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 2P 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 X2P3/2 and A2P1/2 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 Rn77+ cation

Let us proceed with the isoelectronic, but heavier system: the Fluorine-like (9 electrons), highly charged Rn77+ 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, X2P3/2, and the first excited state, A2P1/2, 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

  1. Why is the MFSSO term more important for the ligher element (F) than for the heavy Rn77+ ?

  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.