Q. How to calculate the
ground and excited states of molecules having open-shell ground state? With
ionization or with electron-attachment?
A. There are
four approaches for calculating open-shell systems depending on the choice of
the reference orbitals and the operators used in the SAC-CI wave function.
Case
1: ROHF SCF (N) - SAC (N+1) - SAC-CI,
CationDoublet (N)
Case
2: ROHF SCF (N) - SAC (N-1) - SAC-CI, AnionDoublet
(N)
Case
3: RHF of Cation (N-1) - SAC (N-1) - SAC-CI, AnionDoublet
(N)
Case
4: RHF of Anion (N+1) - SAC (N+1) - SAC-CI,
CationDoublet (N)
where
(N), (N-1) and (N+1) denotes the number of electrons. Since these types of
calculations give almost the same results, we do not have special
recommendation. We can choose the reference orbitals by specifying the charge
and the multiplicity just one line above the “Gaussaian molecular
specification”. The choice “CationDoublet” or “AnionDoublet” should be
specified in the SAC-CI part of input.
As described
above, there are two ways to construct the open-shell ground-state wave
function by the SAC-CI method: via CationDoublet (1 and 4) or AnionDoublet (2
and 3). The former removes one electron from the (N+1)-electron closed-shell
SAC wave function, and the latter attaches one electron to the (N-1)-electron
closed shell SAC wave function. Therefore, the way to obtain the excited states
is different in these two cases.
In the
molecules having the open-shell ground-state, we can classify the one-electron
excited states into three.
Case
A: Excitation from the doubly occupied MOs to SOMO
Case B:
Excitation from the SOMO to the unoccupied MO
Case
C: Excitation from the doubly occupied MOs to the unoccupied MO
We
note here that even such one-electron excited states should be treated as a
two-electron process in the SAC-CI calculation, because of the open-shell
ground state. In the “CationDoublet” (1 and 4) case, the wave function of the
case A can be described by one-electron ionization from the SAC state having
(N+1)-electrons. Cases B and C, however, should be described by two-electron
process: one-electron ionization and one-electron excitation. In the
“AnionDoublet” (2 and 3) case, only case B can be described by one-electron
attachment to the SAC state having (N+1)-electrons. The cases A and C should be
described by two-electron process: one-electron attachment and one-electron
excitation.
One-electron
process from the SAC closed-shell state can be treated by the SAC-CI SD-R method, while two-electron process
from the closed-shell state requires SAC-CI general-R method. In closing, we give a Table to summarize the above
discussion.
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Table.
SAC-CI approach to the excited state for the open-shell ground state
molecules. |
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Construction of the
open-shell ground-state wave function |
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Case 1 |
Case 2 |
Case 3 |
Case 4 |
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Type
of the
excited states |
Case A |
One-elec. process (SD-R) |
Two-elec. process (general-R) |
Two-elec. process (general-R) |
One-elec. process (SD-R) |
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Case B |
Two-elec. process (general-R) |
One-elec. process (SD-R) |
One-elec. process (SD-R) |
Two-elec. process (general-R) |
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Case C |
Two-elec. process (general-R) |
Two-elec. process (general-R) |
Two-elec. process (general-R) |
Two-elec. process (general-R) |
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