Electrochemistry from the atomic scale, in the electronically grand-canonical ensemble

Abstract

The ability to simulate electrochemical reactions from first-principles has advanced significantly in recent years. Here, we discuss the atomistic interpretation of electrochemistry at three scales: from the electronic structure to elementary processes to constant-potential reactions. At each scale, we highlight the importance of the grand-canonical nature of the process and show that the grand-canonical energy is the natural thermodynamic state variable, which has the additional benefit of simplifying calculations. We show that atomic forces are the derivative of the grand-potential energy when the potential is fixed. We further examine the meaning of potential at the atomic scale and its link to the chemical potential and discuss the link between charge transfer and potential in several situations.

Controlled Electrochemical Barrier Calculations without Potential Control

“…For simplicity, here, we only consider the states discussed until now, i.e., the forward barrier with respect to the proton in the interfacial water layer. The interested reader finds the same analysis for an initial state that considers the proton in the bulk electrolyte in the Supporting Information. , …”

“…Considering that canonical forces F⃗ at a constant charge q are identical to the grand canonical forces scriptF⃗ at the respective constant electrode potential U ( q ), − scriptF⃗ ( r⃗ , U false( q false) = const . ) = F⃗ ( r⃗ , q = const . ) Using the grand canonical energy and forces in combination with a potentiostat provides all relevant quantities that are necessary for common geometry optimizations or transition state search algorithms, which evaluate the PES of the system.…”

“…However, the equivalence of forces of the canonical and grand canonical PES (eq ) implies that a geometrically stationary point r⃗ *, i.e., a local extremum or a saddle point, determined in one PES is also a stationary point in the other PES. − This allows evaluating stationary points in the cPES at certain excess charges q i using structure relaxations or transition state search algorithms yielding r⃗ i * and U i * , Legendre transform their energies E i * = E ( q i , r⃗ i * ) to the grand canonical energies E i * = scriptE ( U i * , r⃗ i * ) , and obtain the identical states with identical grand canonical energies as if we searched for r⃗ i * and E i * at U i * in the gcPES. However, the electrode potentials U i * corresponding to the excess charges q i will differ for the different stationary states IS, TS, and FS.…”

“…The interested reader finds the same analysis for an initial state that considers the proton in the bulk electrolyte in the Supporting Information. 18,40 First, we compare the inter-and extrapolated results obtained from multiple canonical NEB calculations using a parabolic fit, which are shown in Figure 4 as dark and light blue solid lines, with the potentiostat-derived reference results (diamonds).…”

“…The nonlinearity originates from the TS shifting toward the energetically less favored state (in the case of the forward barrier, the IS for low potentials, and vice versa for the reverse barrier). 18,40,50,51 Of the approximate expansion-type methods based on a single canonical NEB calculation, unsurprisingly, the Hessianbased EGC method performs the best (dashed orange lines). It essentially re-creates the extrapolated results that require multiple canonical NEB calculations, indicating that the individual electronic and geometric responses of the respective states determine the nonlinear potential dependence of the barrier.…”