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.


  1. Why is the MFSSO term more important for the ligher element (F) than for the heavy \(Rn^{77+}\) ?
  2. 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 ?
  3. 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 ?
  4. To “increase” relativistic effects in Flourine, decrease the speed of light in four-component calculations. How does it affect the spin-orbit splitting ?
  5. 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.