Fractional Charge Density Functional Theory Elucidates Electro-Inductive and Electric Field Effects at Electrochemical Interfaces

The application of external voltages to a system in which molecules are immobilized on an electrode allows for the dynamic tuning of the molecular properties. The electrode and the voltage generate two main effects, the electro-inductive effect describing the charging of the molecule and the electric field effect, which can significantly change chemical properties and reactivities. Computational study is critical for understanding this open system. To overcome the challenges in quantitative theoretical prediction associated with its open system nature and complex solution environment, we developed a novel theoretical scheme, the fractional charge density functional theory coupled with a model electrode and in a continuum model of solvent (FC-DFT+), to offer quantitative insights into electro-inductive and electric field effects. Our FC-DFT+ approach reformulates open quantum systems in terms of canonical ensembles, which leverages the efficiency of fixed electron number calculations while relating the fractional number of electrons directly to the applied voltage. It uses the Poisson-Boltzmann equation for a continuum model with boundary conditions consistent with the Gouy-Chapman-Stern theory. With FC-DFT+, we simulated the vibrational frequency shift of self-assembled monolayer formed by 4-mercaptobenzonitrile as a function of applied voltages. Our calculations accurately capture the slope of C≡N vibrational frequency vs. voltage curve, the mysterious frequency flattening behavior, and the dependence on the ionic strength of the solution, observed in recent experiments by Zanni and coworkers. Our analysis revealed that the frequency flattening is a key signature of fractional electron number changes, or the electro-inductive effect, emerging when the number of electrons encounters the HOMO-LUMO energy gap. Consistent with a recent finding, we found that the electro-inductive effect is the dominating factor in the frequency shift, while the field effect plays a smaller but opposing role. Similarly to the case of 4-mercaptobenzonitrile, we found that, for 4-nitrobenzenethiol, the flattening behavior exhibited by NO2 stretching frequency is mainly due to the electro-inductive effect. In contrast, we show that the tendency in forming a Lewis adduct by 4-mercaptopyridine results from the electric field effect, demonstrating the versatile capability of FC-DFT+ in describing electric field-dominating behavior. Our approach is efficient and predictive and should play an important role in developing and understanding a broad spectrum of chemical applications at electrochemical interfaces.