DFT calculations of cohesive energies

Introduction

Material properties very often depend directly or indirectly on the structure of a substance. It is therefore extremely important to know the structure and to understand why a particular structure is adopted. For solids, the cohesive energy can be very instructive in this context. It is the energy which is needed to separate a solid into isolated, neutral atoms. Cohesive energies of solid elements typically range between 1 and 10 eV/atom except for the noble gases which have smaller cohesive energies [1]. The cohesive energy represents a measure for the bonding strength in a solid and can be used, for instance, to identify the equilibrium structure of a crystal at low temperature, since the structure with the highest cohesive energy will be adopted.
 
In this case study, DFT (density functional theory) calculations on the cohesive energy of magnesium (Mg) will be presented. The calculations have been performed using the ABINIT [2] module of Scienomics MAPS platform [3]. In the first part, we will demonstrate the validation of the parameter set for the DFT calculations. In the second part, the results of the cohesive energy calculations for the experimental and geometry optimized structure of the Mg unit cell will be discussed.
 
Convergence of energy cutoff and the number of k-points

In electronic structures calculations of solids, the energy cutoff and the number of k-points in the Brillouin zone are two important parameters which have to be selected very carefully. Best practices in this field usually demand a convergence study of the quantity of interest with respect to these parameters prior to production runs. Therefore, the convergence of the total energy with respect to the energy cutoff and the number of k-points has been checked. For this purpose, the total energy of the Mg unit cell has been calculated and the energy cutoff and number of k-points, respectively, have been systematically increased. The generalized gradient approximation (GGA) was used together with the Perdew-Burke-Ernzerhof (PBE) density functional and the respective PAW (Projector Augmented-Wave) pseudopotential generated by AtomPAW [4] as shipped with MAPS/ABINIT. The structure of the Mg unit cell has been taken from MAPS structure library. The cell lengths of this structure are a = 3.201 Å and c = 5.210 Å which corresponds to the experimental values.
 
The results of the convergence study are illustrated in Figure 1 and 2. Figure 1 shows the convergence of the total energy with respect to the energy cutoff. Convergence is reached with an energy cutoff of 1000 eV providing an accuracy of the absolute energy of better than 0.001 eV (0.1 kJ/mol).
 

c16-f1-mg-convergence-in-ecutoff
Figure 1: Convergence of the total energy of Mg with respect to the energy cutoff. The convergence behavior of the total energy with respect to the number of k-points is depicted in Figure 2. An accuracy of the absolute energy of better than 0.001 eV (0.1 kJ/mol) is achieved when using a 8x8x8 k-point mesh.

c16-f2-convergence-in-k-points
Figure 2: Convergence of the total energy of Mg with respect to the number of k-points.

Calculation of the cohesive energy

For calculating the cohesive energy, the total energy of the bulk and of a free atom is required. The total energy of the bulk has been calculated using the GGA approach together with the GGA-PAW pseudopotential generated by Holzwarth using AtomPAW [4] as shipped with MAPS/ABINIT. Based on the above discussed convergence study, the energy cutoff has been set to 1000 eV and a 8x8x8 k-point grid was used for Brillouin zone integration.
 
In order to calculated the energy of the isolated Mg atom, MAPS crystal builder (see Figure 3) has been used to place a single Mg atom into a cubic box with a cell length of a = b = c = 10 Å.
 

Figure 3: MAPS interface showing the unit cells of the bulk and the isolated atom, respectively, and MAPS Crystal builder. The total energy of the free atom has then been obtained from a calculation at the G point using the same energy cutoff as chosen for the bulk (1000 eV).
Figure 3: MAPS interface showing the unit cells of the bulk and the isolated atom, respectively, and MAPS Crystal builder.
The total energy of the free atom has then been obtained from a calculation at the G point using the same energy cutoff as chosen for the bulk (1000 eV).

The cohesive energy Ecoh is defined as:

Ecoh = (S Eatom - Ebulk) / N

with Ebulk as the total energy of the bulk, Eatom as the total energy of the atom, and N as the number of atoms in the units cell which is in the present case 2. With this, the cohesive energy of Mg is obtained as 1.49 eV. This value is in good agreement with the experimental value of 1.51 eV [1] and calculations of Da Silva at al. [5].
 
For comparison, the cohesive energy has also been calculated using the local density approximation (LDA) using the PW92 functional together with the corresponding PAW pseudopotential generated by Holzwarth using AtomPAW [4] as shipped with MAPS/ABINIT. The same parameters, i.e. an energy cutoff of 1000 eV and a 8x8x8 k-point grid for Brillouin zone integration, have been applied. Using the LDA approach, the cohesive energy is obtained as 1.77 eV and thus overestimated to some extent, which is, however, a commonly observed finding [5].
 
In addition, to illustrate the sensitivity of the cohesive energy on the structure, the Mg unit cell has been geometry optimized using the GGA approach and the PBE density functional. The cell parameters of the optimized structures are a = 3.347 Å and c = 5.435 Å, respectively, and are slightly larger compared to the experimental values, but the ratio c/a of the geometry optimized structure is with 1.624 the same as for the experimental structure. For the geometry optimized structure, a cohesive energy of 1.51 eV is obtained which is slightly larger compared with the value obtained for the experimental structure and is in excellent agreement with the experimental data. The results are summarized in Table 1.
 

Table 1: Cohesive energies in eV of Mg.
Table 1: Cohesive energies in eV of Mg.

Since Mg is a closed-shell system, the electronic energy of the Mg atom in the ground state can be obtained rather straightforwardly from standard DFT calculations. It should be stressed, however, that this task can be more challenging for open-shell systems, especially transition metals, rare earths, or actinides, which may demand a more advanced treatment such as the inclusion of spin-polarization effects.
 
Conclusion

We have presented DFT calculations on the cohesive energy of magnesium. The calculated cohesive energies are in very good agreement with literature data. They are useful for studying the binding strength in crystal structures and can help to gain information about structural preferences of solids.
 
References:

  1. C. Kittel, in: Introduction to Solid State Physics, 7th ed., John Wiley & Sons, Inc., 1996.
  2. X. Gonze et al., Computer Phys. Commun., 180, 2582-2615, 2009.
  3. MAPS, Version 3.4, Scienomics, Paris, France, 2014.
  4. N. Holzwarth et al., Computer Phys. Commun., 135, 329–347, 2001.
  5. MAPS, Version 3.3.2, Scienomics, Paris, France.
  6. J.L.F. Da Silva et al. Surface Science, 600, 703-715, 2006.