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

  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.