The iterative procedure to optimize the hexagonal structure

This procedure is taken from a tutorial using the Excinting code and didn't tested using the wien2k code.


In the next, we illustrate the iterative procedure for performing the optimization of the crystal structure of hexagonal Beryllium. The only relevant parameters in this case are the volume of the unit cell and the c/a ratio.

STEP1: Optimizing the volume

At the first step, we optimize the energy with respect to the volume. In order to generate input files for a series of volumes you have to use the script OPTIMIZE.job.


The ....... file, for instance, contains all final fit parameters and pressures which are calculated from equation of state fitting procedure. Also, the script generates a plot, PostScript (BM_eos.eps) or PNG (BM_eos.png) file, which looks like the following.

On this plot, you can also find the optimized values of the parameters appearing in the equation of state (minimum energy, equilibrium volume, bulk modulus, and bulk modulus pressure derivative).

Please note:

1. The bulk modulus and bulk modulus pressure derivative which are derived here have to be interpreted only as fitting parameters. This is due to the fact that in this tutorial we are changing the volume, however, we are keeping all other lattice parameter constant. In order to obtain the real physical bulk modulus and bulk modulus pressure derivative, one has to fully optimize at each given volume with respect to all other lattice and internal parameters (e.g., in the case of hexagonal beryllium, with respect to the c/a ratio, too).
2. The visual analysis of the plots is very important. The user should always check them at each step. In particular, if the minimum lies ouside the displayed region, the calculation should be restarted with more appropriate values of the initial parameters (e.g., volume or c/a ratio). The optimal situation is when the minimum of the energy curves is located in the middle of the investigated region.
3. If the difference in energy between the calculated points and the fit is larger than the final required accuracy in the energy, the calculation should be restarted with more appropriate computational parameters. In particular, one should consider the number of k points , the value of RKmax, and the accuracy in the calculated total energy .

At this point, you have performed the first optimization step by varying only the volume. In order to be prepared for the next step, you should create a struct file with the new optimized value. This file will be used as the input file in the next step.

STEP2: Optimizing the c/a ratio

In order to perform the next optimization step, you have to run again OPTIMIZE.job with the c/a optimization.

In this case, the optimization is performed using a fourth-order polynomial fit for calculating the minimum energy.


At this point, you have completed the second optimization step (by varying only the c/a ratio). Similar to the previous step, you should create a struct file using the optimized value of c/a.  

STEP3: Optimizing again the volume

Notice: Due to the fact that the volume and c/a ratio have been already optimized once, we can choose smaller range of distortion for the next steps.


you will get the following plot.




At this point, you have optimized the volume for the second time. You have the same thing as in the first step and create the struct file for the next step.

STEP4: Optimizing again the c/a   ratio

Proceed in a similar way to to STEP2.  


At this point, the second optimization of the c/a ratio has been completed and the new optimized structure has been saved in tmp file.

Reaching convergence

In the following table you find a summary of the result of the first 4 optimization steps.

Step Vmin [Bohr3] (c/a)min Emin [Ha] 0 103.2857 1.50000 -29.3607371 1 106.5871 " -29.3609600 2 " 1.52219 -29.3610633 3 106.3591 " -29.3610639 4 " 1.52256 -29.3610642

As you can see from the previous table, at the 4-th iteration you reached the following convergence.

• The equilibrium volume is converged within 2×10^-1 Bohr³.
• The c/a ratio is converged within 4×10^-4.
• The energy at the minimum seems to be converged within 3×10^-4 mHa. Indeed, such a small value should be considered an artifact of the optimization procedure, which assumes that the calculated total energies are exact. However, the accuracy in the determination of the minimum energy cannot be smaller than the accuracy of the total energy in a single SCF calculation. For the calculations performed in this tutorial, total energies are calculated with the default value of the accuracy, i.e., 10^-3 mHa.

Note: 1 Hartree= 2 Rydenberg


Reference: http://exciting-code.org/carbon-general-lattice-optimization