The origin of this difference is very simple. The gaussian basis set has the tendency to overlocalised the electronic cloud around the nuclei due to their mathematical description which centre the atomic basis set around one defined atom; however the plane wave have the tendency to over delocalise the electronic cloud (originally they are used to describe metallic system).

You must also add the fact that with the gaussian basis set you do not used the spilling parameter it is a direct projection on the atomic orbitals; however with the plane wave, the spilling paraemeter is a sort of the cutoff radius to do some projection on s, p, d types orbitals that's why you do not obtain a sum equal to zero.

The consequence is that for MgO which is an ionic crystal the lowding charge with the gaussion basis set is Mg2+ O2- and with a plane wave basis set the lowding charge can tend to Mg+1 O-1. There is a Phys. Rev. B published on this topic cited in the references of Quantum Espresso on the implementation of the Lowding charges in QE and the explanation of the influence of the spilling parameter.

When plane waves are used molecular or crystal orbitals are linear combinations of extended plane waves and not possible to separate out atomic contributions. This problem does not exist when atom-centered gaussian basis is used (g09). In order to get Lowdin charges in QE the molecular or crystal orbitals are projected on the atomic centered pseudo-atomic wavefunctions defined by the pseudopotential used. However, this basis set is "less complete" than the plane wave basis and there is always a missing charge when you project, i.e. parts of the plane waves that cannot be expressed in terms of the atom centered basis. Thats why spilling parameter, measuring exactly the loss in electron charge due to the projection, is not zero.

According to QE manual you can look at Sanchez-Portal et al., Sol. State Commun. 95, 685 (1995).

Dear Philippe and Nektarios

Thanks for your consideration and references.

Everything is right, I mean the pevious two responses. When you are in calculation, follow the one method you have chosen - plane set or localised set basis set. Otherwise, you'll get condrarictory results.

Using plane waves you can not really project eigenfunctions and therefore charge on a given atom. The projection is only approximate and can be fine-tuned using the "atomic" radius parameter. That is why you can get charge spilling and non-conservation. For localized orbitals, since they can well be assigned to a given atom, it is less problematic projecting and getting the charge on that atom, and the way it is setup, you always get charge conservation.

Given the small numbers you provided for charge spilling, the PW calculation is giving reasonable results. You can expect this from any PW (or non LCAO) calculation.

You can also use a localized orbital basis set code to get the Löwdin charges, it is more intuitive this way since you work directlyu with Lowdin charges in this case.

Dear Professor Esfarjani , Prof. Dappe and Prof. Tulub,

Thanks for your time and instructive comments.

The real reason for different numbers is that partial atomic charges are NOT observable. So, the values depend upon the method used for their definition. It is impossible to tell which method gives better result. You can only compare the numbers obtained by THE SAME method.

Dear Jeremy,

all of my results obtained with DFT.

Thanks Dear Jeremy,

OK I checked these references,

I will say you about my results and hope you have enough time for further argument.

thanks again