Spin-orbit states from the COSCI method¶
This tutorial demonstrates the importance of the effective mean-field spin-orbit screening on spin-orbit states of open-shell systems. Several two-component Hamiltonians are employed.
Spin-orbit states of the F atom¶
In the DIRAC test we calculate the energy difference between spin-orbit splitted states of the \(^{2}P\) state of Fluorine,
using the COSCI wavefunction and with several different Hamiltonians.
All input files for download (together with output files) are in the corresponding test directory of DIRAC, test/cosci_energy.
The following table shows the energy difference betweem \(X ^{2}P_{3/2}\) and \(A ^{2}P_{1/2}\) states:
Hamiltonian |
Splitting/cm-1 |
|---|---|
DC |
434.511758 |
BSS+MFSSO |
438.792872 |
BSS_RKB+MFSSO(*) |
438.793184 |
DKH2+MFSSO |
438.792782 |
BSSsfBSO1+MFSSO |
438.868634 |
DKH2sfBSO1+MFSSO |
438.868738 |
BSSsfESO1+MFSSO |
438.866098 |
DKH2sfESO1+MFSSO |
438.866201 |
BSS |
583.459766 |
BSS_RKB(**) |
583.459995 |
DKH2 |
583.459700 |
BSSsfESO1 |
583.533060 |
DKH2sfESO1 |
583.533187 |
BSSsfBSO1 |
583.535908 |
DKH2sfBSO1 |
583.536036 |
DC2BSS_RKB(DF) |
585.906861 |
(*) Known as X2C. (**) Known as X2C-NOAMFI.
Calculated values can be devided into two categories: those with the mean-field spin-orbit term (MFSSO) and those without. Results matching the four-component Dirac-Coulomb (DC) Hamiltonian are those containing the MFSSO screening term.
Spin-orbit states of the \(Rn^{77+}\) cation¶
Let us proceed with the isoelectronic, but heavier system: the Fluorine-like (9 electrons), highly charged \(Rn^{77+}\) cation (Z=86).
All input files for download (together with output files) are in the corresponding test directory of DIRAC, test/cosci_energy.
Calculated energy differences between the ground, \(X ^{2}P_{3/2}\), and the first excited state, \(A ^{2}P_{1/2}\),
are in the following table:
Hamiltonian |
Splitting/eV |
|---|---|
DC |
3700.081 |
BSS+MFSSO |
3796.844 |
DKH2+MFSSO |
3777.837 |
DC2BSS_RKB(DF) |
3810.190 |
BSS |
3808.859 |
BSS_RKB (*) |
3810.273 |
DKH2 |
3790.044 |
DKH2sfBSO1+MFSSO |
4047.324 |
DKH2sfBSO1 |
4056.349 |
(*) Known as X2C-NOAMFI.
Excercises¶
Why is the MFSSO term more important for the ligher element (F) than for the heavy \(Rn^{77+}\) ?
The one-electron spin-orbit term, SO1, is sufficient for representing spin-orbital effects in the Flourine atom, but not of the Rn^{77+} cation. Why ?
For the Flourine atom, increase the speed of light (.CVALUE) in four-component calculations to emulate non-relativistic description. What is the effect on the spin-orbit splitting ? What artificial value of the speed of light generates the DC-SCF energy identical with nonrelativistic SCF energy up to 5 decimal places ?
To “increase” relativistic effects in Flourine, decrease the speed of light in four-component calculations. How does it affect the spin-orbit splitting ?
Change the symmetry from D2h to automatic symmetry detection in the F mol file and add molecular spinors analysis to the input file (**ANALYZE). Identify molecular spinors (orbitals) of Flourine according to the extra quantum number in linear symmetry.