[Wien] real basis projection for f-orbitals

[Chang-Jong Kang]

Dear Prof. Blaha and Wien2k users,

I would like to decompose f-orbitals into real basis like fz3, fxz2, fyz2,
fxyz, fz(x2-y2), fx(x2-3y2), and fy(3x2-y2) by using "qtl" problem.

First, I set qsplit is 2 (real basis) in case.inq file and then run "x qtl"
without the spin-orbit coupling in order to see the crystal field splitting.

When I saw "case.qtl" file, I found that the decomposed f-orbitals were
named as A2, x(T1), y(T1), z(T1), ksi(T2), eta(T2), and zeta(T2).

My first question is these labeling are correct?

In my knowledge,
A2       = fxyz
x(T1)    = fx3
y(T1)    = fy3
z(T1)    = fz3
ksi(T2)  = fx(z2-y2)
eta(T2)  = fy(z2-x2)
zeta(T2) = fz(x2-y2)

but real basis are represented as sums of spherical harmonics (l, m) like
fz3         = (3, 0)
fxz2       = (3, 1) - (3, -1)
fyz2       = (3, 1) + (3, -1)
fxyz       = (3, 2) - (3, -2)      <---- This one is exact same as A2 in
the cubic harmonic set.
fz(x2-y2) = (3, 2) + (3, 2)
fx(x2-3y2) = (3, 3) - (3, -3)
fy(3x2-y2) = (3, 3) + (3, 3)

The real basis set is different from the cubic basis set, so that these
labeling like A2, x(T1), y(T1), ... is not correct in my thought.

Second question is related to the print order of decomposed f-orbitals in
"case.qtl" file.

When I draw the decomposed f-orbital labeled as z(T1) in "case.qtl" file, I
realized that this was not z3, but exactly same as A2, that is xyz
f-orbital.

I think print order is not correct in case of "qsplit = 2".

In my feeling, the print order is z(T1), x(T1), y(T1), A2, zeta(T2),
ksi(T2), and eta(T2) in "case.qtl" file if these labeling were correct.

Any comments would be appreciated.


Best wishes,

Chang-Jong Kang
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