Incorporating Electrolyte Correlation Effects into Variational Models of Electrochemical Interfaces

We propose a way for obtaining a classical free energy density functional for electrolytes based on a first-principle many-body partition function. Via a one-loop expansion, we include coulombic correlations beyond the conventional mean-field approximation. To examine electrochemical interfaces, we integrate the electrolyte free energy functional into a hybrid quantum-classical model. This scheme self-consistently couples electronic, ionic, and solvent degrees of freedom and incorporates electrolyte correlation effects. The derived free energy functional causes a correlation-induced enhancement in interfacial counterion density and leads to an overall increase in capacitance. This effect is partially compensated by a reduction of the dielectric permittivity of interfacial water. At larger surface charge densities, ion crowding at the interface stifles these correlation effects. While scientifically intriguing already at planar interfaces, we anticipate these correlation effects to play an essential role for electrolytes in nanoconfinement.

The goal of computing the grand canonical free energy and therefore the partition sum from Eq. ( 2) is, as mentioned in the main text, in general impossible to do exactly.It was, however, shown by Netz and Orland, that within the functional integral formalism, we can use the loop expansion method to transform the functional integral Eq. ( 2) into a variational functional for the electrostatic potential.
It is possible to show that the electrostatic potential is the functional average over the auxillary field, ϕ(r) ≡ i⟨ψ(r)⟩, (S.1) because ϕ solves the Poisson equation for the ensemble averaged particle density as shown in Eq. ( 58) of the article by Netz and Orland.S1 The average, ⟨...⟩ = 1 Z Dψ(...)e −βS [ψ] , (S.2) is a functional average with the action, Eq. (3), as probability distribution.The electric potential within functional integration can be calculated by adding a fictitious source term to the action in Eq. ( 2), that couples an auxiliary charge density to the auxiliary field.Accordingly, the grand canon- which is only equal to the old one for ρ aux = 0. Taking the derivative with respect to ρ aux allows one to calculate the electric potential through The first equal sign is the reason why the electrostatic potential can be called a mean-field.
The Legendre transformation, that changes the active variable in the calculation of the partition function Z[ρ aux ] from ρ aux to ϕ, gives the so called generating functional, effective action or thermodynamic potential, Note that until now, the formalism does not involve approximations, rendering the theory exact.However, the functional integral is still impossible to solve.
In the one-loop expansion, the action is expanded around the mean-field ϕ in fluctuations δψ until second order, where it was used that the linear variation vanishes because the mean-field ϕ is up to oneloop order also the saddle-point of the action.In the last line, a general Gaussian functional integral was performed, neglecting the constant factor.By re-exponentiating, one arrives at the last line.A comparison of the left and right hand side of Eq. (S.8) yields the result for the generating functional in one-loop approximation in Eq. ( 4).
Eqs. ( 4) and ( 5) constitute the transition from a thermodynamic equilibrium function to a free energy functional of a varying field, ϕ, where only the physical value of the grand canonical free energy, Ω C [ϕ], is given by its evaluation at the field configuration ϕ that satisfies the variational principle Eq. (4).
To obtain a variational functional for the particle densities, we artificially make the chemical potentials (fugacities) in the action Eq. ( 3) spatially dependent and thus transform the grand canonical free energy from a function to a functional This creates an additional degree of freedom in the choice in chemical potentials, which we S-4 fix later by using the variational principle.This allows us to show that derivative, is nothing but the functional average of the particle densities.If we look a the variational equation of the grand potential with respect to the particle density together with Eqs. ( 1) and ( 6), we find In order to find the true density distribution, we know that the chemical potentials must be constant and equal to the externally applied chemical potentials, µ j (r) = μj , (S.12) which means that the variational derivative above for the particle density is lifted to a variational principle, In summary, the true particle density and potential distributions, according to equilibrium thermodynamics, are the ones with makes the functional Ω[n j , ϕ] stationary which is equivalent to Eq. ( 14).

S-5 2 Derivation of the chemical potentials
For the charge carriers, the particle density can be obtained as, where the mean-field contribution is, = −e βµ i (r) Λ −3 i e −q i βϕ(r) , (S.15) and the one-loop contribution is, Together this gives the charge density, (S.17) Inverted yields the chemical potential, where we define the scaling parameter, 3 Derivation of the second variational derivative Starting from the action, Eq. ( 3), the first variational derivative is exactly the Euler-Lagrange equation, which reads, The second variational derivative can be computed in a similar fashion, using a generalized Euler-Lagrange equation or simply by using the rules of functional differentiation which S-7 yields If we now evaluate the second variational derivative at the electric potential Eq. (S.1), like it is stated in the result for the grand canonical free energy Eq. ( 4), and insert our results for the chemical potentials Eqs. ( 8) and ( 9), we obtain the differential equation for the correlation function Eq. ( 12).

Details on the correlation parameters
The solution of the differential equation Eq. ( 12), for spatially independent fields, follows closely the works of Levy et al. who studied Coulombic correlations in bulk electrolytes.S2 The equal point Green's function and its Laplacian can be obtained by, assuming spatially constant fields, and Fourier transforming Eq. ( 12), which allows us to compute, and where a maximum wavelength cutoff k max = 2π/Λ B was introduced to fix the divergence of the Green's function at equal argument.This large-momentum cutoff corresponds to a small-distance cutoff.
The solution for the equal point correlation functions inserted into Eqs.( 10) and ( 11), we obtain the correlation parameters where ϵ(r) is a prefactor in Eq. ( 12) and λ D = ϵ(r)/(β i=a/c q 2 i n i (r)) is a spatially dependent Debye length and Λ B is the small distance cutoff.

Details of free energy functionals
The metal free energy functional is decomposed in three terms, with T e being the kinetic energy from Thomas-Fermi theory, S3 which depends on the electron density and its gradients with U ex as well as U c are exchange and correlation terms from the Perdew-Burke-Ernzerhof functional, which is a generalized gradient approximation, S3,S4 which have the form, Here, e au = e 2 0 /(4πϵ 0 a o ) is the atomic energy and a 0 the Bohr radius.
The interaction free energy is, where ω j is a positive number describing the repulsion force.The two length scales σ 1 j and σ 2 j describe range of interaction.The function dM(r) is the closest distance to the metal at position r, which is negative when r is in the metal and positive if it is outside of the metal.
Excess free energies describe non electrostatic interactions such as hard-core repulsion and vary in complexity.In this model we account for steric effects using the excess free energy of an ideal lattice gas, where all particles occupy a lattice with density n max .S6,S7 For the steric the proper electrostatic field configuration makes the functional Ω C [ϕ] stationary.If one now inserts the partition function from Eq. (S.3) into the definition of the generating functional Eq.(S.5) and exponentiates both sides, one arrives after a shift of integration at the important relation, S.34) U c = e au a −3 0 (u c + θ c n e a 3 energy and u c is an interpolation of the volumetric correlation energy of an uniform electron gas.The factors s ∼ |∇n e |/n 4/3 e and t ∼ |∇n e |/n 7/6 e are corrections for the gradient terms in the kinetic-and exchange correlation energies.Thus the parameters θ T , θ ex and θ c are tuning the strengths of the gradient coefficients.These parameters need to be determined through simulation or experimental data, e.g. for an AG(111)-KPF 6 aqueous interface.S5 where W j is a parametrizable Lennard Jones potentials of the form,W j (r) = ω j • σ 1 j dM(r) • Θ(dM(r)) + σ 2