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I noticed that in ORCA forum there is a user asked how to calculate transition dipole moments between excited states with TDDFT using ORCA

I'm pretty new to TD-DFT and I'm currently working on silicon quantum dots passivated with hydrogen (ranging from 40 to 250 atoms), trying to determine the transition dipole moments between electronically excited states. I didn't find any information about how to proceed in order to obtain them (I just got the absorption spectrum, which only shows the transition between the ground state and each individual excited state). Is this feasible or did I miss something ?

Unfortunately, current ORCA doesn't have this feature. However, using Multiwfn, this can be easily done. Below is my reply, I copy it here since some other Multiwfn/ORCA users may have the same problem.

Current latest version, namely version 3.6(dev) of Multiwfn is able to calculate transition dipole moment between excited states based on ORCA output.

First, conduct a CIS or TDA-DFT calculation using keywords like below, assume that file name is test.inp

! PBE0 def2-SVP nopop pal4

%tddft

nroots 4

tprint 1E-8

endThen use such as "orca_2mkl test -molden" command to convert test.gbw to test.molden.input

Finally, boot up Multiwfn and input

test.molden.input // The .molden input file

18 // Electronic excitation analyses

5 // Calculate transition electric dipole moments between all excited states

test.out // The ORCA output file

1 // Output transition dipole moments to screen

Now you will find below result on screen.Ground state dipole moment in X,Y,Z: -0.271701 -0.314645 0.344277 a,u,

Transition dipole moment between excited states (a.u.):

i j X Y Z Diff.(eV) Oscil.str

1 1 -1.7862676 -0.2640772 0.2405912 0.00000 0.00000

1 2 0.1976941 0.0854227 0.0346682 0.70800 0.00083

1 3 2.1779748 -0.1165127 0.1223195 1.38100 0.16146

1 4 -0.4226960 0.0657206 -0.0047136 1.73500 0.00778

2 2 -2.6132918 -0.1285522 0.0996265 0.00000 0.00000

2 3 0.0625351 0.3407469 -0.0481496 0.67300 0.00202

2 4 0.2904364 -0.1027734 -0.2020122 1.02700 0.00341

3 3 -2.9400691 -0.2538387 0.1676538 0.00000 0.00000

3 4 0.4640359 0.0322024 -0.0096710 0.35400 0.00188

4 4 2.3204590 1.0665381 -0.2011480 0.00000 0.00000The i and j are excited state indices, the line such as i=2 and j=2 corresponds to electronic dipole moment of excited state 2.

This feature of Multiwfn should work perfectly for CIS and TDA-DFT. However, for the TDHF or TDDFT, the result may be not always reliable, since in this case, current version of ORCA does not output excitation configuration coefficients and de-excitation configuration coefficients separately, which are needed for strictly evaluating transition dipole moments.

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**polinalisin****Member**- Registered: 2019-01-28
- Posts: 4

First of all, thanks for the great work!

I have two questions regarding the transition dipole moments from the excited states: how do you calculate them? Do you integrate the corresponding transition dipole density?

And is it possible to obtain a cube file for a transition density between any excited states using Multiwfn?

Thanks in advance,

Polina

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Dear polinalisin,

1 The transition dipole moment calculated in aforementioned way is based dipole moment integral between basis functions and configuration coefficients, the calculation is purely analytic.

2 Currently Multiwfn is only able to calculate and export cube file of transition density between ground state and an excited state (via hole-electron analysis module of main function 18). However, extending the code to support calculating and exporting cube file of transition density between two excited states is not difficult.

Best regards,

Tian

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Dear polinalisin,

I just slightly updated the Multiwfn 3.6(dev) on the website, I am glad to inform you that now the transition density cube file between two excited states can be generated. Below is an example for N-phenylpyrrole system.

Boot up Multiwfn and input

examples\excit\N-phenylpyrrole.fch

18

9 // Generate and export transition density matrix

2 // Generate transition density matrix between (TDM) two excited states

examples\excit\N-phenylpyrrole.out

2,3 // Assume that you want to analyze S2-S3 transition

[Press ENTER directly]

y // Symmetrize the resulting TDM in usual way

y // Export wavefunction information including the newly generated TDM to TDM.fch in current folder

Reboot Multiwfn and input

TDM.fch

200

16 // Generate natural orbitals based on the density matrix in .fch/.fchk file

SCF // We input this because the "Total SCF Density" field in the TDM.fch currently correspond to S2-S3 TDM

y // Export new.molden, which contains natural orbitals corresponding to S2-S3 TDM, and then let directly load it

0 // Return to main menu

5 // Calculate grid data

1 // Electron density

2 // Medium quality grid

2 // Export cube file

Now the density.cub in current records transition density between S2 and S3.

PS: This example has also been added to the manual as Section 4.18.2.3.

Best regards,

Tian

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**polinalisin****Member**- Registered: 2019-01-28
- Posts: 4

Dear Tian,

thank you very much for the quick reply and updating of your program! Could you please share or add to the manual the formula, which you have used to generate the TDM between two excited states? As far as I see, the MO and atomic basis expansions in Sections 3.21.1.1 and 3.21.2 correspond only to the ground-to-excited state case. Or do I miss something?

Best regards,

Polina

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Dear Polina,

I have updated the 3.6(dev) manual on the Multiwfn website, in Section 3.21.9 you will find the formula used to evaluate transition density matrix between two excited states.

Best regards,

Tian

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**polinalisin****Member**- Registered: 2019-01-28
- Posts: 4

Dear Tian,

thank you again for the answer.

However, when I tried to derive the formula myself, I ended up with a similar, but slightly different expression. Namely, the MO expansion coefficients (C_mu,i) are grouped together corresponding either to two occupied or two virtual MOs:

C_mu,a*C_nu,b for i==j, a!=b

-C_mu,i*C_nu,j for i!=j, a==b

It seems also logical for me, because if we calculate the integral between two (CIS) excited state wavefunctions (similar like in Section 3.21.1.1 Theory 4), it will fall apart into integrals between singly excited configurations, and then only the terms with either the same MOs or different by only one MO will not be zero. In the singly excited configurations this means that either they have same occupied MO and different virtuals or different occupied and same virtual. In both cases the integral will reduce to the product of either two occupied or two virtual MOs.

And some more questions about the configuration coefficients: in the case of TDDFT calculation do I understand right, that the final expansion coefficient you use to calculate TDM is just the sum of the excitation and de-excitation coefficients for the corresponding pair of orbitals? At least it follows from the expansion on page 178 of the 3.6(dev) manual. Do you normalize or orthogonalize this coefficients somehow? Of the main interest for me are the coefficients provided by Gaussian09.

Thank you in advance,

Best regards,

Polina

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Dear Polina,

Your equations are right, and they are what I actually used to code the Multiwfn, I will rectify the typo in the manual, sorry for inconvenience.

For TD case, the coefficient of excitation and de-excitation configurations are not simply summed up, and thus renormalization is not needed. In the calculation of TDM between two excited states, the coupling terms between excitation and de-excitation configurations are ignored. In addition, when calculating the terms between de-excitation configurations, they are additionally multiplied by -1.

Best regards,

Tian

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**polinalisin****Member**- Registered: 2019-01-28
- Posts: 4

Dear Tian,

thank you again, this helped me a lot.

Best regards,

Polina

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