HOMO-LUMO in SF-TDDFT method

How To
#1

Hi all
I want to know about the HOMO-LUMO in the SF-TDDFT method.
At E.S-1 (which I am considering S0-state) optimization:
There are two singly occupied molecular orbitals (considering alpha spin). The lower orbital (singly occupied molecular orbital i.e., 70 alpha orbital) - I am considering HOMO, and the upper orbital (singly occupied molecular orbital i.e., 71 alpha orbital) - I am considering LUMO. Am I right here?

This is from my S0-state-optimized file (E.S-1 optimization).

        SF-DFT Excitation Energies

(The first “excited” state might be the ground state)

Excited state 1: excitation energy (eV) = -2.1399
Total energy for state 1: -899.13106438 au
<S**2> : 0.0520
S( 2) → S( 1) amplitude = 0.9887 alpha

Excited state 2: excitation energy (eV) = 0.4697
Total energy for state 2: -899.03516342 au
<S**2> : 2.0108
S( 1) → S( 1) amplitude = -0.6612 alpha
S( 2) → S( 2) amplitude = 0.7322 alpha

Excited state 3: excitation energy (eV) = 1.5292
Total energy for state 3: -898.99622825 au
<S**2> : 0.1426
S( 1) → S( 1) amplitude = 0.6886 alpha
S( 2) → S( 2) amplitude = 0.6411 alpha

Excited state 4: excitation energy (eV) = 2.0673
Total energy for state 4: -898.97645400 au
<S**2> : 1.0676
D( 68) → S( 1) amplitude = 0.5027
S( 2) → V( 1) amplitude = 0.8377 alpha

Similarly, I am considering the situation of HOMO-LUMO at the S1 state optimization (at E.S-3).

This is from my S1-state-optimized file (E.S-3 optimization).

        SF-DFT Excitation Energies

(The first “excited” state might be the ground state)

Excited state 1: excitation energy (eV) = -1.0045
Total energy for state 1: -899.07962046 au
<S**2> : 0.0845
S( 2) → S( 1) amplitude = 0.9750 alpha
S( 2) → V( 2) amplitude = -0.1749 alpha

Excited state 2: excitation energy (eV) = 0.6387
Total energy for state 2: -899.01923483 au
<S**2> : 1.4809
D( 68) → S( 1) amplitude = -0.1512
S( 1) → S( 1) amplitude = 0.9533 alpha
S( 2) → S( 2) amplitude = -0.1880 alpha

Excited state 3: excitation energy (eV) = 0.6604
Total energy for state 3: -899.01843757 au
<S**2> : 0.6532
S( 1) → S( 1) amplitude = 0.1945 alpha
S( 2) → S( 2) amplitude = 0.9046 alpha
S( 2) → V( 1) amplitude = -0.3398 alpha

Excited state 4: excitation energy (eV) = 2.3028
Total energy for state 4: -898.95808083 au
<S**2> : 1.0486
D( 64) → S( 1) amplitude = -0.3618
D( 65) → S( 1) amplitude = -0.5534
D( 66) → S( 1) amplitude = -0.7330

#2

HOMO and LUMO are just labels, and perhaps not that useful for an open-shell system. I think some would object to using “HOMO” for a singly-occupied MO (SOMO), but really these are just words. What’s important are the excitation amplitudes. The S(1) and S(2) in the output indicate the singly-occupied MOs, i.e., your orbitals 70 and 71.

#3

Thanks @jherbert for the clarification.
But if anybody asks for information about which transitions correspond to each state?
we can see in E.S-1 S( 2) → S( 1) amplitude = 0.9750 alpha
S( 2) → V( 2) amplitude = -0.1749 alpha is happening.

But If we see E.S-3 S( 1) → S( 1) amplitude = 0.1945 alpha
S( 2) → S( 2) amplitude = 0.9046 alpha
S( 2) → V( 1) amplitude = -0.3398 alpha
Can you please explain in E.S-3 what these transitions signify?

#4

In spin-flip methods, you are starting with a high-spin reference state and your target SF states have multiconfigurational wave functions. The transitions you see are spin-flipping transitions.

The first state you get is a singlet. Looking at the dominant S(2) → S(1) amplitude, it looks like this state is a closed-shell singlet, i.e., this state would be well described by a single closed-shell configuration in which the S(1) orbital is doubly occupied and S(2) is empty.

The third state you get is spin-contaminated but <S^2> suggests that it is a singlet. The S(2) → S(2) spin-flip transition suggests that it is a open-shell singlet, i.e., the dominant configuration has singly occupied S(1) and S(2) orbitals with these two electrons having opposite spin relative to each other. I think it is likely a result of spin-contamination but the spin complement of this dominant configuration has a much different weight (0.1945 vs 0.9046).

Therefore, the third state can be described as a single excitation from S(1) → S(2) relative to the first state (ground state) by comparing the respective dominant configurations.

#5

Thanks, @kaushik for the explanation.
According to my understanding of the output S(1) and S(2) indicate the singly-occupied MOs, i.e., my orbitals 70 and 71.

Or it indicates S(1) = singly occupied alpha and S(2) = single occupied beta.

I converted my writing method to the output S(1) and S(2) transitions into Singly-occupied alpha and Singly-occupied beta.

