[Wien] Spin-polarization and spin-orbit coupling

[pieper]

Reading your questions I get the impression that you miss a crucial
point of what and how Density Functional Theory (DFT) does. Therefore
I would like to extend on what Peter Blaha said. Maybe it helps to
clarify some of his answers.

Pauli's Hamiltonian represents the energy of an electron in some
electromagnetic field (your external magnetic field).

DFT aims to find the ground state of a system of many interacting
Fermions (electrons) represented by any Hamiltonian for the
interaction you care to throw at it. The mean interaction energy of a
given electron spin with all others in the system is the same as its
energy in some fictitious internal field.

DFT works via a subtle reinterpretation of the role of the charge
density, which has far reaching consequences. In 1964 Hohenberg and
Kohn proved a fascinating property of the charge density: This single,
real valued function of (without spin) one spacial variable uniquely
determines the many particle wave function of interacting electrons in
an external Coulomb potential (due to, in this case, nuclear
charges). Calculation of the spacial charge density from the wave
function by an expectation value <psi|psi> is not a one way
process. If two solutions of Schroedingers equation for the
interacting many particle system in a given external Coulomb potential
give the same charge density, then the two wave functions with all
their N space variables for N electrons are the same. One can, in
principle, go back: the charge density determines the wave
function - and with it everything, including the energy.

Furthermore, the ground state charge density can be determined by a
variational principle and its Euler-Lagrange equations (the Kohn-Sham
equations). They can be solved very effectively in a self consistent,
iterative process: start from some charge densitiy (the closer to the
solution the better to avoid local minima). Then calculate the
potentials - solve the Kohn-Sham equations - calculate the charge
density - compare it to previous ones. If the difference is larger
than some threshold, mix up a variation of it and calculate the
potentials ...

The most simple case is if your favorite interaction Hamiltonian has
Coulomb potentials from local charges and no spin
contribution. Spin-up and -down charge densities are identical (Local
Densitsity Approximation, LDA, for a certain type of electric 
potential).

If there are exchange interaction energies present (products of the
spin operators of pairs of interacting electrons) the densities of the
two spin directions become different (Local Spin Density
Approximation, LSDA, the electric potential stays the same). Such a
Heisenberg exchange interaction does, however, not depend on the
oriention of interacting spins in the crystal lattice. Only the
strength of the equivalent internal field has a meaning, not its
direction.

In contrast, a spin-orbin coupling in the Hamiltonian does depend on
the orientation of the spin moment in the lattice. With the
interaction energy depending on spin orientation, so does the
equivalent internal field. The system is magnetically anisotropic. You
have to specifiy the orientation of the moment to determine the energy
of the graound state.

There is, of course, a huge number of books an reviews on DFT. The UG
cites enough stuff to keep you busy for a long time. Personally, I
like a review of A. Becke: THE JOURNAL OF CHEMICAL PHYSICS 140, 18A301
(2014)


---
Dr. Martin Pieper
Karl-Franzens University
Institute of Physics
Universitätsplatz 5
A-8010 Graz
Austria
Tel.: +43-(0)316-380-8564


Am 2019-09-24 08:23, schrieb Luigi Maduro - TNW:
> Dear WIEN2k users,
> 
> I have three questions concerning the inclusion of spin in a material
> in WIEN2k.
> 
> The three questions concern the two terms where a spin-dependent term
> appears in the Pauli Hamiltonian for magnetic systems, which are:
> 
> Question 1)
> In the Pauli Hamiltonian a term appears which is a dot product of the
> spin-matrices of the system and an effective magnetic field.
> 
> The effective magnetic field is a summation of an external magnetic
> field and an exchange-correlation term. The exchange-correlation term
> B_xc, is expressed as a derivative of the density w.r.t. the
> magnetization (in the LDA framework) and that B_xc is parallel to the
> magnetization density vector. If I understand correctly then the
> material of interest is magnetic when B_xc is nonzero.
> 
> When doing a spin-polarized calculation, what happens then to the
> external magnetic field term? Is the external magnetic field term set
> to zero?
> 
> Question 2)
> The other term in the Pauli Hamiltonian is the spin-orbit coupling
> (SOC) term, which is proportional to (1/r x dV/dr ) (dV/dr = the
> derivative of the potential w.r.t. the radial coordinate).
> 
> When doing a calculation including SOC the script init_so asks for the
> magnetization direction (in hkl).
> 
> In a non-spin polarized calculation with SOC the magnetization
> direction has no meaning, is this correct?
> 
> Question 3)
> 
> If the system of interest is a magnetic system then a spin-polarized
> calculation with SOC should give me 1) the strength of the
> magnetization along the chosen magnetization axis, and 2) the spin-up
> and spin-down density of states (DOS) along the chosen axis. But a
> spin-polarized calculation without SOC will not give me the spin-up
> and spin-down DOS. Is this correct?
> 
> Cheers,
> 
> Luigi Maduro
> 
> PhD candidate
> Kavli Institute of Nanoscience
> 
> Department of Quantum Nanoscience
> 
> Faculty of Applied Sciences
> 
> Delft University of Technology
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