One of the earliest attempts at virtual space localization was through projected atomic orbitals (PAOs). Since the atomic orbital space spans the same subspace as the complete MO space, a complete, localized, and linearly-dependent description of the virtual space can be determined from the atomic orbitals and occupied MO coefficients. PAOs have a more local character compared to canonical virtual molecular orbitals. The following equations represent how PAOs are formed by projecting out the occupied MO space from the complete AO space: