Abstract
This post calculates the reaction energy and reaction barrier height for C-atom coupling on a Fe 100 surface. It also assesses the effect of using more atomic layers in the energy calculations.
Introduction
Iron is an attractive heterogeneous catalyst used in processes, such as N2 hydrogenation and CO hydrogenation along with the flame synthesis of CNTs. An important class of reactions in these processes is the C1 +C1 coupling with double-carbon species (C2) being produced (C1 +C1 -> C2)[1]. In this post, spin-polarized density functional theory calculations are performed to characterize the energetics of C+C -> CC reaction on the Fe (100) surface using the Vienna Ab initio Simulation Package (VASP)[2,3]. Along with the calculation of the reaction energy and barrier energy for the reaction, the effect of the number of layers in the slab on the energy calculations is also assessed here.
Methodology
The plane wave DFT calculations are used to evaluate the reaction energy as well as the barrier for the C+C -> CC reaction on Fe 100 surface. A 3×3 Fe 100 surface is used for these calculations with 4, 6, and 8 atomic layers (see Fig. 1). All the calculations are performed using the Vienna Ab initio Simulation Package (VASP)[2,3]. The projector augmented wave (PAW) [4] method is used for core-valence treatment. The exchange and correlation energies are calculated using the Perdew, Burke, and Ernzerhof (PBE) [5] functional described within the generalized gradient approximation (GGA) [6]. Electronic convergence tolerance of 1E-05 is used for all the calculations. The core electrons are treated using pseudo-potential with a core radius of 2.3 Bohrs (1.22 Angstrom) generated with a panel of 8 valence electrons ( 3d6 4s2). All the calculations are performed as spin-polarized calculations.
Cut-off Energy and K-Point Optimization
Fig. 1: A 3×3 Fe 100 surface with 4 layers.
The essential step before performing a plane-wave basis set calculation is to optimize the k-points and cut-off energy. This process is performed as follows:
K-point
Table 1: K-point grid vs the number of K-points used
| K-point grid | K-points used |
|---|---|
| 3x3x1 | 5 |
| 4x4x1 | 8 |
| 5x5x1 | 13 |
| 6x6x1 | 18 |
| 7x7x1 | 25 |
| 8x8x1 | 32 |
The numerically determined optimal lattice parameter for Fe bcc crystal structure is 2.835 Angstrom which agrees reasonably with the experimental value of 2.856 Angstrom[7]. The K-point optimization is performed at this lattice parameter with a 3×3 Fe (100) surface with 4 atomic layers containing 36 Fe atoms in the slab (see Fig 1).
Fig.2 k-point convergence. The Y-axis is the energy difference between successive calculations. The red line represents the convergence criteria.
Fig.2 shows the absolute change in the consecutive total energy (|ΔE|) with respect to the number of irreducible k-points. The corresponding K-point grid is shown in Table 1. As a convergence criterion, |ΔE| less than 0.003 eV is used for the K-point convergence. This corresponds to the optimal grid of 7x7x1.
Cut-off energy
Fig.3 Cut-off energy convergence. The Y-axis is the energy difference between successive calculations. The red line represents the convergence criteria.
In Fig. 3, the absolute change in the consecutive total energy (|ΔE|) is plotted as a function of the cut-off energy. As a convergence criterion, it is assumed that the total energy is converged if |ΔE| is less than 0.003 eV. From Figs. 3 and 4, it is clear that the cut-off energy of 550 eV sufficient. A lattice parameter of 2.835 Angstrom and K-point grid of 7x7x1 is used for this convergence.
Reaction Energetics
The most stable adsorption sites for atomic carbon as well as carbon dimer are determined using the same slab to determine the reaction energy. Table 2 lists the relative energies for these structures.
Table 2: Relative Energies for different C and CC adsorption sites on Fe 100 surface
| C-adsorption | Top | Bridge | Hollow | CC-adsorption | Vertical | Horizontal |
|---|---|---|---|---|---|---|
| Relative Energy (eV) | 2.9 | 1.9 | 0.0 | 1.2 | 0.0 |
To locate the transition state, a series of 4 linearly interpolated images were formed between the reactant and the product state. The nudged elastic band method (NEB) [8] is used to locate the transition state. Once an approximate transition state is obtained, it is further refined using the climbing image nudged elastic band method (CI-NEB) [9]. The intermediate state is assumed to reach the transition state when the tangential force on the highest energy image is less than 0.05 eV/Angstrom. Vibrational frequency calculations are then performed using IBRION = 5 in VASP to confirm that the single imaginary frequency is obtained along the reaction pathway.
Results
Figure 4: The reaction energetics of C + C -> CC on Fe 100 surface and its comparison with values from Yin et al.[1]. The grey colored spheres represent carbon atoms and the orange-colored ones represent iron atoms.
Additionally, similar calculations are also performed with 6 and 8 atomic layers to asses the effect of the number of layers on the reaction energetics. Table 3 lists the reaction energies as well as barrier heights for the different number of slabs. It shows that the effect of including twice the atomic layers results in the change of energies within 0.1 eV at the cost of a significant increase in the computation time. In theory, including more number of layers would make the system closer to the real conditions. However, the energy difference between 6 and 8 layer reaction calculations is already 0.03 eV, which is just ~2% of the reaction energy. It can be concluded that for similar calculations, using 4 atomic layers should be sufficient to be within ~10% of the converged energy value with respect to the number of layers if the energies follow a similar trend as shown here. However, it should be noted that more calculations are required to confirm this conclusion with even more atomic layers, which is beyond the scope of this post.
Table 3: Reaction energies and barrier heights for C+C -> CC reaction on Fe 100 surface with different number of atomic layers in the slab.
| Number of layers | Reaction energy (eV) | Barrier Energy (eV) |
|---|---|---|
| 4 | 1.63 | 2.14 |
| 6 | 1.69 | 2.19 |
| 8 | 1.72 | 2.22 |
Conclusions
In this post, the reaction energy and barrier are calculated for the C + C -> CC reaction on a Fe 100 surface. The transition state calculations performed using the CI-NEB[8] method implemented in VASP[2,3] shows a good agreement with the literature study[1]. The post also investigates the effect of including more atomic layers in the energy calculations. It can be concluded that adding more atomic layers for such calculations should be carefully decided based on the desired accuracy and available computational power.
References
[1] Yin, J., He, Y., Liu, X., Zhou, X., Huo, C.F., Guo, W., Peng, Q., Yang, Y., Jiao, H., Li, Y.W. and Wen, X.D., 2019. Visiting CH4 formation and C1+ C1 couplings to tune CH4 selectivity on Fe surfaces. Journal of Catalysis, 372, pp.217-225.
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[6] J.P. Perdew, J.A. Chevary, S.H. Vosko, K.A. Jackson, M.R. Pederson, D.J. Singh, C. Fiolhais, Atoms, Molecules, Solids, And Surfaces – Applications of the Generalized Gradient Approximation for Exchange and Correlation, Phys. Rev. B, 46 (1992) 6671-6687.
[7] Greenwood, Norman Neill, and Alan Earnshaw. Chemistry of the Elements. Elsevier, 2012.
[8] G. Henkelman, H. Jónsson, Improved tangent estimate in the nudged elastic band method for finding minimum energy paths and saddle points, The Journal of Chemical Physics, 113 (2000) 9978-9985.
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