[Wien] Questions about forces on nuclei in the presence of spin orbit coupling

[Zhiyong Zhu]

Dear Prof. Blaha,
Dear Wien2k users and authors,

Happy New Year!

I am using wien2k(_11.1) to do atomic relaxations in some materials.

I have some questions about the force calculations (in particular the Pulay
correction
of valence electron force) in the presence of spin orbit coupling.

As shown in the Wien2k webpage and maillist, "*In spin-orbit calculations
forces are
not yet implemented*". The forces on nuclei (even after Pulay correction)
are
meaningless when the spin-orbit coupling is switched on, even though forces
are produced without any warnings.

However, I don't understand exactly the reason why the forces are
meaningless.
As I guess, (not for sure), the reason is that the force calculations in
wien2k is based
on the paper by Yu, et al ( PhysRevB.43.6411(1991)All-electron and
pseudopotential
force calculations using the linearized-augmented-plane-wave method), in
which,
however, only the nonrelativistic case is treated.

If I am right, then my questions are:

1)  Has the scalar relativistic effect (i.e. the mass and Darwin terms)
already been
considered in the force calculations in wien2k? If yes, what is the
difference between
the cases with and without scalar relativistic effect? Is there any
additional force,
compared with the formula in PhysRevB.43.6411(1991), due to the presence of
scalar relativistic effect?

2)  What is the meaning of "*In spin-orbit calculations forces are not yet
implemented*"?
As I understand, "*not yet implemented*" means that the additional forces
due to the presence
of spin-orbit coupling are not calculated at all. If this is true, then the
forces calculated with and
without spin-orbit coupling should be same. However, I find that the forces
on nuclei are changed
very much when spin-orbit coupling is switched on.

3)  Is there any physical meaning, (although perhaps not the real physics
or not the
correct physics), of the forces on nuclei (after Pulay correction) in the
presence of
spin-orbit coupling? Take a nonmagnetic calculation as an example. As I
understand,
(not for sure), without spin-orbit coupling, the Pulay correction from the
valence
electrons is calculated using the wavefunctions and energy eigenvalues from
non-spin-polarized lapw1 calculations. With spin-orbit coupling, two sets
of data
(wavefunctions and energy eigenvalues) are obtained from lapw1 and lapwso
calculations: one is for spin up and one is for spin down. Then the Pulay
correction
of forces from valence electrons is calculated using these two sets of data.
This means that the additional forces due to spin-orbit coupling are caused
by
the modification of wavefunctions due to spin-orbit coupling. Is my
statement
correct?

4)  If the forces calculated with spin-orbit coupling are indeed incorrect
and meaningless,
can we use PORT option in case.inM when doing atomic relaxation (min_lapw)?
This
question arises because I read in the wien2k user guide that "It (PORT)
minimizes the
total energy and NOT the forces (using the forces as derivative of E vs.
atomic positions).".
As I understand, the PORT method will use "total energy", instead of "force
on nuclei",
as a criterion to find the equilibrium positions of atoms, i.e., the PORT
method will find
an energy minimum by a real displacement of atoms, rather than by find a
structure in
which forces on atoms are zero. This means that the forces on nuclei is NOT
the essential
ingredient in the PORT method, and PORT method is still valid (as long as
the total
energy are calcuated correctly) even in the presence of spin-orbit
coupling. Is my
ratiocination correct?

5) Could you please also explain to me the meaning of some variables in the
source code
of lapw2? In the SUBROUTINE FOMAI1 (fomai1.f), there are four variables
relating to the
Pulay correction from the valence electrons: fsph, fsph2, forb, and fnsp.
In the SUBROUTINE
fsumai1 (fsumai1.F) there is one variable: fsur.  In the SUBROUTINE FOMAI2
(fomai2.F), there
is one variable: fvdrho. As I understand, (not for sure), "fvdrho" comes
from Eq. A8 in
PhysRevB.43.6411(1991), "fsur" comes from Eq. A12 in the paper, and "fnsp"
comes from
Eq. A20 in the paper. Am I right? Then my question is which formula are
related to "fsph",
"fsph2", and "forb"?

Thank you very much.

Best Regards,
Zhiyong Zhu
Post Doctoral Fellow
*King Abdullah University of Science and Technology* (*KAUST*)
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