When he returned to Pasadena in the fall of 1930, Pauling returned to the problem of the tetrahedral carbon atom. That year, a young American physicist names John C. Slater had found an important simplification of the Schrödinger wave equation that made it possible to better picture carbon's four binding electrons. Spurred by Slater's work, Pauling picked up his pen and started making calculations again in earnest. In order to match the chemists' reality of a carbon tetrahedron, the physicists' two sets of electron subshells had to be broken and mixed together somehow in a new, equivalent form. The central problem was finding appropriate mathematical approximations of the wave function, shortcuts that would make manageable the equations for combining the subshells' wave functions.

For weeks through the fall, though, none of Pauling's shortcuts worked. Then, on a night in December 1930, sitting at his desk in the study of his home, he tried one more approximation. This time in trying to combine the two subshells' wave functions, he chose to ignore a part of the mathematics called the radial function, a simplification that Slater's papers indicated might work. By stripping away that layer of complexity, Pauling was surprised to find that "the problem became quite a simple one from the mathematical point of view" — at least, for a Sommerfeld-trained quantum physicist.

He could now, with the right coefficients, combine the wave functions of the physicists' two carbon subshells into a mathematical description of a new hybrid form: four equal orbitals oriented precisely at the angles of a tetrahedron. Not only that, but his new hybrid orbitals were more highly directed away from the nucleus, capable therefore of overlapping more with the orbitals of other electrons from other atoms. And here was a basic insight: The greater the overlap of orbitals from two atoms, the more exchange energy was created and the stronger the bond.

Pauling had a sudden rush of energy. From the principles and equations of quantum mechanics he had formed a tetrahedral carbon atom. The calculated angles between bonds were right; the bond lengths looked right; the energy required to change the electron subshell orbitals into their new shapes was more than accounted for by the energy of the electron exchange. He had solved a paradox, reconciling the physicists’ and chemists’ views of carbon.