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#3-1 How to Calculate Cohesive Energy

Viewed : 3911 times,  2019-01-01 02:00:00
The energy for bonding can be calculated using the difference between a bonded structure and a dissociated one. This energy, which is necessary to separate an atom from the solid, is called cohesive energy. [1]

The cohesive energy can be used for determining thermodynamic stability of the material.[2]

It can be calculated by getting the total energy of the bulk structure and the single atom, and their difference with the plane-wave DFT.



1. Calculation of DFT Energy for Silicon bulk

First, you should prepare silicon bulk model for DFT energy calculation.

We can simulate bulk structure by modeling small unit with Periodic Boundary Condition (PBC).

We can find the conventional cell, the generally known shape, and the primitive cell (unit cell), the minimum unit of crystal, from silicon bulk. By performing DFT simulation with them, it is possible to calculate properties of large structure such as bulk crystal with a small amount of computation resource.

For easy modeling, Click 'PRESET' icon in the manipulator, and select 'Silicon (unit cell)'.

To perform DFT energy calculation, add the Quantum Espresso module and connect that to the structure builder module.

To use the previous setting which achieved convergence, set the scripting option to General, and set the input parameters. In the previous examples, Weekly tip #1 and Weekly tip #2, we had obtained the proper input parameter set of single crystal silicon: cutoff energy of 40–240 Ry, and 6x6x6 k-points grid.


Set the job name and click the 'Start Job!' button to start the calculation.

It takes about several tens seconds or minutes to make an SCF calculation for the structures having a good symmetry structure such as silicon single crystal.

The progress of the submitted job is displayed in the dashboard of the menu bar on the upper section.

Once the calculation is complete, come back to the Quantum Espresso module and click the 'Update' button or refresh the page to check whether the end message is normally displayed.


Once the green finish message is displayed, the calculation job is has been finished normally.

Add the energy module and connect to the Quantum Espresso module to check the final energy value. The silicon conventional cell, the calculation model, is consisted of 8 atoms. Therefore, this energy is for eight atoms.

👉 Check the Calculation Results



2. Calculation of DFT Energy for Silicon Single Atom

Next, we need to perform DFT calculation for a single atom to calculate the cohesive energy. As like bulk calculation, you should prepare the structure to calculate for the single atom.

You can add a new structure builder, but another structure can be added by opening the Structure List ( button) on the upper right of the visualizer of the existing structure builder. Click the green + button to add a new structure, and create a cell with only one atom in the crystal builder menu.



Click the 'CRYSTAL' icon to select the crystal builder menu. Make the single atom cell with referring the following inputs.

👉 Check the Calculation Results

It is important to set the length of the cell to a large value.

In the periodic-boundary condition, it is assumed that the structure seen in the structure builder is repeated infinitely.

Thus, an empty space (vacuum) should be created to separate the repeating nearby atoms and prevent them from interacting with each other.

If there is much vacuum, the interaction between atoms decreases, but the calculation time increases. Thus, it tends to set the vacuum width to as much as 10–15 Å. This example uses 12 Å.


For the energy calculation, add a new Quantum Espresso module and connect that to the structure builder and set the input script.
For cutoff energy, 40–240 Ry, which was used in the bulk calculation and be decided for a convergence setting in the Weekly tip #1, needs to be used.
However, as the single atom DFT energy is not related to k-points for the fully converged case, gamma (1x1x1 grid, the one k-point) can be set for a fast calculation.

Note that you need to considering spin polarization in the case of a single atom.

In general, every electrons is pair in the bulk material, so the Quantum Espresso assumes that the electron pair as one bundle. Therefore, for the single atom, which having a single electron due to the Hund's rule, the spin-up / down must be considered as a different state to obtain a correct result.

It can set from the 'Number of electron spin' tag.

Set the job name and start the job.

Once the calculation is complete, click the 'Update' button or refresh the page, and check if the calculation is normally finished. If so, add an energy module to check the energy.



3. Calculation of Cohesive Energy

In the memo module, you can use the cell reference and basic arithmetic operations, by writing directly the cell address. Add a memo module, and paste the final energy value of the silicon bulk and the atom to compare the data easily.

Calculate the energy for each atom, and find the difference to get the Ecoh/atom of the silicon.

👉 Check the Calculation Results

The cohesive energy calculated is 4.59 eV/atom, and the experimental value is 4.63 eV/atom.[3] This shows a difference of about 0.04 eV.

If the spin is not considered, the calculation will have much error. Thus, it is good to consider the spin polarization when calculating the energy of a single atom.



4. Perform Convergence Test with Cohesive Energy

In previous weekly tips, it mentioned that performing a convergence test using Ebond vs Ecut may be efficient for some structures.

Refer to the following post to check the convergence test result with cohesive energy.



5. Cohesive Energy for Heusler Alloy

Heusler alloy is having chemical formula of X2YZ. In this time, the cohesive energy can be calculated by the following equation.[2]



[1] Parrill A. L., Lipkowitz, K. B. (Eds.). (2016). Reviews in Computational Chemistry, Volume 29. 44-47, Wiley
[2] Khandy, S. A., Islam, I., Gupta, D. C., Khenata, R., & Laref, A. (2019). Lattice dynamics, mechanical stability and electronic structure of Fe-based Heusler semiconductors. Scientific reports, 9(1), 1-8.
[3] Yin, M. T., & Cohen, M. L. (1982). Theory of static structural properties, crystal stability, and phase transformations: Application to Si and Ge. Physical Review B, 26(10), 5668–5687. doi:10.1103/physrevb.26.5668