# Grand canonical electronic density-functional theory: algorithms and applications to electrochemistry

## TL;DR

This paper introduces two algorithms for grand-canonical density-functional theory that enable more efficient and realistic modeling of electrochemical systems at fixed potentials, with applications to copper deposition on platinum.

## Contribution

The paper presents two novel algorithms, GC-SCF and GC-AuxH, for solving Kohn-Sham equations in the grand canonical ensemble directly at fixed potential, improving computational performance.

## Key findings

- GC-AuxH exhibits reliable exponential convergence.
- Algorithms enable realistic electrochemical modeling.
- Application to copper deposition reveals chloride desorption as key process.

## Abstract

First-principles calculations combining density-functional theory and continuum solvation models enable realistic theoretical modeling and design of electrochemical systems. When a reaction proceeds in such systems, the number of electrons in the portion of the system treated quantum mechanically changes continuously, with a balancing charge appearing in the continuum electrolyte. A grand-canonical ensemble of electrons at a chemical potential set by the electrode potential is therefore the ideal description of such systems that directly mimics the experimental condition. We present two distinct algorithms, a self-consistent field method (GC-SCF) and a direct variational free energy minimization method using auxiliary Hamiltonians (GC-AuxH), to solve the Kohn-Sham equations of electronic density-functional theory directly in the grand canonical ensemble at fixed potential. Both methods substantially improve performance compared to a sequence of conventional fixed-number calculations targeting the desired potential, with the GC-AuxH method additionally exhibiting reliable and smooth exponential convergence of the grand free energy. Finally, we apply grand-canonical DFT to the under-potential deposition of copper on platinum from chloride-containing electrolytes and show that chloride desorption, not partial copper monolayer formation, is responsible for the second voltammetric peak.