Fig. 2
From: Understanding the apparent fractional charge of protons in the aqueous electrochemical double layer
Electron density analysis for the solvated proton. Plane-integrated electron density distribution as a function of the unit cell z-coordinate calculated with PBE (black), HSE06 (red), and EXX1.0 (blue) exchange-correlation functionals. The top and bottom panels show H3O+ in the outer Helmholtz plane (first solvent layer) and third solvent layer, respectively. The plane-integrated charge for the solvated proton is fractional in the first solvent layer with values of +0.64, +0.69, and +0.71 e at the PBE, HSE06, and EXX1.0 levels of theory. These charges approach unity with values of +0.86, +0.88, and +0.91 e, respectively, for the PBE, HSE06, and EXX1.0 functionals where the proton is in the third solvent layer. The charges on the proton obtained for each structure with all three functionals agree within 0.1 e
