# The First Derivative of TDM for the cartersian coordinate

Hi,

I am a new user of Psi4. Recently, I have been trying to calculate the first derivative of the transition dipole moment for the coordinate based on the lowest triplet state.

Here is the configuration I’ve used for the calculation:

memory 6000 MB
molecule an {
0 1
6 -2.23467E-16 3.649499278 0.711281052

1 -2.81423E-16 -4.59598235 1.245038098
1 -2.81423E-16 -4.59598235 -1.245038098
1 4.56189E-16 -2.46806719 -2.491034191
}

set {
basis 6-31G(d)
reference rhf
tdscf_states 3
tdscf_triplets only
tdscf_tda true
save_jk true
}

energy(‘pbe0’, return_wfn=True)
frequencies (‘pbe0’)
tdscf(wfn)
res = tdscf_excitations(wfn, states=3, TDSCF_TDM_PRINT=[‘E_TDM_LEN’,‘M_TDM’])

However, I was unable to find the first derivative of the transition dipole moment for the coordinate.

My question is:

Is it possible to obtain the first derivative of the transition dipole moment for the coordinate based on the lowest triplet? If yes, how can I do it, and where can I find this information?

Thanks a lot.

Psi4 doesn’t support this.

DFT transition dipole moments come from the dot product of the dipole moment integrals and the transition density matrix. The hard part is the transition density matrix derivative that shows up in the chain rule. Getting the transition density matrix on its own requires the solution of the TDDFT equations (second order derivatives). Getting derivatives of that is probably going to require third order derivatives, which Psi4 doesn’t have implemented at current. We’d love to support this (and TDDFT gradients), but we don’t have the manpower for that right now, sorry.

What about numerical derivatives using nuclear displacements?

Good question! That can probably be hacked together, if that’s something the OP wants. Back when I was with Francesco’s group, we did something similar to get derivatives of regular dipole moments for vibrational intensities.

Many thanks for the reply. The results from the numerical derivatives method for obtaining the derivative of the transition dipole moment are not accurate enough. Therefore, I am wondering if it is possible to compute the derivative of the TDM using the Hessian matrix. Both the TDM and Hessian matrix are readily obtainable. However, I am uncertain whether this approach is feasible and how to realize it.

Your assistance would be greatly appreciated. Thank you once again for your help.

Can you explain what you mean by using the Hessian matrix here?

Also, what precision do you need to compute the derivatives of the transition dipole? I would think you could converge the states tightly enough to do this, but I don’t know what your application is.

Thank you for your help. Well, it’s a long story. Actually, what I need is the Herzberg-Teller correction for the transition dipole moment, which is crucial for my project. Currently, I’ve found that only fcclass can handle this, and it requires the transition dipole moment and its first derivative with respect to Cartesian coordinates, which fcclass automatically extracts from FCHK files of Gaussian and outputs from Psi4.

The challenging part is that the transition process for my project is from T1 to S0, and Gaussian cannot provide the TDM due to a lack of consideration for SOC. Therefore, I considered using Psi4, but it seems Psi4 cannot handle the first derivative of TDM, at least not currently. Therefore, I am trying to prepare the data myself. The TDM can be easily obtained, but the first derivative of TDM is difficult for me. I’m not well-versed in theory, and I don’t know how to calculate the first derivative of TDM manually. Your help would be greatly appreciated.

I can’t offer anything on the SOC component, but I think the numerical differentiation would just require you to compute the TDM at displaced geometries. For a central-difference formula, the numerical error would on the order of the square of the displacement size, and you could try different displacements (and convergence criteria for the TD-DFT or whatever method you’re using) to minimize the error.