Understanding excited states degeneracies in TD-DFT calculations

Hello,

Having run a TD-DFT calculation on the Si4GeH12 molecule, we are currently trying to rationalize the results and understand in particular how do the degeneracies of the excited states work.

This is the input file of the calculation:

$molecule
   0 1
Si         1.38455       -1.38455        1.38455
Si         1.38455        1.38455       -1.38455
Si        -1.38455        1.38455        1.38455
Si        -1.38455       -1.38455       -1.38455
Ge        -0.00000       -0.00000       -0.00000
H          2.24907        2.24907       -0.54291
H          0.54291        2.24907       -2.24907
H          2.24907        0.54291       -2.24907
H         -0.54291        2.24907        2.24907
H         -2.24907        0.54291        2.24907
H         -2.24907        2.24907        0.54291
H          2.24907       -0.54291        2.24907
H          2.24907       -2.24907        0.54291
H          0.54291       -2.24907        2.24907
H         -2.24907       -0.54291       -2.24907
H         -2.24907       -2.24907       -0.54291
H         -0.54291       -2.24907       -2.24907
$end

$rem
   MEM_TOTAL            2000
   JOBTYPE              sp
   EXCHANGE             b3lyp
   BASIS                def2-tzvp
   CIS_N_ROOTS          4
   SCF_ALGORITHM        diis_gdm
   SYM_TOL              4
   CIS_SINGLETS         true
   CIS_TRIPLETS         true
   CALC_SOC             true
   STS_MOM              true
   PRINT_ORBITALS       true
   MOLDEN_FORMAT        true
   IQMOL_FCHK           true
$end

And here is the relevant part of the output file:

