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§ Phase viii · Molecules · Chapter 02

sp hybrids

Carbon has four valence orbitals: one 2s and three 2p. Read them off the hydrogen-atom solutions and they look terrible for chemistry. The s is round, the three p's stick out along x, y, and z at right angles to each other, and the s sits at a lower energy than the p's. So you would expect carbon to form three perpendicular bonds and one weaker bond pointing nowhere in particular. Methane begs to differ. A young Caltech chemist named Linus Pauling figured out, in the spring of 1931, how to make the math agree with the molecule.

The story starts inside a problem that should not have been a problem. By 1930, organic chemists had spent a hundred years building up the structural picture of methane, ethane, ethylene, benzene, and ten thousand other carbon compounds. They knew that carbon, in nearly every organic molecule, made exactly four bonds. They knew, from heats of combustion and from substitution patterns, that the four bonds of methane were equivalent to each other. And they knew, from van ‘t Hoff and Le Bel’s 1874 stroke of geometric insight, that the four bonds pointed toward the corners of a regular tetrahedron, with every H-C-H angle equal to the same 109 degrees and change. That number was not negotiable. It was carved into the existence of chiral molecules, into the optical rotations of sugars and tartrates, into the entire architecture of biology.

Then quantum mechanics arrived, and in 1927 Walter Heitler and Fritz London showed how Schrödinger’s equation explained the bond in H₂. Two hydrogen 1s orbitals share their electrons, the lower combination drops in energy, and you have a molecule. Beautiful. But carbon is the next monster. Its ground-state configuration is 1s² 2s² 2p², a filled 2s pair and two lonely electrons in two of the three 2p orbitals. Two unpaired electrons say two bonds. Methane says four. Worse, the three 2p orbitals point along x, y, and z at exactly 90 degrees to each other, and the 2s is a sphere with no direction at all. If you took those four orbitals straight off the hydrogen-atom solution, you would predict that carbon makes three perpendicular bonds plus one fourth bond in some arbitrary direction. The bonds would not be equivalent. The angles would be 90 degrees, not 109.47. Every page of structural chemistry would be a lie.

Something obvious was missing. In 1931 a 30-year-old assistant professor at Caltech, fresh off a European tour with Sommerfeld, Bohr, and Schrödinger, decided that the trouble was not with carbon but with the way chemists were reading the atomic orbitals. The atomic orbitals, Pauling argued, are the right basis for an isolated atom drifting in vacuum. They are the wrong basis for an atom about to bond. Inside a molecule, what counts is not what each electron looked like before but what combinations of orbitals will give the molecule its lowest total energy. And those combinations, he showed, are mixtures of the originals: a quarter s plus three quarters p, repeated four times in four different directions.

Here is the move, stripped to a sentence. Take the four atomic orbitals of carbon’s valence shell, 2s, 2p_x, 2p_y, and 2p_z, and build four new orbitals that are linear combinations of them. The Schrödinger equation is linear, so any linear combination of solutions for an isolated atom is itself a perfectly good orbital. The question is only which combination to use. Pauling’s prescription: pick the four combinations that point along the four vertices of a regular tetrahedron, so that the molecule can place a hydrogen at the end of each one and minimize its total energy. Call those combinations sp³ orbitals, because each is built from one part s and three parts p.

The numbers fall out cleanly. If the s orbital contributes a coefficient of 1/2 to each hybrid and the three p orbitals contribute coefficients of plus or minus 1/2 with the signs picked to point in four different directions, you get four equivalent orbitals at exactly the tetrahedral angle. Each hybrid is one-quarter s in character and three-quarters p. Each points along one of the body diagonals of a cube. The angle between any two of them is the arccosine of -1/3, which works out to 109.4712 degrees. The number is not approximate. It is the exact opening angle of a regular tetrahedron, sitting inside a perfect cube, and it falls out of geometry the moment you decide that the four hybrids are required to be equivalent.

That same logic, applied to a different mix of orbitals, predicts the other two flat shapes of organic chemistry. If carbon mixes its 2s with only two of the three 2p orbitals (say 2p_x and 2p_y) and leaves 2p_z alone, you get three equivalent sp² hybrids that all sit in the xy plane and point 120 degrees apart from each other, like the spokes of a triangle. The leftover 2p_z stays perpendicular to that plane, pointing up and down. This is the geometry of ethylene, H₂C=CH₂. Each carbon makes three sp² bonds (two to hydrogens and one to the other carbon) at 120 degrees, and the two unhybridized 2p_z orbitals on the two carbons overlap sideways to form a second bond, called a π bond, above and below the molecular plane. The double bond, the planarity, and the rigid 120-degree angles all fall out of one geometric choice.

