Response to "Comment on 'Nonlinear dielectric response of polar liquids"' [J. Chem. Phys. 144, 087101 (2016)].

The problem raised in the Comment is to how correctly calculate the Piekara coefficient linking the fieldinduced alteration of the material dielectric constant to squared Maxwell electric field E, ∆ǫE = ǫE−ǫs = −aE ; ǫE is the field-dependent dielectric constant and ǫs is its E = 0 value. Derivations of such relations are typically based on statistical perturbation theories expanding the average dipole moment of the sample 〈M〉E in powers of the perturbation Hamiltonian H ′ = −M · E0. Here M is the dipole moment of a macroscopic sample and E0 is the field of external charges applied along the z-axis of the laboratory frame (〈. . . 〉E is the average in the presence of the field, 〈. . . 〉0 is the average in the absence of the field, and 〈M〉0 = 0). The main difficulty of applying such fluctuation formulas is that the dielectric constant ǫE−1 = 4π〈M〉E/(V E) is given in terms of the Maxwell field, while perturbation expansions produce powers in E0 (V is the sample volume). Therefore, arriving at the material dielectric constant from fluctuation relations requires connecting E to E0, which depends on the chosen geometry of the sample. In order to avoid the asymmetry of the response to E0, early theories employed spherical samples polarized by the cavity field of the surrounding dielectric. In contrast, slab geometry was used in Ref. 5. The Piekara coefficient is then proportional to 〈M z 〉0−3〈M 2 z 〉 2 0, while the expression (1/5)〈M〉0 − (1/3)〈M 〉0 was previously derived for spherical samples. The two equations are equal when the spherical symmetry of the sample is applied to M, as is pointed out in the Comment. Correspondingly, the Piekara coefficient at constant volume (subscript “V”) becomes aV = (βǫ 2 s∆ǫ 2 s/8πρ)BV , where β = 1/(kBT ), ∆ǫs = ǫs − ǫ∞ and ρ = N/V is defined by the volume V and the number of molecules N . The reduced cumulant BV = N [1− κV ] is given through either Mz or through M as follows