>  ---------------------------------------------------
>          TDDFT/TDA Excitation Energies              
>  ---------------------------------------------------
> 
>  Excited state   1: excitation energy (eV) =    5.7487
>  Total energy for state  1:                 -3242.16571604 au
>     Multiplicity: Triplet
>     Trans. Mom.:  0.0000 X   0.0000 Y   0.0000 Z
>     Strength   :     0.0000000000
>     D(   48) --> V(    1) amplitude =  0.7197
>     D(   48) --> V(    5) amplitude = -0.2270
>     D(   49) --> V(    2) amplitude =  0.4381
>     D(   50) --> V(    3) amplitude = -0.4381
> 
>  Excited state   2: excitation energy (eV) =    5.7487
>  Total energy for state  2:                 -3242.16571604 au
>     Multiplicity: Triplet
>     Trans. Mom.:  0.0000 X   0.0000 Y   0.0000 Z
>     Strength   :     0.0000000000
>     D(   48) --> V(    2) amplitude =  0.4381
>     D(   49) --> V(    1) amplitude =  0.7197
>     D(   50) --> V(    4) amplitude = -0.4381
> 
>  Excited state   3: excitation energy (eV) =    5.7487
>  Total energy for state  3:                 -3242.16571604 au
>     Multiplicity: Triplet
>     Trans. Mom.:  0.0000 X   0.0000 Y   0.0000 Z
>     Strength   :     0.0000000000
>     D(   48) --> V(    3) amplitude = -0.4381
>     D(   49) --> V(    4) amplitude = -0.4381
>     D(   50) --> V(    1) amplitude =  0.7197
>     D(   50) --> V(    6) amplitude =  0.2484
> 
>  Excited state   4: excitation energy (eV) =    5.8757
>  Total energy for state  4:                 -3242.16105030 au
>     Multiplicity: Triplet
>     Trans. Mom.:  0.0000 X   0.0000 Y   0.0000 Z
>     Strength   :     0.0000000000
>     D(   48) --> V(    3) amplitude = -0.5677
>     D(   49) --> V(    4) amplitude =  0.5677
>     D(   50) --> V(    5) amplitude = -0.5688
> 
>  Excited state   5: excitation energy (eV) =    5.8757
>  Total energy for state  5:                 -3242.16105030 au
>     Multiplicity: Triplet
>     Trans. Mom.:  0.0000 X   0.0000 Y   0.0000 Z
>     Strength   :     0.0000000000
>     D(   48) --> V(    2) amplitude =  0.5677
>     D(   49) --> V(    5) amplitude = -0.3313
>     D(   49) --> V(    6) amplitude = -0.4654
>     D(   50) --> V(    4) amplitude =  0.5677
> 
>  Excited state   6: excitation energy (eV) =    5.8757
>  Total energy for state  6:                 -3242.16105030 au
>     Multiplicity: Triplet
>     Trans. Mom.:  0.0000 X   0.0000 Y   0.0000 Z
>     Strength   :     0.0000000000
>     D(   48) --> V(    5) amplitude =  0.2374
>     D(   48) --> V(    6) amplitude = -0.5197
>     D(   49) --> V(    2) amplitude =  0.5677
>     D(   50) --> V(    3) amplitude =  0.5677
> 
>  Excited state   7: excitation energy (eV) =    5.9270
>  Total energy for state  7:                 -3242.15916397 au
>     Multiplicity: Triplet
>     Trans. Mom.:  0.0000 X   0.0000 Y   0.0000 Z
>     Strength   :     0.0000000000
>     D(   48) --> V(    4) amplitude = -0.5681
>     D(   49) --> V(    3) amplitude = -0.5681
>     D(   50) --> V(    2) amplitude =  0.5681
> 
>  Excited state   8: excitation energy (eV) =    6.0509
>  Total energy for state  8:                 -3242.15460988 au
>     Multiplicity: Triplet
>     Trans. Mom.:  0.0000 X   0.0000 Y   0.0000 Z
>     Strength   :     0.0000000000
>     D(   48) --> V(    3) amplitude =  0.3412
>     D(   49) --> V(    4) amplitude =  0.3412
>     D(   50) --> V(    1) amplitude =  0.6274
>     D(   50) --> V(    6) amplitude = -0.5918
> 
>  Excited state   9: excitation energy (eV) =    6.0509
>  Total energy for state  9:                 -3242.15460988 au
>     Multiplicity: Triplet
>     Trans. Mom.:  0.0000 X   0.0000 Y   0.0000 Z
>     Strength   :     0.0000000000
>     D(   48) --> V(    2) amplitude = -0.3412
>     D(   49) --> V(    1) amplitude =  0.6274
>     D(   49) --> V(    5) amplitude = -0.4843
>     D(   49) --> V(    6) amplitude =  0.3448
>     D(   50) --> V(    4) amplitude =  0.3412
> 
>  Excited state  10: excitation energy (eV) =    6.0509
>  Total energy for state 10:                 -3242.15460988 au
>     Multiplicity: Triplet
>     Trans. Mom.:  0.0000 X   0.0000 Y   0.0000 Z
>     Strength   :     0.0000000000
>     D(   48) --> V(    1) amplitude =  0.6274
>     D(   48) --> V(    5) amplitude =  0.5408
>     D(   48) --> V(    6) amplitude =  0.2471
>     D(   49) --> V(    2) amplitude = -0.3412
>     D(   50) --> V(    3) amplitude =  0.3412
> 
>  Excited state  11: excitation energy (eV) =    6.1093
>  Total energy for state 11:                 -3242.15246491 au
>     Multiplicity: Triplet
>     Trans. Mom.:  0.0000 X   0.0000 Y   0.0000 Z
>     Strength   :     0.0000000000
>     D(   48) --> V(    4) amplitude =  0.3359
>     D(   49) --> V(    3) amplitude =  0.4688
>     D(   50) --> V(    2) amplitude =  0.8048
> 
>  Excited state  12: excitation energy (eV) =    6.1093
>  Total energy for state 12:                 -3242.15246491 au
>     Multiplicity: Triplet
>     Trans. Mom.:  0.0000 X   0.0000 Y   0.0000 Z
>     Strength   :     0.0000000000
>     D(   48) --> V(    4) amplitude =  0.7353
>     D(   49) --> V(    3) amplitude = -0.6586
> 
>  Excited state  13: excitation energy (eV) =    6.3587
>  Total energy for state 13:                 -3242.14329933 au
>     Multiplicity: Singlet
>     Trans. Mom.: -0.0000 X  -0.5116 Y  -0.0000 Z
>     Strength   :     0.0407751903
>     D(   48) --> V(    1) amplitude =  0.8228
>     D(   48) --> V(    5) amplitude = -0.3168
>     D(   49) --> V(    2) amplitude = -0.3107
>     D(   50) --> V(    3) amplitude =  0.3107
> 
>  Excited state  14: excitation energy (eV) =    6.3587
>  Total energy for state 14:                 -3242.14329933 au
>     Multiplicity: Singlet
>     Trans. Mom.: -0.5116 X   0.0000 Y  -0.0000 Z
>     Strength   :     0.0407751913
>     D(   48) --> V(    2) amplitude = -0.3107
>     D(   49) --> V(    1) amplitude =  0.8228
>     D(   49) --> V(    5) amplitude =  0.2837
>     D(   50) --> V(    4) amplitude =  0.3107
> 
>  Excited state  15: excitation energy (eV) =    6.3587
>  Total energy for state 15:                 -3242.14329933 au
>     Multiplicity: Singlet
>     Trans. Mom.: -0.0000 X  -0.0000 Y   0.5116 Z
>     Strength   :     0.0407751956
>     D(   48) --> V(    3) amplitude =  0.3107
>     D(   49) --> V(    4) amplitude =  0.3107
>     D(   50) --> V(    1) amplitude =  0.8228
>     D(   50) --> V(    6) amplitude =  0.3467
> 
>  Excited state  16: excitation energy (eV) =    6.4214
>  Total energy for state 16:                 -3242.14099440 au
>     Multiplicity: Singlet
>     Trans. Mom.:  0.0000 X   0.0000 Y   0.0000 Z
>     Strength   :     0.0000000000
>     D(   49) --> V(    2) amplitude =  0.7013
>     D(   50) --> V(    3) amplitude =  0.7013
> 
>  Excited state  17: excitation energy (eV) =    6.4214
>  Total energy for state 17:                 -3242.14099440 au
>     Multiplicity: Singlet
>     Trans. Mom.:  0.0000 X  -0.0000 Y  -0.0000 Z
>     Strength   :     0.0000000000
>     D(   48) --> V(    2) amplitude =  0.7013
>     D(   50) --> V(    4) amplitude =  0.7013
> 
>  Excited state  18: excitation energy (eV) =    6.4214
>  Total energy for state 18:                 -3242.14099440 au
>     Multiplicity: Singlet
>     Trans. Mom.:  0.0000 X   0.0000 Y  -0.0000 Z
>     Strength   :     0.0000000000
>     D(   48) --> V(    3) amplitude = -0.7013
>     D(   49) --> V(    4) amplitude =  0.7013
> 
>  Excited state  19: excitation energy (eV) =    6.4868
>  Total energy for state 19:                 -3242.13859182 au
>     Multiplicity: Singlet
>     Trans. Mom.: -0.0000 X  -0.0000 Y   0.6058 Z
>     Strength   :     0.0583334991
>     D(   50) --> V(    1) amplitude =  0.4893
>     D(   50) --> V(    6) amplitude = -0.8192
> 
>  Excited state  20: excitation energy (eV) =    6.4868
>  Total energy for state 20:                 -3242.13859182 au
>     Multiplicity: Singlet
>     Trans. Mom.: -0.6058 X   0.0000 Y  -0.0000 Z
>     Strength   :     0.0583335041
>     D(   49) --> V(    1) amplitude =  0.4893
>     D(   49) --> V(    5) amplitude = -0.6704
>     D(   49) --> V(    6) amplitude =  0.4773
> 
>  Excited state  21: excitation energy (eV) =    6.4868
>  Total energy for state 21:                 -3242.13859182 au
>     Multiplicity: Singlet
>     Trans. Mom.: -0.0000 X  -0.6058 Y  -0.0000 Z
>     Strength   :     0.0583335053
>     D(   48) --> V(    1) amplitude =  0.4893
>     D(   48) --> V(    5) amplitude =  0.7485
>     D(   48) --> V(    6) amplitude =  0.3420
> 
>  Excited state  22: excitation energy (eV) =    6.4896
>  Total energy for state 22:                 -3242.13849021 au
>     Multiplicity: Singlet
>     Trans. Mom.:  0.0000 X   0.0000 Y  -0.0000 Z
>     Strength   :     0.0000000000
>     D(   50) --> V(    5) amplitude = -0.9881
> 
>  Excited state  23: excitation energy (eV) =    6.4896
>  Total energy for state 23:                 -3242.13849021 au
>     Multiplicity: Singlet
>     Trans. Mom.:  0.0000 X   0.0000 Y  -0.0000 Z
>     Strength   :     0.0000000000
>     D(   49) --> V(    5) amplitude = -0.5756
>     D(   49) --> V(    6) amplitude = -0.8086
> 
>  Excited state  24: excitation energy (eV) =    6.4896
>  Total energy for state 24:                 -3242.13849021 au
>     Multiplicity: Singlet
>     Trans. Mom.: -0.0000 X   0.0000 Y   0.0000 Z
>     Strength   :     0.0000000000
>     D(   48) --> V(    5) amplitude =  0.4124
>     D(   48) --> V(    6) amplitude = -0.9028

