[Pw_forum] On the argument charge for l or d orbital of pseudo potential

[yukihiro_okuno at fujifilm.co.jp]

Dear PWSCF users and developer.

I have primitive question of the pseudo charge density.

I'm now tring to construct the atomic pseudo charge density of each orbit,

 calculated the pseudo charge like

n(r) = |psi(r)|^2 + \sum_{i,j}<psi|beta_{i}>Q_{i,j}(r)<beta_{j}|psi>.

where psi is pseudo wave function defined in UPF file.

And I found the strange results which come from the argument

charge Q_{i,j},  in the p (l=1) or d(l=1) pseudo wave function

case.

There are  charge density peak around  r=0 ( peak at r=0 ) from the
argument

charge Q_{i,j}.  I think p wave and d wave wave function must have node at

r = 0, but the charge density from Q has peak around r = 0.

We can see this fact from the values of UPF file simply.

For example Zn.pz-van_ak.UPF file it was downloaded from PWSCF site,

The d wave  pseudo wave function psi  has angular momentum l = 2 and

the arguement wave function Q_{ij}   only comes frol l(i)=l(j)=2 cases (
beta's angular

momentum l = 2)  case and,   the relevant Q's index  values in UPF file
with l(i)=l(j)=2 are

 i, j = 5, or 6 .

And at  the radical point near the origin

 r(2) =   1.4123607*10^{-6}  ( very near the r(1)=0 ),

and  the relevant values Q_{i,j} are in UPF file

r^2Q_{5,5}(r) = 2.983286310*10^{-10} ,

r^{2}Q_{6,6}=3.54725669 * 10^{-10}

r^2Q_{6,5} = -3.2531434588 * 10^{-10}.

(The Q function in UPF file is difened as r^{2}Q{r},  multipled by r^{2} ,
I had cheked this

definition in the PWSCF source code and former mail posting

http://www.democritos.it/pipermail/pw_forum/2008-November/010758.html
http://www.democritos.it/pipermail/pw_forum/2009-January/011188.html

)

And we can see, for example, The argument charge density Q_arg around r(2),
are

 Q_{arg} = r^{2{Q_{5,5}/(4.0*pi*r^{2}) = 11.9.....  and Q_{5,6},  and
Q_{6,6} are same order,

and these values are not cancelled by the factors, <psi| beta> .

Same problem occurs in  O.pz-rrkjus.UPF  file with pseudo wave function

with l=1, I had cheked many times also checked in the Vanderbilt's pseudo
potential

generation program.

Why the argument charge has its value around r (peak at the  r = 0) in the

case of l=1, or l=2 case?


Sincerely.

Yukihiro Okuno.