Examples of open-shell systems¶
Here we provide few examples of open-shell systems.
Boron atom: open-shell SCF and small CI relativistic calculations¶
An example of one electron in two spinors is the Boron atom.
We seek to get the first excited state, 2P3/2, of B (X 2P1/2), and find out the 2P1/2-2P3/2fine structure splitting.
We select the Dirac-Coulomb Hamiltonian and neglect the (SS|SS) class of integrals.
Complete open-shell CI¶
Possible way to get the value is performing small CI over the active open-shell space, containing six 2p orbitals, which are split due to the spin-orbital interaction. Note, that for pedagogical purposes we have employed several computational symmetries.
Starting from the average SCF occupation (one electron distributed over 2p1/2 and 2p3/2 shells),
pam --mol=B.sto-3g.lsym.mol -inp=B.2Pav.dc_rkb.scf_2fs_cosci.inp
pam --mol=B.sto-3g.D2h.mol --inp=B.2Pav.dc_rkb.scf_2fs_cosci.inp
for one-fermion symmetries,
pam --mol=B.sto-3g.C2v.mol --inp=B.2Pav.dc_rkb.scf_1fs_cosci.inp
we get the value of 17.179488 cm-1 and in the outputs we see the proper degeneracy of resulting total CI energies - 2 and 4.
By averaging the COSCI energies of the ground and the states we get the total SCF energy of the average occupation
((2*(-24.249116730099) + 4*(-24.249038454591)))/6 = -24.249064546427 SCF energy: -24.249064546425263
To apply the MO reordering scheme, first, one has to obtain the MO coefficients file of the ground state:
pam --outcmo --mol=B.sto-3g.D2h.mol --inp=B.2P12.dc_rkb.scf_2fs.inp pam --outcmo --mol=B.sto-3g.C1.mol --inp=B.2P12.dc_rkb.scf_1fs.inp
and use them for calculation of the excited state:
pam --incmo --mol=B.sto-3g.D2h.mol --inp=B.2P32.dc_rkb.scf_2fs_reord_ovlsel.inp pam --incmo --mol=B.sto-3g.C1.mol --inp=B.2P32.dc_rkb.scf_1fs_reord_ovlsel.inp
For the linear symmetry one can employ the M_J specified occupation of shells both for the ground and the first excited state. Only the preDHF reordering works in this case, not the overlap selection.
pam --outcmo --mol=B.sto-3g.lsym.mol --inp=B.2P12.dc_rkb.scf_2fs_mj.inp pam --incmo --mol=B.sto-3g.lsym.mol --inp=B.2P32.dc_rkb.scf_2fs_reord_mj.inp
Finally, in the output one can see proper order of spinors for the excited state - first 2p3/2 spinors (four-fold degeneracy), and then the 2p1/2 spinors (two-fold degeneracy):
* Fermion symmetry E1u * Open shell #1, f = 0.2500 -0.28691011961454 ( 4) * Virtual eigenvalues, f = 0.0000 0.12724113329935 ( 2)
Regarding total energies we get
SCF energy 2_P1/2: -24.249116730391307 SCF energy 2_P3/2: -24.249038454664991 FSS : (-24.249116730391307+24.249038454664991)*219474.631280634 cm-1 = 17.179536 cm-1
what is very close to the above given COSCI value of 17.179488 cm-1.
How to handle open-shell systems using CC methods¶
Coupled-Cluster Singles, Doubles and Noniterative triples (CCSD(T)) and Fock-space Coupled-Cluser methods are powerful correlation methods in the DIRAC program suite.
Coupled Cluster (CC) methods in DIRAC are most powerful ab-initio correlation methods working upon two/four-component Kramers unrestricted spinors. They serve as widely employed relativistic analogue with respect the nonrelativistic realm.
Therefore we feel it is important to present the user few hints on how to fully exhaust CC capabilities for practical calculations.
Even at the Coupled Cluster correlated level the user has certain variability in choosing the occupation of spinors. Thus he may alter the electronic state of a system.
We give you few hints how to employ them in correlated open-shell calculations.
Example 1: FO molecule
Fock space CCSD method¶
The starting system is always closed shell. One can add one or two-electrons to the N-electron system to iterate into N+1 and/or N+2 systems.