Methodology

3.2.1 DFT Calculations

The CASTEP suite [35], part of the Materials Studio software package [36] was used for the calculations. The RPBE functional [37] was used, hence GGA was used to calculate the exchange- correlation energy. A Gaussian smearing width of σ = 0.1 eV was utilized in all calculations. The ion- electron interactions were approximated using ultrasoft pseudopotentials with core corrections and a cutoff energy was set at 400 eV. A five-layer slab with three layers relaxed was used with an optimized vacuum spacing of 10 Å between surfaces. The k-point grid was generated using the Monkhorst-Pack[38] procedure with a k-point spacing of <0.03 Å^-1. Spin-polarization was allowed for all calculations

The integral adsorption energy is the average adsorption energy of a configuration and is calculated as follows:

𝐸_{𝑎𝑑𝑠,𝐶𝑂}=^{𝐸(𝐹𝑒+𝑛𝐶𝑂)−𝐸(𝐹𝑒 𝑆𝑙𝑎𝑏)}

𝑛 − 𝐸_𝐶𝑂 (3.1)

Where E(Fe+nCO) is the energy of n CO molecules on an iron surface, E(Fe Slab) is the energy of a clean iron surface and ECO is the energy of CO in the gas phase

The differential adsorption energy can be used to describe the adsorption of individual adsorbates of a particular configuration as is calculated as follows:

𝐸̅_{𝑎𝑑𝑠,𝐶𝑂}= 𝐸_{(𝐹𝑒+𝑛𝐶𝑂)}− 𝐸(𝐹𝑒+(𝑛−1)𝐶𝑂)− 𝐸_𝐶𝑂 (3.2)

35 The interaction between the CO molecules and the iron surface caused a shift in the position of the iron atoms. This shift in positions occurs mainly with the first Fe layer even though three layers are relaxed. If the configuration is frozen and the adsorbate(s) removed, we can compare this ‘deformed’

structure with the clean Fe (100) surface. The energetic contribution due to this shift in metal surface was considered part of the lateral interactions. To measure this quantity, the adsorbates were removed and a single point energy calculation was completed. The deformation energy was then the difference in energy between the clean optimised Fe surface and the single point energy calculation.

The deformation of the iron surface was then calculated as follows:

𝐸_{𝑑𝑒𝑓𝑜𝑟𝑚} = 𝐸[(𝐹𝑒+𝑛𝐶𝑂)−𝑛𝐶𝑂]− 𝐸_{(𝐹𝑒 𝑆𝑙𝑎𝑏)} (3.3)

Where E[(Fe+nCO)-nCO] is the energy of the geometry optimized iron/CO surface having removed the CO molecules and E(Fe Slab) is the energy of a clean geometry optimized iron surface. ¹¹

The mapped difference in the electron density for the configurations considered is also produced. This is achieved by subtracting the electron density of the clean surface from the electron density of the slab with adsorbed molecules as shown in Figure 2-9. The result is electron density of the adsorbed CO and the electron density changes it induces on the metal. It is expected that the electron density attributed to the electrons not involved in the interactions, for the most part core electrons, would cancel each other out and only deviations in what was classically know as frontier orbitals [25,39–41]

will be seen. The changes to the metal are seen as a change in colour from dark blue to teal, which indicates the repositioning of electrons relative to the clean surface.

3.2.2 Energetic breakdown

In order to find an energetic breakdown for the lateral interactions, all the terms included in the Hamiltonian need to be considered. For CASTEP [35] the resulting energies are the kinetic energy, Hartree energy, local and non-local pseudopotential energies, exchange-correlation energy, Ewald energy and non-Coulombic energy.

The energy contribution to the total electronic energy of each of the terms was considered and the change in each energy on adsorption was considered. This was calculated using the same procedure to calculate the integral adsorption energy was calculated. This means:

𝐸𝑖,𝑎𝑑𝑠=^{𝐸𝑖,(𝐹𝑒+𝑛𝐶𝑂)−𝐸}𝑖,(𝐹𝑒 𝑆𝑙𝑎𝑏)

𝑛 − 𝐸𝑖,𝐶𝑂 (3.5)

Where 𝑖 is either the kinetic energy, Hartree energy, local or non-local pseudopotential energies, exchange-correlation energy, Ewald energy or non-Coulombic energy.

3.2.3 Unique configurations

Due to computational time and the resources available, DFT calculations for periodic systems were traditionally calculated on p(2x2) unit cells for Fe (100) [4,8,34,43]. As a result, the number of geometric configurations per unit cell was limited and only coverages of 0.25 ML, 0.5 ML, 0.75 ML, and 1.0 ML have typically been calculated [3,4,34]. In principle, at a coverage of 𝜃 = 𝑛/𝑚 where 𝑛 is the number of adsorbates and 𝑚 is the total number of adsorption sites, the total number of configurations is given by _{𝑛!(𝑁−𝑛)!}^𝑁! . However, since the different configurations are periodic, all rotations, translations and reflections yield equivalent configurations. Hence, at a p(2x2) cell for a coverage of 0.5 ML there are only 2 unique configurations instead of the 6 possible configurations.

11 He deformation energy will always be positive since the clean slab is the minimum of the Fe (100) surface

36 The equivalence between various configurations is illustrated for a coverage of 2/9 on a p(3x3) cell in Figure 3-1. Therefore, only unique configurations should be considered.

Figure 3-1: Equivalent configurations of a 2/9 ML coverage of CO on Fe (100) for a p(3x3) unit cell By eliminating equivalent configurations, the total number of unique configurations in a particular unit cell size can be calculated¹². This is shown in Figure 3-2. Note only adsorption in the hollow site was considered for this study. For the p(2x2) unit cells a total of 16 configurations are possible but only 6 are unique. For the p(3x3) unit cells a total of 512 configurations are possible but only 27 are unique.

For the p(4x4) unit cells a total of 65536 configurations are possible but only 805 are unique. Note that this assumes that CO is only tilted in the hollow and does not consider different orientations at higher coverages, this will of course further increase the possible configurations.

Advancements in computing power have made calculations on larger unit cells feasible. For the purposes of this study the adsorption of CO on Fe (100) was investigated on p(4x4) unit cells.

12 This is valid assuming a maximum coverage of one adsorbate per p(1x1) cell, which is the case for CO adsorption on Fe (100) [46]

37 Figure 3-2: Unique configurations for CO adsorbed in the hollow site on Fe (100) for a) p(2x2) unit cells, b) p(3x3) unit cell and c) p(4x4) unit cell.

1 1

2

1 1

0 1 2 3

0 1/4 2/4 3/4 1