# How to calculate the energy of a surface?

I want to study the interaction of a surface with some molecules (green house effect gases).

To obtain the binding energy, the procedure is to calculate the energy of both systems separately $$E_\mathrm{surface}$$ and $$E_\mathrm{gas}$$, and to calculate the energy of the complex $$E_\mathrm{surface+gas}$$. Then, the binding energy can be calculated as:

$$E_\mathrm{bind} = E_\mathrm{surface+gas} - E_\mathrm{surface} - E_\mathrm{gas}$$

My question is: how to calculate the energy of a surface for the terms $$E_\mathrm{surface}$$ and $$E_\mathrm{surface+gas}$$?

• If there's already a supercell to calculate $E_{surface+gas}$, $E_{surface}$ is result of the calculation with the supercell of just the surface. The first calculation needs a slab model for the surface along with the gas, while the second one only needs the model for the surface. May 15, 2020 at 6:14
• @Mythreyi, sure, but how to calculate the energy surface? Just like any other system? Or I have to do in a different way?
– Camps
May 15, 2020 at 13:52
• Yes, and no. The process is different because we usually get the surface energy from two separate calculations: one with a slab model of the surface (which will have two exposed "surfaces" due to periodic boundary conditions) and one calculation for getting bulk energy per atom or formula unit. I will try to add an answer elaborating this. Meanwhile, this book has a good introductory chapter on surface calculations. May 15, 2020 at 18:46

For calculating the energy of a surface, we need to split the contribution of the surface and the contribution of the bulk to the total energy.

$$E_\mathrm{total} = E_\mathrm{surface} + E_\mathrm{bulk}$$

The term $$E_\mathrm{total}$$ is obtained from a calculation on a simulation cell that models the surface (let this be Calculation 1). The term $$E_\mathrm{bulk}$$ is obtained from a calculation that only contains the material in bulk (let this be Calculation 2).

Since the simulation cells are different, we must normalise the contribution of the energy. This is usually done by dividing the calculated energy by the number of atoms (or formula units) in the simulation cell. If $$n_2$$ is the number of atoms (or formula units) in the simulation cell in Calculation 2, the energy contribution per atom (or formula unit) be $$E'_\mathrm{bulk} = \frac{E_\mathrm{bulk}}{n_2}$$

Now, we can subtract the contribution of the bulk and obtain the surface contribution. If $$n_1$$ is the number of atoms (or formula units) in the simulation cell in Calculation 1 (see point 1 below), the surface contribution will be: $$E_\mathrm{surface} = E_\mathrm{total} - n_1 E'_\mathrm{bulk}$$

Surface energy is also usually normalised with the area. Hence, if $$A$$ is the area of the surface (see point 2 below), the final expression will become $$E'_\mathrm{surface} = \frac{1}{A} \left( E_\mathrm{total} - n_1 E'_\mathrm{bulk} \right)$$

Points to note:

1. Even though some of the atoms in Calculation 1 are part of the "surface", we must still include them in $$n_1$$ while calculating the bulk contribution because the surface energy is defined as the excess energy due to a surface. So the energy in the bulk is simply the reference energy using which we calculate the contribution of the surface.
2. When slab models are used to model the surface, and if the energy calculations use periodic boundary conditions, then the contribution to area $$A$$ will actually be 2 times the area of one surface.
3. The only input we need here is the energy, so irrespective of what atomistic modelling method we use to calculate the energy, this procedure can be used to get the surface energy.
• How do you calculate the formation energy of an impurity in the bulk with respect to the slab? Aug 18, 2023 at 15:44
• @NanaKofiBoakye: This question doesn't deal with impurities, only surfaces. Can you ask a separate new question specifying what you'd like to calculate? It's not clear from your comment as to what you're interested in. Aug 18, 2023 at 17:04
• I want to calculate the segregation energy of two different impurity atoms from the bulk to the surface. Any formula for such a calculation? Aug 18, 2023 at 17:24