Example: Spin-Orbit coupling: Bi and Bi2¶
Application of the Spin-Orbit relativistic option (using double-group symmetry) to Bismuth (atom and dimer).
For comparison with the full double-group calculation, the ‘standard’ unrestricted calculation on Bismuth is carried out, using the scalar relativistic option.
A net spin polarization of 3 electrons is applied (key charge).
$ADFBIN/adf <<eor
title Bi unrestricted
BeckeGrid
quality good
End
relativistic scalar ZORA
ATOMS
Bi 0.000000 0.000000 0.00000000
end
Basis
Type TZ2P
Core None
end
unrestricted
charge 0 3
xc
GGA becke perdew
end
end input
eor
The CHARGE key, in conjunction with the UNRESTRICTED key is used to specify that 3 electrons must be unpaired (second value of the CHARGE key), while the system is neutral (first value of the CHARGE key).
Next we do a Spin-Orbit calculation on the Bismuth atom.
Note that it is a ‘unrestricted’ run using the noncollinear approximation, and SYMMETRY NSOYM. The electronic charge density is spin-polarized.
$ADFBIN/adf <<eor
title Bi spinorbit unrestricted noncollinear
BeckeGrid
quality good
End
relativistic spinorbit ZORA
ATOMS
Bi 0.000000 0.000000 0.00000000
end
symmetry nosym
unrestricted
noncollinear
Basis
Type TZ2P
Core None
end
xc
GGA becke perdew
end
end input
eor
Because an all electron basis set is used, the bond energy is huge, due to the very large higher order spin-orbit effect on the 2p orbitals.
Bi2 dimer
Now we turn to the dimer Bi2 : a series of Single Point calculations, all with the same inter atomic distance.
First the scalar relativistic run.
$ADFBIN/adf <<eor
title Bi2, scalar relativistic
BeckeGrid
quality good
End
relativistic scalar ZORA
ATOMS
Bi 0.0 0.0 1.33
Bi 0.0 0.0 -1.33
end
Basis
Type TZ2P
Core None
end
xc
GGA becke perdew
end
end input
eor
mv tape21 t21Bi2
The calculated scalar relativistic atomization energy will be close to 2.74 eV. This is the bond energy of the dimer minus 2 times the bond energy of the unrestricted scalar relativistic atom.
The result file tape21 is used as reference in subsequent calculations: run the spin-orbit case starting from the just completed dimer calculation as a fragment. The resulting ‘bonding energy’, ie the energy w.r.t. the scalar relativistic ZORA dimer, gives directly the effect of the full-relativistic ZORA versus the scalar relativistic ZORA option: the energy is lowered by huge amount, again mainly due to the large spin-orbit effect on the 2p orbitals.
$ADFBIN/adf <<eor
title Bi2 from fragment Bi2, with SpinOrbit coupling
PRINT SpinOrbit
BeckeGrid
quality good
End
relativistic spinorbit ZORA
ATOMS
Bi 0.0 0.0 1.33 f=Bi2
Bi 0.0 0.0 -1.33 f=Bi2
end
fragments
Bi2 t21Bi2
end
xc
GGA becke perdew
end
end input
eor
rm TAPE21 logfile
If one looks at the SFO analysis in the output of this calculation, one can observe that a first-order spin-orbit splitting of the scalar relativistic fragment orbitals is a good approximation to many of the calculated valence spinors.
A final consistency check: run the spin-orbit dimer from single-atom fragments. The bonding energy should equal the sum of the bonding energies of the previous two runs: scalar relativistic dimer w.r.t. single atom fragments plus spin-orbit dimer w.r.t. the scalar relativistic dimer.
$ADFBIN/adf <<eor
title Bi2 from atomic fragments, SpinOrbit coupling
PRINT SpinOrbit
BeckeGrid
quality good
End
relativistic spinorbit ZORA
ATOMS
Bi 0.0 0.0 1.33
Bi 0.0 0.0 -1.33
end
Basis
Type TZ2P
Core None
end
xc
GGA becke perdew
end
end input
eor
The calculated spin-orbit coupled relativistic atomization energy will be close to 2.18 eV. This is the bond energy of the dimer minus 2 times the bond energy of the unrestricted non-collinear spin-orbit coupled relativistic atom. Note that one has to subtract huge numbers.