Fig. 5: Justification of the constant-ε method based on MS-DFT.

a (upper panels) The plane-averaged electrostatic potential differences \(\Delta {\bar{v}}_{H}\) calculated for the H-down water molecule on the Au (left) and graphene (right) under Φ = −2.0 V within MS-DFT. (lower panels) The \(\Delta {\bar{v}}_{H}\) curves calculated within MS-DFT at Φ = −2.0 V (red solid lines) are compared with those obtained from the constant-electric field method at ε₀=−0.25 VÅ⁻¹ (blue dotted lines) and −0.20 VÅ⁻¹ (green dotted lines). b The plane-averaged charge density differences \(\Delta \bar{\rho }\) calculated within MS-DFT at Φ = −2.0 V (red solid lines) and constant-electric field method at ε₀ = −0.25 VÅ⁻¹ (blue dashed lines) and −0.20 VÅ⁻¹ (green dotted lines) for the H-down water on Au (left panel) and graphene (right panel). The 2D contour plots of Δρ obtained from MS-DFT calculations at Φ = −2.0 V and the corresponding constant-\({\varepsilon_0}\) DFT calculations (ε₀ = −0.25 VÅ⁻¹ for Au and −0.20 VÅ⁻¹ for graphene).