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A Density Functional Theory based Molecular Dynamics (DFTMD) perspective on computational electrochemistry: What can be done? Contents Electrochemistry has played a crucial role in the development of the atomistic picture of chemistry in the 19th century. More than a century later electrochemistry has returned to the forefront of research. The main reason for this renewed interest is practical. Many devices for conversion and storage of energy in sustainable energy schemes involve an electrochemical process. This has stimulated a parallel development in computation. Most of this effort is focused on the materials chemistry of electrodes and the mechanistic detail of electro and photoelectro-catalysis. The aim is to improve efficiency and durability by "computational" design. This is a very active and productive field of research. However, electrochemical interfaces are highly complex heterogeneous systems and there remain a number of fundamental questions in physical electrochemistry, in particular concerning the structure and effect of electrical double layers. Even as recent as ten years ago it was in practice not feasible to address these questions using electronic structure calculation methods. It is now, and this is the motivation behind current and future projects in computational electrochemistry in our group. Our background is computational solution chemistry. Fluctuations and dynamics are crucial in our view, at least for the liquid half of the interface, which is why DFTMD is our method of choice. Electrodes are interfaces between electronic and ionic conductors converting one type of charge transport into the other. Experimental electrochemistry has developed a range of current-voltage measurement techniques to probe this process. Atomistic modelling is not yet ready for this challenge, or at least we are not. Electrochemical interfaces also act as capacitors, which can be charged (electrified). This can be studied under open circuit conditions (zero current). Here computational methods have more of a chance and considerable progress has been made in the calculation of open circuit electrode potentials. What is actually computed are ionization potentials or electron affinities relative to some suitable reference. This works quite well for metal electrodes. The popular model system is the Pt(111)-water interface for which there is also a wealth of experimental data available. One of the successes is the computation of the potential of an uncharged Pt electrode (the potential of zero charge) which is in good agreement with experiment. These calculations also showed the importance of the modulation by solvent fluctuations which makes the point that sampling of the thermal motion is necessary. This is a strong justification for the use of DFTMD methods.
The calculation of the potentials of semiconductor electrodes is more involved than for metals. The one-electron states responsible for electronic transport are delocalized similar to metals (as long as the gap is not too large). However, unlike the Fermi level of metal electrodes, the band edges (ionization potential or electron affinity) of a semiconductor are not electrode potentials. An electrode potential is an electrochemical potential consisting of a standard potential and an activity term depending on concentration of redox active species (doping in semiconductors, ions in solution). Standard potentials are (reversible) ionization potentials. We can calculate ionization potentials. However, DFTMD model systems are too small for explicit modelling of finite concentration effects. The activity terms will have to be added in "by hand" similar to quantum chemical calculations of thermochemical constants, which makes the calculation more indirect. For the same reason the modelling of space charge is also out of reach of DFTMD. Space charge layers are too wide. This essentially limits the study of semiconductor electrodes to flatband conditions (zero net electronic charge). Note, however, that metal oxide electrodes can also be charged by protonation or deprotonation even at the DFTMD length scale (see below). Excess electrons or holes in semiconductors have a tendency to localize in defects, such as vacancies or impurities. Ionizable defects are dopants and determine the properties of interest in semiconductor device physics. Ionizable sites on surfaces play an equally important role in catalysis. Ionic semiconductors, such as metal oxides, display a particular large variety of charged defects most of which are also ionizable (redox active). What is needed for the calculation of the (standard) potentials of these defects is the adiabatic or reversible ionization potential including the relaxation of the atomic structure. Again DFTMD is a very suitable method for this purpose capable of also accounting for the contribution of thermal (free energy) effects.
Water (or an aqueous solution) is the other half of an electrochemical interface. Water is a molecular liquid, a disordered system stabilized by thermal fluctuations. It is the system DFTMD was invented for. From an electronic structure perspective water is a large gap (8.7 eV) insulator. Similarly, ionizable aqueous solutes can be considered as an example of charge localization. As discovered by the computational solid state physicists already more than two decades ago, DFT, in the common generalized gradient approximation (GGA), has a very bad track record for the calculation redox potentials of defects in semiconductors (called charge transition levels in semiconductor defect physics). Errors of more than 1 eV are not uncommon. The error was traced back to the infamous band gap problem, more precisely the mixing of the localized defect states with the misaligned delocalized band states. This is a manifestation of the enhancement of the delocalization error in extended systems. After a bit of a detour we realized that the DFTMD calculation of redox potentials in homogeneous solution suffers from the same problem. It is in fact one of the more extreme examples. Methods favoured in quantum chemistry to treat solvation, such as implicit solvent models or QM/MM, are much less affected by the delocalization error, because the band states have been eliminated. The band gap error is a steep hurdle in application of all-atom DFTMD methods, the price one has to pay for insisting on a "fundamental" approach. The point we want to make is, of course, that these problems must ultimately be faced at electrochemical interfaces, because the essence of the exchange between electronic and ionic current taking place there is precisely the interaction between the localized states in the electrolyte and extended electronic states of the electrode. Fortunately, hybrid functionals, adding a fraction of exact exchange, lead to a significant improvement. Application of hybrid functionals in extended systems is expensive. However, there has been substantial progress in the implementation of exact exchange methods in systems under periodic boundary conditions reducing the computational cost. Recent hybrid DFT results for redox potentials in solution and for charge transition levels in solids using these methods are encouraging. There is indeed a strong parallel between electrode potential calculations in solution and semiconductors. The electronic structure issues are the same. The main difference is that solvent reorganization energies have a significant entropic component which must be computed by free energy perturbation methods. This calculation is demanding but doable as we have shown. The time for all-atom DFTMD computational electrochemistry has come.
Transition metal oxides are exceptionally complex and versatile solids (and, hence, of course a challenge for computation). They can be metals, semiconconductors or insulators. They can be antiferromagnetic or ferroelectric and undergo transitions between these various states (multiferroics). A key question is how this variability in electronic properties enhances (or frustrates) the catalytic properties of their surfaces. Transition metal oxide/water interfaces are also of special interest from a physical chemical perspective. Like main group metal oxides, transition metal oxide surfaces act as amphoteric acids exchanging protons with the aqueous solution. The surface charge density that can be built up in this way can be very high (an order of magnitude higher compared to the electronic charge of metal electrodes). The ionic strength of the electrolyte under these conditions is equally high and the dimensions of the compact double layers compensating for the protonic charge are (just) within the DFTMD length scale. This is an area where electrochemistry and colloid science overlap. We regard this as an unique opportunity for DFTMD as our recent results demonstrate (see publications by topic). The main area of application we are aiming for is (photo)electrolysis (in particular water splitting) under strongly alkaline conditions.
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