First-Principles Simulation of the Active Sites and Reaction Environment in Electrocatalysis - Fuel Cell Catalysis: A Surface Science Approach - page 99

were modeled by shifting the energetics and/or the d-electron band of the electrode by
a constant term given by neU, where n is the number of electrons and U is the potential
[Anderson and Ray, 1982]. These calculations demonstrated the potential-
dependent variation of reaction and adsorbate energies; however, the use of small
metallic clusters and a limited solvation model have hindered its application. More
recently, Anderson introduced the local reaction center model, which allows the
calculation of potential-dependent activation barriers and methods to begin to intro-
duce electrolyte [Anderson et al., 2005].
Approaches developed to maintain the integrity of the electrode structure (a semi-
infinite planar electrode) while integrating electrochemical ideas were later introduced
by Nørskov and co-workers [Nørskov et al., 2004], in parallel with the concepts devel-
oped by Anderson [Sidik and Anderson, 2006; Vayner et al., 2007]. These approaches
also consider the variation in reaction energies to be linear in the potential; however,
the improved description of the electrode leads to more reliable results. Furthermore,
an implicit reference potential was cleverly introduced via the equivalence between
the energy of H 2 and the energies of 2H þ þ 2e 2 implied by the use of the normal
hydrogen reference electrode. These models remain limited, however, as charges
are not directly applied to the slab, and hence exploration of the thermodynamics of
slab charging and the kinetics of electron transfer cannot readily be investigated.
A framework for probing these latter effects was laid out in a theoretical discussion
by Alavi and Lozovoi [Lozovoi et al., 2001]. This framework depended on the concept
of a reference electrode, against which the potential of a charged slab could be
measured. Filhol and Neurock built upon this model by developing a more ambitious
model for studying aqueous electrochemical interfaces with periodic DFT [Filhol
and Neurock, 2006; Taylor et al., 2006c]. The resulting technique allows the
simulation of electrochemical interfaces in a way that parallels the potentiostatic con-
trol of experimental electrochemical systems. The methodology, referred to as the
double-reference method, employs two internal potential references within the calcu-
lation. The first is a vacuum reference that relates the interfacial potential to the vacuum
scale. The second is an aqueous reference that ties the vacuum reference to a tunable
potential that is varied by the addition or removal of electrons from the unit cell (i.e.,
system charging). The theoretical underpinning of the double-reference method is
discussed briefly below. A more complete discussion of the approach and the
theory behind it is given in Taylor et al. [2006b]. An alternative DFT-based, surface
slab representation of charged surfaces by Otani and Sugino [2006] that removes
the periodic boundary condition (PBC) in the surface normal direction to allow for
calculations of charged surfaces is an encouraging approach for the consideration of
electrochemical systems. Otani and Sugino couple the results from conventional
periodic plane-wave electronic structure calculations for surfaces with a Poisson
solver. This offers considerable flexibility in the treatment of electrochemical systems,
although its implementation is nontrivial. Using the methods that we have developed
to describe electrochemical systems within a PBC model, much of the physics
proposed by Otani and Sugino is captured, without resorting to some of the continuum
assumptions in their model (such as that water close to the interface has a dielectric
constant of 78).