Vibrational frequencies and stark tuning rate with continuum electro-chemical models and grand canonical density functional theory

TL;DR

The work addresses incorporating electrochemical potential into density functional theory by exploiting grand-canonical (fixed-potential) formulations and Legendre transforms to relate grand potential to Helmholtz free energy. It combines a continuum SCCS solvent–electrolyte model with fixed-potential self-consistency and proves that atomic forces are identical in the two ensembles, while the force-constant matrix carries a correction term due to electron-number variation, enabling GC vibrational frequencies to be predicted from canonical data. Using CO on Pt(111) as a testbed, the authors demonstrate sizable differences for perpendicular vibrational modes between ensembles that diminish with increasing surface area, and validate the analytical corrections against direct GC finite-difference calculations. They also show that implicit-solution parameters (dielectric constant, interfacial distances) critically affect Stark tuning rates, suggesting careful calibration (and possible future hybrid explicit-implicit schemes) to achieve quantitative agreement with experiment.

Abstract

Simulating electrochemical interfaces using density functional theory (DFT) requires incorporating the effects of electrochemical potential. The electrochemical potential acts as a new degree of freedom that can effectively tune DFT results as electrochemistry does. Typically, this is implemented by adjusting the number of electrons on the solid surface within the Kohn-Sham (KS) equation, under the framework of an implicit solvent model and the Poisson-Boltzmann equation (PB equation), thereby modulating the potential difference between the solid and liquid. These simulations are often referred to as grand-canonical or fixed-potential DFT calculations. To apply this additional degree of freedom, Legendre transforms are employed in the calculation of free energy, establishing the relationship between the grand potential and the free energy. Other key physical properties, such as atomic forces, vibrational frequencies, and Stark tuning rates, can be derived based on this relationship rather than directly using Legendre transforms. This paper begins by discussing the numerical methodologies for the continuum model of electrolyte double layers and grand-potential algorithms. We then show that atomic forces under grand-canonical ensemble match the Hellmann-Feynman forces observed in canonical ensemble, as previously established. However, vibrational frequencies and Stark tuning rates exhibit distinct behaviors between these conditions. Through finite displacement methods, we confirm that vibrational frequencies and Stark tuning rates exhibit differences between grand-canonical and canonical ensembles.

Figures (4)

• Figure 1: Atomic structure of the selected test systems (round spheres) together with the continuum interfaces adopted for the solvent, in panel (a), and the electrolyte, in panel (b). For the solvent's continuum interface (left), the reported yellow and purple iso-surfaces correspond to $\rho_{min,\epsilon}=1.2\times 10^{-4}$a.u. and $\rho_{max,\epsilon}=2.2\times 10^{-3}$a.u., lying approximately at a distance from the Pt surface of 2.0$\AA$ and 3.1$\AA$, respectively. For the electrolyte's continuum interface, the yellow and violet iso-surfaces correspond to $\rho_{min,\gamma}=3\times 10^{-5}$a.u. and $\rho_{max,\gamma}=4\times 10^{-4}$ a.u., lying approximately at a distance from the Pt surface of 2.6$\AA$ and 3.8$\AA$, respectively.
• Figure 2: Vibrational frequencies of selected modes as a function of the Fermi energy. (a) Modes in the xy plane (parallel) for 1/4 CO coverage; (b) Modes in the z direction (perpendicular) for 1/4 CO coverage; (c) z-direction modes for 1/9 CO coverage; (d) z-direction modes for 1/16 CO coverage. Frequencies computed under grand-canonical and canonical ensemble using finite-difference methods are shown in blue and red colors respectively. Grand-canonical frequencies computed from canonical data using are shown in green symbols. Linear fits are used to extract Stark tuning rates from all three data sources.
• Figure 3: Comparison of vibrational properties computed under canonical and grand-canonical ensemble for Pt(111) surfaces of different coverage (1/4, 1/9, and 1/16). (a) shows the average vibrational frequency differences between the two ensembles. Orange bars represent values directly calculated via finite-difference (FD) methods, while green bars are obtained by applying Eq. (\ref{['eq:cmatrix_gc_ef']}) to canonical results to compute grand-canonical frequencies. (b) presents the Stark tuning rates, extracted from the slopes of vibrational frequencies as a function of the Fermi energy, under both canonical (red bars) and grand-canonical (blue bars) conditions.
• Figure 4: Comparison of Stark tuning rates calculated by different solvation and electrolyte environment parameters. (a) Stark tuning rates with solvation permittivity =6. (b) Stark tuning rates with shorter solvation and electrolyte interface distance. (c) Stark tuning rates with longer solvation and electrolyte interface distance.