Based on the results in the file, the following state levels are given for the S0-optimized geometry:

E.S-1 (S0) 0.9887 bHOMO->aLUMO 2.1399 eV
E.S-2 (T1) -0.6612 aHOMO->aLUMO 2.6096 eV
0.7322 bHOMO->bLUMO
E.S-3 (S1) 0.6886 aHOMO->aLUMO 3.6691 eV
0.6411 bHOMO->bLUMO

and for the S1-optimized geometry:

E.S-1 (S0) 0.9750 bHOMO->aLUMO 1.0045 eV
-0.1749 bHOMO->bLUMO+2
E.S-2 (T1) 0.9533 aHOMO->aLUMO 1.6432 eV
-0.1880 bHOMO->bLUMO
E.S-3 (S1) 0.9046 bHOMO->bLUMO 3.3073 eV
-0.3398 bHOMO->bLUMO+1

aHOMO means alpha-spin HOMO.
bHOMO means beta-spin HOMO.

Which one is correct? I think S(1) and S(2) indicate the singly-occupied MOs, i.e., my orbitals 70 and 71.

#6

Maybe one comment here: the difficulty with SF-TDDFT is that it prints excitations with respect to this artificial high-spin reference.

One thing you can do is activate
state_analysis = true

This will produce output that is more consistent with what you’d find in a standard TDDFT calculation.

#7

Hi, I tried with the state_analysis = true keyword.
For example: If see the output for optimized E.S-3 (S1-state)

================================================================================
Excited State Analysis

Spin-flipped State 1 (Reference) :

NOs (alpha)
  Occupation of frontier NOs:
     0.0006   0.9996
  Number of electrons: 70.000000
NOs (beta)
  Occupation of frontier NOs:
     0.0004   0.9994
  Number of electrons: 70.000000
NOs (spin-traced)
  Occupation of frontier NOs:
     0.0294   1.9705
  Number of electrons: 140.000000
  Number of unpaired electrons: n_u =  0.08668, n_u,nl =  0.00718
  NO participation ratio (PR_NO):  4.071328

Spin-flipped State 2 :

NOs (alpha)
  Occupation of frontier NOs:
     0.0515   0.9597
  Number of electrons: 70.000000
NOs (beta)
  Occupation of frontier NOs:
     0.0403   0.9485
  Number of electrons: 70.000000
NOs (spin-traced)
  Occupation of frontier NOs:
     0.9897   1.0104
  Number of electrons: 140.000000
  Number of unpaired electrons: n_u =  2.04013, n_u,nl =  2.00312
  NO participation ratio (PR_NO):  2.123805

Spin-flipped State 3 :

NOs (alpha)
  Occupation of frontier NOs:
     0.0423   0.9604
  Number of electrons: 70.000000
NOs (beta)
  Occupation of frontier NOs:
     0.0396   0.9577
  Number of electrons: 70.000000
NOs (spin-traced)
  Occupation of frontier NOs:
     0.9747   1.0256
  Number of electrons: 140.000000
  Number of unpaired electrons: n_u =  1.99362, n_u,nl =  1.99875
  NO participation ratio (PR_NO):  2.092046

Spin-flipped State 4 :

NOs (alpha)
  Occupation of frontier NOs:
     0.0122   0.9886
  Number of electrons: 70.000000
NOs (beta)
  Occupation of frontier NOs:
     0.0114   0.9878
  Number of electrons: 70.000000
NOs (spin-traced)
  Occupation of frontier NOs:
     0.9990   1.0125
  Number of electrons: 140.000000
  Number of unpaired electrons: n_u =  2.06406, n_u,nl =  2.00422
  NO participation ratio (PR_NO):  2.157916

and one thing is this:

                          SA-NTO Decomposition

Spin-flipped State 2 :

Decomposition into state-averaged NTOs
Alpha spin:
  H- 2 -> L+ 0:  0.9520 ( 90.6%)
                 omega =  91.0%
Beta spin:
  H- 0 -> L+ 1: -0.1967 (  3.9%)
                 omega =   4.0%

Spin-flipped State 3 :

Decomposition into state-averaged NTOs
Alpha spin:
  H- 2 -> L+ 0:  0.1943 (  3.8%)
                 omega =   3.8%
Beta spin:
  H- 0 -> L+ 1:  0.9612 ( 92.4%)
  H- 1 -> L+ 1: -0.1685 (  2.8%)
                 omega =  95.3%

Spin-flipped State 4 :

Decomposition into state-averaged NTOs
Alpha spin:
  H- 1 -> L+ 0:  0.9632 ( 92.8%)
  H- 0 -> L+ 0:  0.1765 (  3.1%)
                 omega =  96.2%
Beta spin:
                 omega =   0.1%

In SA-NTO Decomposition, what H- 2 → L+ 0 signifies?

#8

Hi, this part refers to the state averaged NTOs. So, you’d have to look at the file sa_nto.mo . And then the transition would correspond to the HOMO-2 to LUMO transition in this file.

But maybe these state-averaged NTOs are a bit more difficult to interpret. If you are interested in the orbitals involved, then it may be easier to look at the indiviual *_nto.mo files for the different states.