Semi-infinite surfaces for adsorption: a valid approach?

Question

I'm currently trying to surface reaction of small molecules on metal oxides in VASP. Several papers I've read have approached surface energy calculations in a variety of ways. I first began looking at how surface energies are obtained without any adsorbates considered. This seemed like a logical starting point since I assumed the surface energy would be calculated via $E(\text{Surface+Adsorbate})-(E(\text{Surface})+E(\text{Adsorbate}))$, so I would need to find the clean surface energy to start with.

The best approach to reach a converged seems to have received some debate. Fiorentini and Methfessel [1] finds that the widely used expression:

$$\sigma=\lim_{N\to\infty}\frac{1}{2}(E_\text{slab}^N-NE_\text{bulk})\tag{1}$$

is poor at reaching a stable converged surface energy, where N represents the number of slab layers. Instead they find the expression (linear fit to slab energies):

$$E_\text{slab}^N\approx2\sigma+NE_\text{bulk}\tag{2}$$

reaches a stable convergence. A later study [2] finds the first expression to be adequate at reaching surface energies only when large enough k-point set is used.

When it comes to adsorption energies on surfaces I mostly find that researchers [3] [4] approach the calculations using a semi-infinite slab where the top layers are allowed to relax and an arbitrary 1 or 2 layers are frozen below. However, I've yet to find a critical evaluation of such approach. Is it worth performing a series of convergence tests on the number of layers frozen, as well as the number of layers themselves? I can imagine this would become quite time consuming.

Alternatively, I have seen others suggest that a better approach is using a symmetrical slab model. That is, putting the same adsorbate on the "bottom" side of the slab in exactly the same geometry as the top. Again, I would like to hear people's thoughts on this choice of method, and whether this approach has more 'validity' than the semi-infinite approach. Any paper recommendations welcome to, I found the Fiorentini and Methfessel paper in a discussion on the VASP forum.

1. Fiorentini, V., & Methfessel, M. (1996). Extracting convergent surface energies from slab calculations. Journal of Physics Condensed Matter, 8(36), 6525–6529.

2. Da Silva, J. L. F., Stampfl, C., & Scheffler, M. (2006). Converged properties of clean metal surfaces by all-electron first-principles calculations. Surface Science, 600(3), 703–715.

3. Lischka, M., & Groß, A. (2003). Hydrogen on palladium: a model system for the interaction of atoms and molecules with metal surfaces. Recent Developments in Vacuum Science and Technology, 661(2), 111–132.

4. Mamun, O., Winther, K. T., Boes, J. R., & Bligaard, T. (2019). High-throughput calculations of catalytic properties of bimetallic alloy surfaces. Scientific Data, 6(1), 1–9.

Answer 1

Is it worth performing a series of convergence tests on the number of layers frozen, as well as the number of layers themselves? I can imagine this would become quite time consuming.

Yes, it is generally considered a valid approach, and if you are new to a given material, you should be running such convergence tests both with regards to the number of layers in the slab and the number of layers allowed to move. Of course, you have to keep some layers fixed otherwise it will not be representative of a bulk surface. At the same time, it can be too rigid if you only let the top layer move. For people that work in this field, they often have a good handle on how many layers to use based on prior work, so you will likely not see a convergence test in every paper. However, it is best practice. As a side-note, for high-throughput calculations (such as the Sci. Data paper you mentioned), there is a more significant balance that needs to be had with regards to the accuracy-cost tradeoff since they are purposefully trying to study as many materials as possible.

Alternatively, I have seen others suggest that a better approach is using a symmetrical slab model. That is, putting the same adsorbate on the "bottom" side of the slab in exactly the same geometry as the top. Again, I would like to hear people's thoughts on this choice of method, and whether this approach has more 'validity' than the semi-infinite approach.

Generally, the reason for doing this is to avoid the presence of a fictitious dipole moment in your slab. It is a separate issue from the number of layers question you asked. My view is that this is a bit of an archaic route. In VASP and other codes, you can use a dipole correction to offset any dipole present in an asymmetrical slab model. Refer to the IDIPOL flag for more details.

In general, the answers to many of your questions are discussed in the "Modeling materials using density functional theory" ebook by Kitchin, found here.

Answer 2

If you need an adsorption energy, then Andrew Rosen's answer gives good points. Symmetric adsorbate deposition will require about twice thicker slab, than single sided adsorption. So it is not really good idea today. Additionaly it is worth noting that you need to converge not only with slab width, but also with slab size along surface dimensions.

You also mentioned about surface energy. It is completely different thing. You do not need to reach convergence of surface energy if you are interested in adsorption energy only. Error cancellation can work on you here.