Dear Shruthi,

No, the energy convergence with respect to k-point sampling is not monotonic, in fact it is typically oscillatory. The reason is that the discrete k-point sampling is approximating an integral over the Brillouin zone; for example the energy of a particular Kohn-Sham state is strictly:

E_b = \int_{BZ} \epsilon_b(k) d^3 k

i.e. you integrate the (continuous) band-energy, epsilon, over the 3D Brillouin zone, and that gives you the total energy of that state.

Since we don't know the analytic form of the band-energy across the Brillouin zone (that's what you're computing!) we have to do the integral numerically, and that means some kind of finite sampling. For some k-point meshes you are underestimating the integral, so the energy is too low, and for some k-point meshes you are overestimating the integral, so the energy is too high.

You are almost certainly using a Monkhorst-Pack (or Chadi-Cohen) grid for the sampling, and I would encourage you to look at the original papers where you can see exactly what they are trying to do. In essence, we can take inspiration from a tight-binding model, where the band-energies are cosinusoidal in k, and use that to generate an efficient sampling scheme. You may also find it useful to investigate the concept of the "Baldereschi point" -- the single best k-point to sample the band-energies at, and the optimal centre for a Monkhorst-Pack grid from a sampling perspective.

All the best,

Phil Hasnip

The energy will change as a function of the k-point sampling. So, as you increase the k-point sampling you will reach a point where the energy does not change, i.e. an horizontal line, at this point you got the convergence. Take a look to the Payne's paper:

Article Iterative Minimization Techniques for Ab Initio Total-Energy...

Regards

Thank you Sinhue Lopez for the reference. I will go through it.

Actually the problem that I am facing is increase in energy at higher k meshes. Isn't it necessary to have a monotonic decrease in energy? Of course the energies are getting saturated at higher k meshes but at higher energy.

e.g. 1x1x1 : -16.83 eV

2x2x2 : -12.09 eV

3x3x3 : -12.02 eV

4x4x4 : -12.07 eV

5x5x5 : -12.09 eV

6x6x6 : -12.087 eV

7x7x7 : -12.087 eV

Is this trend in energies accepted?

Dear Shruthi,

No, the energy convergence with respect to k-point sampling is not monotonic, in fact it is typically oscillatory. The reason is that the discrete k-point sampling is approximating an integral over the Brillouin zone; for example the energy of a particular Kohn-Sham state is strictly:

E_b = \int_{BZ} \epsilon_b(k) d^3 k

i.e. you integrate the (continuous) band-energy, epsilon, over the 3D Brillouin zone, and that gives you the total energy of that state.

Since we don't know the analytic form of the band-energy across the Brillouin zone (that's what you're computing!) we have to do the integral numerically, and that means some kind of finite sampling. For some k-point meshes you are underestimating the integral, so the energy is too low, and for some k-point meshes you are overestimating the integral, so the energy is too high.

You are almost certainly using a Monkhorst-Pack (or Chadi-Cohen) grid for the sampling, and I would encourage you to look at the original papers where you can see exactly what they are trying to do. In essence, we can take inspiration from a tight-binding model, where the band-energies are cosinusoidal in k, and use that to generate an efficient sampling scheme. You may also find it useful to investigate the concept of the "Baldereschi point" -- the single best k-point to sample the band-energies at, and the optimal centre for a Monkhorst-Pack grid from a sampling perspective.

All the best,

Phil Hasnip

Thank you Philip James for making that point clear. So I hope that is not a bug in my calculations. I will go through the original paper by Monkhorst-Pack ; I'm using MP scheme for k point sampling.

But the TOTEN in the OUTCAR file is the free energy of the system rather than the band energies, isn't it?

Dear Shruthi,

I agree with Philip, convergence is usually oscillating, sometimes decreases a lot for 2x2x2 and then increases until asymptotic convergence.

I would also recomend David Sholl's book about DFT (https://onlinelibrary.wiley.com/doi/book/10.1002/9780470447710). You will find why, depending on the symmetry of the crystal (irreducible k-points), increasing the mesh by one (3x3x3 to 4x4x4 for example) will make no difference (same energy). That is if you use ISYM=1, to use the crystal symmetries to avoid equivalent k-points calculations in the Monkhorst-Pack scheme.

TOTEN ("free energy TOTEN" or "ion-electron TOTEN" in the OUTCAR, or "F= " in the standard output) is the total ion-electron free energy of your crystal at T=0, but its meaning depends on the ensemble (canonical, microcanonical) for molecular dynamics. You have ETOTAL= TOTEN+EKIN+ES+EPS, which are different energies available that are different from the "energy without entropy", the actual potential energy of the system (you have to add the kinetic energy of the ions separately in MD). The bands energies and occupations are listed in the table entitled " band No. band energies occupation" in the OUTCAR.

Felipe.