Photodeactivation Channels of Transition Metal Complexes …
271
left-side term contributes more to the intensity borrowing. Equation (3) is a sum-overstates expression that can be solved using the quadratic response (QR) methodology
[86], which replaces (3) by solutions of sets of linear equations, and which combined with TD-DFT calculations makes tractable the phosphorescence calculations
in TMCs. Additionally, phosphorescence decay rates are accessible with other alternative approaches, namely self-consistent SOC-TD-DFT and perturbative SOC-TDDFT (pSOC-TD-DFT). In general, perturbative schemes yield systematically lower
rates than self-consistent approaches [87].
As stated in Sect. 1, the calculations of ES → GS nonradiative rates still pose
significant challenges. Two main processes, i.e., k IC and k ISC , contribute to the global
nonradiative rates in TMCs (we recall that IC and ISC processes might be indistinguishable in the case that strong spin–vibronic coupling applies). The expressions for
k IC and k ISC require a more elaborated treatment as compared to their radiative counterparts. Second-order perturbation theory (i.e., Fermi-Golden rule like) is usually
a good approximation for the calculations of the nonradiative rates within the local
harmonic approximation, where distortions, displacements, as well as Duschinsky
rotation effects of the PESs are taken into account. The effective couplings (nonadiabatic electronic coupling and SOCs) are often calculated at a particular nuclear
geometry (typically the ES and/or GS minimum). Evaluation of k IC and k ISC can be
done in the energy or in the time domain, the latter being preferred for its reduced
associated computational cost. Among the time-domain approaches, I highlight the
TVCF formalism of Shuai and coworkers [84, 88] and the time-dependent integration
schemes developed by Marian and coworkers [89]. Importantly, vibronic contributions to the k ISC values can also be computed [88]. The nonradiative rate calculations
might be sensitive to the computed relative energies between the final and initial
states and to the geometric rearrangements of the involved PES, and thus, these calculations may require a prior assessment of the chosen electronic structure methods
to describe both the ES and GS properties. Additionally, further problems may arise
from (i) the use of the harmonic approximation, which might be inappropriate for
strongly distorted ESs and (ii) the interference with other close-lying ESs. Despite
these limitations, these approaches have been successfully applied for many TMCs,
importantly for Ir(III) complexes [37, 90] (see an example in Sect. 3.1), but also
in [Fe(bpy) 3 ]
2+ [91]. In these examples, the computed rates are on the same order
than the measured ones. Finally, these schemes can be extended to include thermal
effects and high-order spin-vibronic interactions, the latter ones being more relevant
in cases of negligible SOCMEs between the initial and final states.
2.3 ES Reaction Dynamics Methods
As detailed in Sect. 1, an alternative approach to ES decay rate theories to attain timeresolved information are ES reaction dynamic methods. These methods provide a
solution to the time-dependent Schrödinger equation and do treat the nuclear motion
over a PES in an explicit manner. Hence, they generally yield an accurate picture of
271
left-side term contributes more to the intensity borrowing. Equation (3) is a sum-overstates expression that can be solved using the quadratic response (QR) methodology
[86], which replaces (3) by solutions of sets of linear equations, and which combined with TD-DFT calculations makes tractable the phosphorescence calculations
in TMCs. Additionally, phosphorescence decay rates are accessible with other alternative approaches, namely self-consistent SOC-TD-DFT and perturbative SOC-TDDFT (pSOC-TD-DFT). In general, perturbative schemes yield systematically lower
rates than self-consistent approaches [87].
As stated in Sect. 1, the calculations of ES → GS nonradiative rates still pose
significant challenges. Two main processes, i.e., k IC and k ISC , contribute to the global
nonradiative rates in TMCs (we recall that IC and ISC processes might be indistinguishable in the case that strong spin–vibronic coupling applies). The expressions for
k IC and k ISC require a more elaborated treatment as compared to their radiative counterparts. Second-order perturbation theory (i.e., Fermi-Golden rule like) is usually
a good approximation for the calculations of the nonradiative rates within the local
harmonic approximation, where distortions, displacements, as well as Duschinsky
rotation effects of the PESs are taken into account. The effective couplings (nonadiabatic electronic coupling and SOCs) are often calculated at a particular nuclear
geometry (typically the ES and/or GS minimum). Evaluation of k IC and k ISC can be
done in the energy or in the time domain, the latter being preferred for its reduced
associated computational cost. Among the time-domain approaches, I highlight the
TVCF formalism of Shuai and coworkers [84, 88] and the time-dependent integration
schemes developed by Marian and coworkers [89]. Importantly, vibronic contributions to the k ISC values can also be computed [88]. The nonradiative rate calculations
might be sensitive to the computed relative energies between the final and initial
states and to the geometric rearrangements of the involved PES, and thus, these calculations may require a prior assessment of the chosen electronic structure methods
to describe both the ES and GS properties. Additionally, further problems may arise
from (i) the use of the harmonic approximation, which might be inappropriate for
strongly distorted ESs and (ii) the interference with other close-lying ESs. Despite
these limitations, these approaches have been successfully applied for many TMCs,
importantly for Ir(III) complexes [37, 90] (see an example in Sect. 3.1), but also
in [Fe(bpy) 3 ]
2+ [91]. In these examples, the computed rates are on the same order
than the measured ones. Finally, these schemes can be extended to include thermal
effects and high-order spin-vibronic interactions, the latter ones being more relevant
in cases of negligible SOCMEs between the initial and final states.
2.3 ES Reaction Dynamics Methods
As detailed in Sect. 1, an alternative approach to ES decay rate theories to attain timeresolved information are ES reaction dynamic methods. These methods provide a
solution to the time-dependent Schrödinger equation and do treat the nuclear motion
over a PES in an explicit manner. Hence, they generally yield an accurate picture of