their secrets in code. Four families of spectral lines (sharp, principal, diffuse, fundamental) labeled with their first initials by Victorian spectroscopists, survived the revolution of 1925 and quietly attached themselves to the shapes of the wavefunctions that produced them. This is how a 19th-century shorthand became the modern alphabet of chemistry.

If you have ever heard a chemist say “carbon’s 2p electrons” or read a periodic-table column described as “the d block,” you have used a vocabulary that predates the Schrödinger equation by half a century. The letters s, p, d, f were not invented by quantum theorists. They were the trade jargon of late-Victorian spectroscopists who spent their evenings staring at the glow of sodium and potassium through prisms, cataloguing what they saw. The lines came in four reliable families. One family looked clean and isolated. Another was always the strongest. A third looked smeared. A fourth turned up so far in the infrared that it took heroic optics to catch. The spectroscopists, being practical people, named them after how they looked.

By the 1880s, working in places like the Cavendish Laboratory and the Sorbonne, observers had begun to notice that the line series in alkali metals (lithium, sodium, potassium, rubidium, caesium) all came in the same four flavors. The clean lines they called the sharp series. The brightest, most prominent ones, including sodium’s famous yellow doublet, were the principal series. The fuzzy-looking ones were the diffuse series. And the long-wavelength quartet, the one buried deepest in the infrared and the hardest to pry out of the data, they called the fundamental series, partly out of habit and partly because the spacing of its lines looked closer to a pure Rydberg formula than anything else they had.

It was Hertha Ayrton, Heinrich Kayser, Carl Runge, and a generation of optical chemists in Britain and Germany who hammered the catalogue into shape. The spectra were beautiful and the labels were stuck. A chemist of 1895 could say “the principal doublet of sodium” and any colleague in Europe knew exactly which two lines were meant. None of them had any idea what was inside the atom doing the shining. The names were a triumph of pattern recognition without an underlying picture.

Then came 1925 and the revolution. Within eighteen months, Heisenberg in Göttingen and Schrödinger in Zürich produced two different-looking but mathematically equivalent versions of a new mechanics. By the end of 1926, the equation for the hydrogen atom had been solved exactly, and the answer was a family of wavefunctions labeled by three integers. The principal quantum number n said which shell. The orbital angular-momentum quantum number ℓ said how much rotational motion the electron carried. And the magnetic quantum number m said how that rotation was oriented in space. The previous chapter explained where these integers come from. The question that landed on chemists’ desks in 1927 was: what should we call them?

The labels were already there, waiting in the spectroscopy catalogues. Atoms in their ℓ = 0 state emitted the sharp series when they fell into it from above. Atoms in their ℓ = 1 state were the source of the principal series. Atoms in ℓ = 2 produced the diffuse series; atoms in ℓ = 3 produced the fundamental. The transitions had been numbered for decades by the lower state of the jump. Friedrich Hund, working at Göttingen, and Robert Mulliken, working at Chicago, formalized the convention almost immediately. The orbital with ℓ = 0 became the s orbital. ℓ = 1 became p. ℓ = 2 became d. ℓ = 3 became f. For ℓ values beyond three (which start appearing in heavy actinides and matter for the chemistry of plutonium and americium) the convention shifts to alphabetical order: g, h, i, k. The letter j is skipped on the same instinct that skips it in serial numbers, to avoid confusing it with i.

So when a chemist writes the ground-state configuration of carbon as 1s² 2s² 2p², they are saying three things in one line. The leading number is n, the shell. The letter is ℓ, the shape. The superscript is the count of electrons in that subshell. The notation is so compact that it hides how much physics is buried inside. The 1s² means: two electrons sitting in the n = 1, ℓ = 0 orbital, with opposite spins (the Pauli principle, which we have already met, demands they pair). The 2p² means: two electrons distributed among the three p orbitals at n = 2. Hund’s rule, which we will get to a few chapters from now, says they will sit one each in two of those three p orbitals with parallel spins. This is why carbon famously prefers four bonds and tetrahedral geometry. The electron count is the seed of the chemistry.