As you can see, we now have 12 roots (NRoots was altered from 4 to 12) and most, but not all, of the excited states appear to have a 3-fold degeneracy, even the singlets. Is that normal? What might be the rationale behind those results?

Also, in the output file, just before the start of the TDDFT/TDA calculation, I have the following lines which might be related:

>  Q-Chem warning in module 0, line  198:
> 
>  OriOrb: Failure to resolve orbital degeneracies.

But I can’t find anything about this “OriOrb” thing…

Thank you in advance for your help!

Best regards!

Iacobellis Nicolas, Ph.D. Student

1 Like

I think the results are pretty normal for such a highly symmetric molecule. I want to note that the excited state degeneracy of both the singlets and the triplets results from the Td point group symmetry of the molecule, not from angular momentum coupling. Here the excited states should belong to the A1,2, E or T1,2 representations of the Td group, which corresponds to 1-, 2- and 3-fold degeneracy, respectively.

1 Like

Note that your HOMO and your LUMO+1 are triply-degenerate, by symmetry. I think this is why the requested number of roots was altered.

The OriOrb warning has to do with attempting (and failing) to apply a quadrupole field to determine symmetry labels for orbitals in infinite point groups. Would be nice if this could be explained in the manual.

1 Like

Thank you to both of you, your answers were really helpful!