Photodeactivation Channels of Transition Metal Complexes …
269
The most rigorous description of SOC requires the use of the four-component Dirac
equation, from which SOC naturally emerges. However, four-component calculations are hardly tractable even for the smaller molecular systems; and therefore, for
practical applications, the spin–orbit operator is often added ad-hoc to nonrelativistic electronic structure theories. The most used approximated SOC operators arise
from the formulation of the one- and two-electron SOC Breit–Pauli operator [76].
As in TMCs the one-electron term dominates, effective one-electron SOC operators are commonly used for (i) the calculation of SOC matrix elements (SOCMEs)
between the involved states, and (ii) related spin–orbit properties. The calculation
of SOCMEs is nowadays available with multiple electronic structure theories, often
being the results not extremely sensitive to the chosen level of theory [75].
A final aspect to be mastered by the theoretical investigations is the calculation
of the geometrical rearrangements occurring during the course of photodeactivation.
The calculations of (i) stationary point geometry optimizations in the ES PES and
(ii) ES vibrational levels are considerably more expensive than those for the GS [30].
In practical terms, these calculations require the implementation of analytical first
and second derivatives within the electronic structure theories of choice. As those are
available within TD-DFT, this latter method is often used for such purposes and has
found many applications in TMCs photochemistry. However, due to its single reference character, special caution should be taken in the surroundings of surface crossings, such as CoIns. Alternatively, in these situations, the use of multi-configuration
wave function approaches is best suited. Recent implementations of analytical first
derivatives within some of these methods (such as for CASPT2 [77] and MC-pDFT
[78]) should alleviate not only to perform nonadiabatic molecular dynamics simulations but also the optimization of CoIns with the latter methods. Among other
critical points of relevance, minimum energy crossing points (MECP) between states
of different multiplicity are highlighted. Efficient algorithms to locate MECPs are
available since the late 1990s [79]. MECPs are not only relevant in two- and multistate reactivity but also in TMCs photochemistry (see, e.g., for [Ru(bpy) 3 ]
2+ [80] and
for cyclometalated Ir(III) complexes [36, 37]). In the case that the MECP involves
the lower-lying states of their given multiplicity, these calculations can be performed
with DFT. The recent automated algorithms to search for MECP (and/or minimum
energy CoIns) geometries developed by Maeda and coworkers [81, 82] should help
to automatically map out the intricate PES of TMCs, as recently shown for a Re(I)
complex [83].
2.2 ES Decay Rate Theories
As discussed in Sect. 1, ES decay rate formalisms are best suited to assess the
competition between radiative and nonradiative deactivation channels. In the context
of TMCs, these formalisms are often used to model the long-lived phosphorescence
arising from T 1 . The validity of these formalisms might be questioned for these
systems if near-degeneracy situations between different low-lying ES apply, or in the
269
The most rigorous description of SOC requires the use of the four-component Dirac
equation, from which SOC naturally emerges. However, four-component calculations are hardly tractable even for the smaller molecular systems; and therefore, for
practical applications, the spin–orbit operator is often added ad-hoc to nonrelativistic electronic structure theories. The most used approximated SOC operators arise
from the formulation of the one- and two-electron SOC Breit–Pauli operator [76].
As in TMCs the one-electron term dominates, effective one-electron SOC operators are commonly used for (i) the calculation of SOC matrix elements (SOCMEs)
between the involved states, and (ii) related spin–orbit properties. The calculation
of SOCMEs is nowadays available with multiple electronic structure theories, often
being the results not extremely sensitive to the chosen level of theory [75].
A final aspect to be mastered by the theoretical investigations is the calculation
of the geometrical rearrangements occurring during the course of photodeactivation.
The calculations of (i) stationary point geometry optimizations in the ES PES and
(ii) ES vibrational levels are considerably more expensive than those for the GS [30].
In practical terms, these calculations require the implementation of analytical first
and second derivatives within the electronic structure theories of choice. As those are
available within TD-DFT, this latter method is often used for such purposes and has
found many applications in TMCs photochemistry. However, due to its single reference character, special caution should be taken in the surroundings of surface crossings, such as CoIns. Alternatively, in these situations, the use of multi-configuration
wave function approaches is best suited. Recent implementations of analytical first
derivatives within some of these methods (such as for CASPT2 [77] and MC-pDFT
[78]) should alleviate not only to perform nonadiabatic molecular dynamics simulations but also the optimization of CoIns with the latter methods. Among other
critical points of relevance, minimum energy crossing points (MECP) between states
of different multiplicity are highlighted. Efficient algorithms to locate MECPs are
available since the late 1990s [79]. MECPs are not only relevant in two- and multistate reactivity but also in TMCs photochemistry (see, e.g., for [Ru(bpy) 3 ]
2+ [80] and
for cyclometalated Ir(III) complexes [36, 37]). In the case that the MECP involves
the lower-lying states of their given multiplicity, these calculations can be performed
with DFT. The recent automated algorithms to search for MECP (and/or minimum
energy CoIns) geometries developed by Maeda and coworkers [81, 82] should help
to automatically map out the intricate PES of TMCs, as recently shown for a Re(I)
complex [83].
2.2 ES Decay Rate Theories
As discussed in Sect. 1, ES decay rate formalisms are best suited to assess the
competition between radiative and nonradiative deactivation channels. In the context
of TMCs, these formalisms are often used to model the long-lived phosphorescence
arising from T 1 . The validity of these formalisms might be questioned for these
systems if near-degeneracy situations between different low-lying ES apply, or in the