Assessing excited state lifetime with TD-DFT

Question

My question is brief — is it possible to estimate excited state lifetimes using computational chemistry (DFT/TD-DFT) tools? I am aware this may be a rough approximation (e.g., neglecting non-radiative pathways) but it is still better than nothing. I have substances with experimentally measured values, and while I do not need to reproduce exact results, obtaining a trend that aligns with the experimental data would be sufficient.

I appreciate any (ideally step-by-step) help, thanks! PS: I am using ORCA.

Answer 1

Yes, and you can even do that if you don't neglect the non-radiative pathways.

1. If non-radiative pathways can be neglected, and contributions from molecular vibration can also be neglected, then the lifetime can be calculated from the Fermi's golden rule from the transition dipole moment, the excitation energy, and the refractive index of the solution. Almost all programs that support TDDFT will give the transition dipole moment, and when they don't, they give the oscillator strength that can be easily converted to the transition dipole moment. So basically a TDDFT calculation plus a simple calculation by hand will give the results. Details are given in the answers of another question on this site.

2. If non-radiative pathways can be neglected, but contributions from molecular vibration cannot be neglected (due to the Herzberg-Teller effect), then you have to calculate the lifetime from the path integral formalism, for example using the ESD module of ORCA. This requires the ground state and excited state Hessians, but under certain circumstances the latter can be omitted (as in the VG method), which introduces some approximations. The ORCA manual includes tutorials for calculating fluorescence and phosphorescence rates.

3. If non-radiative pathways cannot be neglected, you have to also calculate the total non-radiative relaxation rate. In general, this includes internal conversion (IC), intersystem crossing (ISC) and photolysis. The excited state lifetime is then the inverse of the sum of all excited state relaxation rates (radiative and non-radiative), not just the radiative one.

   • IC rates can be computed by ORCA 6.0.0 or later versions (provided that the excited state relaxes directly to the ground state), but not earlier versions. If you are studying a high excited state that can undergo IC to a lower excited state, then you have to use other programs, e.g. BDF.
   • ISC rates can be calculated by the ESD module of ORCA.
   • Photolysis rates can be calculated by transition state theory through the Eyring equation, if the activation energy is not too low, and there are no other factors that render transition state theory inaccurate. The procedure is identical to calculating activation Gibbs free energies of ground state reactions, i.e. one locates the excited state equilibrium geometry and excited state TS, performs frequency calculations on both of them, and subtract their Gibbs free energies. For very fast photolysis rates, nonadiabatic molecular dynamics is frequently necessary for accurate results.

4. If the excited state of interest is in equilibrium with another state (e.g. in TADF systems where usually the $S_1$ and $T_1$ states are in equilibrium), then the excited state lifetime can usually only be obtained by a kinetic simulation (after all related rate constants have been calculated), and is not simply the inverse of a sum of rate constants. However, sometimes the kinetic simulation may be bypassed by approximations such as the steady state approximation, yielding a closed-form expression of the excited state lifetime that can be calculated by hand.

For a recent example of excited state lifetime calculation, please refer to our paper on copper(II) porphyrin.