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.