In the late 1930s, calculating how an electron behaves inside a real crystal was, in practice, impossible. The most direct method — expanding the electron's wave function in a series of plane waves, as Bloch's theory predicts for a periodic potential — converged with insurmountable slowness: near the atomic nuclei, where valence electrons must oscillate as rapidly as the inner ("core") electrons to remain orthogonal to them, thousands of plane-wave terms were needed to obtain a reasonable approximation. Predictive calculation of energy bands in real metals and semiconductors was, for this reason, practically stalled. Conyers Herring, then a National Research Council fellow at MIT, solved the problem in 1940 with an idea as simple as it was powerful: instead of directly expanding the valence wave function in plane waves, he constructs the expansion by combining plane waves with the already-known wave functions of the core electrons, choosing the coefficients so that the result is automatically orthogonal to those inner states. These "orthogonalized plane waves" (OPW) absorb from the outset the rapid oscillatory behavior near the nucleus, so that the series converges with a manageable number of terms. The method, published in a single paper in Physical Review, became the standard band-structure calculation tool for the following two decades, and also laid the conceptual groundwork from which the pseudopotential method (Phillips and Kleinman, 1959) would derive in 1959, today the dominant technique in computational materials calculation and predictive semiconductor design.