To picture what these orbitals actually look like in space, you have to picture two factors multiplied together. The radial part R(r) controls how the probability density depends on the distance from the nucleus. The angular part Y(θ, φ) controls how it depends on direction. The previous chapter gave you the radial story: each (n, ℓ) pair has a particular pattern of radial nodes that the Laguerre polynomials carry. The angular factor is the new business of this chapter. It is described by the spherical harmonics Y_ℓ^m(θ, φ), a family of standing waves on the surface of a sphere. The s, p, d, f labels are simply the chemist’s name for which spherical harmonic you are looking at.

The s orbital is the simplest of the lot. With ℓ = 0 you have no angular dependence at all. Y_0^0 is a constant. The wavefunction depends only on r, so the electron density is perfectly spherical, a fuzzy ball centered on the nucleus. The 1s of hydrogen has no radial nodes and no angular nodes; it is the densest, most compact wavefunction the atom owns. The 2s has one radial node (a spherical shell where the probability dips to zero) but it is still spherical overall. The 3s has two radial nodes, and so on. Because the s orbital has no centrifugal barrier (the angular-momentum term ℓ(ℓ + 1)/r² is zero), it is the only orbital that has nonzero probability density at the nucleus itself. This is why s electrons in heavy atoms get strongly relativistic corrections, why mercury is a liquid at room temperature, and why gold is yellow. Penetration to the nucleus matters.

The p orbitals are next. At ℓ = 1 the spherical harmonics come in three flavors. Plotted as electron probability densities, each one looks like a dumbbell: two lobes of density on opposite sides of the nucleus, separated by a flat plane where the wavefunction is exactly zero. There are three independent dumbbells, naturally oriented along the three Cartesian axes, and chemists call them p_x, p_y, and p_z. The plane through the nucleus perpendicular to each dumbbell is called an angular node: a place where the wavefunction changes sign as you cross it. For a p orbital you get one angular node, a flat plane. Cross the nucleus and the sign flips. The shape is what makes p orbitals strongly directional, which is why molecules with p-bonding (everything organic, basically) have such definite geometries. A pure p_z bond points up the z-axis and nowhere else.

The d orbitals add another layer. At ℓ = 2 the count is five, and four of them look like cloverleaves: d_xy, d_xz, d_yz, and d_x²-y², each with four lobes arranged in a plane. The fifth, d_z², stubbornly refuses to look like the others. It has two big lobes along the z axis (like a p_z) plus a doughnut of density around the equator. Generations of students have asked why d_z² is different. The honest answer is that it is not, really. There are six “natural” d-shape building blocks (d_xy, d_yz, d_xz, d_x²-y², d_y²-z², d_z²-x²) but only five are linearly independent. The combination d_z² is shorthand for what is really 2z² minus x² minus y², a linear combination that picks out a perfectly valid orthogonal mate to the four cloverleaves. The torus is the price you pay for keeping a symmetric set of five.

The f orbitals carry ℓ = 3 and come in seven varieties. Their shapes are intricate (eight-lobed, multi-cusped affairs that look like exotic pollen grains) and they matter most for the lanthanides and actinides, the two long rows at the bottom of the periodic table. Neodymium magnets, the brilliant colors of europium-doped phosphors in old cathode-ray TVs, the chemistry of uranium and plutonium, all of it is f-orbital chemistry. The reason these elements get a row of their own at the bottom of the table is that the 4f and 5f subshells fill in a region of the table where the principal quantum number has already moved on to the next shell, so they create columns that do not fit anywhere else. We will come back to this in the next chapter, when we read the periodic table itself as a consequence of orbital filling.

