Part 2: Choosing a reference

The first thing we need to do is decide on a reference set of species; all the DFT-computed energies will then be mapped into formation energies, which are essentially energy differences with respect to these reference species. To decide how many reference species we need and which these species should be the following rules are useful:

  1. The pristine catalytic phase (facet, nano-cluster etc. with nothing adsorbed) has to be included in the reference set.
  2. The rest of the reference species will be gas phase species, and we need as many such species as the number of different atoms encountered in our species.
  3. The reference species must have linearly independent compositions. This means that we should not be able to write a reaction that produces one (or more) reference species from other reference species.

In our case, we have a Pt surface as the catalyst and C-, H-, and O-containing species. According to rule 1, Pt(111) will be in our reference set. In terms of the gas species of rule 2, perhaps the easiest choice of a reference set is the one used in thermodynamics: in our case, that would be atomic C and the diatomics O2 and H2. Such a basis might not be very convenient, however, in practical terms. For instance, the triplet state of O2 is not accurately captured by DFT, so this might not be a good reference species. Also, it is frequently convenient to have the reactant gas molecules in the basis so that the corresponding bound species’ formation energies are equal in magnitude to the binding energies with opposite sign. So in our case, we will choose as our reference the gas species CO, H2O and H2. It can be verified that rule 3 holds true. An alternative option would be CO, H2O and CO2. An invalid option would be CO, CO2 and O2: these molecules’ compositions are not linearly independent (we can write the CO oxidation reaction), and also H-containing molecules remain unreferenced.

In the following, we will work with the reference set {Pt(111), CO, H2O, H2}.

To better understand the purpose of the “reference set”, an analogy with the idea of a “basis set” would perhaps be useful: one can think of the set of reference species as a basis (in terms of stoichiometric compositions), out of which we can “construct” any other species we are interested in. The construction process would involve first determining which and how many basis/reference molecules we need and then simply rearranging atoms in order to form the species of interest. These ideas can be formalised mathematically using vectors to represent the composition of a species in terms of its constituent elements. Given a basis on the space of species’ compositions, any other species can be represented as a linear combination of the basis vectors.

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