  Cuprate high-temperature superconductors exhibit a pseudogap in the normal
state that decreases monotonically with increasing hole doping and closes at x
\approx 0.19 holes per planar CuO2 while the superconducting doping range is
0.05 < x < 0.27 with optimal Tc at x \approx 0.16. Using ab initio quantum
calculations at the level that leads to accurate band gaps, we found that
four-Cu-site plaquettes are created in the vicinity of dopants. At x \approx
0.05 the plaquettes percolate, so that the Cu dx2y2/O p{\sigma} orbitals inside
the plaquettes now form a band of states along the percolating swath. This
leads to metallic conductivity and below Tc to superconductivity. Plaquettes
disconnected from the percolating swath are found to have degenerate states at
the Fermi level that split and lead to the pseudogap. The pseudogap can be
calculated by simply counting the spatial distribution of isolated plaquettes,
leading to an excellent fit to experiment. This provides strong evidence in
favor of inhomogeneous plaquettes in cuprates.