There is a subtlety hidden in the way we just drew p_x, p_y, p_z. The Schrödinger equation in spherical coordinates does not directly produce orbitals labeled by Cartesian axes. It produces orbitals labeled by m, the magnetic quantum number, with values m = -ℓ, -ℓ + 1, …, ℓ - 1, ℓ. For ℓ = 1 the natural states are Y_1^\{-1\}, Y_1^0, Y_1^\{+1\}. The middle one, m = 0, is real and looks like p_z. The other two are complex-valued; their angular dependence carries the factor e^\{±iφ\}, which is a phase that rotates around the z axis. You can take their probability density and it looks like a doughnut around the z axis, but the wavefunction itself is complex. Chemists rarely want a complex wavefunction. They want a shape to put on a model and draw with arrows in a textbook. So they take real linear combinations: p_x is (Y_1^\{-1\} minus Y_1^\{+1\}) divided by an i times root two, and p_y is (Y_1^\{-1\} plus Y_1^\{+1\}) divided by root two. The recombined orbitals are real, they point along the x and y axes, and they are still solutions of the hydrogen Schrödinger equation. The price of the realness is that p_x and p_y are no longer eigenstates of L_z; they are superpositions of m = +1 and m = -1. The chemist gets a picture they can draw; the physicist gives up a quantum number.

Y_1^0(θ, φ)    = √(3/4π) cos θ
Y_1^{+1}(θ, φ) = -√(3/8π) sin θ e^{+iφ}
Y_1^{-1}(θ, φ) = +√(3/8π) sin θ e^{-iφ}

p_x = (Y_1^{-1} - Y_1^{+1}) / √2 ∝ sin θ cos φ = x/r
p_y = i (Y_1^{-1} + Y_1^{+1}) / √2 ∝ sin θ sin φ = y/r

d_z²       ∝ 3z² - r²     (from Y_2^0)
d_xz       ∝ xz           (from Y_2^{±1})
d_yz       ∝ yz           (from Y_2^{±1})
d_x²-y²    ∝ x² - y²      (from Y_2^{±2})
d_xy       ∝ xy           (from Y_2^{±2})

When you put it all together, the s, p, d, f labels carry a remarkable amount of information in remarkably few letters. The letter tells you the shape and the angular-node count. The principal quantum number n tells you the shell. The electron count tells you how full the subshell is. From those three pieces of data you can read off the geometry of bonds, the magnetic behavior, the chemistry. The 3d incomplete shells of iron, cobalt, and nickel are why those metals are ferromagnetic. The 4f shell of neodymium is why neodymium magnets are the strongest in everyday use. The 2p shell of carbon, with its two electrons hungry for four bonds, is the reason organic chemistry exists. The 1s shell of hydrogen, fed in pairs into a million molecules, is the reason water is liquid and life is wet. Every fact in chemistry, when you trace it back far enough, lands on a few labels invented by spectroscopists who could not have explained why their data came in four series.

The labels were a piece of pre-quantum jargon that the new mechanics decided to keep. There were people in the late 1920s who tried to introduce a cleaner notation. Some books used “ℓ = 0, 1, 2, 3” all the way through. Others tried Greek letters. None of it stuck. The Cavendish spectroscopists had been so consistent in their cataloguing, and the chemistry community had been so steeped in their labels, that no one wanted to relearn the alphabet. So we still write “the 4f orbitals of neodymium” with a letter whose origin most chemists no longer know, in a notation set down by men who would have been startled to discover that the shapes they were unwittingly labeling were the wavefunctions of an electron.

What you now have is the chemist’s bilingual dictionary. Hand them an atomic configuration like [Ne] 3s² 3p² and you can read it back as a complete physical specification: ten core electrons in the neon configuration plus two more in the 3s subshell (spherical, no nodes) and two distributed among the three 3p orbitals (dumbbells along x, y, z, one of each occupied). That is silicon. From “[Ne] 3s² 3p²” you can predict that silicon will sit one column to the right of carbon in the periodic table, that it will form four bonds with tetrahedral geometry, that it will crystallize in the diamond structure, and that it will be a semiconductor. The notation is a compressed prediction of nearly every fact you know about that element. In the next chapter we will turn this compression engine all the way up: the entire periodic table, all 118 elements, falls out of the rules for filling these s, p, d, f orbitals in order.