[Pw_forum] Projection of plane wave functions into atomic orbitals

[Stefano de Gironcoli]

> I have a question concerning the problem of projection of PW wave 
> function
> on atomic orbitals.
> example 8 shows the case of bulk Ni with LDA pseudo pot NiUS.RRKJ3.UPF
> which uses two wave functions s and d (see below)
>
>     2    6             Number of Wavefunctions, Number of Projectors
>  Wavefunctions         nl  l   occ
>                        4S  0  1.00
>                        3D  2  9.00
>
> The result of the projection with projwfc.x gives the following 
> decomposition:
>
>  Atom #   1: total charge =   9.8849, s, p, d, f =   0.9347  0.0000  
> 8.9501
>      Spilling Parameter:   0.0115
>
> My question is the following: where have the p electrons gone?


The code projects the calculated wavefunctions on the atomic 
wavefunctions that are present in the pseudopotential file. In this case 
the 4s and 3d.
There is no 4p wavefunction in the pseudopontential and there will not 
be any 4p projected density of states in the projected DOS.
 
The fact that the spilling parameter is not that bad indicates that 
these wavefuntions are not really needed in order to describe the 
occupied manyfold.
If some important atomic wavefunction were missing the spilling 
parameter would be much worse.

It is possible that if the 4p wavefunctions were present in the 
pseudopotential somewhat different  projected densities of states would 
result... (together with  a smaller spilling parameter).
This is because the 4p states of an atom are NOT ORTHOGONAL to the 4s/3d 
states of neighboring atoms so that part of the DOS that would be 
interpreted as 4p when this state is present are interpreted as 4s of 3d 
if it is absent...
This would imply that some redundancy (overcompleteness) is present in 
the atomic-wavefunction basis-set and the choice operated by the code 
(orthogonalizing the atomic wavefunctions before making the projection 
ro get the so called Lowdin projections) is only one of the infinite 
(all arbitrary) ways this redundancy is removed.

Stefano de Gironcoli