# 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.