I am looking for software that can calculate the first-order non-adiabatic coupling vector $$\vec \tau_{ij}=\langle \varphi_i|\nabla_R\varphi_j\rangle$$ between excited electronic singlet states using DFT/TDDFT. The software I am aware of(Gaussian, Turbomole, Dalton<-I'm not 100% sure about this one, Molpro only MSCF/CI), are either limited to calculations for transitions between electronic ground and excited state or don't feature DFT/TDDFT(Bagel).

Does someone know a program/software that features the calculation of non-adiabatic coupling vectors between excited states, using DFT/TDDFT? Or is there perhaps a fundmental reason why this combination cannot be used?

The application are small organic molecules < 100 atoms, no solids or crystals.


It seems that the commercial software Q-CHEM can calculate nonadiabatic coupling vectors also between excited states, using TDDFT or spin-flip TDDFT. But It would be good to have at least one free/non-commercial alternative.


1 Answer 1


The BDF program (https://bdf-manual.readthedocs.io/en/latest/; the English manual is under ongoing translation from Chinese. For a purely English but older version, see is the first program to support nonadiabatic coupling vectors between TDDFT excited states. Although it is not free, the interested user can apply for a free trial by registering an account at https://iresearch.net.cn/web/personal-space/activity-page (in Chinese, but very machine-translatable). A few advantages of BDF's excited state-excited state nonadiabatic coupling functionality include:

  1. The nonadiabatic coupling vectors can be calculated at either the TDA or full TDDFT levels.
  2. Both singlet states and triplet states are supported. Therefore, internal conversions from e.g. the T2 state to the T1 state can be calculated as well.
  3. BDF offers two ways to calculate the nonadiabatic coupling vectors: via equation-of-motion (EOM) theory or via time-dependent perturbation theory (TDPT). The TDPT approach is the most rigorous, since it provably gives the exact nonadiabatic coupling vectors when the DFT functional is exact and when the exact frequency-dependent response kernel is used. The EOM nonadiabatic coupling vectors do not have this property, but usually differ negligibly from the TDPT ones, and have much better numerical stability. By comparison, many other programs employ the auxiliary wave function approaches or the Chernyak-Mukamel formula; the former are not rigorous (because the rely on making analogies between Kohn-Sham and wavefunction theory, and the analogies are not fully justified), while the latter requires extremely large basis sets for fully converged results. See my paper for detailed discussions.
  4. BDF features a seamless interface to the MOMAP program, which can calculate the internal conversion rate between two electronic states using the harmonic approximation to the potential energy surfaces, using the nonadiabatic coupling vectors calculated by BDF.
  • $\begingroup$ Is the auxiliary wave function approach also know as pseudo-wavefunction approach or is this what is sometimes called quadratic response theory? I am not very familiar with the details of DFT/TDDFT, and the terms are used in the Q-Chem manual on page 599, manual.q-chem.com/5.2/qchem_manual_5.2.pdf $\endgroup$
    – Hans Wurst
    Apr 21, 2023 at 13:41
  • 1
    $\begingroup$ @HansWurst "Auxiliary wavefunction approach" is synonymous with "pseudo-wavefunction approach" (as can be seen by the references cited in my paper that study auxiliary wavefunction approaches), while "quadratic response theory" is synonymous with "TDPT" in my paper. $\endgroup$
    – wzkchem5
    Apr 21, 2023 at 15:01

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