Push the mixing further. If carbon mixes its 2s with only one 2p orbital (say 2p_x) and leaves 2p_y and 2p_z alone, you get two equivalent sp hybrids pointing 180 degrees apart along the x axis, perfectly linear. The two leftover 2p orbitals stick out perpendicular to that axis. This is the geometry of acetylene, HC≡CH. Each carbon makes two sp bonds (one to hydrogen and one to the other carbon) along the same straight line, and the two pairs of unhybridized p orbitals on the two carbons form two π bonds wrapped around the molecular axis like the wires of a coaxial cable. Three of the most important shapes in organic chemistry, the tetrahedron, the trigonal triangle, and the line, come out of mixing one s with three, two, or one p.

A practical test that this is not just a tidy story: take the s character in each scheme and ask what fraction it is. In sp³ it is 1/4 = 25 percent. In sp² it is 1/3 = 33 percent. In sp it is 1/2 = 50 percent. As the s fraction rises the orbitals get pulled closer to the nucleus (s orbitals have higher density at r = 0 than p orbitals), so the bond gets shorter and stiffer, and the carbon gets more electronegative. Predictions: C-H bonds in methane (sp³) should be the longest and weakest, C-H bonds in ethylene (sp²) intermediate, C-H bonds in acetylene (sp) the shortest and strongest. Pauling published numbers, and decades of measurement vindicated them. Methane’s C-H bond is 1.09 angstroms long with a stretch frequency near 2950 cm⁻¹. Ethylene’s is 1.085 angstroms at 3030. Acetylene’s is 1.06 angstroms at 3330. The order is exactly what the s-content prediction says it should be. The model is doing real work.

hᵢ = a · s + b · (lᵢ · p)

a² + b² (lᵢ · lⱼ) = 0

cos(θᵢⱼ) = -a² / b² = -cos²θ / sin²θ = -cot²θ

Pauling’s framework is not the last word on bonding, and he knew it. Robert Mulliken, working at the University of Chicago in the same decade, was building a different language called molecular orbital theory, in which electrons live in orbitals that belong to the whole molecule rather than to any one atom. The two descriptions sometimes disagree on details (Mulliken’s MO picture is better for diatomic gases and excited states; Pauling’s valence-bond picture is better for ground-state organic molecules and bond drawing) and a complete chemist uses both. The next chapter, the hydrogen molecule, is where the two pictures meet head-on. But for the molecules of life and most of synthetic chemistry, the hybrid orbital is still the way people draw, sketch, and think. Take any organic chemist into a coffee shop and hand them a napkin; what they will draw is sp³ carbons with tetrahedral angles and sp² carbons with planar triangles. It is the working language of the field, and it is Pauling’s.

Step back and look at what the move actually was. Before 1931, the question “why is methane a tetrahedron” had two kinds of answer. There was the geometric answer (van ‘t Hoff and Le Bel had simply demanded a tetrahedron because optical isomers require it) and there was the orbital answer (carbon has one s and three p, take them straight). The first answer was unphysical, an empirical fit dressed up as a postulate. The second answer was physical but wrong, because it predicted three perpendicular bonds plus an awkward fourth. Pauling’s contribution was to notice that the orbitals you write down for an isolated atom are not engraved on the atom. They are a basis, a coordinate system, and you can rotate the basis without changing the physics. Once you give yourself permission to mix the basis before you bond, the geometry that chemistry was begging for falls out as the lowest-energy choice. The tetrahedron is not a coincidence. The tetrahedron is the answer that minimizes the energy when you let the orbitals reshape themselves to point at the partners that are already waiting.

Pauling himself did not stop with chemistry. By the 1950s he had turned his attention to atmospheric fallout from nuclear weapons tests, calculating, in a few brisk papers, that the world’s accumulated test fallout would over a human lifetime cause a measurable number of leukemia deaths and birth defects. He took his case directly to the public. In 1958 he and his wife Ava Helen presented a petition signed by more than nine thousand scientists from 49 countries to the United Nations, calling for an end to atmospheric testing. The petition helped pressure the 1963 Partial Test Ban Treaty into existence. Pauling received the 1962 Nobel Peace Prize the year that treaty was signed, becoming the only person in history to win two unshared Nobels (Marie Curie’s two were shared and in different fields). The State Department took away his passport for a while. The McCarthy era was not kind to him. He kept working.

The Caltech administration once warned him that his peace activism was embarrassing the institute, and would he please stop. He resigned and moved to the Center for the Study of Democratic Institutions in Santa Barbara. He kept publishing. In his later years he became famous, and to many chemists infuriating, for advocating very large doses of vitamin C as a treatment for the common cold and (in a more controversial campaign) cancer. The clinical evidence has not aged well. The bonding theory has. Pauling died in 1994 at the age of 93, half a century after the original 1931 paper, with the book that he wrote in his thirties still on every organic chemist’s shelf.