:orphan: 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 :math:`^{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, :file:`test/cosci_energy`. The following table shows the energy difference betweem :math:`X ^{2}P_{3/2}` and :math:`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 :math:`Rn^{77+}` cation ------------------------------------------------- Let us proceed with the isoelectronic, but heavier system: the Fluorine-like (9 electrons), highly charged :math:`Rn^{77+}` cation (Z=86). All input files for download (together with output files) are in the corresponding test directory of DIRAC, :file:`test/cosci_energy`. Calculated energy differences between the ground, :math:`X ^{2}P_{3/2}`, and the first excited state, :math:`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 :math:`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 (:ref:`GENERAL_.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 (:ref:`**ANALYZE`). Identify molecular spinors (orbitals) of Flourine according to the extra quantum number in linear symmetry.