Why the KS-Hamiltonian is not Hermite?

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Why the KS-Hamiltonian is not Hermite?

Date: 2024/06/08 10:58
Name: Liang Liu <liangliu@mail.sdu.edu.cn>: Hello!
I made a minimal Fe-Pt molecular with considerations on spin-orbital-coupling (SpinP_switch==3). Then I computed the KS-Ham. with analysis_example. According to the instruction, the hopping matrix between up and down spin tunnels is Hks[2] +/- I*(Hks[3]+iHks[2]). So, I think Hks[2] should be a symmetric matrix to make the KS-Ham. be Hermite.
Below is part of the Hks[2]:

Kohn-Sham Hamiltonian spin=2
global index=1 local index=0 (global=1, Rn=0 [0 0 0])
0 0 0 2.286268081E-05 ...
0 0 0 1.702758878E-01 ...
0 0 0 0 ...
2.287425318E-05 -1.717342946E-01 0 0 ...
...

Apparently there are both symmetric elements: Hks[2][0,3]=Hks[2][3,0] and anti-symmetric elements: Hks[2][1,3]=-Hks[2][3,1]. The latter one is to violate the Hermite condition of Ham., why?

Here is my the input file:
#######################################################
#
# File Name
#

System.CurrrentDirectory ./ # default=./
level.of.stdout 1 # default=1 (1-3)
level.of.fileout 2 # default=1 (0-2)
System.Name Pt
DATA.PATH ../DFT_DATA19
HS.fileout on # on|off, default=off
#
# Definition of Atomic Species
#

Species.Number 2
<Definition.of.Atomic.Species
Fe Fe8.0H-s1p1d1 Fe_PBE19H
Pt Pt7.0-s1p1d1 Pt_PBE19
Definition.of.Atomic.Species>

#
# Atoms
#

Atoms.Number 2
Atoms.SpeciesAndCoordinates.Unit Ang # Ang|AU
<Atoms.SpeciesAndCoordinates # spin-chg spin-ang orb-ang cons +U enhance
1 Pt 0 0 0 8.0 8.0 0.0 0.8 0.0 0.0 0 off
2 Fe 2.4 0 0 9.5 6.5 0.0 0.8 0.0 0.0 0 off
Atoms.SpeciesAndCoordinates>
Atoms.UnitVectors.Unit Ang # Ang|AU
<Atoms.UnitVectors
20 0 0
0 20 0
0 0 20
Atoms.UnitVectors>

#
# SCF or Electronic System
#
scf.restart off
scf.XcType GGA-PBE # LDA|LSDA-CA|LSDA-PW|GGA-PBE
scf.SpinPolarization NC # On|Off|NC
scf.SpinOrbit.Coupling on # On|Off, default=off
scf.ElectronicTemperature 300.0 # default=300 (K)
scf.energycutoff 250.0 # default=150 (Ry)
scf.maxIter 400 # default=40
scf.EigenvalueSolver Cluster # DC|GDC|Cluster|Band
scf.Kgrid 1 1 1 # means n1 x n2 x n3
scf.Generation.Kpoint regular # regular|MP
scf.Mixing.Type rmm-diisk # Simple|Rmm-Diis|Gr-Pulay|Kerker|Rmm-Diisk
#scf.Init.Mixing.Weight 0.01 # default=0.30
#scf.Min.Mixing.Weight 0.01 # default=0.001
#scf.Max.Mixing.Weight 0.1 # default=0.40
scf.Mixing.History 50 # default=5
scf.Mixing.StartPulay 15 # default=6
scf.criterion 1.0e-9 # default=1.0e-6 (Hartree)
scf.lapack.dste dstevx # dstevx|dstedc|dstegr,default=dstevx

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Re: Why the KS-Hamiltonian is not Hermite? ( No.1 )

Date: 2024/06/08 13:33
Name: Naoya Yamaguchi

Hi,

>Apparently there are both symmetric elements: Hks[2][0,3]=Hks[2][3,0] and anti-symmetric elements: Hks[2][1,3]=-Hks[2][3,1]. The latter one is to violate the Hermite condition of Ham., why?

There remains an ambiguity in your explanation, but if `Hks` is a Hamiltonian stored in a scfout file, it is in a local form (equation (28) in https://www.openmx-square.org/tech_notes/tech1-1_2.pdf), i.e., independent of wavenumber, and the matrix elements of the Hamiltonian for neighboring atoms when focusing on one atom. In this form, the indices of the first and second atoms are not directly interchangeable, and it is not appropriate to represent them in a two-dimensional matrix (although there is no multidimensional array in C). Based on equation (28), the correct correspondence should be found.

Regards,
Naoya Yamaguchi

Re: Why the KS-Hamiltonian is not Hermite? ( No.2 )

Date: 2024/06/08 18:02
Name: Liang Liu <liangliu@mail.sdu.edu.cn>

Yes, you are right. I am very grateful for your helps!

Best regards,
Liang Liu

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