TD-DFT with SOC: Kramers pairs

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Quartet state with SOC

$COMPASS
Title
 CS molecule
Basis
 cc-pvdz
Geometry
 C 0. 0. 0.
 S 0. 0. 1.1151
END geometry
$END

$XUANYUAN
scalar
heff
0
soint
hsoc
0
$END

$SCF
UKS
DFT
SVWN5
charge
1
spin
2
$END

$TRAINT
utddft
orbi
hforb
$END

$tddft
imethod
2
isf
1
itda
1
idiag
1
istore
1
nexit
2 2 2 2
$END

$TDDFT
isoc
2
nfiles
1
ifgs
0
imatsoc
-1
$END

The quartets are split into Kramers pairs by SOC interaction with higher quartet states.

 *** List of SOC-SI results ***

  No.      ExEnergies            Dominant Excitations         Esf        dE      Eex(eV)     (cm^-1)

--------> Quartet: Sigma^+

    1       8.3501 eV    99.9%  Spin: |S+,1>    1-th   A1    8.3516   -0.0015    0.0000         0.00
    2       8.3501 eV    99.9%  Spin: |S+,4>    1-th   A1    8.3516   -0.0015    0.0000         0.00
    3       8.3512 eV   100.0%  Spin: |S+,3>    1-th   A1    8.3516   -0.0004    0.0011         8.50
    4       8.3512 eV   100.0%  Spin: |S+,2>    1-th   A1    8.3516   -0.0004    0.0011         8.50

--------> Quartet: Delta

    5       8.8949 eV    50.0%  Spin: |S+,4>    2-th   A1    8.9015   -0.0066    0.5448      4393.72
    6       8.8949 eV    50.0%  Spin: |S+,1>    2-th   A1    8.9015   -0.0066    0.5448      4393.72

    7       8.8988 eV    49.9%  Spin: |S+,3>    2-th   A1    8.9015   -0.0027    0.5486      4425.14
    8       8.8988 eV    49.9%  Spin: |S+,2>    2-th   A1    8.9015   -0.0027    0.5486      4425.14

    9       8.9030 eV    49.9%  Spin: |S+,2>    1-th   A2    8.9015    0.0015    0.5529      4459.10
   10       8.9030 eV    49.9%  Spin: |S+,3>    1-th   A2    8.9015    0.0015    0.5529      4459.10

   11       8.9075 eV    49.9%  Spin: |S+,1>    1-th   A2    8.9015    0.0061    0.5574      4495.86
   12       8.9075 eV    49.9%  Spin: |S+,4>    1-th   A2    8.9015    0.0061    0.5574      4495.86

--------> Quartet: Pi

   13       9.3335 eV    48.2%  Spin: |S+,1>    1-th   B1    9.3505   -0.0170    0.9834      7931.49
   14       9.3335 eV    48.2%  Spin: |S+,4>    1-th   B1    9.3505   -0.0170    0.9834      7931.49
   15       9.3446 eV    48.7%  Spin: |S+,2>    1-th   B1    9.3505   -0.0059    0.9945      8021.16
   16       9.3446 eV    48.7%  Spin: |S+,3>    1-th   B1    9.3505   -0.0059    0.9945      8021.16
   17       9.3555 eV    40.3%  Spin: |S+,2>    1-th   B2    9.3505    0.0050    1.0054      8108.95
   18       9.3555 eV    40.3%  Spin: |S+,3>    1-th   B2    9.3505    0.0050    1.0054      8108.95
   19       9.3671 eV    49.7%  Spin: |S+,1>    1-th   B2    9.3505    0.0166    1.0170      8202.80
   20       9.3671 eV    49.7%  Spin: |S+,4>    1-th   B2    9.3505    0.0166    1.0170      8202.80

--------> Quartet: Sigma-

   21       9.4651 eV    98.1%  Spin: |S+,3>    2-th   A2    9.4627    0.0025    1.1150      8993.28
   22       9.4651 eV    98.1%  Spin: |S+,2>    2-th   A2    9.4627    0.0025    1.1150      8993.28
   23       9.4653 eV    98.8%  Spin: |S+,4>    2-th   A2    9.4627    0.0026    1.1151      8994.24
   24       9.4653 eV    98.8%  Spin: |S+,1>    2-th   A2    9.4627    0.0026    1.1151      8994.24

--------> Quartet: Pi

   25      15.9875 eV    50.0%  Spin: |S+,4>    2-th   B2   16.0102   -0.0227    7.6374     61599.53
   26      15.9875 eV    50.0%  Spin: |S+,1>    2-th   B2   16.0102   -0.0227    7.6374     61599.53
   27      16.0026 eV    44.1%  Spin: |S+,2>    2-th   B2   16.0102   -0.0076    7.6525     61721.52
   28      16.0026 eV    44.1%  Spin: |S+,3>    2-th   B2   16.0102   -0.0076    7.6525     61721.52
   29      16.0177 eV    50.0%  Spin: |S+,3>    2-th   B1   16.0102    0.0076    7.6676     61843.52
   30      16.0177 eV    50.0%  Spin: |S+,2>    2-th   B1   16.0102    0.0076    7.6676     61843.52
   31      16.0329 eV    50.0%  Spin: |S+,4>    2-th   B1   16.0102    0.0227    7.6827     61965.52
   32      16.0329 eV    50.0%  Spin: |S+,1>    2-th   B1   16.0102    0.0227    7.6827     61965.52

Similarly, in the case O2 ground state, the ZFS can be computed as follows (K*Psi = Psi):

  No.      ExEnergies            Dominant Excitations         Esf        dE      Eex(eV)     (cm^-1)
--->
    1      -0.0001 eV   100.0%  Spin: |Gs,2>    0-th  B1g    0.0000   -0.0001    0.0000         0.00
--->
    2      -0.0001 eV   100.0%  Spin: |Gs,1>    0-th  B1g    0.0000   -0.0001    0.0000         0.11
    3      -0.0001 eV   100.0%  Spin: |Gs,3>    0-th  B1g    0.0000   -0.0001    0.0000         0.11
--->

The spin-spin interaction will be implemented in future!