The GW approximation is one of the most successful methods for computing electronic excitation energies — ionization potentials, electron affinities, band gaps. Its quasiparticle energies agree with experiment for a wide range of materials. Yet GW total energies show significant delocalization errors: the energy does not increase correctly as charge is spread across multiple sites.
This paper identifies why. The functional derivative of the GW correlation energy with respect to electron number has a discontinuity at integer particle counts — the derivative jumps as the system gains or loses a single electron. This derivative discontinuity is analogous to the well-known discontinuity in the exact exchange-correlation energy of density functional theory, but it appears here in the many-body perturbation theory framework where it was not previously recognized.
The discontinuity resolves the contradiction. Accurate quasiparticle energies and poor total energies coexist because the quasiparticle spectrum reflects the derivative (which is correct, including the jump), while the total energy reflects the integral (which misses the jump's contribution). The derivative is well-behaved; the integral accumulates errors from the missing discontinuity.
The practical consequence concerns chemical potentials calculated in the random phase approximation. Direct differentiation of the RPA energy and evaluation via the functional derivative both give the same result — provided the discontinuity is properly accounted for. Ignoring it produces the delocalization error.
The lesson: a function can be almost everywhere smooth and still contain a jump at the most important point. The physics lives at the discontinuity — the threshold where one electron becomes two.