The Shell Model

In the Shell model, atomic interactions are represented by potentials between each pair of atoms in the system. Electronic polarization of the atoms is implemented via the Dick-Overhauser model [45], in which an atom is considered as a charged core connected by a harmonic spring to a massless charged shell (see figure 3.2). The equilibrium distance between the core and shell is a representation of the electronic polarization of that atom. This is important, as there are no electrons in the SM, all atoms are effectively represented by point charges and the shells approximate the effects of electron density flow on the atomic interactions.

**Figure 3.2:** Schematic diagram of core and shell atomic representation.
$\includegraphics{figures/shell.eps}$

The interactions between cores and shells are controlled by empirical potentials whose parameters are fitted to achieve the best possible comparison with experiment or ab initio techniques. The potentials are usually derived from three interactions: (i) electrostatic coulomb interactions between the atoms (cores and shells), (ii) van der Waals interactions and (iii) short-range repulsive interactions. The charge-charge electrostatic interaction between atoms $i$ and $j$ is given as the sum of four terms:

$\begin{displaymath} V^{elec}_{i}=\sum ^{n}_{j}\frac{q_{i}q_{j}}{4\pi \varepsilon... ...on _{0}\left\vert \mathbf{r}_{si}-\mathbf{r}_{cj}\right\vert } \end{displaymath}$

(3.1)

where $i\neq j$ , $n$ is the number of atoms, $q_{i}$ is the shell charge of atom $i$ , $Q_{i}$ the core charge of atom $i$ , $\mathbf{r}_{si}$ is the position vector of the shell of atom $i$ and $\mathbf{r}_{ci}$ is the position vector of the core of atom $i$ . Buckingham two-body potentials were used throughout this study to represent the non-coulombic short-range interactions between the shells. These potentials have the following form:

$\begin{displaymath} V^{short}_{i}=\sum _{j}^{n}\left( -C\left\vert \mathbf{r}_{s... ...t \mathbf{r}_{si}-\mathbf{r}_{sj}\right\vert }{\rho }}\right) \end{displaymath}$

(3.2)

where $C$ , $A$ and $\rho$ are parametrized constants specific to each pair of shells $i$ and $j$ , and $i\neq j$ . Note that for some atoms there is no shell and all references to distance apply to the position of the core instead. The first term in equation (3.2) represents the attractive van der Waals interaction and the second term the short-range repulsion due to electron cloud overlap. For shells there is also an additional contribution to the interaction due to the elastic force in the spring connecting core and shell. This force is equal to $k\delta r_{i}$ , where $k$ is the parametrized spring constant between a core-shell pair and $\delta r_{i}$ is the distance between the centres of core and shell for atom $i$ . The spring interaction between the cores and shells is given by:

$\begin{displaymath} V^{spring}_{i}=\frac{1}{2}k\delta r_{i}^{2} \end{displaymath}$

(3.3)

Combining equations (3.1), (3.2) and (3.3)gives the total energy of the system as:

$\begin{displaymath} E={1\over 2}\sum _{i}^{n}\left[ V^{elec}_{i}+V^{short}_{i}+2V^{spring}_{i}\right] \end{displaymath}$

(3.4)

This can then be minimized with respect to core and shell positions to find the equilibrium geometry of relaxed atoms in the system. Usually certain atoms within the tip-surface unit cell remain frozen to represent the interface between the macroscopic and microscopic features, e.g. in fig. 3.1 region I would be relaxed and region II would be frozen. For infinite systems the unit cell is repeated, according to the system lattice vectors, across space until the atomic interactions converge to the desired accuracy. For bulk samples the cell is repeated in three dimensions, but for surfaces two possibilities exist. One method for calculating surfaces is by cutting the infinite bulk system and creating a series of slabs which are infinite in two dimensions, but separated in the third dimension by a large vacuum gap. These slabs are a good model of a surface if the gap is large enough that there is no significant interaction between the slabs. Another method for calculating surfaces is just to repeat the cell in two dimensions, directly generating a real infinite surface. Note that the electrostatic interaction converges conditionally for an infinite system and methods, such as Ewald summation [46], must be used to calculate this contribution.