Dear Maximo,

In general, I would recommend the method of Cococcioni et al. but it requires your DFT program to apply their "Hubbard-alpha" potential. This is certainly possible in Quantum Espresso and there are on-line tutorials on how to do it, and I've recently implemented it in my own code (CASTEP) but I'm not sure about other DFT programs.

However you compute U and J (usually just U_eff = U-J) it is not a straightforward process because you have to converge your calculations very well with respect to cut-off energy, k-point sampling and supercell size. For Cococcioni's method you also have to perform very precise calculations because the U_eff (=U-J) is computed using numerical derivatives of your calculation result.

The value you get for U_eff will depend strongly on the implementation of DFT+U in the program; specifically, the basis set used to apply the Hubbard operator. Comparing results between different basis sets can easily give U_eff values which differ by a factor of 2 or more. For this reason you have to be careful when using values published in the scientific literature; ideally you would only use values computed with the same program you are using, but failing that you should at least use ones computed with the same U method and basis set.

The main aim of the Hubbard U is to correct for the effects of self-interaction, in particular that localised states (usually d- and f-states) are too close to the Fermi energy. With that in mind, it is usually sufficient to use a U_eff value which pushes them away from the Fermi level so that they do not hybridise incorrectly with the bonding states; the precisely "correct" value of U_eff is not necessary to capture this important effect. You should not fit to the band-gap of the material, as this will not be correct for the correct U_eff, and trying to get the right value will usually lead to an overly large U_eff and hence bands which are too flat. If you do want to fit something to get U_eff I expect the band-width would be a reasonable choice (though this does ignore the fictitious nature of the Kohn-Sham eigenvalues and associate them directly with electrons or quasiparticles).

Hope that helps.

Phil Hasnip

(CASTEP developer)

Dear Maximo,

In general, I would recommend the method of Cococcioni et al. but it requires your DFT program to apply their "Hubbard-alpha" potential. This is certainly possible in Quantum Espresso and there are on-line tutorials on how to do it, and I've recently implemented it in my own code (CASTEP) but I'm not sure about other DFT programs.

However you compute U and J (usually just U_eff = U-J) it is not a straightforward process because you have to converge your calculations very well with respect to cut-off energy, k-point sampling and supercell size. For Cococcioni's method you also have to perform very precise calculations because the U_eff (=U-J) is computed using numerical derivatives of your calculation result.

The value you get for U_eff will depend strongly on the implementation of DFT+U in the program; specifically, the basis set used to apply the Hubbard operator. Comparing results between different basis sets can easily give U_eff values which differ by a factor of 2 or more. For this reason you have to be careful when using values published in the scientific literature; ideally you would only use values computed with the same program you are using, but failing that you should at least use ones computed with the same U method and basis set.

The main aim of the Hubbard U is to correct for the effects of self-interaction, in particular that localised states (usually d- and f-states) are too close to the Fermi energy. With that in mind, it is usually sufficient to use a U_eff value which pushes them away from the Fermi level so that they do not hybridise incorrectly with the bonding states; the precisely "correct" value of U_eff is not necessary to capture this important effect. You should not fit to the band-gap of the material, as this will not be correct for the correct U_eff, and trying to get the right value will usually lead to an overly large U_eff and hence bands which are too flat. If you do want to fit something to get U_eff I expect the band-width would be a reasonable choice (though this does ignore the fictitious nature of the Kohn-Sham eigenvalues and associate them directly with electrons or quasiparticles).

Hope that helps.

Phil Hasnip

(CASTEP developer)

Dear Philip,

Thanks your answer is very complete. I forget mention that I using VASP. I'm not sure if in VASP is implemented Hubbard-alfa, but in most of publications based on this code mention that get U value from method of Cococcioni et al, that is based on Simplified (rotationally invariant) approach (Dudarev et al) method that is implemented in VASP.

This method used U_eff = U - J,  Cococcioni set J = 0 and some other use a constant value like J = 1eV. Those assumption not cause others problems?

I'll try to fit to band-width. Before fit to band-width, I need converge my calculations to cut-off energy, k-point sampling and supercell size, before, after or both? When you mention supercell size, you refer only to lattice constant, not ions and cells relaxation?

I read that I can fit to lattice constant, too. During this fit I need allow full relaxation, or only lattice change?

Dear Maximo,

I think most programs these days will use the method of Dudarev et al. as it simplifies everything, whilst retaining the essential physics.

What you compute is basically U_eff so it doesn't really matter whether (or how) you partition that into separate U and J components.

Fitting to any data has issues. The main things to bear in mind are:

1. How accurate are the values you're fitting to?(Are they high-level theoretical results, or experimental results?)

2. Would the true XC functional get this property right?

In other words, the key question is: "Is the discrepancy between DFT and the target value primarily due to spurious self-interaction of localised atomic states in the calculation?" If the answer is "yes" then DFT+U can correct it, and it might be a good candidate for fitting to; however if the answer is "no" then you cannot use it.

Now you could ask at this point whether the property I suggested (band-width) satisfies the criteria above -- strictly speaking the answer is "no", but because we know these states are in fact localised we can use it to approximate U_eff. More importantly, using too large a U_eff will over-localise the states and lead to an unphysical flattening of the appropriate bands which, unlike fitting to many other properties, will make the fit worse.

The convergence with supercell size is because in the Cococcioni method you vary the Hubbard alpha term for one particular atom in the simulation cell. Because this atom has an interaction with its periodic image, you need to do this in a large enough simulation cell that this periodic image interaction is negligible. This means you need a supercell for the Hubbard U calculation even for perfect bulk material, because the method breaks the equivalence between atoms of the same element; for the same reason, it also lowers the symmetry of your calculation so you'll often have more k-points than you usually expect.

Hope that helps,

Phil

Dear Philip,

I'm using the complete unit cell (56 atoms), should be use more large cell, like two unit cell (112) or more?

Based in your experience what you suggest about relaxation during fit U_eff value:

Relax before, after, before and after, or maybe also during U_eff calculation? or relax before, during and after?

You recommend allow full relaxation (ions, shape cell, volume cell)?

I appreciate your help!

Thanks again!