# #1 Convergence Test: Cutoff Energy Optimization for Silicon Bulk

Viewed : 2102 times,  2019-01-01 00:00:00

Convergence test is a way of optimizing the input script for a simulation to use limited computational resources efficiently. It is essential to find a proper initial setting for research to decide the accuracy and reliability of the simulation.

Density Functional Theory (DFT) obtains a result by using an iterative method, the self-consistent field (SCF) method. SCF determines ground-state electron density by calculating the energy difference with slightly changing the electron density. If the energy reaches the global minimum with achieves convergence, the calculation is finished. The convergence means the difference between the previous step and the energy is within less than a specific value (convergence threshold).[1]

Computers determine whether convergence is achieved using the set threshold. Thus, setting smaller threshold values produces more accurate results. However, an accurate input setting takes a longer time to calculate. Therefore, a convergence test is required to find a setting optimized for a model.

There are many convergence test methods. We would like to introduce two methods for two weeks. The first one is to check the energy convergence by changing the cutoff energy, and the second one is to check the energy by changing k-points.

1. How to set the cutoff energy

Add the Quantum Espersso module and connect with the Structure Builder. For the detail, please refer to the following documentation page.

👉[Docs] Work | To connect modules

In the Quantum Espresso module, you can find the Ecut(wfc) and Ecut(rho) two keywords in relation to cutoff energy, under the &SYSTEM namelist.

Ecut(wfc) indicates the kinetic energy cutoff of wave functions, and Ecut(rho) means that of charge density (Unit: Ry) [2].

Ecut keyword is to set the number of plane waves used to calculate the DFT. If the value is high for the calculation of plane waves, the accuracy of the calculation is improved, but it takes a longer time to calculate it. Thus, setting the maximum kinetic energy cutoff to adjust the number of plane waves accurately finds the balance point between the number of plane waves and the calculation time [3].

In theory, the charge density of the norm-conserving pseudopotential is proportional to the square of the wave function, and the charge density cutoff vector needs twice the wave function cutoff vector for an accurate calculation in the Fourier space. Thus, the Ecut(rho) should be exactly four times higher than the Ecut(wfc) [4]. However, in the case of ultrasoft or PAW-based pseudopotential, it is recommended to set the Ecut(rho) to six to eight times higher than the Ecut(wfc) considering the augmentation charge [5]. The following example is tested with Ecut(rho) = 6 × Ecut(wfc).

2. Convergence Test

👉 Check the Calculation Results

It is easy to compare the energy change if you add the data set to the Energy module. As shown in the graph, the higher the Ecut(wfc), the smaller the energy difference is, resulting in convergence. It may be proper to set the Ecut(wfc) to 40–50 Ry with an appropriately small energy fluctuation to economize computational resources. Occasionally, some pseudopotential authors write 'suggested cutoff energy' when creating their pseudopotential files. By referring to this value, the cutoff energy can be selected.

3. How to determine cutoff energy based on the Pseudopotential

You can check the input/output file on the 'Data' page after starting the calculation. In here, the pseudopotential file (*.UPF) is the data which approximates the potential of the element, contains useful information for simulation. For the detail of pseudopotential, please refer to Weekly tip #19.

Some pseudopotentail file has the 'Suggested cutoff energy' information. This is what the author who created the potential wrote down the appropriate cutoff energy when performing the calculation using the pseudopotential file. It is useful to determine the cutoff energy.

The Si pseudopotential file used in this example recommends setting Ecut(wfc) to at least 44 Ry. This is similar to the conclusion Ecut(wfc)=40~50 Ry, figured from the example result. Therefore, in the convergence test, the convergence test can be performed to a minimum by selecting a value around the 'Suggested cutoff energy' value.

In this weekly tip, we have learned about the importance of the convergence test and how to perform the convergence test. To obtain more reliable results, it is recommended to check whether the energy has completely converged through a convergence test.

Start a simulation of interesting systems with the convergence test, in MatSQ!

[1] Slater, J. C. (1951). A simplification of the Hartree-Fock method. Physical review, 81(3), 385.
[2] https://www.quantum-espresso.org/Doc/INPUT_PW.html
[3] Payne, M. C., Teter, M. P., Allan, D. C., Arias, T. A., & Joannopoulos, A. J. (1992). Iterative minimization techniques for ab initio total-energy calculations: molecular dynamics and conjugate gradients. Reviews of modern physics, 64(4), 1052.
[4] Kohanoff, J. (2006). Electronic structure calculations for solids and molecules: theory and computational methods. Cambridge University Press.
[5] Kresse, G., & Joubert, D. (1999). From ultrasoft pseudopotentials to the projector augmented-wave method. Physical Review B, 59(3), 1758.; Fujiwara, T., & Hoshi, T. (1997). Optimized ultrasoft pseudopotentials. Journal of the Physical Society of Japan, 66(6), 1723-1729.