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.
For more information, see Refs. [Ilias2001], [Ilias2007].
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.