Vienna Ab-initio Simulation Package

As mentioned previously VASP has been simulation package which was used for the various DFT calculations performed in this thesis. VASP has the ability pseudopo-tentials, or the Projector Wave Method (PAW) with a plane wave basis set.[41–47]

The code has been used for simulations of materials, interactions between catalyst and substrates, and quantum mechanical Born-Oppenheimer molecular dynamics.

The Kohn-Sham equations for calculating the ground state properties of a system are solved using an efficient iterative matrix diagonalisation method, with Pulay/Broy-den charge Pulay/Broy-density mixing. This leads to accurate calculations for transition metal systems, and by using the forces on the atoms allows for optimisation of the geometry of the system.

The PAW method in VASP is implemented in VASP in order to correctly simulate first-row transition metal elements with minimal effort, and also provides access to the full wave function when generating the density functional. In the PAW method the core electrons are considered to be frozen, and only the valence electrons are considered. The PBE functional has been used for all calculations performed in this thesis, the PAW functionals for PBE were taken from the VASP database.

Figure 2.12: Flow chart of DFT optimisation in VASP

The optimisation of the density functional in VASP consists of two loops (figure 2.12). The density and wave function are optimised in the inner loop, and then the forces and atom positions are optimised in the outer loop. Inside the inner loop the method used to optimise the wave function is set by the ALGO tag. They can be optimised using a blocked algorithm (ALGO = normal ), a residual minimisation scheme - direct inversion of the iterative subspace, known as RMM-DISS (ALGO = Very Fast ), or a mixture of the two algorithms (ALGO = Fast ). For all of the work performed in this thesis ALGO = Fast has been used.

For the optimisation of the outer loop (movement of ion positions), the IBIRON

Chapter 2 Kinetic Analysis and Modelling in Heterogeneous Catalysis

tag is used to specify which method to use. For all of the calculations in this thesis IBIRON = 2 has been used, which uses a conjugate gradient method for the minimisation of the forces and atomistic positions, which is the recommended setting in VASP. The conjugate gradient method is a very common optimisation method for linear equations and is well established in the literature.[48]

Figure 2.13: Schematic showing how to define a unit cell simplified to two dimensions

When studying a material it is not possible to simulate the almost infinite number of electrons required to replicate the material properties on the macroscopic scale, and hence a method of reducing the number of atoms in the system is required, and this is performed using Periodic Boundary Conditions. In order to replicate the ma-terial in three dimensions the unit cell needs to be defined (see figure 2.13), which is the simplest form of the 3D periodic structure, and the properties of the system are calculated using that unit cell. This reduces the total number of atoms used in the calculation. The actual calculation of these properties involves the complex trans-formation of the unit cell into reciprocal space and then into k -space. The process is described very well in Computational Materials Science: An Introduction[49]:

• The solid is reduced into a supercell consisting of several unit cells and is expanded to infinity by the Periodic Boundary Conditions

• The supercell is transformed to reciprocal space and is contained within a first Brillouin Zone

Figure 2.14: Treatment of solids using Periodic Boundary Conditions [49]

• The wave functions are mapped using two vectorsk and G which are the wave vector and reciprocal lattice vector respectively.

• The first Brillouin Zone is then converted into it’s simplest form known as the irreducible Brillouin Zone

• The irreducible Brillouin Zone is then mapped using a grid ofk-points and then by integration/summation/extrapolation of these points all of the properties of the infinite system can be obtained.

When replicating the true plane wave of the solid in reciprocal space, a periodic wave function and a plane wave function are summed together to generate a sim-ulated wave function. When the number of plane waves summed to the periodic function rises to infinity then the simulated wave function can be said to be a true replication of the real wave function. Unfortunately, summing an infinite number of wave functions is impossible, and hence a cutoff for the energy of the plane waves is included in calculations with the ENCUT tag in VASP. It can be considered that the higher energy (and higher frequency) wave functions have little effect on the overall shape of the simulated function, and after a certain point the energy of the system will converge upon a certain value. Practically this is performed by a plane

Chapter 2 Kinetic Analysis and Modelling in Heterogeneous Catalysis

wave cutoff test.

Figure 2.15: Convergence in plane wave energies for a Zr₂O₄ unit cell

As it can be see in figure 2.15 the total energy of the system converges at> 500 eV but the increase in run time is still linear. Alongside setting the cut off energy for the plane wave, the total number of k-points to be used in the k-grid also needs to be set. This is performed via a similar method where instead of varying the plane wave cutoff energy, the size of the k-point grid is varied instead. The k-point grid is assigned using the KPOINTS file in VASP, and the total number of k-points per lattice vector is correlated to the size of that particular lattice vector (i.e. The larger the unit cell in theab or c direction the less k-points required). A k-point expansion test is usually performed on the unit cell of the bulk structure being studied, so that the correct properties can be calculated. For larger systems using anything other than a singlek-point in each direction becomes far too computationally expensive as the reciprocal space becomes very small and a singlek-point is more than sufficient to describe the Brillouin Zone. As it can be seen in figure 2.16 setting the correct k-grid has a large effect on the computation time of the calculation, with the run time increasing exponentially with increasing k-grid density.

Figure 2.16: Convergence in k-point cutoffs for a Zr₂O₄ unit cell

The final step in setting up the calculation is setting up the partial occupancies of the orbitals, and this is performed using the ISMEAR tag in VASP. The calculations in this thesis all use either metallic smearing IMSEAR = 1 which uses the method of Mathfessel-Paxton in order to set the occupancies, or Gaussian smearing ISMEAR

= 0 for semiconductors and metallic nanoparticles. The width of the smearing is defined by the SIGMA tag, with a standard setting of 0.2 eV used for metallic smearing, but in the case of Gaussian smearing a value of ≤ 0.05 is often required in order to get the correct occupancy of the orbitals.